epoxy matrix. Microphotography was made using FEI Phenom. Ultimate tensile strength (UTS) of single glass monofiber (Figure 1):. Axial UTS11 = u100000.
page.1
Multiscale Homogenization of Fiber Reinforced Composite Materials Using Distributed Computing Mechanics of Composites (MECHCOMP2014), Symposium 17.4. Finite Element Analysis of Composite Materials & Structures, Stony Brook University, 08-12 June, 2014
Prof. Yu.I. Dimitrienko*, PhD. A.P. Sokolov* *Department of Computational Mathematics and Mathematical Physics, Scientific-educational Center of Supercomputer Modeling and Software Engineering, Bauman Moscow State Technical University, Moscow, Russia
08-12 June USA Long Island, New York 2014
1/30
page.2
Contents
1
Objectives and subtasks
2
Introduction
3
Mathematical model Mathematical model of multiscale hierarchy structure (MHS) L˜pq task statement
4
Development of calculation subsystem in DCS GCD Strength-elastic properties of composites analysis method
5
Typical computational results obtained using DCS GCD
6
Conclusion
7
Contacts
2/30
page.3
Objectives and subtasks
Main objective The main objective of this research was to develop a special software for obtaining effective elastic and strength properties of real composite materials with arbitrary type of reinforcement. Main subtasks. Test new graph-based program implementation of computational subsystem of Distributed Computational System GCD (DCS GCD); Make test calculations of determination of elastic-strength properties of several types of composites; Validate results1 .
1
Not presented here.
3/30
page.4
Introduction
Problems of solving real practical tasks Incomplete initial data set
Incomplete initial data set2 . Inaccurate initial experimental data.3 . Postprocessing validation of calculation results.
σ 22
σ11
σ 33 Figure 1: Glass monofibers of thinkness ≈ 8𝜇m in composition with epoxy matrix. Microphotography was made using FEI Phenom.
Ultimate tensile strength (UTS) of single glass monofiber (Figure 1): Axial UTS11 = 𝜎u100000 a could be found from experiment. Transverse UTS22 = 𝜎u010000 and UTS33 = 𝜎u001000 b could not be found from experiment. a 𝜎u100000 is the maximum resistance to fracture when 𝜎11 increasing while loading in axial direction of glass monofiber in its local physical coordinate system. b UTS in normal direction to the axis of fiber
2
Some initial data cannot be found by experiment. Unknown technological modes which were used while production the composite samples results in poor quality of experimental data. 3
4/30
page.5
Introduction
Problems of solving real practical tasks Large dimension of finite approximation of real practical tasks
Large dimension of practical problems results in increasing of calculation time. Large dimension also results in necessity of meeting special hardware requirements (RAM requirements, number of CPUs, multiprocessor architectures usage).
Figure 2: Shear stress state task solution. Shear in OXZ plane of unit cell of 3D-reinforcement composite. Finite element approximation: ≈ 21 million tetrahedron elements. System of Linear Algebraic Equations (SLAE): ≈ 11 million equations. Hardware: Single CPU Intel PC 3 GHz, 8 cores at CPU, ≈ 4 GB RAM. Calculation time: ≈ 4 hours. Program implementation: standart C++ with OpenMP. Conjugate Gradient (CG) algorithm was used to solve resolution SLAE of FEM. This task was stated and solved for estimating calculation time. 5/30
page.6
Introduction
Features of mathematical modeling of composite materials Models, numeric methods and program implementation
Mathematical models that were used: for stress-strain state tasks: Linear elastic, Nonlinear plastic, Thermoelastic material models4 : isotropic, orthotropic, anisotropic. different strength criteria (failure functions) were used for every component: tensor polynomial, Malmeister-Woo, Tsai-Woo, Goldenblat-Kopnov, maximum stress and etc.).
Numerical methods that were used: Finite Element Method (FEM). Asymptotic Averaging Method (Bahvalov N.S., Pobedria B.E.). Multiscale homogenization (Dimitrienko Yu.I., Chiang C. Mei). Hook-Jives linked with Monte-Carlo methods for optimisation and inversed problems.
Features of program implementation of Distributed Computational System GCD (DCS GCD): C/C++ programing language. Three-tired client-server architecture of the system. Special remote database for initial, intermediate and result data. Model-View-Controller principles. Component Object Model principles. 4
These models were used for components of composites.
6/30
page.7
Introduction
Task statement Overview
Initials Composite material under research (Fig. 3). Scheme of reinforcement of corresponding composite material given by its unit cell (Fig. 4). The set of elastic, thermal and strength properties of each component of composite (could be incomplete): 1 2 3
Young Modulus Ei ; Poisson’s ratios 𝜈ij ; Thermal conductivities 𝜆i and expansion coeff. 𝛼i ; Strength data: UTS, proportional limit and other.
Figure 3: Microphotos of composites of woven type.
Assumption Inner reinforcement structure of composite has periodic multiscale hierarchy type and it can be described by unit cells. To develop and to find 1
2 3
Relational database structure to store calculation results. Special extension modules to DCS GCD. Effective(homogenized) properties of composite (elastic, thermal, strength).
Figure 4: Example of 3D models of unit cells. 7/30
page.8
Introduction
Task statement Details and assumptions
The full set A𝛼 of initial mechanical properties of every component 𝛼 of composite under research is defined as a set of scalars: Young Modulus Ei𝛼 ; Poisson’s ratios 𝜈ij𝛼 , Shear moduli Gij𝛼 ; 𝛼 Thermal conductivities 𝜆𝛼 i and expansion coeff. 𝛼i ; Strength data: UTS, proportional limit stress and other; i, j ∈ {1, 2, 3} indices of axis in 3D case.
We suppose that in real practical task initial data is defined as subset B ⊆ A. It means that some data can be unknown. Special format was used to designate strength properties in the most general 3D case in 6-dimension space of stress tensor: 𝜎xLMNPST , where: L, M, N, P, S, T ∈ {0, E , C }, E – Extension, C – Compression, 0 – absence of loads, x ∈ {e, u, y }, where e – elastic, u – ultimate tensile, y – yield5 .
Scheme of reinforcement have to be defined as a set of unit cells in multiscale case: each cell for every scale level of the composite model and for every heterogeneous composite component.
5 For example: 𝜎uE 00000 is Ultimate Tensile Strength in case of extension loading along the OX axis according to physical coordinate system of component of composite.
8/30
page.9
Mathematical model
Mathematical model of multiscale hierarchy structure (MHS)
Mathematical model of multiscale hierarchy structure (MHS) Elastic case
Asymptotic averaging method was used for modeling effective properties of composites (single scale homogenization) at every scale level. Mathematical model of the effective elastic properties presented here6 . Special additional program module was developed to be effectively used in case of incomplete initial data set. Monte-Carlo methods and stochastic analysis were used to identify unknown initial data. Supercomputer program implementation was developed and integrated into server tier of DCS GCD.
6 Mathematical models of effective thermoelastic, strength and thermo-strength properties were developed and implemented in C++ in DCS GCD. Not presented here.
9/30
page.10
Mathematical model
Mathematical model of multiscale hierarchy structure (MHS)
Mathematical model of multiscale hierarchy structure (MHS) Elastic case. Asymptotic averaging method (single scale homogenization)
The model of composite is a set of unit cells from top (first) scale level, where each of them consists from unit cells from second scale level and etc. to the bottom (nth) scale level.
Every scale level is described by its characteristic dimension. Each of them meets the next relationship: lN
