Figure 4 Maximum pull-out force recorded for NiTi wires with embedded .... traffic load from a wheel ( , assumed to be distributed on a circular area of ... Ï at the bottom of the pavement for two extreme cases of subgrade reaction modulus, ...... In this chart, the curves represent the values of for a low value of (50 lb/in3) and a.
EVALUATION OF DAMAGE AFFECTING THE MECHANICAL BEHAVIOR OF COMPOSITE MATERIALS by Federico C. Antico
A Dissertation Submitted to the Faculty of Purdue University In Partial Fulfillment of the Requirements for the degree of
Doctor of Philosophy
Lyles School of Civil Engineering West Lafayette, Indiana August 2018
ii
THE PURDUE UNIVERSITY GRADUATE SCHOOL STATEMENT OF COMMITTEE APPROVAL
Dr. Pablo Zavattieri, Chair Lyles School of Civil Engineering Dr. Jason Weiss School of Civil & Construction Engineering, Oregon State Unviersity Dr. Jan Olek Lyles School of Civil Engineering Dr. Marisol Koslowski School of Mechanical Engineering
Approved by: Dr. Dulcy Abraham Head of the Graduate Program
iii
To my beloved wife and my children Julia, Emilia, Lucas and Juan José To my parents and my brother Nancy, Carlos and Gerardo
iv
ACKNOWLEDGMENTS
I want to express my deepest gratitude to my wife Julia Wiener for her love, support and advice during this journey. It would have been impossible for me to finish without her on my side. I thank Professors Pablo Zavattieri, Jason Weiss, Jan Olek, and Marisol Koslowski for being my examination committee members. I acknowledge Professor Zavattieri for challenging me to be an independent and resilient researcher. I thank Professor Weiss for igniting the interest on a topic that was unfamiliar to me when I started my Ph.D. program. I thank also Professor Olek for his support and valuable comments that have made this thesis more profound. This research was partially supported by the Indiana Department of Transportation under the Joint Transportation Research Program, Grant SPR‐3403. This support is greatly acknowledged. I also thank the School of Engineering of Universidad Adolfo Ibáñez for their support. Thanks Professor Marcos Actis for the professional opportunities and for transferring me the passion to mentor motivated students. Family, friends and colleagues have been of great support during preparation of this thesis; special thanks to Juan Pablo, Lourdes & Elena Sesmero, Karla & Gale Pitner, Carlos Ruestes, Inés & Patricio Wiener, Fernando Cordisco, Gerardo Araya-Letelier, Igor De la Varga, Yaghoob Farnam and Robert Spragg. Last but not least, thanks be to God
v
TABLE OF CONTENTS
TABLE OF CONTENTS ................................................................................................................ v LIST OF TABLES ....................................................................................................................... viii LIST OF FIGURES ....................................................................................................................... ix ABSTRACT .................................................................................................................................. xv INTRODUCTION ................................................................................................ 1 1.1 Motivation ........................................................................................................................... 1 1.2 Goals and objectives ........................................................................................................... 2 1.3 Organization of the thesis ................................................................................................... 3 ADHESION OF NICKEL–TITANIUM SHAPE MEMORY ALLOY WIRES TO THERMOPLASTIC MATERIALS: THEORY AND EXPERIMENTS ................................. 5 2.1 Introduction ......................................................................................................................... 5 2.2 Research significance.......................................................................................................... 5 2.3 Pull-out test ......................................................................................................................... 9 2.3.1 Test configuration .......................................................................................................... 9 2.3.2
Test results ................................................................................................................... 12
2.3.3
Analytical models......................................................................................................... 13
2.3.4
Finite element models .................................................................................................. 15
2.3.5
Results and discussion ................................................................................................. 19
2.4 Concluding remarks .......................................................................................................... 27 QUANTIFYING THE RISK OF MICROCRACKING FOR CONCRETE PAVEMENTS UNDER TRAFFIC LOADING AT EARLY AGES USING TESTING AND SEMI-ANALYTICAL MODELS ................................................................................................ 30 3.1 Introduction ....................................................................................................................... 30 3.2 Research approach ............................................................................................................ 30 3.2.1 Experimental methodology .......................................................................................... 30 3.2.2 Analytical solution of 1 at an elastic pavement as result of traffic loads .................. 33 3.2.3 Experimental results of 1 at an elastic pavement as result of traffic loads ................ 34 3.2.4 Finite element model (FEM) of the APT pavement .................................................... 35
vi 3.3 Results and discussion ...................................................................................................... 36 3.3.1 t measurements and mechanical linear response of thick pavements ........................ 36 3.3.2 Finite element simulations to study strain on pavements tested at APT ...................... 38 3.3.3 Effect of E t and h on I f r .................................................................................... 40 3.3.4 Comments on the permanent strain, p ....................................................................... 43 3.4 Proposal for a time early opening to traffic (EOT) criterion ............................................ 43 3.5 Final comments and conclusions ...................................................................................... 44 TENSILE STRENGTH REDUCTION OF MORTAR CAUSED BY SOLIDIFICATION AND NONUNIFORM RESTRAINED DRYING SHRINKAGE .............. 47 4.1 Introduction ....................................................................................................................... 47 4.2 Research approach ............................................................................................................ 47 TENSILE STRENGTH REDUCTION OF MORTAR CAUSED BY SOLIDIFICATION AND NONUNIFORM RESTRAINED DRYING SHRINKAGE .............. 53 5.1 Understanding conditions for brittle and quasibrittle behavior of mortar and concrete ... 53 5.2 Constitutive model ............................................................................................................ 54 5.2.1 The solidification model .............................................................................................. 54 5.2.2 Stochastic, time-dependent damage model .................................................................. 56 5.2.3 Time-dependent cohesive law ...................................................................................... 57 5.2.4 Computational aspects of the cohesive model ............................................................. 59 5.2.5 Stochastic effects modeling.......................................................................................... 59 5.3 Boundary conditions of mortar at early ages .................................................................... 62 5.3.1 Drying shrinkage characterization ............................................................................... 62 5.3.2 Drying shrinkage characterization ............................................................................... 64 5.3.3 Solidification characterization ..................................................................................... 65 5.3.4 Damage characterization .............................................................................................. 66 RESULTS FROM MODEL OF MORTAR SLABS ACCOUTING FOR SOLIDIFICATION AND DAMAGE SUBJECT TO DRYING SHRINKAGE .......................... 68 6.1 Stresses relaxation induced by distributed damage .......................................................... 68 6.2 Residual tensile strength induced by distributed damage ................................................. 73 NUMERICAL MODEL OF DAMAGE-MECHANICAL BEHAVIOR OF RESTRAINED MORTAR RINGS AT EARLY AGES .............................................................. 76
vii 7.1 Introduction ....................................................................................................................... 76 7.2 Sensitivity analysis............................................................................................................ 79 SUMMARY AND CONCLUSIONS ................................................................. 87 REFERENCES ............................................................................................................................. 89 VITA ........................................................................................................................................... 102
viii
LIST OF TABLES
Table 1 Test matrix with mean forces recorded for each treatment for pull-out specimens with l e = 90mm. The untreated-wire specimens require the lowest pull-out force, while those treated with 5%PPPA require the highest pull-out forces. The maximum value achieved with the untreated NiTi wires is even lower than the maximum values from tests of the treatedwires. ............................................................................................................................... 12 Table 2 Summary of the tests performed according to the loading time and pavement thickness. ......................................................................................................................................... 30 Table 3 Viscoelastic coefficients for sensitivity analysis extracted from [104] ........................... 65 Table 4 Aging, mechanical and damage parameters used in this work ....................................... 66
ix
LIST OF FIGURES
Figure 1 SMA-reinforced thermoplastic composites for actuation devices. (a) Resulting extruded film with embedded NiTi wires using two different thermoplastic vulcanizate. (b) Composite material made from overmolding the extruded film using a standard injection molding process (a). This technique enables better control of the position of of the NiTi wires. Bending of the composites is achieved due to shape recovery of the prestrained NiTi wires located off of the neutral plane. (c-d) cyclic experiments showing bending actuation of SMA/thermoplastic composites. Shape recovery can be obtained by temperature induced martensitic to austenitic transformation (e.g., via electrical current). The white background shows the expected deflection (vertical axis) of the composite as a function of the current (horizontal axis) that flows through the SMA [1], [15]................................................................................................................ 7 Figure 2 Typical load-displacement curve obtained from the experiments. Inset Figure: Pull-out specimen containing an embedded NiTi wire [1] ....................................................... 10 Figure 3 Schematics of the wire pull-out process at different stages of the pull-out test as described in Figure 6.(a)-(b) Process of debonding during the pull-out force: if the wire is in the twinned martensitic phase, then it first undergoes a stress-induced detwinning with strains up to 4%. The white arrow in the NiTi wire denotes the applied force. Debonding starts in (I). A gap is left in the wake of the debonding front caused by the thinning of the NiTi wire. (c) debonding propagates towards the other end of the specimen until (d) complete debonding is attained at (IV). Insets beneath each figure shows the location where the debonding process zone takes place relative to the entire specimen [1] ................................................................................................................ 11 Figure 4 Maximum pull-out force recorded for NiTi wires with embedded length le = 70 mm (in green squares) and le = 90 mm (in red circles). The mean values for each treatment is denoted with symbols and the standard deviation is indicated with the error bars. The tensile stress is obtained from Equation (9) for all the experimental points. The horizontal dashed line denotes the manufacturer recommended tensile stress of 180 MPa [1] ....................................................................................................................... 13
x Figure 5 (a) Schematic of the axisymmetric FEM geometry. The light region denotes the polymer matrix, the dark region denotes a NiTi wire. Here, le = 70 mm. r is the radius of the wire, R is the maximum radius of the TPO adopted, w is the radial dimension of the TPO relative to the outer radius of the wire, a is the crack length. (b) Experimental uniaxial stress-strain curve (Dotted line) and exponential hardening law for NiTi (Solid line; Equation 10). (c) Schematics of the initially elastic constitutive cohesive law employed to describe the interface behavior. The red double arrow indicates any irreversible post-peak unloading-loading cycle [1] .................................................... 19 Figure 6 Fmax vs. G c . Solid red line: FEM accounting for inelastic behavior of NiTi considering residual stresses. Gray dashed line: FEM accounting for inelastic behavior of NiTi assuming no residual stresses. Green long-dashed line: FEM calculation assuming a linearly elastic behavior for the NiTi. Blue dashed-dot line: LEFM analytical prediction. The horizontal dashed lines denote the minimum and maximum values recorded in Figure 15 and Table 1 (from out pull-out tests). Circular, triangular and square symbols indicate discrete values of Fmax obtained from each set of simulation respectively [1] ..................................................................................................................................... 20 Figure 7 Effect of the increment of residual stresses in the change of the maximum force reached by the pull-out specimen considering different relations between the manufacture and the operative temperatures (0, 100, 150, 250, 350oC). Also, the dotted line shows an approximate case value obtained by simulations where it was assumed linear response of the wire. Based on this the debonding energy was set in order to match the maximum force obtained by the simulations assuming a nonlinear constitutive model of the wire. Good agreement was observed between LEFM calculations and simulations assuming wire linear response [1] ............................................................................................... 23 Figure 8 Gc calculated for NiTi wires with embedded length le = 70 mm (in red squares) and le = 90 mm (in red circles). The standard deviation is used for the error bars. These results were obtained using the FEM simulations considering residual stresses (Figure 18). For comparison purposes, the same values obtained with the LEFM model are included as the filled gray circles [1] ............................................................................................. 26
xi Figure 9 G I and GII vs. distance from crack tip for three different levels of residual stresses (ΔT = 0 C – no residual stresses, ΔT = 100 C and ΔT = 150 C). Gc represents critical strain energy release rate of the interface [2] ........................................................................ 27 Figure 10 A schematic illustration showing the location of the thermocouples, linear variable differential transformers (LVDTs) and pairs of strain gages (at the top and the bottom of the pavement) oriented longitudinal and transverse direction. b and c. Strain development at the bottom of the pavement, of thickness h and width w, when the traffic load from a wheel ( , assumed to be distributed on a circular area of radius “a”) is moving on top and off a strain gage. The stress profile in the longitudinal direction x z , due to traffic loading beneath the wheel, is depicted as well as 1 at the bottom
of the pavement [4], [64] ............................................................................................ 33 Figure 11 Measured t at 1 in. from the bottom with respect to wheel position. a) Lane 2 (opening at 350 psi for a 10 in. thick pavement) and b) Lane 3 (opening at 350 psi for a 5 in. thick pavement). c) t versus number of passes for one of the thick (10 in.) pavements [4], [64] ....................................................................................................................... 37 Figure 12 Measured t at 1 in. from the bottom with respect to wheel position. a) Lane 2 (opening at 350 psi for a 10 in. thick pavement) and b) Lane 3 (opening at 350 psi for a 5 in. thick pavement). c) t versus number of passes for one of the thick (10 in.) pavements [4], [64] ....................................................................................................................... 40 Figure 13 1 at the bottom of the pavement for two extreme cases of subgrade reaction modulus, k 50 lb in3
and 400 lb in3 . For each k , 1 is determined for E t at 8 h (solid lines)
and for E t at 7 d (dashed lines). Experimental values of 1 are shown in symbols for h 5 and 10 in. Horizontal dashed lines indicate f r at different ages [4], [64] ........ 42
Figure 14 Evolution of 1 , at the bottom of the pavement, vs. age up to 7 d for two different pavement thicknesses: (a) 10 in., and (b) 5 in. 1 estimated using the analytical and the experimental approximation on both cases. The main source of error in the calculation of 1 using equation 4 comes from the uncertainty of the position of the strain gages. The 5 in. pavements are the most sensitive to this uncertainty. Horizontal dashed lines indicate f r at different ages [4], [64] ......................................................................... 42
xii Figure 15 Proposed criterion for determination of time of EOT[64]........................................... 44 Figure 16 Maximum pull-out force recorded for NiTi wires with embedded length le = 70 mm (in green squares) and le = 90 mm (in red circles). The mean values for each treatment is denoted with symbols and the standard deviation is indicated with the error bars. The tensile stress is obtained from Equation (9) for all the experimental points. The horizontal dashed line denotes the manufacturer recommended tensile stress of 180 MPa [1] ....................................................................................................................... 48 Figure 17 Solidification concept. From left to right it is showed how differential blocks of hydrated cement engage to the preexisting load-bearing material as tensile displacement is applied. If the material experiences mechanical loading while it solidifies, each block of solidified material will experience different levels of strain .................................. 55 Figure 18 Scheme representing the time evolution of the time-dependent cohesive zone model relative to final parameters of the time-dependent cohesive law. ............................... 58 Figure 19 Example of Weibull distribution of interface Tmax in a typical finite element mesh (a) histogram of Tmax To for the case where
m 10, (b) different intensities of gray indicate
value of Tmax To . (c) Time evolution of Tmax t for a specific time evolution of To , considered in this work to model tensile strength at the element level. Tmax t is represented for different Weibull parameters ( m 5, 10 and 50). A typical distribution of f Tmax To at a specific age is depicted for the Weibull distribution with
m5
as an
example ....................................................................................................................... 61 Figure 20 Non-uniform moisture profile considering that the outer surface in contact with the environment corresponds to the normalized position equal to 1 and the farthest point located at position equal to 0 ...................................................................................... 63 Figure 21 Representation of creep response at different ages using solidification model using a fixed aging law and different sets of viscoelastic properties obtained from two different mixes and creep tests started at different ages. The black line corresponds to simulation of the response of a creep tests started at 1 day, considering viscoelastic properties for the reported Type II [104] ........................................................................................... 64 Figure 22 Aging laws corresponding to Eqs. 30a-b considering the parameters listed in Table 4 ..................................................................................................................................... 66
xiii Figure 23 (a) Schematic of the 2D FEM geometry of restrained mortar. (b) Schematics of the initially elastic constitutive cohesive law employed to describe the interface behavior. The red double arrow indicates any irreversible post-peak unloading–loading cycle. The gray are defined by the irreversible and the unloading lines corresponds to the dissipated cohesive energy .......................................................................................... 68 Figure 24 (a) Tensile stress development caused by nonuniform drying shrinkage for different values of m (5, 20, 50, 1000), (b)-(c) tensile stress map (right) and damage distribution map (left) for m = 5 at 10, 14 and 20 days. As age evolves, the damage zone increases ..................................................................................................................................... 69 Figure 25 1D scheme of the simplified model to describe the effects of nonuniform shrinkage in a solidifying material. represents the general displacement of the vertical bar ...... 71 Figure 26
(a)-(b) Model steps towards the estimation of tensile strength with a previous restrained/unrestrained conditions to induce distributed damage, (c) numerical reduction of strength ( % f t T ) vs. age. Error bars represent numerical results using different random distributions of
ft
. (d) Experimental results [4] of the effects of
developed microscopic damage on strength relaxation caused by drying under restrained conditions ................................................................................................... 74 Figure 27 (a) Typical restrained ring test configuration, (b) the FE model of the ring will represent a slice of the middle plane of the restrained ring, (c) dimensions of the ring test numerical model: a = 140mm, b = 150mm, c = 250mm ............................................ 77 Figure 28 (a) Hoop stress vs. radius at 2, 10 and 20 days, (b) Peq t and E vs. Age, (c) and (e) damage development at 10 and 20 days respectively when VE is removed, (d) and (f) damage development at 10 and 20 days respectively when solidification is considered. m=
20 and To = 2 MPa is considered for all the cases ............................................... 80
Figure 29 (a) Hoop stress vs. radius at 2, 10 and 20 days and (b) Peq t and cumulative damage vs. Age for
m=
5, 20 50 respectively. (c) and (e) damage development at 10 and 20 days
respectively for for
m=
m = 50 (d)
and (f) damage development at 10 and 20 days respectively
20. Equation 30a and VE 1 are considered for all the cases .......................... 82
xiv Figure 30 (a) Hoop stress vs. radius at 2 and 10days, (b)
Peq t and relative cumulative damage vs.
age for different aging rates. (c) and (d) damage development at 2 and 14 days respectively for 0.3 1/day. VE set 1 and
m=
20 are considered for all the cases ....... 84
Figure 31 Hoop stress vs. radius at 30 days. Equation 30a and VE 1 are considered for all the cases ............................................................................................................................ 85
xv
ABSTRACT
Author: Antico, Federico, C. PhD Institution: Purdue University Degree Received: August 2018 Title: Evaluation of Damage Affecting The Mechanical Behavior of Composite Materials Committee Chair: Pablo Zavattieri Laboratory mortar samples and large structures of concrete under tension most likely behave as brittle. Contrary to what is expected from a brittle material, the previously mentioned structures could: 1) develop distributed damage, 2) relax tensile stresses and 3) reduce its macroscopic tensile strength when subject to time-dependent tensile stress gradients. Time-dependent tensile stress gradients can be found in large concrete structures such as pavements and in laboratory samples like the restrained ring test. In those cases, tensile strength reduction means that the deterministic tensile strength obtained from standard tests, e.g. split tensile test, is over predicting its value. Up to now, there is no complete explanation on the mechanisms responsible for the quasibrittle behavior of laboratory size mortar and large structures of concrete under tension caused by timedependent loading conditions. Mortar and concrete are part of the family of cementitious materials which together represent the most manmade produced material worldwide. Therefore, the main motivation of this research is to provide new insights that help bridge some of the remaining gaps in the literature. I aim to contribute to understand the age-dependent behavior and degradation of mortar and concrete at early ages. Based on a review and critical analysis of the state of the art in terms of experimental, analytical and numerical characterization, I proposed and develop of a phenomenological model, where solidification and a cohesive zone model with stochastic strength distribution are implemented in a finite element code to capture the behavior of laboratory size restrained mortar subject to nonuniform shrinkage.
1
INTRODUCTION
1.1
Motivation
Laboratory mortar samples and large structures of concrete under tension most likely behave as brittle. Contrary to what is expected from a brittle material, the previously mentioned structures could: 1) develop distributed damage, 2) relax tensile stresses and 3) reduce its macroscopic tensile strength when subject to time-dependent tensile stress gradients. Time-dependent tensile stress gradients can be found in large concrete structures such as pavements and in laboratory samples like the restrained ring test. In those cases, tensile strength reduction means that the deterministic tensile strength obtained from standard tests, e.g. split tensile test, is over predicting its value. Up to now, there is no complete explanation on the mechanisms responsible for the quasibrittle behavior of laboratory size mortar and large structures of concrete under tension caused by timedependent loading conditions. Mortar and concrete are part of the family of cementitious materials which together represent the most manmade produced material worldwide. Therefore, the main motivation of this research is to provide new insights that help bridge some of the remaining gaps in the literature. I aim to contribute to understand the age-dependent behavior and degradation of mortar and concrete at early ages. Based on a review and critical analysis of the state of the art in terms of experimental, analytical and numerical characterization, I proposed and develop of a phenomenological model, where solidification and a cohesive zone model with stochastic strength distribution are implemented in a finite element code to capture the behavior of laboratory size restrained mortar subject to nonuniform shrinkage. Addressing the linear-nonlinear behavior of materials through appropriate models to obtain the effective properties exceeds the civil engineering field. For example, the fracture behavior affecting wire pull-out forces acting on shape memory alloy wire-reinforced polymer matrix composites which are used mainly as actuators in automotive applications. Two opposite analytical approaches based on: 1) strength of materials and 2) linear elastic fracture mechanics (LEFM), have been applied indistinctively to convert pull-out forces to interfacial strength or adhesion
2 energies. Questions still remain over which of the two approaches mentioned earlier provide a closer prediction of the fracture properties of these composite materials.
1.2
Goals and objectives
The goals of this research are: 1.
Contribute to the damage-mechanical description of materials with nonlinear and agedependent properties by means of a more fundamental understanding of size effects and physical mechanisms involved.
2.
Lead future lines of research in the area of multiscale analysis of materials.
In order to achieve these goals, I propose a set of experimental and semi-analytical strategies summarized in the following list of objectives: 1.
Address the conditions at which this distributed damage happens in pavements opened to traffic at early ages and restrained laboratory size mortar.
2.
Determine if strength reduction and stress relaxation in mortar is caused by distributed damage and if this happens in the actual material to the same extent by means of experimental data to compare with these numerical results obtained from the developed model of restrained mortar presented in this work.
3.
Determine the level of confidence of the cohesive zone model to be employed for damage distribution scenarios such as the one expected in restrained mortar.
4.
Address conditions at which stress relaxes and strength reduces due to distributed damage in mortar under restrained conditions.
5.
Explain the role of solidification, nonuniform drying shrinkage, sample geometry and degree of restraint for stress relaxation and strength reduction due to distributed damage.
6.
Understand limitations (if any) associated with the use of experimental parameters used in the cohesive zone model with stochastic distribution of strength to model restrained mortar.
7.
Regarding the damage-mechanical analysis of wire-reinforced laboratory size composites, determine if the mechanical interaction between the matrix and the wire is the cause of an interface behavior which depends on the cohesive energy.
3 1.3
Organization of the thesis
The thesis is divided in eight chapters. The first chapter of this thesis presents the introduction and motivations for this study, goals and objectives. Chapter 2 presents a combined analytical-numerical study aimed at enhancing adhesion between a NiTi wire and a thermoplastic polyolefin (TPO) matrix in which it is embedded. The extent to which each treatment increased the pull-out force was quantified. Existing theoretical models of wire pull-out based upon strength of materials and linear elastic fracture mechanics are reviewed. Results from a finite element model (FEM), wherein the NiTi/TPO matrix interface is modeled with a cohesive zone model, suggest that the interface behavior strongly depends on the cohesive energy. The FEM model properly accounts for energy dissipation at the debonding front and inelastic deformation in a NiTi wire during pull-out. We demonstrate that residual stresses from the molding process significantly influence mode mixity at the debonding front. This work was already presented in two publications [1], [2]. The purpose of Chapter 3 is two-fold: 1) to examine the current opening strength requirements for concrete pavements (typically a flexural strength from beams, f r ) and 2) to propose a criterion based on the time-dependent changes of I f r which accounts for pavement thickness and subgrade stiffness without adding unnecessary risk for premature cracking. An Accelerated Pavement Testing (APT) facility was used to test concrete pavements that are opened to traffic at an early age to provide data that can be compared with an analytical model to determine the effective I f r based on the relevant features of the concrete pavement, the subgrade, and the traffic load. It is anticipated that this type of opening criteria can help the decision makers in two ways: 1) it can open pavement sections earlier thereby reducing construction time and 2) it may help to minimize the use of materials with overly accelerated strength gain that are suspected to be more susceptible to develop damage at early ages than materials that gain strength more slowly. This work was already presented in two publications [3], [4]. Chapters 4 and 5 present a review of brittle/quasibrittle behavior of mortar and concrete, and what is know about the mechanisms responsible for these behaviors. It includes a discussion regarding the counterintuitive behavior of some structures of concrete and mortar. A description of the model
4 of solidification and a damage with stochastic strength distribution was implemented in a finite element code to capture the previously behavior of restrained mortar is presented. Chapter 6 presents an explanation of the reasons why stress relaxation and strength reduction of mortar subject to nonuniform shrinkage occurs. It includes a set of numerical examples of laboratory size mortar under restrained conditions in order to support the previous explanation. Chapter 7 presents examples of the restraint ring test to show the value of the proposed model to illustrate the effect that solidification and strength distribution have on distributed damage that causes stress relaxation and strength reduction of mortar under restrained conditions. A sensitivity analysis of the relevant time-dependent properties of mortar is presented and its impact on its performance presented. Chapter 8 presents the final comments, recommendations, conclusions and proposed future work related to the analysis and modeling of laboratory size mortar under restrained conditions and a nonuniform drying shrinkage.
5
ADHESION OF NICKEL–TITANIUM SHAPE MEMORY ALLOY WIRES TO THERMOPLASTIC MATERIALS: THEORY AND EXPERIMENTS
2.1
Introduction
As part of the learning process related to this thesis, this chapter describes the development and implementation of a damage model to address damage evolution in fiber composite materials. Specifically, the mechanics of debonding of the latter subject to pullout forces is analyzed using both, analytical and numerical approaches. In particular for this thesis, the modeling of damagemechanical behavior of shape memory alloy (SMA) wire reinforced polymer matrix composites worked as a way to test an initial version of the cohesive zone implemented in the finite element code (FEAP) that later will be used to model cementitious materials.
2.2
Research significance
Shape memory alloy (SMA) wire reinforced polymer matrix composites for actuators have gained significant importance during the last decade [5]–[8]. The capabilities of these smart composites, especially for their impact resistance, high-temperature mechanical properties, shape control, fatigue resistance, and vibration and buckling response, have been demonstrated in Refs.[9]–[13]. Considerable effort has been focused on applying knowledge acquired in laboratory settings directly to industrial applications[11]. Shape recovery makes these alloys good candidates for actuators in transportation industries since they can be activated via applied stress and temperature change and do not substantially contribute to vehicle mass. Typical automotive applications employ thermoplastic polymers, such as Thermoplastic Vulcanizate (TPV) and Thermoplastic Olefins (TPO), which require manufacturing processes such as injection molding, compression molding or extrusion techniques. Although, the high temperatures and pressure levels required for these processes make the embedding of SMA wires particularly challenging, a recent work has shown that an efficient procedure to fabricate these composites is achievable by a combination of extrusion and injection molding [14]. Figure 1a shows composite films made of a Nickel-titanium (NiTi) wire-reinforced thermoplastic vulcanizate using extrusion molding. A subsequent step where the film is over molded using standard injection molding yields a composite material with
6 the NiTi wires located at very specific locations, typically eccentric from the neutral plane of the composite (Figure 1b). These types of composites are ideal for bending actuation in automotive applications. Bending can be achieved by inducing shape recovery to pre-strained NiTi wires that are embedded in the polymeric matrix at different locations through the thickness of a thin specimen. If the NiTi wire is bonded to the matrix, it transmits forces to the matrix inducing a distributed bending moment to the entire composite. Figure 1c and 1d show the type of bending actuation that can be obtained from these materials under thermal cyclic loading. Non-trivial challenges associated with actuator fabrication processes and a lack of optimal component designs are barriers to wide-scale implementation of SMA materials in transportation industries in general. A significant barrier is load and displacement transfer from NiTi shape memory wires to a polymer matrix; this is primarily controlled by chemical and mechanical bonding at the interface between these two materials. Nickel-titanium shape memory alloys typically have low surface energies which limits their ability to effectively bond to other materials. To make matters worse, TPV and TPO also have very low surface energies. Poor interfacial adhesion due to the lack of sufficient chemical and/or mechanical bonding between NiTi and polymeric materials results in premature failure and insufficient actuation response.
7
Figure 1 SMA-reinforced thermoplastic composites for actuation devices. (a) Resulting extruded film with embedded NiTi wires using two different thermoplastic vulcanizate. (b) Composite material made from overmolding the extruded film using a standard injection molding process (a). This technique enables better control of the position of of the NiTi wires. Bending of the composites is achieved due to shape recovery of the prestrained NiTi wires located off of the neutral plane. (c-d) cyclic experiments showing bending actuation of SMA/thermoplastic composites. Shape recovery can be obtained by temperature induced martensitic to austenitic transformation (e.g., via electrical current). The white background shows the expected deflection (vertical axis) of the composite as a function of the current (horizontal axis) that flows through the SMA [1], [15]
Much of the extensive literature on shape memory materials focuses on quantifying material properties [16] with minimal attention to adhesion in actuators. Poor interfacial bonding due to low surface energies can be improved through mechanical interlocking [6], [17]–[22], chemical treatments [23]–[25], or by simply neglecting adhesion and facilitating the sliding of the SMA material and anchoring it at the ends of the specimen (e.g. by inserting it in tubes with internal low adhesion/friction surfaces) [26]. Adhesion enhancements via mechanical interlocking in SMA
8 actuators have been investigated with only the most rudimentary means such as hand sanding, sandblasting or wire twisting to alter otherwise smooth surface [17]–[19]. While these approaches help to alter the smooth surface of the SMA and improve mechanical interlocking between the wire and the matrix, these methods are either impractical or prone to leave impurities and debris. The existing literature does not address repeated actuation [13] or fatigue of devices using SMA. However, the degradation of polymers through repetitive heating of a SMA has been identified as a critical obstacle to increasing the durability of these materials [20], [27]. Chemically functionalizing NiTi wire surfaces to enhance adhesion to polymeric materials is attractive from the standpoint that additional mechanical fixtures can be avoided. This idea, which is the basis of aluminum sheet conversion coating treatments with chemical coupling agents, was applied by Smith et al. [25] to improve adhesion between NiTi wires and Plexiglass. They noted improvements in adhesive strength of up to 100% with silane coupling agents. While this is encouraging, silane coupling agents must be hydrolyzed before use, and the hydrolyzed solutions have finite lifetimes. Chemical coupling agents function in several ways. They can directly bond to a polymer matrix by reacting with functional groups in the polymeric matrix; they can incorporate themselves into the polymer matrix due to the similarity of their structure to that of the polymer; they can interact through van der Waals bonding with the polymeric matrix. Oxide surface functionalization using a coupling agent typically requires hydroxyl groups (-OH) on the NiTi surface. Phosphorus-containing compounds [28] and organosilane-coupling agents react with oxide/hydroxide groups and bind to metal oxide surfaces via a condensation reaction. A heating step may be employed after treatment with the coupling agent in order to assure complete condensation. In this chapter, we explore two experimental methodologies for improving adhesion between a NiTi SMA in wire form and a polymeric matrix [29]. These are: (i) functionalizing a NiTi wire surface with a chemical conversion coating or coupling agent; (ii) application of a surface microgeometry to a NiTi wire surface for enhancement of mechanical interlocking between the wire and the polymer matrix. Details about the surface preparation and wire embedding process can be found in [29]. For this purpose, theoretical and computational models are developed and employed to examine and quantify NiTi wire/ TPO matrix interactions obtained from mechanical pull-out tests of wire-polymer adhesion due to the surface treatments.
9 2.3
Pull-out test
In this work, the pull-out force is defined as the applied force to initiate debonding between a NiTi wire and TPO matrix. This test methodology has been used to measure interfacial adhesion in fibrous composites, for example, and the mechanics of debonding is well understood [30]. The experimental data used for this study considered the wires are tested at room temperature [15]. It is believed that this type of tests represents the worst-case scenario for the interface since the wire thinning caused by the stress-induced martensitic detwinning of the NiTi, contributes to an opening mode in the interface helping prevent any additional energy dissipation mechanisms that may be triggered by mechanical interlocking due to the roughness of the interface.
2.3.1 Test configuration The inset in Figure 2 is a schematic of the pull-out test and typical force-displacement curve. This configuration ensures that the specimen does not experience lateral compressive stress.
A
monotonically increasing force, F , is applied to the crimp at the end of the wire until debonding is triggered. This test effectively captures the peak (or maximum force, Fmax ) to pull the wire out of the polymer. Each specimen (i.e. a NiTi wire embedded in TPO) was first cut through its gage section exposing one end of the wire. The specimens were cut to have the same embedded wire length l e = 90 mm. However, a few samples with l e = 70 mm were also measured. The free end of the NiTi wire was loaded with a monotonically increasing force. The following final set of TPO specimens with embedded NiTi wires were investigated in the pull-out tests: (1) As-received wires (5 specimens with l e = 90mm and 7 specimens with l e = 70 mm). (2) Hand-sanded wires (4 specimens with l e = 70 mm and 4 specimens with l e = 90mm). (3) Pre-treated wires with acid only (oxide/hydroxide layer, see [1]) (5 specimens with l e = 90mm). (4) Wires with fresh oxide/hydroxide layer followed by treatment with 1% phenylphosphonic acid (PPPA) (8 specimens with l e
= 90mm and one specimen with l e = 70
mm). (5) Wires acid etched and then treated with 5% PPPA (8 specimens with l e = 90mm). Three distinct regions can be observed in the force-displacement response: an initial portion in which the force increases to point I, a plateau in the force between points II and III, and a final
10 decaying portion down to zero force. During the experiments, the translucency of the TPO allowed observation of debonding of the NiTi wire from the polymer matrix [1]. Wire deformation and debonding started at the wire/matrix interface from the force application end before the force achieved its maximum value in Figure 2. The axial deformation observed in the wire at force levels equal or greater than Fmax indicates that the wire was undergoing detwinning. Finally, the wire breaks free of the TPO matrix (point IV) as the debonding process becomes unstable. Thinning of a wire in the transversal direction also contributed to wire debonding from the TPO.
Figure 2 Typical load-displacement curve obtained from the experiments. Inset Figure: Pull-out specimen containing an embedded NiTi wire [1]
The progression of the debonding process is schematically depicted in Figure 3. Each inset in Figure 3 shows the relative position of the image above it with respect to the entire NiTi/TPO specimen). The region where a NiTi wire debonded from the TPO is referred to as the debonding process zone (denoted by the light gray region labeled “Debonding process zone” in Figure 3a).
11 Figure 3b depicts the debonding process zone advancing deeper into the TPO as the applied pullout force increases during loading stages I-II. Meanwhile, large axial deformation in the wire is still observed near the debonding process zone. The debonding front finally reaches the specimen end opposite that of the applied force as shown in Figure 2 (inset) and 3c, with complete debonding shown in Figure 3d.
Figure 3 Schematics of the wire pull-out process at different stages of the pull-out test as described in Figure 6.(a)-(b) Process of debonding during the pull-out force: if the wire is in the twinned martensitic phase, then it first undergoes a stress-induced detwinning with strains up to 4%. The white arrow in the NiTi wire denotes the applied force. Debonding starts in (I). A gap is left in the wake of the debonding front caused by the thinning of the NiTi wire. (c) debonding propagates towards the other end of the specimen until (d) complete debonding is attained at (IV). Insets beneath each figure shows the location where the debonding process zone takes place relative to the entire specimen [1]
12 2.3.2 Test results Table 1 reports the pull-out forces obtained in the experiments. Due to the large number of experiments, the table only shows mean force values associated with each treatment. The mean maximum pull-out forces and equivalent tensile stress (with their standard deviations) were plotted in Figure 4. The red dots denote those pull-out specimens with l e = 90mm, whereas the green dots are associated with the shorter pull-out specimens ( l e = 70mm). This data shows that the maximum mean pull-out force for all of tests was within the standard deviation regardless of NiTi wire surface treatment. However, as seen in Table 1, the wires treated with 5% PPPA required the largest pull-out force achieved in a single test. In contrast, the untreated wires required the smallest pull-out forces, although a single specimen with an untreated wire reached a maximum force close to that of the treated wires. The forces required with the treated wires were, in general, higher than those associated with the untreated wires. The pull-out results suggest that the various treatments lead to an improvement in adhesion of the NiTi wires to the TPO matrix.
Table 1 Test matrix with mean forces recorded for each treatment for pull-out specimens with l e = 90mm. The untreated-wire specimens require the lowest pull-out force, while those treated with 5%PPPA require the highest pull-out forces. The maximum value achieved with the untreated NiTi wires is even lower than the maximum values from tests of the treated-wires. Number of Mean Force ± Min. Force Max. Force specimens tested
stand. dev.
recorded
recorded
( l e =90mm)
(Newtons)
(Newtons)
(Newtons)
Untreated
5
16.2 ± 5.1
6.9
22.2
Hand-sanded
4
22.7 ± 2.0
19.2
24.3
Acid
5
21.0 ± 2.8
18.9
26.4
1% PPPA
8
22.0 ± 2.6
18.1
25.7
5% PPPA
8
22.4 ± 3.1
18.0
28.0
Treatment
13
Figure 4 Maximum pull-out force recorded for NiTi wires with embedded length le = 70 mm (in green squares) and le = 90 mm (in red circles). The mean values for each treatment is denoted with symbols and the standard deviation is indicated with the error bars. The tensile stress is obtained from Equation (9) for all the experimental points. The horizontal dashed line denotes the manufacturer recommended tensile stress of 180 MPa [1] Wire pull-out forces typically need to be interpreted through appropriate models to obtain the effective wire/matrix properties. Previous studies have used analytical models based on strength and linear elastic fracture mechanics (LEFM) to convert pull-out forces to interfacial strength or adhesion energies [16], [19], [21]. In the following section, we briefly review relevant analytical models and discuss the extent to which they apply to the pull-out tests in this study. Both analytical expressions based on linear elastic fracture mechanics (LEFM) and strength of materials are presented in Section 2.5.1. Finally a finite element (FE) model that represents the mechanical response of the pull-out specimens under quasi-static loading is used to analyze the debonding process (Section 2.5.2).
2.3.3 Analytical models According to Penn and Lee [31], wire debonding can exhibit two distinct behaviors: (i) ductile behavior, where the interface has a uniform stress distribution, i.e. a large fracture process zone, and yielding of the matrix is the main mechanism of deformation; (ii) brittle behavior, where the interface has a small debonding process zone compared with the dimensions of the specimen (which is described by LEFM), and a non-uniform stress distribution along the NiTi wire/matrix
14 interface. When ductile interface debonding takes place, Fmax is proportional to the embedded wire length Fmax l e . Accounting for the contact area between the wire and a polymeric (e.g. TPO) matrix gives
(4) where
is the wire radius, and iy is the shear strength of the wire/matrix interface or polymeric
matrix (whichever is smaller). In brittle interfaces, where plastic deformation around the interface is limited, Fmax is not proportional to le because of the nonlinear shear stress distribution around the interface: this tends to very large values near the ends of the wire/matrix interface. Following Piggott [32], Penn and Lee [31] showed that if a failure criterion based on the energy of the interface is assumed, then the following expression provides a valid estimate of the critical strain energy release rate, G c [J/m2], or work of separation, in terms of Fmax
(5) where, for the present case,
Ef
is Young’s modulus of a NiTi wire, s l e a r ,
of a debonded initial region, le is the embedded wire length when a 0 , and
n
a
is the length
is defined as
(6) where E m are m are the Young’s modulus and Poisson’s ratio of the matrix respectively, and R is the radius of a cylinder of polymeric material that surrounds the wire. To estimate a lower bound for G c (See Equation 5), R is approximated as the minimum dimension of the cross section of the specimen (in the present case, this is 4 mm which corresponds to the thickness of the specimen). For very short values of le,
ns
75 mm, average perimeter > 1m) are in the range of an element expected to have a brittle failure as described earlier. Yet, previous works stated that if the restraint provided by the inner ring is high enough, mortar will start developing microscopic damage and eventually radial cracks. In general, those cracks occur in an unstable fashion [76], [85], [96], [97]. Related to the latter, using an estimation of the stress development over time in the mortar mix using continuum (no damage) linear elastic models with homogenous properties, Weiss and coworkers made observations on the effect of microcracking on tensile stress development in restrained mortar rings by means of the residual interface pressure between the steel ring and the mortar, Pres (see Figure 16e) [97]. The advantage of this approach is that Pres only requires the elastic properties of the elastic ring (which is typically steel) and radial deformation extracted from strain gages in the elastic ring. On the contrary to what it should be expected from a brittle behavior, the information provided by Weiss and coworkers indicate that experimental results present substantial differences with the analytical estimation of the elastic stress exerted by the mortar ring and residual pressure, Pres , see Figure 16d [97]. The previous description adds other challenges in the identification of brittle or quasibrittle behavior of mortar. For the particular case of the ring test, as a time-dependent test, other mechanisms should be taken into account (i.e. aging and viscoelasticity) as part of the stress relaxation observed in restrained ring tests. This was suggested in a previous work and is the main subject in this work[76]. Overall, these contradicting evidence on the behavior of mortar and large scale concrete under tension suggest that there are intrinsic time and history dependent mechanisms combined with a time-dependent strain field in direct relation to its stress and strain history of these materials. In other words, at early ages (days and/or months after casting) when cement is still actively hydrating, the development of damage, tensile stresses and tensile strength could be affected by the restrained of mortar and its rheological, aging behavior and defect distribution.
52 In this work, an analysis of the mechanisms involved in damage development of restrained mortar under tension and a numerical approach is presented to describe the tensile stress relaxation and strength reduction at early ages. The role of the time and history dependent mechanical behavior of mortar by means of the approach known as solidification of non-aging constituents (solidification) is considered in this study for its physical understanding of the aging-viscoelastic phenomena of cement based materials and its simplicity to be applied as an analytical and numerical tool to predict stress development in mortar and concrete [98]–[101]. As for the damage formation, a statistical, time-dependent cohesive zone model is implemented, whose spatial distribution depends on a Weibull distribution and gain of strength is considered as hydration evolves in time. Different geometries representing mortar under restrained conditions are presented to demonstrate that distributed damage formation at early ages might affect the stress profiles of restrained mortar subject to drying shrinkage at early ages. This work suggests that tensile stress gradients are responsible for distributed damage caused by the combination of the solidification behavior of mortar under nonuniform drying shrinkage. The results from this work will enforce what was previously referenced about the importance of considering solidifying behavior of mortar and observations regarding stress distribution and microscopic damage on restrained mortar. Overall, this work shows that under specific conditions, the expected brittle behavior of laboratory size mortar (or large scale concrete) under tension could be quasibrittle as a results of the time-dependent behavior combined of this material combined with a nonuniform shrinkage.
53
TENSILE STRENGTH REDUCTION OF MORTAR CAUSED BY SOLIDIFICATION AND NONUNIFORM RESTRAINED DRYING SHRINKAGE
5.1
Understanding conditions for brittle and quasibrittle behavior of mortar and concrete
As described by Bazant et al. [102], the basic creep behavior of concrete is the time-dependent strain of mortar or concrete caused by sustained stress in absence of moisture movements and it is hypothesized to reside in the response of the calcium silicate hydrate (C-S-H) particles to stress [58]. As a consequence of hydration C-S-H particles engage to the preexisting load-bearing matter volume and the creep response of mortar and concrete is age dependent [17], [30]. From the macroscopic point of view, the rise of the elastic modulus and strength (both compressive and tensile) of the material with age is proportional to the incremental growth of load-bearing matter [31]. Based on the latter description, restrained concrete subject to volumetric changes, such as drying conditions, are subject to both: 1) the time-dependent strain and 2) rise of the elastic modulus and strength in time having a direct impact on the mechanical response of it. In particular, the previous time-dependent behavior of C-S-H under stress is most likely described the solidification theory proposed by Bazant and coworkers [98]. The solidification theory proposes that the material parameters for creep are age-independent but the volume fraction of the ageindependent material increases with age [98]. This model provides good prediction of creep response at early ages (i.e. less than 28 days) on mixtures with water to cement ratios above > 0.4 [98], [100]. In addition, the solidification theory has the advantage of decoupling viscoelastic and aging behaviors to address the importance of each one in the mechanical response of cementitious materials [103]. On the other hand the solidification model requires the characterization the nonaging creep using a rate-type description consisting of first-order differential equations as indicated by Bazant and coworkers [102]. For this purpose, the proposed viscoelastic model is based on a classical Maxwell chain model with age independent properties to represent the linear creep combined with aging the behavior of mortar as an aging viscoelastic assuming no damage that for the rest of this work will be identified as the solidification model [104]. Challenges remain on how to characterize experimentally the viscoelastic parameters for the solidification model.
54 The typical test to characterize the viscoelastic parameters is the creep test but the intrinsic nature of mortar and concrete makes it impossible to isolate the aging response of the creep response. Bazant and Wu [104] presented and extensive compilation of aging-viscoelastic responses of different mixtures of cementitious materials using creep tests at different ages assuming that: 1) the stress is less than about 0.5 of strength in order to avoid significant microcracking that could affect the mechanical response, and 2) the water content is constant. Later in this work an analysis of this problem will be presented. Lastly, for a more accurate representation of the real phenomena, strength of mortar at early ages will be considered as time-dependent as described in many previous works [76].
5.2
Constitutive model
5.2.1 The solidification model Currently, solidification model proved to be effective describing mechanical behavior of mortar under restrained conditions at stress levels below 50% where the presence of significant microcracking is not likely to occur [105]. For these reasons, the constitutive behavior of a material like mortar, with no damage, is represented by the solidification theory in this work. The aging part modeled by the solidification theory, [106], is described as a succession of stress-free hydrated cement volume fraction, i , at the moment of engagement to the preexisting load-bearing material, t , as shown in Figure 17.
55
Figure 17 Solidification concept. From left to right it is showed how differential blocks of hydrated cement engage to the preexisting load-bearing material as tensile displacement is applied. If the material experiences mechanical loading while it solidifies, each block of solidified material will experience different levels of strain
The viscoelastic behavior of the non-aging blocks is represented by the classical Maxwell chain model [107], [108]. For the 1D example showed in Figure 17, the incremental stress of relaxation
t associated to a block of solidified matter that engages to the preexisting load-bearing material is described by Eq. (19). t
t i E t t 't 'd
(19)
ti
Where i and Et t ' are the differential volume of solidified matter and the relaxation modulus of solidified matter that engages to the preexisting load-bearing material at time t i , from setting, respectively; t ' is mechanical the strain rate. The multiaxial representation of the viscoelastic phenomena is commonly expressed in terms of the volumetric and deviatoric stress, pt , st and strain, t , et , respectively. Materials, like mortar, are particularly sensitive to drying shrinkage [109]–[111]. Consequently, solidification theory applied to mortar would require accounting for volumetric changes, associated to drying shrinkage, represented by sh t ' . Therefore, the analytical solution for this problem may be described as:
56 t
sij t Gt t 'eij t ' t 'dt '
(20)
0
t
. pt K t ' 3sh t ' t 'dt ' 0
(21)
Where, Gt t ' and K are the shear relaxation and bulk modulus respectively, and
.
eij t ' , t '
and
sh t ' are the deviatoric, volumetric and free-strain as a function of time, t ' , respectively. This
constitutive model assumes that only the deviatoric stress and strain are involved in the viscoelastic process and therefore K and Poisson ratio are constant parameters of the model. For the latter, many works have analyzed the effects and measurement of changes of Poisson ratio in time and the general conclusion is that Poisson ratio can be assumed to remain constant in time in general for concrete and mortar [112]–[117]. Equation (20) and (21) represent a generalization of a Maxwell material and Gt t ' can be discretized by means of a Prony Series [104], [112], [118]– [122] : Gt Go o
N
t
exp i
i 1
(22)
i
where, Go is the relaxation modulus @ t 0 and the pairs i , i represents the relaxation shear modulus coefficient and the relaxation time respectively[45], [104], [107], [123].
5.2.2 Stochastic, time-dependent damage model A cohesive zone model (CZM) is used to model distributed damage to evaluate its impact on stress and strength development of a brittle-solidifying material like mortar under tension using an explicit time integration scheme to integrate the system of ordinary differential equations. As a heterogeneous material, mortar is made of fine aggregate and cement paste acting as inclusions and matrix respectively. Damage is assumed in this work to occur at interfaces of bulk elements modeled through cohesive zone elements; no distinction is made between matrix and aggregate damage. This is acceptable if the intent of this model is to predict macroscopic performance of a composite (engineering perspective)[124].
57 The CZM has been widely used in concrete, mortar and other brittle materials to address crack initiation and propagation [53], [125]–[127]. As for the author knowledge, CZM was used previously to address the effect of distributed damage caused by drying conditions on stress development considering a numerical model representing asphalt as a two-phase system [128]. In a similar way, CZM has been also used to quantify the effect of distributed damage on the residual reaction force of a solder bump subjected to cyclic loading conditions represented by a two-phase numerical model as well [129]. In a previous work, Zavattieri used a stochastic CZM to model damage distribution effects in relation to grain boundary strength and toughness [53].
5.2.3 Time-dependent cohesive law A potential-based triangular-shaped, initially-elastic cohesive zone model is implemented in this work to model distributed damage initiation and propagation prior to macroscopic crack formation under tension. The cohesive model formulation is based on a non-dimensional effective displacement jump un n that accounts for current normal displacement jump, u n , and the normal displacement jump at failure, n . The critical effective displacement jump, cr t , at which the time-dependent normal traction of the cohesive zone model, Tn t , matches the time-dependent tensile strength of mortar at the interface, Tmax t , defines the beginning of the softening regime (see Figure 18). Tn t is expressed as: Tn t
1 * un Tmax t * n t 1 cr t
(23)
Where * maxmax , , assuming initially max cr if cr and max if cr . Figure 18 shows and example of the initial, ti , and final, t f , loading and post peak variation of Tn t as a function of .
58
Figure 18 Scheme representing the time evolution of the time-dependent cohesive zone model relative to final parameters of the time-dependent cohesive law.
The cohesive model of this work is based on two material parameters, Tmax t and fracture energy, or work of separation, in mode I, GC , and are related through n t described by Eq. (24). n t
2Gc
Tmax t
(24)
For small mortar structures (e.g. ring samples), D lFPZ , failure is dominated by strength rather than fracture energy [130]. This is related to the relative size of the fracture process zone and the characteristic geometrical length scale of the problem [86], [131]. According to this, the damage model in this work will account for time-dependent behavior of Tmax t to address its impact on damage initiation and growth, while GC will be kept constant.
59 5.2.4 Computational aspects of the cohesive model The slope of the initially elastic behavior of the cohesive model, s , defined between 0 cr is selected such that the wave speed of the material defined by the bulk elements is not affected by the interface elements (see Figure 18). As
s
grows, the wave speed of the numerical model with
interface elements approaches the speed of the numerical model without interfaces. According to the traction-separation law of the cohesive element in this work, s
Tmax t n t cr t
s
is given by: (25)
As described by Espinosa and Zavattieri [53], this slope works as a penalty parameter. The authors proposed a maintain fixed
s
by adjusting cr t to the changes of Tmax t and n t ruled by the
time-dependent mechanical behavior of mortar as presented in the scheme representing the time evolution of the time-dependent cohesive zone model described previously is depicted in Figure 18. Failure under tensile loading is typical mode of failure of mortar [132]. For this reason, the cohesive implemented in this work was allowed to address damage only in the normal direction. As for the shear (mode II) shear strength, max , is max Tmax and only failure in mode I is allowed.
5.2.5 Stochastic effects modeling Mortar microstructure is a random distribution of unhydrated and hydrated cement particles with different phases, air voids, chemically and not chemically combined water and aggregates. This complex and heterogeneous microstructure of different material phases and types are expected to affect damage-mechanical macroscopic behavior and properties. As in any composite material, modeling of damage of mortar require a representation of its defects and heterogeneities which affect its local mechanical properties [133]–[135]. The stochastic distribution of phases and particles leads to consider a statistical space variation of the strength of mortar. For these reasons, the stochasticity of the micro damage process distribution within the spatial domain of the numerical model is considered in this work. Specifically, previous studies [136], [137] offered theoretical justification of the use of Weibull theory representing microscopic flaws or microcracks and its impact on the macroscopic strength response of materials. Strength distribution of ductile materials is expected to be narrow compared to the strength distribution of brittle/quasibrittle, e.g.
60 cementitious materials, is very broad with a large tail on the hit-strength side that can be explained by a statistical distribution called the Weibull distribution. This distribution is ruled by the Weibull parameter,
m,
which is a measure of the variability of the strength of the material. Generally,
m=
3–10 for the case of brittle ceramic samples [54], for cementitious materials like mortar and concrete, this range is extended up to values between 20-50 [138]. The Weibull distribution provides a stochastic representation of defects of brittle materials that has been studied in depth previously and it is representative of the damage behavior of mortar and concrete [138]. The statistical distribution of Tmax t of mortar is represented in this work using the Weibull distribution theory [139]. The cumulative failure probability, pTmax , of the selected test specimen as a function of the average instantaneous failure strength, Ts , and
m
is described by Eq.(26).
pTmax 1 e V Tmax t VsTs t
m
(26)
Where V is the volume of two triangular elements attached to the cohesive zone element and V s is the volume of the specimen used to determine Ts . The scaling factor, V Vs , in Eq. 26 was explained by Zhou and Molinari [127] as there is a size dependence of Tmax t at the element level to the size of the sample at which Ts was obtained from experiments. For the rest of this work, the average strength at the element level, TsVs V , is defined as To Tmax t in this work represents the strength at the element level (triangular elements of 1 mm side in this work).
Figure 19 Example of Weibull distribution of interface Tmax in a typical finite element mesh (a) histogram of Tmax To for the case where
m 10,
(b) different intensities of gray indicate value of Tmax To . (c) Time evolution of Tmax t for a specific time evolution of To , considered in this work to model tensile strength at the element level. Tmax t is represented for different Weibull parameters ( m 5, 10 and 50). A typical distribution of f Tmax To at a specific age is depicted for the Weibull distribution with m 5 as an example
61
62 The stochastic spatial distribution of Tmax t depends on the evolution of To and
m . As an example,
Figure 19c shows a scheme of the time evolution of Tmax t for a specific time evolution of To at the element level and for different Weibull parameters ( m 5, 10 and 50) showing that as reduces, the spread of Tmax t is larger. On the contrary, as
m
m
increases, the spread of Tmax t tends
to the deterministic solution of the problem, To . Figure 19 also shows the distribution of the Weibull probability density function, f Tmax To , corresponding to
m 5, for a specific age. Figure
19b presents an example where each interface element is assigned with a different value of Tmax To . Also, the histogram for Tmax To is shown in Figure 19a.
5.3
Boundary conditions of mortar at early ages
5.3.1 Drying shrinkage characterization The damage-mechanical model presented will be used to study the role of shrinkage, sh t , on the mechanical response and strength evolution of restrained mortar. sh t in mortar at early ages associated to autogenous, thermal and drying shrinkage has been studied by Weiss and co-workers [76], [85], [96], [97], [140]–[146] and Grasley [101], [147]. These previous works showed how sh t is mostly related to the evolution of the moisture profile, associated to an exposed surface
under controlled environmental conditions. Weiss showed that the relationship between sh t and relative humidity, RH 50% , can be considered as linearly proportional [141]. According to this, sh t and RH in this work are related by a constant shrinkage coefficient, shconst , as described
in Equation 27: sh t shconst.RH
(27)
In particular, the role of nonuniform shrinkage profiles, caused by nonuniform drying conditions, on the development of stress gradients is of special interest for being frequent within structures. Experimental observations indicate that stress development and AE emission when substantial cracking had not occurred in restrained mortar is likely due to moisture gradients that cause microcracking. Until now the impact of microcracking on stress and strength development of mortar have not been addressed. Different experiments on mortar under restrained conditions attempted to reproduce nonuniform shrinkage profiles using linear and nonlinear geometries to
63 address its mechanical implications on mortar [4], [148]. In the numerical models in this work, a experimental drying profile similar to the ones measured in experiments on relatively small volumes of mortar will be modeled in two different numerical cell geometries. It is considered that the outer surface of each sample is exposed to drying (see Figure 20, normalized position equal to 1). The later causes drying shrinkage to occur more rapidly on the outer surface of the numerical cell compared to the inner surface (see Figure 20, normalized position equal to 0) that loses water humidity more slowly[146]. Earlier works considered that a diffusion-controlled process is similar to that used in thermal analysis [146][96]. In addition, Moon et al. fitted experimentally the relative humidity distribution, RH , depending linearly to the diffusion coefficient [146] which is valid specially for early ages [146]. According to this, a linear diffusion function (Equation 28) is used in this work, similar to physical and analytical tests performed in previous works [101], [146]: r RH (r , t ) RH I RH I RH S erfc 2 Dc .t
(28)
Where, RH (r, t ) is the relative humidity at a depth r , measured from the drying surface,
t
is the
drying time, erfc is the complementary error function, RH I = 100% is the internal relative humidity, RH S = 50% is the relative humidity at the outer surface of the specimen, and Dc = 1.65x10 9 m 2
s
is the diffusion coefficient of mortar. Figure 20 presents the spatial evolution of
RH ( x, t ) for different ages up to 20 days.
Figure 20 Non-uniform moisture profile considering that the outer surface in contact with the environment corresponds to the normalized position equal to 1 and the farthest point located at position equal to 0
64 5.3.2 Drying shrinkage characterization As mentioned earlier, the solidification theory of hydrating cement has the advantage of decoupling viscoelastic and aging behaviors from each other, [103], as long as no significant microcracking occurs [113]. Experimentally, the non-aging material properties for the viscoelastic model of a material are extracted from a creep test [104]. The more sensitive the test is to a specific parameter, the more effective to use test to address those parameters. For this purpose, the first part of this section test the sensitivity of the creep test to different sets of viscoelastic parameters using an aging law in the solidification model. Bazant and Wu [104] presented a compilation of aging-viscoelastic responses of different mixtures of mortar and concrete using creep tests at different ages. For this analysis two mixes using with similar proportions using different cements (i.e. general purpose Type I cement and a low hydration heat, Type IV cement) were selected [104]. Specifically, viscoelastic parameters from the creep tests of each mix, started at 1 and 28 days were selected to use as an input of the solidification model [104].
Figure 21 Representation of creep response at different ages using solidification model using a fixed aging law and different sets of viscoelastic properties obtained from two different mixes and creep tests started at different ages. The black line corresponds to simulation of the response of a creep tests started at 1 day, considering viscoelastic properties for the reported Type II [104]
65 Figure 21 presents the experimental measurement of creep compliance reported by Bazant and Wu [104]. The three shadows describe the envelope of creep compliances of different viscoelastic sets reported earlier. As presented in Figure 21, different concrete mixes can register significantly different creep responses (up to 25% difference for the selected input parameters) indicating that the creep test is sensitive to changes of viscoelastic parameters. For the remaining of this work, a set of viscoelastic measurements extracted from typical types of mortar previously reported, [104], are analyzed and used in this work (Equation 22) to provide insights on the importance of the viscoelastic parameters on the mechanical and damage response of restrained mortar. Table 3 presents the coefficients for each set of viscoelastic parameters described in Equation 22:
Table 3 Viscoelastic coefficients for sensitivity analysis extracted from [104] VE Set 1 2 3
Parameter
0
1
2
3
4
5
6
7
i [day]1
0.1117 0.0042 0.0008
0.005 0.1279 0.0795 0.0608
0.05 0.1020 0.0829 0.0678
0.5 0.0916 0.1219 0.1158
5.0 0.1142 0.1921 0.2106
50.0 0.1605 0.2273 0.2652
500.0 0.1817 0.1976 0.2020
5000.0 0.1104 0.0946 0.0770
i
A N-term (N = 8) Prony series is used in this work to represent the viscoelastic behavior of mortar considering the relaxation time of the eight element of the Maxwell chain as infinite (i.e. a spring with no dashpot) as shown in Table 3.
5.3.3 Solidification characterization Evolution of hardened properties, such us damage and mechanical, of mortar are driven by the growth of hydrated cement volume fraction [149]. These properties can be represented by the aging function, t , as described in Equation 29: Bo B t
(29)
Where Bo and B stand for the instantaneous and the asymptotic value of a specific property of mortar respectively. To study the sensitivity of the aging function considered in the solidification
66 theory, three evolutions of t were selected suggested from previous works to describe mortar [96], [101] and presented in Equations 30a and b: 1 exp C1 t t C2 .t 1 C .t 2
(a) (30)
(b)
Figure 22 Aging laws corresponding to Eqs. 30a-b considering the parameters listed in Table 4 Figure 22 shows the evolution of Equation 30a-b using the parameters defined on Table 4. In addition to the previous, Table 4 presents the mechanical and damage parameters used in the example presented in this work later.
Table 4 Aging, mechanical and damage parameters used in this work C1 [101] [1/day] 0.30
C2 [1/day] 2.45 and 1.0
E [96]
GIC
[GPa]
[146] [N/m]
21.37
50.0
5.3.4 Damage characterization The stochastic cohesive zone model implemented for this work requires, the definition of the Weibull parameter,
m , and the average instantaneous strength, T0 t . For the analyzed cases in this
67 work the time dependent behavior of T0 t is described using Equation 30a-b, and used in Equation 26 to describe Tmax t at the element level. T0 t depends on the asymptotic tensile strength recorded at later ages ( T ). In this work restrained mortar will be modeled considering two different geometries (linear, see Figure 23 and ring, see Figure 27) both of them under the nonuniform drying shrinkage profile described in the previous section. As described by Weiss in a previous work [146], tensile stress that develops in a ring under nonuniform drying shrinkage has two components (a) the self-restraint that develops due to non-uniform drying conditions and (b) the restraint that is provided by the steel ring. The tensile stress in the linear geometry subject to the same nonuniform drying shrinkage can be also described with the same components of stress described for the ring previously (considered as a ring with infinite radius). In the linear geometry, tensile stress caused by the external restraint is less than the stress caused by external the restraint in ring assuming both (linear element a ring) have the same thickness. Therefore, lower values of tensile stress are expected to be achieved in the linear element with respect to the ring. In order to achieve similar levels of damage the proposed values of T at the element level that will be used for the linear element will be half of the values of T at the element level considered for the ring (i.e. T = 1 x10-3 and 2x10-3 GPa respectively). Regarding Weibull parameter,
m,
it was experimentally observed scatter of the estimated value
obtained from multiple specimens [150]. For instance, Zech and Wittman [151] estimate plain concrete using middle range laboratory specimen and estimated a value of
m
m
for
= 12. Later,
Bazant and Novak [152], extended Zech and Wittman analysis using data from multiple previous studies to account for the influence of
m
on the flexural strength for a wide range of specimen
sizes. Bazant and Novak combine an empirical estimation of the deterministic flexural strength of concrete beams associated with the size of the tested samples to the statistical distribution of flexural strength using a Weibull type distribution [153], [154]. From this analysis they show that due to the limited experimental data of Zech and Wittman [151], their estimation of
m
could only
be associated to laboratory specimens only. In their study, Bazant and Novak indicate that
m
should larger than 20 and less than 25 to provide the best fitting for the strength of plain mortar and concrete [152].
68
RESULTS FROM MODEL OF MORTAR SLABS ACCOUTING FOR SOLIDIFICATION AND DAMAGE SUBJECT TO DRYING SHRINKAGE
6.1
Stresses relaxation induced by distributed damage
As mentioned earlier, mortar is not expected to develop distributed damage that relaxes average macroscopic stresses and reduce strength. Previous to this work, experimental observations of damage-mechanical behavior of mortar at early ages subject to drying shrinkage showed that distributed damage occurs and strength is reduced by its presence [4], [64], [148]. The purpose of this chapter is to address possible effects on tensile stress and strength development caused by distributed damage induced by restraining uniaxial a slab of mortar. A 2D numerical cell of a slab will be first, subject to the proposed time-dependent nonuniform shrinkage described in the previous section (see Figure 23) and second, the numerical cell will be subject to a displacement controlled loading condition to address the residual tensile strength of the model.
Figure 23 (a) Schematic of the 2D FEM geometry of restrained mortar. (b) Schematics of the initially elastic constitutive cohesive law employed to describe the interface behavior. The red double arrow indicates any irreversible post-peak unloading–loading cycle. The gray are defined by the irreversible and the unloading lines corresponds to the dissipated cohesive energy
69 In this analysis, mortar is considered to behave mechanically as Equation 30a, previously proposed by Grasley and D´Ambrosia [101]. The viscoelastic properties considered for this analysis correspond to set 1, a general purpose mortar with Type I cement, presented in Table 3. The numerical cell is exposed to nonuniform shrinkage from one surface and is sealed on the opposite side ( r = 0 and R respectively in Figure 23). Shrinkage is represented by sh t as a function of RH (r , t ) as described in a previous section (Equation 10 and 11 respectively). This moisture profile
was selected as it is representative of the experimental moisture content in mortar slabs reported in previous works [101], [155]. The following results will present the effect of variation of
m
on
the macroscopic response of mortar. To is considered as described by Equation 30b ( C2 =2.45 1/day) [85]. As To is fixed in this analysis to study the effects of of mortar, Ts is function of
m
m
on the macrsocopic response
according to the size effects of the stochastic distribution described
in Eq. 26.
(a) Figure 24 (a) Tensile stress development caused by nonuniform drying shrinkage for different values of m (5, 20, 50, 1000), (b)-(c) tensile stress map (right) and damage distribution map (left) for m = 5 at 10, 14 and 20 days. As age evolves, the damage zone increases
70
(b)
(c)
(d)
Figure 24 Continued
Figure 24a shows average z (see Figure 23) for both time and space distributions during the shrinkage evolution in the numerical cell. Figure 24a shows results considering different combinations of
m
to express its effect on tensile stress time-space development. The results of
the simulations with damage in this section are compared with simulations without damage (black lines in Figure 24a) to study the overall effect of damage on tensile stress. We first analyze the mechanical behavior presented in Figure 24a corresponding to the model without damage. The tensile stress peak moves opposite direction to the drying surface as drying progresses. Grasley [156] developed an analytical solution of restrained mortar rings under drying conditions and concluded that the role of solidification in the peak stress movement correspond to the new stressfree material is added after the initial application of shrinkage at the surface causing that a gradual decay of tensile stress on the surface [101]. In addition, stress-free material is also added away from the drying surface that when subject to drying shrinkage applied after the initial application
71 of shrinkage (at the surface), causes tensile stress to move inwards (see Figure 24a) and relax tensile stress outward (near the drying surface).
Figure 25 1D scheme of the simplified model to describe the effects of nonuniform shrinkage in a solidifying material. represents the general displacement of the vertical bar
Using the principles of solidification model proposed by Bazant [103] a mathematical description can be obtained for the basic simplified model of solidification process subject to nonuniform shrinkage as shown in Figure 25. The analytical solution of this model is complex and for the sake of simplicity will only describe its results qualitatively to understand the effect of solidification on stress and damage distribution. Figure 25 shows a system with four nonaging blocks, with cross section areas A1 A2 A3 A4 , attached to a spring ( k ) that represents the mechanical restraint of the system. At time t o , sh acts on the block 1 causing high tensile stress on it and at the same time a low stresses on the rest of the block opposes to the deformation of the system. At this step, the stress on block 2 to 4 are small compared to the tensile stress on block 1 due to A1 A2 A3 A4 . The description of the block model in its first step is what is observed as a result of restraint of the numerical model presented in Figure 24a when sh starts acting on the material near de drying surface. Back to the block model, at t1 all the blocks have higher elastic modulus ( E(t1 ) E(t o ) ) because of solidification and sh is applied also in block 2. At this time, the stress in block 2
72 increases (tensile) and is higher than the initial tensile stress on block 1. Also, at t1 block 1 is subjected to a combination of the stress caused by the initial shrinkage and the restraint coming from the rest of the blocks. Blocks 3 and 4 now stretch as a result of the shrinkage on blocks 1 and 2. The latter is a simplified description of the phenomena that occurs when combining nonuniform drying shrinkage with a solidifying material as presented in Figure 24 when an infinite degree of restraint is applied to a solidifying material. Part of this was already discussed and is equivalent to the peak tensile stress movement inwards as shrinkage evolves described by Grasley [101]. Overall, the stress distribution in the simplified model depicted in Figure 25 will depend mainly on the combination solidification rate, k and the cross section area of the blocks. In any case, the stress of the block 1 (representing the portion of material near the drying surface in the numerical model) at t1 will always be lower than the stress of the peak stress at t t1 . If damage is added to the simplified model in Figure 25, as tensile stress progresses inwards some parts of the material will reach tensile strength. At this moment, exceeds cr , and the interface elements are assumed to start developing damage (see Eq. 23). Figure 24b-c show the spread evolution from the drying surface (left) to the sealed surface (right) and level of cohesive energy that occurs as drying shrinkage progresses at 10, 14 and 20 days. Soon after that a piece of material started to damage, the nonuniform shrinkage combined with solidification will relax stress in the same piece of material where previously damage was triggered and slowing down the evolution cohesive energy dissipation there. At that moment, another piece of material even further from the drying surface will reach cr and the process will be repeated at different locations. This explanation is shows that strength does not have to be necessarily stochastically distributed to achieve stress reduction. Proof of this is shown in Figure 24a for the showing results of stress reduction even for nearly deterministic strength distributions ( m 1000 ). As
m
reduces, spread of
strength increases (time and space), allowing more spread of damage and eventually have multiple places where energy dissipation localizes relaxing even more the tensile stress of the model with respect to the model without damage as stress profiles corresponding to
m 5 and 20 are shown in
Figure 24a. The damage-mechanical description is this paragraph is most likely the mechanism responsible for causing stress reduction due to damage in the numerical model presented in Figure 24 and Figure 25. To establish whether this damage-mechanical response can be achieved only due the combination of non-drying shrinkage and solidification, the time-dependent stress profile
73 shown in Figure 24a was mimicked using a specific strain field for that purpose and applied to the same numerical cell of Figure 23 considering linear elasticity. The latter could be an indication that the same stress reduction can be achieved with the latter conditions, indicating that distributed damage and stress reduction could be due to the induced stress gradients. Previous works [157], [158] showed that materials expected to behave as brittle, such as glass, can be toughen by inducing stress gradients, e.g. using a tempered glass treatment.
6.2
Residual tensile strength induced by distributed damage
The reduction of stress caused by the stress gradients induced by a nonuniform shrinkage profile and solidification have an impact on strength development as described in previous sections regarding to experimental observations [4]. To address this, the material shown in Figure 24, up to different times where a displacement controlled loading condition takes place to address the residual strength of the numerical model (see Figure 26a-b). In contrast, the same model is tensile tested at the same times as the restrained cases with no previous restraint and shrinkage development (unrestrained case in Figure 26c). Using the same inputs used in the previous section to model stress relaxation caused by distributed damage and stress gradients, the simulations in this section mimic similar restrained and unrestrained experimental tests performed in a previous work [4] (and presented in Figure 16c) that studied the effects of developed microscopic damage on strength caused by drying under restrained conditions. The main difference of the simulations presented in this section with respect to a previous experimental work [4] is that tensile strength is extracted from simulations while flexural strength was extracted from experiments. Flexural strength,
fr
, is not interchangeable with the tensile strength,
ft ,
as the geometry of the sample
influences the strength due to micro-cracking and pre-critical crack growth. For a typical bending test specimen [71] (typical ASTM standard 6 x 6 in. [152 x 152 mm] cross sectional testing geometry)
fr ft
ranges from 1.0 to 1.2.
74
Figure 26 (a)-(b) Model steps towards the estimation of tensile strength with a previous restrained/unrestrained conditions to induce distributed damage, (c) numerical reduction of strength ( % f t T ) vs. age. Error bars represent numerical results using different random distributions of f t . (d) Experimental results [4] of the effects of developed microscopic damage on strength relaxation caused by drying under restrained conditions
75 Figure 26c presents the calculated values of
% f t T
from simulations accounting for restrained
and unrestrained conditions. Two set of results corresponding to different Weibull distributions are presented ( m 5 and 20). Results with higher values of
m
(e.g. 50) showed little reduction of
strength caused by restrained conditions (not showed in Figure 26c). Under restrained conditions, more damage is developed causing higher reduction of % f t T
ft
as age evolves. The difference of
between restrained and unrestrained results is similar for both cases ( m 5 and 20) as it
can be observed in Figure 26c. The simulation presented a reduction of conditions at 7 days of 3.4 and 9.4 % and at 14 days of 19.4 – 25.5 % (for
ft
under restrained
m 20 and 5 respectively)
with respect to the unrestrained simulations. In Figure 26c, the overall reduction of strength when m5
(restrained and unrestrained) with respect to the strength when
m 20
corresponds to the
size effect between To and Ts described earlier. The results presented in Figure 26 from simulations representing mortar under restrained and unrestrained conditions demonstrate the role of distributed damage in a reduction of strength. For reference purposes only, the experimental reduction of
fr
reported from experiments at 7 and 14 days (maturity index of 3.4 and 4 °C-hours
respectively in Figure 26d) was approximately 8 – 36 % [4] similar to the percentage values presented in this work regarding reduction of
ft
at the same ages in Figure 26c.
76
NUMERICAL MODEL OF DAMAGE-MECHANICAL BEHAVIOR OF RESTRAINED MORTAR RINGS AT EARLY AGES
7.1
Introduction
The restrained ring test is an accepted experimental tool [159] to address mechanical and fracture behavior of mortar because of its controlled restrained conditions. The test sample of the ring test consists of fresh mixed mortar or concrete cast and compacted around a metallic ring (see Figure 27a). As mortar deforms due to shrinkage, the inner ring restraints it inducing stresses. If the restraint provided by the inner ring is high enough, mortar will start developing microscopic damage and eventually radial cracks will form [97]. Kim recently showed, through a modified ultrasound test, that distributed damage is present in mortar at early ages subject to drying shrinkage before macroscopic damage occurs [160]. Experimentally, it has been observed that fracture of the restrained ring tests forms radially, in a 2D plane [96], [161], [162]. Stress development in the ring can be represented by a plane strain condition (see Figure 27b) [163]. The cohesive zone model used in this work assumes plane strain conditions as presented by Espinosa and Zavattieri [53] in a previous work. In the same way as the models presented in the previous section, the solidification-damage model described earlier will be used in a 2D plane strain finite element (FE) model in an attempt to make an approach to the mechanical behavior of mortar rings under restrained conditions and distributed damage. For this purpose, a slice of the restrained ring test is modeled (see Figure 27c). Both, steel and mortar rings are modeled with 3-node triangular, constant strain, elements (see Figure 27c). Experimentally, friction between steel and the mortar is prevented by adding a layer of oil between them prior to casting. According to the later, the finite element model of this work has implemented contact elements between steel and mortar that prevents friction between these two bodies (Figure 27c). Finally, four-node, zero-thickness cohesive elements with stochastic strength distribution described in previous sections will be to represent the fracture behavior of mortar under tension [47]–[49], [53], [105], [164].
77
Figure 27 (a) Typical restrained ring test configuration, (b) the FE model of the ring will represent a slice of the middle plane of the restrained ring, (c) dimensions of the ring test numerical model: a = 140mm, b = 150mm, c = 250mm
For this analysis, it is considered that the outer ring of mortar (see Figure 27a) is exposed to the nonuniform drying profile showed in Figure 20 (Eqs. 27 and 28). The later causes drying shrinkage to occur more rapidly on the outer surface of the ring compared to the inner surface that loses water humidity more slowly[146]. Hossain and Weiss [96] suggested that the residual interface pressure between the steel ring and the mortar, Pres , can be computed as a function of the average measured strain in the steel in the circumferential direction, st R 0 at RIS , as shown in Eq. 31: IS
Pres st
R IS
ES
R
2
OS
RIS 2
2 ROS
2
0
(31)
Where ROS b and RIS c are the external and internal radius of the steel ring respectively (see Figure 27b), E S is the elastic modulus of the steel. As discussed by Hossain and Weiss [96],
78 Equation 31 illustrates that Pres can be determined using only the mechanical properties and geometry of the steel ring and st RIS using the electrical resistance strain gages. The only two assumptions are that Pres is uniform and that the steel ring remains elastic all times. According to this, the recorded Pres could be contrasted with the integrated pressure, P , over the circumference of the inner ring of the mortar (using as the angle that describes the circumference) in contact with the steel, called Peq , in the finite element model of a restrained ring using the solidification theory to account for the aging-viscoelastic behavior at early ages of mortar. Peq is estimated in the numerical model as presented in Eq. 32 considering quasistatic equilibrium
between the pressure exerted by the mortar ring and the reaction of the steel ring (assumed as linear elastic) at the interface of both, through the whole external area of the steel ring ( ).
Peq
1
P .d
(32)
The adapted model of mortar including solidification effects and a statistical damage model permit mapping the circumferential stress profiles, r, t , at different ages. The following example in this work will demonstrate through a sensitivity analysis, accounting for different viscoelastic, aging and Weibull parameters, the importance of damage in the relaxation of both r, t and Peq . If a single strength law is considered (no statistical distribution), damage will concentrate in a single line starting from the outer ring of mortar and moving inward as time evolves up to failure where the total dissipated energy per unit length will be. Gc RO RI . Where, RO a and RI b are the outer and inner radius of mortar respectively (see Figure 27b). In other words, Gc RO RI represents the dissipated energy per unit length associated to the localization of a single crack (opened crack). In this work, the time evolution of cumulative cohesive energy per unit length,
G a
i i
, is calculated where a i is the length of the cohesive element with an energy release rate
Gi . The cumulative damage will be analyzed using the ratio E
G a / G RO RI . i i
c
79 7.2
Sensitivity analysis
In his work, Grasley presented an analytical of r, t accounting for solidification without damage [101]. In this section, the same ring test configuration to the one proposed by Grasley [101] is presented to account for solidification and damage development on stress relaxation. Figure 28a shows the VE sensitivity analysis (for sets 1, 2 and 3 presented in Table 3), using Equation 30a for solidification (same as the linear case of the previous section) and
m
=20. Figure
28a presents r, t radial distribution at 2, 10 and 20 days. Included in this figure as a reference, the evolution of r, t assuming no damage (dashed-dotted line) is presented considering the set 1 of VE properties presented in Table 3.
80
(a)
(b)
(c)
(d)
(e)
(f)
Figure 28 (a) Hoop stress vs. radius at 2, 10 and 20 days, (b) Peq t and E vs. Age, (c) and (e) damage development at 10 and 20 days respectively when VE is removed, (d) and (f) damage development at 10 and 20 days respectively when solidification is considered. m = 20 and To = 2 MPa is considered for all the cases
81 Similar to the results in the previous section, Figure 28a shows the stress gradients, induced by nonuniform shrinkage and solidification, moves r, t peak inward (i.e. from the outer radius of the ring) as age progresses. In particular, Figure 28a shows that the overall evolution of r, t occurs almost the same regardless the selected set of VE in this work. Regarding the effect of distributed damage on r, t , the peak of r, t for the solidifying case with no damage is higher than the solidifying cases accounting for time-dependent damage (using VE properties 1 to 3 presented in Table 3) at 10 and 20 days (5% and 10% stress reduction at peak respectively). Figure 28b shows the evolution of Peq t and E with respect to age. It can be observed that the reductions of r, t (Figure 28a) are caused by a 0.2 to 0.3 % of Gc RO RI at 20 days as showed in Figure 28b. As age evolves, more elements in the model reach values of r, t similar to Tmax t , this is consistent with the rapid growth of cohesive energy (Figure 28b) that starts accumulating within 6-9 days. Even though onset of damage occurred at slightly different ages (between 6 and 9 days) depending on the VE set (Figure 28b), the relaxation of r, t spread for all the cases within a radius of 190 and 230 mm at 20 days (Figure 28a). Similarly to r, t , Peq t shows no significant sensitivity to the proposed changes in VE properties (1% maximum change at 20 days). Yet, if VE is removed from the solidification model, the response of ring is different. Figure 28a shows an example of r, t at 20 days of the response of the solidification model (without VE, green line) with damage. Although the mechanical response of the ring is not sensitive to changes of VE parameters, it is important to describe the mechanical response of the restrained ring considering VE because it affects the stress distribution. On the other hand, a reduction of Peq t is observed in Figure 28b, when damage is consider in the model, regardless the selected VE set at 20 days. At this age, for the same VE set of parameters (VE 1) it was observed a reduction of Peq t of 8% when distributed damage is present. In addition, Figure 28c and e present the cohesive energy maps at 10 and 20 days respectively when VE is removed from the model. Figure 28d and f present the cohesive energy maps at 10 and 20 days respectively when VE is considered in the model. Same as stress distribution, the dissipated energy maps are completely different if VE is removed from the model. Distributed energy spreads near the outer and localizes later ring if VE is removed while for the cases
82 considering VE, distributed energy spreads along the radius in a length that is consistent with the length of stress reduction presented in Figure 28a at 10 and 20 days.
(a)
(b)
(c)
(d)
(e)
(f)
Figure 29 (a) Hoop stress vs. radius at 2, 10 and 20 days and (b) Peq t and cumulative damage vs. Age for m = 5, 20 50 respectively. (c) and (e) damage development at 10 and 20 days respectively for m = 50 (d) and (f) damage development at 10 and 20 days respectively for m = 20. Equation 30a and VE 1 are considered for all the cases
83 Figure 29 shows the sensitivity analysis for different values of
m
. The solidification law
considered corresponds to Equation 30a and a specific set of VE properties (1 in Table 3). Similar to the previous analysis, r, t develops in a similar manner to what was described for the sensitivity analysis of the VE properties. Yet, the stress that the plateau of r, t (stress relaxation) caused by distributed damage depends on the value of
m.
For this reason, Figure 29a presents
average values of stress relaxation extracted from the r, t evolution and compared to the maximum stress development on the case with solidification without damage. Figure 29a shows stress relaxation values for both, different ages 2, 10 and 20 days and different this example. As described earlier, r, t relaxation is sensitive to changes in
m= m
5, 20, 50 for
with respect to
the solidifying case without damage. Specifically, when Tmax t has a wide spread ( m 5), r, t relaxes 17 and 36 % average with respect to the “reference case” at 10 and 20 days respectively. For those cases considering
m 20
and 50, it was observed a similar but lower difference (10 %
average) in the peak stress with respect to the “reference case” at 20 days. Same as the linear geometry presented in the previous chapter, stress relaxation in the ring test simulations are a cause of (a) distributed damage caused by the stress gradients generated by the combination of nonuniform drying shrinkage and solidification and (b) the strength distribution. As more stress reduction is observed. From the macroscopic point of view, a reduction
m
reduces,
m of is related
to a wider spread of Tmax (t ) and a reduction of Ts . Moreover, as demonstrated with the linear geometry in the previous section, it is expected that the tensile strength of the mortar ring would also be affected by the distributed damage enforced by the restrained condition that causes the stress relaxation. The relaxation of r, t is caused only by a 0.3 ( m 50 ) to 1 % ( m 5 ) of Gc RO RI at 20 days as showed in Figure 29b. Also about cumulative cohesive energy (see Figure 29b), as the spread of strength is higher (i.e. 11 days for as
m
m
m
reduces) cohesive energy accumulation triggers at early ages (1.5 and
= 5 and 50 respectively). This occurs because lower values of Tmax t are expected
reduces and therefore, more chances that r, t reaches Tmax t at earlier ages. As showed
in Figure 29c-e, the radial spread of damage is wider as 50 respectively). Peq t also reduces as
m
m
reduces (10 mm to 60 mm for
m =5 and
decreases. In particular at 20 days, Peq t considering
84 m 20 and 50 decreases around a 2% with respect to the “reference case”. While, Peq t at 20 days
considering
m=
5 reduces 13% with respect to the case without damage.
Figure 29 c and e present the cohesive energy maps at 10 and 20 days respectively when
m=
Figure 28d and f present the cohesive energy maps at 10 and 20 days respectively when
50.
m=
5.
Figure 29c-f show that distributed energy spreads along the radius in a length that is consistent with the length of stress reduction presented in Figure 29a at 10 and 20 days. As
m
reduces, the
length of spread damage is larger.
(a)
(b)
(c)
(d)
Figure 30 (a) Hoop stress vs. radius at 2 and 10days, (b)
Peq t and relative cumulative damage vs. age
for different aging rates. (c) and (d) damage development at 2 and 14 days respectively for 0.3 1/day. VE set 1 and m = 20 are considered for all the cases
Figure 30 presents the results from the sensitivity analysis of t rate on solidification theory (considering the three aging laws presented in Figure 22 and Equations 30a and 30b considering C2 = 1 and 2.45). VE set 1 and m = 20 are considered for all the cases. The purpose of this analysis
85 is to determine whether there is a rate effect where distributed damage and stress relaxation turns into localized damage (and stress). Figure 30a shows that aging rate increases, the peak of r, t moves inward. Also increasing aging rate also caused r, t near the drying surface to remain with a value of around 30% with respect to peak stress as shown in Figure 22a. It is observed that higher initial rates of t cause higher values of Peq t at early ages (Figure 30b). The case assuming no damage is presented to highlight the effect of accounting for damage in stress development prediction (see dashed-dotted line in Figure 30b). Figure 30b shows that at a certain age (5 days approximately for these simulations), those cases with higher t rates (0.8 and 1.6 1/day, Equation 30b with C2 = 1 and 2.45 respectively) start reducing Peq t below the case with no damage up to 14% at 20 days and %E on high rates reached 2-5%. Figure 30c-d present the cohesive energy maps at 2 and 14 days respectively when Equation 30b with C2 = 2.45 is considered. As the aging rates increase, the distributed energy spreads near the outer and localizes later ring.
Figure 31 Hoop stress vs. radius at 30 days. Equation 30a and VE 1 are considered for all the cases
The results presented in this section correspond to a specific geometrical configuration and level of restraint (degree of restraint (DOR) of 60% approximately, see [143] for more information) of the ring test. In particular about the latter, a question remain on whether the DOR has any role on the development of distributed damage. Two more DOR were tested that are common in the
86 literature for representing an upper and lower bound of the DOR that can be found in real structures: (a) 30% and (b) 75% (equivalent to thickness of the steel ring of 3mm and 19mm) [101], [163], [165]. The numerical results showed that a DOR of 75% is capable of developing distributed damage and stress reduction similar to the presented behavior in this section on moderate DOR (60%), consistent with experimental mechanical response observed in previous works [85], [97]. On the other hand, low DOR (30%) is no capable to develop significant amount distributed damage due to the low levels of stress and therefore, no stress reduction was observed in simulations up to and age of 30 days. This is consistent with previous analytical/experimental works that demonstrate that low DOR stress response have a negligible effect from damage [85], [97].
87
SUMMARY AND CONCLUSIONS
The mechanisms responsible for distributed damage affecting stress relaxation and strength reduction of mortar at early ages are described in this work using a phenomenological model where solidification and a cohesive zone model with stochastic strength distribution were implemented in a finite element code. A set of experiments of restrained mortar were selected to be represented numerically, to study the effects of distributed damage on stress relaxation and strength reduction. Within these test, the restraint ring and a uniaxial geometries showed the value of the proposed model and illustrates the effect that solidification, drying shrinkage and strength distribution have on history-dependent strength development of mortar under tension. Based on these the following conclusions can be mentioned: 1.
The numerical models of mortar developed for this research, show that there are stress gradients (caused in mortar by a combination between solidification, stochastic strength distribution, VE and nonuniform shrinkage) capable of causing distributed damage prior to localization affecting both stress development (see Figure 24, 28, 29, 30, 31) and macroscopic strength development (Figure 26). The latter was observed when the damage zone length covers between a 10% and a 50% of the sample width (see Figure 24b-e, Figure 28d and f as an example). Although the selected inputs for the models correspond to real data of mortar.
2.
A reduction of macroscopic tensile strength (between 10 – 25%) is observed in the numerical model of mortar after being subject to restraint conditions regardless of the value of m (Figure 26c). The range of m used in this work to model mortar includes experimental estimations of m for strength of mortar and concrete. Reduction of macroscopic strength is proportional to the age of restraint mortar regardless of m and Ts (see Figure 26c). The recorded strength reduction from the numerical model is in accordance to reduction of flexural strength reported in a previous work from experiments at early ages (Figure 26d).
3.
Numerical results of restraint mortar showed that stress relaxation (associated to distributed damage) occurs at high DORs (above 60%), Figure 31. This is in accordance with reported experimental results of stress relaxation of restrained ring tests. Simulations with low DOR (e.g. 30%) did not showed stress reduction caused by damage up to an age of 30 days, see
88 Figure 31. The latter is consistent with experimental data previously reported [97]. Restrained ring tests of mortar w/c between 0.3 and 0.5, with 30% DOR do not crack at least during the first weeks after casting and AE is lower (at least 60% less) than AE (prior to localization) of restrained rings with DOR of 60% (that eventually develop visible cracks and a sudden loss of load). 4.
The numerical model tested under a discrete number of VE properties of mortar, show that the effect of different sets of VE of stress relaxation is a small fraction of the effect caused by distributed damage. Yet, VE is important to describe the distribution and magnitude of stress of restrained mortar (see Figure 28a).
5.
The model of restrained ring with nonuniform shrinkage showed that as shrinkage rate is faster damage develops faster at earlier ages. On the contrary, as shrinkage rate decreases, the solidification response causes stress gradients to move inward and delay distributed damage and allowing more spread of damage. As suggested by a previous work [148], this confirm that prevention of moisture loss (i.e. controlling shrinkage rate) is not sufficient to prevent damage and other variables, such as aging rate, should be considered.
Future work will be focused on address limits (if any) of the effects of stress gradients associated to other nonuniform shrinkage profiles, aging development, distributed damage length scales (Figure 29c-e). In addition, more work needs to be done to find a quantitatively relation between AE and E (see Figure 28b, Figure 29b, Figure 30b). Finally, as the intent of the solidification, stochastic strength distribution model in this work was to predict macroscopic performance of a composite (engineering perspective). Future work will study the effects of self-restraint effects (if any) caused by aggregates in laboratory size mortar and large concrete structures on damage and strength reduction. Yet, more experimental work needs to be done to narrow down the mechanical and fracture properties of the constituents and how they interact. I surmise that the presented model considering stochastic strength distribution and solidification could provide more accurate predictions if experimental data coming from the microstructural behavior of mortar and concrete is addressed that allows to model aggregates and matrix-aggregates interfaces accurately.
89
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VITA
Federico C. Antico obtained his M.Sc.E at the school of Civil Engineering of Purdue University and earlier he got his B.S. with a major in light-weight structures design at the school of Aeronautical Engineering of Universidad Nacional de La Plata (UNLP). Federico has been deeply involved in higher education collaborating as teaching assistant, faculty apprentice, tutor and mentor for several undergraduate courses and engineering programs.
Federico´s research interests include early age properties and service life prediction of construction materials improved by the addition of valorized waste. Federico make use of destructive, nondestructive experimental techniques and apply solid and fracture mechanics theories and models to provide smart options to improve the performance of these materials. Specifically, Federico has performed research in areas of smart materials, early age properties of both: non reinforced and reinforced concrete, mortar and adobe. Federico’s teaching interests include engineering materials courses, with a special interest in topics including: durability, failure mechanics, strength evolution of cement-based composites, numerical modeling of materials, and non-destructive testing.
