Atomistic-continuum interphase model for effective

Atomistic-continuum interphase model for effective properties of composite materials containing ellipsoidal nano-inhomogeneities Yao Koutsawaa,∗, Mohamed Cherkaouic , Jianmin Quc , El Mostafa Dayab a

Luxembourg Institute of Science and Technology. 5, Avenue des Hauts-Fourneaux, L-4362 Esch-sur-Alzette, Grand Duchy of Luxembourg b Laboratoire d?Etude des Microstructures et de Mécanique des Matériaux, LEM3, UMR CNRS 7239, Université de Lorraine, Ile du Saulcy, 57045 Metz Cedex 01, France c Unité Mixte Internationale UMI GT CNRS 2958, G.W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0405, USA

Abstract The classical micromechanic is revised to study composite materials containing nano-inhomogeneities. Contrary to the previous works, this paper introduces the concept of interphase instead of sharp interface. From atomistic description, the interphase’s properties are derived in continuum framework. These properties are then incorporated in micromechanics-based interphase models to compute the effective properties of the nanocomposite. This approach bridges the gap between discrete systems (atomic level interactions) and continuum mechanics. An advantage of this approach is that it is developed from earlier models that consider inhomogeneity shape, thereby enabling both the shape and the anisotropy of the nano-inhomogeneity and the nano-interphase to be simultaneously accounted for in computing the overall properties. Keywords: Micromechanics, Interphase model, Nanocomposites, Interfacial excess properties, Effective properties

1. Introduction Currently, due to advances in nanotechnology, many investigations are devoted to nanoscale science and developments of nanocomposites. Nanomaterials in general can be roughly classified into two categories. In one hand, if the characteristic length of the microstructure, such as the grain size of a polycrystal material, is in the nanometer range, it is called a nano-structured material. In the other hand if at least one of the overall dimensions of a structural element is in the nanometer range, it may be called a nano-sized structural element. Thus, this may include nano-particles, nano-films, nano-wires (Alymov and Shorshorov, 1999; Dingreville et al., 2005). In this paper, nanocomposites are defined as either bulk materials that consist of inhomogeneities with at least one dimension within 1 to 100 nm, or a nano-scale structure with inhomogeneities. The latter case, of course, involves nano-scale inho∗

Corresponding author. Tel.: +352 42 59 91 48 79, Fax: +352 42 59 91 555. Email address: [email protected] (Yao Koutsawa ) Preprint submitted to N.A.

July 23, 2015

mogeneities since these inhomogeneities should be about one order smaller than the structure itself. Nanocomposites/nanomaterials are of interest because of their unusual mechanical, thermo-mechanical, electrical, optical and magnetic properties as compared to composites of similar constituents, volume proportion and shape/orientation of reinforcements. The size-dependency in the area of nanotechnology is well known and has been investigated in terms of surface/interface energies, stresses and strains (Dingreville and Qu, 2008; Dingreville et al., 2005; Dingreville and Qu, 2007; Duan et al., 2005a; Sharma and Ganti, 2004). The classical Eshelby’s solution (Eshelby, 1957) of an embedded inclusion neglects the presence of surface or interface energies (stresses, strains) and indeed, the effects of those are negligible except in the size range of tens of nanometers, where one contends with a significant surface-to-volume ratio. Thus, due to the large ratio of surface area to volume in nanosized objects, the behavior of surfaces and interfaces becomes a prominent factor controlling the nano-mechanical properties of nanostructured materials. The reduced coordination of atoms near a free surface induces a corresponding redistribution of electronic charge, which alters the binding situation (Sander, 2003). As a result, the energy of these atoms will, in general, be different from that of the atoms in the bulk. In a similar vein, atoms at an interface of two materials experience a different local environment than atoms in the bulk of the materials, and the equilibrium position and energy of these atoms will, in general, be different from those of the atoms in the bulk. Therefore in the case of nanocomposites the elastic properties of the interface should be given due consideration. There are different ways in which the properties of the surface can be defined and introduced. For example, if one considers an “interface” separating two otherwise homogeneous phases, the interfacial property may be defined either in terms of an interphase, or by introducing the concept of a dividing surface. While “interface” refers to the surface area between two phases, “interphase” corresponds to the volume defined by the narrow region sandwiched between the two phases. In the approach of interface where a single dividing surface is used to separate the two homogeneous phases, the interface contribution to the thermodynamic properties is defined as the excess over the values that would obtain if the bulk phases retained their properties constant up to an imaginary surface (of zero thickness) separating the two phases (Dingreville et al., 2005; Dingreville and Qu, 2007). As pointed out by Dingreville (2007), for realistic bimaterials, there typically exist two distinctive length parameters, namely, the atomic spacing (lattice parameter) d, and the radius of curvature of the interface D, where D is generally several order of magnitude greater than d for most of the problems of engineering interest. Thus, if one measures the characteristic length of these inhomogeneities by D, the radius of curvature of the interface between an inhomogeneity and its surrounding medium, the discrete atomic structure of the material is smeared (homogenized) into a continuum. This is like observing the interface from a far distance so that one cannot see the atomic structure, nor the thickness of the interphase. All one 2

sees is that the properties jump from one bulk value to the other across the interface. Consequently, one may perceive that field quantities (stress, displacement, etc) are discontinuous at the interface when measured by the mesoscopic length scale D (Dingreville, 2007). Several attempts (Sharma and Ganti, 2004; Sharma and Wheeler, 2007; Lim et al., 2006; Huang and Sun, 2007; Duan et al., 2005a) which have been made in analyzing the nanocomposites by considering interfacial effect are based on this viewpoint. Dingreville (2007) develops the interfacial conditions for the displacement, strain and stress fields across the interface of bimaterials and shows that none of the above works has taken the interface effects fully into account. The various solutions for the Eshelby’s nano-inclusion problems that have appeared in the literature recently assume an elastically isotropic surface/interface and are concerned with the case of spherical inhomogeneities problems. Generally, the problem is solved using the generalized YoungLaplace equations for solids (Povstenko, 1993) and the general expressions for the displacements in an infinite region containing a spherical inhomogeneity from Lur’e (1964) in terms of Legendre polynomial of order two. Although Dingreville (2007) establishes the relationship between microscopic properties (measured by d) and mesoscopic jumps of these properties across the interface measured by D by taking into account the “3-D nature” of the surface/interface (Dingreville, 2007; Dingreville and Qu, 2008), the solution of the full boundary value problem remains very complex to solve. The main purpose of this paper is to address the nano-inhomogeneities problem by the interphase approach as it has been suggested earlier by Teik-Cheng (2005). Strictly speaking, in the framework of continuum mechanics, an interface between two dissimilar materials may be considered as a region over which the material properties changes gradually from the bulk property of one material to the other Dingreville (2007). In this paper, this transition region is regarded as the interphase of thickness, t. Field quantities in the continuum framework such as stress, strain and strain energy density may all vary continuously across the interphase. This can be regarded as a magnified version which depicts the interface as an interphase volume, consisting of a gap defined by the interatomic distance between atoms of the phases. The transition from one bulk values to the other may take place over a few layers of atoms (Dingreville and Qu, 2007, 2008). If one is able to characterize this transition region (interphase) by its stiffness tensors, the obtained moduli may then be incorporated, for example, in a developed model for composites with coated inhomogeneities such as (Cherkaoui et al., 1994; Lipinski et al., 2006; Koutsawa et al., 2009). It is obvious that an advantage of this approach is that it is developed from earlier models that consider inclusion shape, thereby enabling both the nano-inhomogeneity shape and the nano-interphase shape to be simultaneously accounted for in computing the overall composite stiffness tensor. The nano-inhomogeneity shape is of importance when dealing with nano-platelet and nano-tube reinforcements. In view of the extremely small nature of interphase thickness, the use of molecular mechanics is of importance. In the following, a general procedure to characterize the interphase from 3

atomistic description is presented and some numerical applications are shown to illustrate the effectiveness of the methodology. In Section 2 the atomistic informations of the interfacial region are used to determined the effective properties of the associated interphase. Section 3 deals with the computation of the effective properties of the nanocomposite. Section 4 presents numerical results to illustrate the effectiveness of the present modeling schemes. 2. Atomistic and continuum description of the interphase 2.1. Atomic level caracterization To evaluate the elastic properties of a given interfacial region from a discrete medium viewpoint, we consider a given interface between two materials A and B. Figure 1 illustrates schematically the two different views based on two different length scales of the nano-inhomogeneities problem. We consider

Figure 1: Concept of interface-interphase for nanocomposites: different views based on different length scales.

a bimaterial system containing N equivalent atoms. The total energy E (n) of the atom n is given by X 1 XX E (n) =E0 + E (r nm ) + E (r nm , r np ) + · · · 2! m6=n

where r nm

m6=n p6=n

(1) X 1 XX nm np nq + ··· E (r , r , · · · , r ) , N! m6=n p6=n q6=n q = (r1nm )2 + (r2nm )2 + (r3nm )2 is the scalar distance between the atom m and the atom n

and E is the interatomic potentials function which may include pair potentials such as the Lennard-Jones

potential as well as multi-body potentials such as the Embedded Atom Method (EAM) potentials. The P (n) . If one considers a single solid total energy of this ensemble containing N such atoms is E = N n=1 E of infinite extent subjected to a macroscopically uniform strain field εij , Johnson (1972) demonstrates

that the elastic stiffness tensor Cijkl of the bulk crystal is pn qn 2 (n) N 1 X X X 1 rj rl ∂ E Cijkl = , N Ω ∂ripn ∂rkqn mn n=1 p6=n q6=n n r 4

(2)

where Ωn is the atomic volume of the atom n. However, when considering an atomic ensemble containing non-equivalent atoms (which is the case for systems containing grain boundaries and interfaces) subjected to a macroscopically uniform deformation, internal relaxations occur (Martin, 1975) and Eq. (2) can be interpreted as a description of the homogeneous elastic response of the ensemble (Dingreville, 2007). To take into account the inner displacements, an atomic level mapping between the undeformed, rîn , and deformed, rin , configurations is defined by n rin − rîn = ε± ˆjn , + ε ˜ ij r ij

(3)

ñij describes the “inner” relaxation where ε± ij corresponds to a homogeneous deformation of atom n and ε (or additional “non-homogeneous” deformation) of atom n with respect to a homogeneous deformation. Note that the positive (or negative) sign should be selected if atom n is in the phase A (or B). The “T” stress decomposition (Qu and Bassani, 1993) can be used to describe the homogeneous deformation of the bimaterial assembly by an in-plane deformation εsαβ and a transverse loading σit , (see appendix Appendix A). Following the appendix Appendix A, one gets ± t s ± ε± ij = Aijαβ εαβ + Bijk σk ,

(4)

with  1 ±  ±  A± , = δ δ − δ δ + γ γ iα 3j 3i jβ ijαβ iαβ 2 jαβ 1 ± ±  ±  Bijk Mjk δ3i + Mik δ3j , = 2

(5)

± ± ± are given in appendix Appendix A. The tensors A± and γiαβ where Mjk ijαβ and Bijk characterize the

homogeneous behavior of the bimaterial. At this point, one can get the difference in position of two atoms, m and n, near their relaxed state as s mn t rimn − rîmn = Amn ˜m ˆjm − εñij rˆjn , iαβ εαβ + Bik σk + ε ij r

mn where Amn iαβ and Bik are defined as (Dingreville, 2007) ±,m ±,n ±,n m ±,m n mn Amn = A + A r ˆ − A r ˆ − A r ˆ iαβ j ijαβ ijαβ ijαβ j ijαβ j ,

±,n m ±,m n ±,m ±,n mn rˆjmn − Bijk rˆj − Bijk rˆj . = Bijk + Bijk Bik

(6)

(7)

(8)

The total strain energy density of the interphase containing N atoms is 1 1 (2) (2) · σ t + εs : A : εs + σ t · B · σ t 2 2 N −1 X + σt · Q : εs + Kn + Dn : εs + Gn · σt : εñ

E =E0 + A

(1)

(1)

: εs + B

n=1

+

N −1 N −1 1 X X n ε˜ : Lmn : ε˜m . 2 n=1 m=1

5

(9)

(1)

Equation (9) shows that the tensors A

(1)

,B

,A

(2)

(2)

,B

and Q describe the homogeneous behavior n n n mn of the assembly upon a deformation configuration εs , σt while the tensors K , D , G and L

represent the components of perturbation response of the system introduced by the non-equivalency of the atomic ensemble such as in grain boundaries or interface and account for the accommodation of internal relaxations upon a deformation configuration εs , σt . Their expressions are derived by Dingreville (2007). The virial stress on atom n is given by n σij =

1 X ∂E nm r . 2Ωn ∂rinm j

(10)

m6=n

n , with respect to r nm near the equilibrium configuration, r Expanding this atomic level stress, σij înm , of i

the bimaterial, one gets n σij

=

n σij nm nm r =ˆ r

n X ∂σij + ∂rknm m6=n

(rknm − rˆknm ) .

(11)

r nm =ˆ r nm

n , takes the following form Dingreville (2007) Making use of Eq. (6), the atomic level stress, σij s,n

t,n

n σij = τijn + C ijαβ εsαβ + M kij σkt +

X

mn m ε˜kl , Tijkl

(12)

m6=n s,n

t,n

mn are known constants given in terms of the interatomic potential E where, τijn , C ijαβ , M kij and Tijkl

and its partial derivative with respect to the interatomic distance r. Derivations and expressions of these tensors are given in Dingreville (2007). In Eq. (12), there are 6N unknowns, ε˜m kl , which describe the internal relaxations. The conditions of mechanical equilibrium and traction continuity across the interface yield σjt,n = σjt .

(13)

Using Eq. (13) in Eq. (12) and some algebra manipulations, the expressions of the 6N unknowns are derived by Dingreville (2007) as s,n s,n t s,n s ε˜s,n αβ = ηαβ − Miαβ σi + Qαβκλ εκλ ,

(14)

t,n t,n t t,n s ε˜t,n i = ηi + Mij σj − Qiαβ εαβ .

(15)

n can be fully determined. The atomic level in-plane stress σ s,n is given as The atomic level stress σij αβ s,n s,n n σαβ = παβ + Cαβκλ εsκλ + Qniαβ σit ,

(16)

where     

n + n = ταβ παβ

PN −1 m=1

nm η t,m + Tαβ3k k

s,m nm m=1 Tαβκλ ηκλ ,

PN −1

PN −1 nm t,m PN −1 nm s,m s,n s,n Tαβ3i Qiκλ + m=1 Tαβµν Qµνκλ , Cαβκλ = C αβκλ − m=1   PN −1 nm P  t,n t,m N −1 nm  Qn M s,m, T m=1 Tαβ3j Mji − iαβ = M iαβ + 6 m=1 αβκλ iκλ

(17)

±,s Similarly, far away from the interface region, the bulk in-plane stress, σαβ , is determined by (see ap-

pendix Appendix A), ±,s ± ± ± ± σit . εsκλ + γiαβ σαβ = Cαβκλ − Cαβ3j γjκλ

(18)

With Eqs. (16) and (18), the interfacial region excess in-plane stress is as Σsαβ =

N A 1 X 0 (2) (1) s,n ±,s = − σαβ Ωn σαβ Γαβ + Γαβκλ εsκλ + Hjαβ σjt , V0 V0

(19)

n=1

where A0 is the area of the interface concerned, V0 is the volume of the associated interphase (interfacial region), and   (1)     Γαβ =

  (2)    Γαβκλ =

N N i 1 X 1 X h n ± n Ωn παβ , Hiαβ = Ωn Qiαβ − γiκλ A0 n=1 A0 n=1 N i h 1 X s,n ± ± ± . − Cαβκλ − Cαβ3j γjκλ Ωn Cαβκλ A0

(20)

n=1

Similarly the transverse excess strain given by Eq. (15), the is determined as ∆tk

N 1 X A0 (1) (2) t s , σ − H ε + Λ Λ = Ωn ε˜t,n = kαβ j αβ kj k k V0 n=1 V0

(21)

where (1)

Λk = −

N 1 X Ωn ηkt,n , A0 n=1

(2)

Λkj =

N 1 X t,n Ωn Mjk . A0

(22)

n=1

The tensors, Γ(1) , Γ(2) , Λ(1) , Λ(2) and H, are the so called interfacial elastic properties. For a given inter-atomic potential function, E (n) , numerical evaluation of the analytical expressions of these tensors requires knowledge of the relaxed state, rˆmn , of the interface. To obtain rˆmn , a preliminary molecular static (MS) simulation may be conducted. This is why the method is called semi-analytical c , of the (Dingreville and Qu, 2007; Dingreville, 2007). One can get now the elastic properties, Cijkl

interphase associated to this interface as it is described in Section 2.2. 2.2. Interphase stiffness tensor Instead of reporting the excess stress and strain to the interface area, A0 , this work attributes then to the volume of the interfacial region named interphase. Thus Eqs. (19) and (21) describe respectively the interphase excess in-plane stress and its transverse excess strain. It is therefore conceivable to attribute to this interfacial region effective elastic properties. To this end, one can make comparison between Eqs. (19) and (A.11) in one hand and Eqs. (21) and (A.10) in this other hand, that is,  A0 (1)   = τˆ s + Cs : εs + γ · σt , Γ + Γ(2) : εs + H · σt V0 A   0 Λ(1) + Λ(2) · σt − H : εs = −M · τ t + M · σt − γ : εs . V0 7

(23)

It follows that Cs =

A0 (2) Γ , V0

γ=

A0 H, V0

M=

A0 (2) Λ . V0

(24)

c , are completely determined from Eq. (24). The 21 components of the interphase stiffness tensor, Cijkl

Thus, one gets from the last equation of Eq. (24), −1 c C3j3k = Mjk =

hA

0

V0

Λ(2)

i−1 jk

,

(25)

c . Next, using the second equation of Eq. (24), one gets a c of Cijkl which gives the 6 components C3k3j c c of Cijkl linear system of 9 equations to solve for the 9 components C3kαβ (2)

c = Hjαβ , Λjk C3kαβ

(26)

c c Finally, the first equation of Eq. (24) gives the 6 components Cαβκλ of Cijkl by c Cαβκλ =

A0 (2) c . Γαβκλ + Hjκλ C3jαβ V0

(27)

The interphase elastic properties are therefore completely determined using Eqs. (25), (26) and (27) and the tensors Γ(2) , H and Λ(2) obtained from MS simulations and the analytical expressions Eqs. (20) and (22). 2.3. Particular case of isotropic interface In the case of isotropic interface, Γ(2) is defined by 2 parameters Ks and µs by (2)

Γαβκλ = (Ks − µs ) δαβ δκλ + µs (δακ δβλ + δαλ δβκ )

(28)

= λs δαβ δκλ + µs (δακ δβλ + δαλ δβκ ) , where, λs and µs can be seen as the Lamé constants of the interface. The Lamé constants of the interphase, λc and µc in this case are as follows A0 µs , µc = V0

2A0 λc = V0

! µs λs . 2µs − λs

(29)

For interphases such V0 /A0 = t, the thickness of the interphase (this is true for rectangular interface or even spherical interface since t is very small), Eq. (29) leads to µs = µc t,

λs =

2µc νc t , 1 − νc

(30)

where νc = 1/[2(1 + µc /λc )] is Poisson coefficient of the interphase. It worthy pointing that, Eq. (30) is similar to Eq. (69) in the work of Wang et al. (2005) or Eq. (7) in the work of Duan et al. (2007) for interface representation of thin and stiff interphase for spherical particles. Note that the result of Wang et al. 8

(2005) is based on the interface stress model of Duan et al. (2005b) which assumes displacement continuity and stress jump across the interface and isotropic interface. The stress discontinuities across an interface are equilibrated by the interface stress through the so-called generalized Young-Laplace equations. The identification of the parameters µs and λs with respect to the interphase parameters µc and λc is related to these features of the interface model of Duan et al. (2005b). The connection between interphase and interface models is then done since, in the case of spherical concentric coating inhomogeneity, the same features (displacement continuity and stress jump across the interface) are observed for thin and stiff interphase. In the other hand, the fully interface approach of Dingreville (2007) assumes the displacement discontinuity and stress discontinuity across the interface. The displacement discontinuity is related to the tensors Λ(1) , Λ(2) and H and the stress jump is the same as in the interface model (Duan et al., 2005a,b; Sharma and Ganti, 2004). Implicitly the results of Wang et al. (2005) assume that the inhomogeneity and the interphase display positive stiffness behavior and thus µs and λs should be always positive whereas the present result suggests the possible presence of negative stiffness (Lakes and Drugan, 2002) region around the nano-inhomogeneity depending on the nature of the interface (µs and λs may be positive or negative). The correspondence between the two results in the case of isotropic (spherical) interfaces leads to very small values of the components of the tensors Λ(1) , Λ(2) and H so that the displacement continuity across the interface can be assumed in the interface approach of Dingreville (2007). Therefore, it is obvious from Eq. (25) or Eq. (26) that the interphase is stiff. The thin assumption is also verified due to the small nature of the interfacial region. 2.4. Nano-particles and negative stiffness behavior Equation (29) shows that the interphase properties, µc and λc , can take positive or negative value depending on the interface elastic properties. This observation is very interesting since the works of Lakes and co-authors (Lakes and Drugan, 2002; Lakes et al., 2001) have shown that included materials possessing negative stiffness behavior can lead to extremely high macroscopic damping properties and high stiffness. Negative stiffness is one way to state that portions of the stress-strain curve of a material have negative values. The existence of such behavior is suggested by the existence of multiple local minimums, or energy wells, predicted by Landau theory for ferroelastic materials (Falk, 1983). Indeed, extreme damping behavior has been experimentally observed in bi-phase materials containing trace elements of single domain crystals undergoing phase transformation (Lakes et al., 2001; Jaglinski and Lakes, 2004). Negative bulk modulus behavior has also been observed in single cells of polymer foams. Negative stiffness behavior is qualitatively well understood to be the material stiffness analogue of the bi-stable force versus displacement curves characteristic of beam buckling or the snap through behavior observed when a lateral force is applied to a post-buckled beam. It is imperative to state that negative stiffness material behavior cannot exist alone in nature as it is inherently instable as it implies that the stiffness tensor of 9

the material is not positive definite. Naturally occurring negative stiffness materials are therefore transitory occurrences at best. However, negative stiffness is not excluded by any physical law. Objects with negative stiffness are unstable if they have free surfaces but can be stabilized when constrained by rigid boundaries as in the case of the buckled tubes studied by Lakes (2001). The sole requirement is that the macroscopic behavior of a heterogeneous system containing negative stiffness elements be described by a positive definite stiffness tensor. Further work has also shown that included phases with negative stiffness may also lead to extreme thermal expansion, and piezoelectricity (Wang and Lakes, 2001), thereby giving further impetus to research the creation of such materials. The ability to create composites containing such phases for practical application is an open, and very active, area of research. The present model shows that a nano-particle embedding leads to local domains of negative stiffness. This is a very promising area of research in material design strategies.

3. Effective elastic constants of the nanocomposite Many micromechanical schemes have been successfully used to obtain effective elastic constants of heterogeneous solids. For a comprehensive exposition, one can refer to the monographs (Aboudi, 1991; Qu and Cherkaoui, 2006). In the present paper, the coated inhomogeneities micromechanical scheme first developed by Cherkaoui et al. (1994) and extended by Lipinski et al. (2006) and Koutsawa et al. (2009) is used to compute the effective properties of the nanocomposite. The two-phases nanocomposite with interfacial effect is transformed to three-phases nanocomposite. The detailed derivation of the effective elastic constants can be found in Refs. (Lipinski et al., 2006; Koutsawa et al., 2009).

4. Numerical simulations and discussions 4.1. Spherical inhomogeneities and isotropic material All the theoretical aspects are hereafter applied to predict effective properties of isotropic elastic composite containing spherical nano-voids. The numerical results are presented for aluminum with bulk modulus and Poisson ratio are respectively k3 =75.2 GPa and ν3 =0.3. In order to show the effectiveness of the models derived herein, the two sets of surface moduli used in Duan et al. (2005a) are considered. As it has been done by Duan et al. (2005a), the free-surface properties are taken from the papers of Miller and Shenoy (2000) and are set to equal to the interfacial properties. These free-surface properties are obtained from molecular dynamic (MD) simulations (Miller and Shenoy, 2000). The elastic properties of the two surfaces named A and B are given in Table 1. With these surface properties, the associated interphase properties, µc and λc (Lamé constants), are determined using Eq. (29). It is assumed a thickness, t, for the interphase in the subsequent numerical calculations. Note that the interphase’s thickness can be evaluated through the so called interfa10

ks (J.m−2 )

µs (J.m−2 )

A [1 0 0]

-5.457

-6.2178

B [1 1 1]

12.9327

-0.3755

Surface

Table 1: Elastic properties of surfaces A and B.

cial “relaxation” tensor Λ(1) (Dingreville and Qu, 2008). The classical coated-inhomogeneities model (Cherkaoui et al., 1994; Lipinski et al., 2006; Koutsawa et al., 2009) is used to calculate the effective properties of the nanocomposite. In what follows, κC and µC represent the classical results without the interfacial effect. The normalized bulk modulus κeff /κC for both surface properties as a function of the void radius is plotted in Fig. 2(a). Figure 2(a) shows that κeff /κC decreases (increases) with an increase of void size due to the surface effect. The variation of the bulk modulus κeff /κC with void volume fraction for two different void radii is shown in Fig. 2(b). The normalized shear modulus µeff /µC calculated 1.3

1.15 A, t = 0.07 nm

1.2

B, R = 5 nm, t = 0.04 nm

B, t = 0.02 nm

A, R = 20 nm, t = 0.07 nm 1.1

1.1

B, R = 20 nm, t = 0.04 nm

1

κeff/κC

eff

κ /κ

C

1.05 0.9

0.8

1

0.7

0.95 0.6

0.5

5

10

15

20 25 30 Void radius R (nm)

35

40

45

0.9 0

50

(a) Void radius for 30%

0.1

0.2

0.3 Void volume fraction

0.4

0.5

0.6

(b) Void volume fraction

Figure 2: Effective bulk modulus as a function of void radius and volume fraction.

for both surface properties as a function of the void radius is shown in Fig. 3(a). The variation of the normalized shear modulus with void volume fraction is shown in Fig. 3(b). Conclusions in Duan et al. (2005a) that is the surface effect is much more pronounced for surface A than for surface B are verified. All the figures presented with the models derived herein are similar to those in Duan et al. (2005a). Thus the results from these first numerical simulations are very encouraging since they show that the present modeling schemes are able to reproduce the results in the work of Duan et al. (2005a). The case of spherical isotropic nano-particle with isotropic interface elastic properties is a particular case of the more general framework in this paper. In order to show the capability of the present models to deal with non spherical shapes and anisotropic materials, other applications are performed in the following. 11

1.05

1.1 1

1

0.9

0.95

µeff/µC

µeff/µ

C

0.8

0.7

0.9

0.6 A, R = 10 nm, t = 0.2 nm 0.85

A, t = 0.2 nm 0.5

B, R = 10 nm, t = 0.02 nm

B, t = 0.02 nm

0.4

0.8

0.3

0.2

5

10

15

20 25 30 Void Radius R (nm)

35

40

45

0.75 0

50

0.1

(a) Void radius for 30%

0.2

0.3 Void volume fraction

0.4

0.5

0.6

(b) Void volume fraction

Figure 3: Effective shear modulus as a function of void radius and volume fraction.

4.2. Ellipsoidal inhomogeneities and isotropic material We suppose now that the previous material (see Section 4.1) contains ellipsoidal nano-voids with semi-axes a, b and c. In what follows, the surface elastic properties presented in table 1 are set to equal to the interfacial properties of the ellipsoidal nano-voids. For the numerical results we take a = 5 nm, b = 3a and c = 5a. In this case, the effective properties corresponding to surfaces A and B are computed for 30% of ellipsoidal nano-voids. The results are presented below in table 2 where Eieff (GPa) , µeff ij (GPa) eff are the effective longitudinal moduli, shear moduli and the Poisson ratios of the nanocomposite and νij

respectively. Surface C corresponds to the effective properties of the same voided composite without surface effect. Obviously, table 3 shows that the effective materials are orthotropic. E1eff

E2eff

E3eff

µeff 23

µeff 13

µeff 12

eff ν23

eff ν13

eff ν12

A

27.39

53.01

57.08

21.25

14.09

13.66

0.28

0.16

0.17

B

29.63

55.27

59.14

21.95

14.92

14.59

0.28

0.16

0.17

C

28.99

54.92

58.94

22.09

14.91

14.39

0.27

0.16

0.17

Surface

Table 2: Effective properties without and with interface effect (GPa).

4.3. Ellispsoidal inhomogeneities and anisotropic interfacial properties To finish we consider a nanocomposite with ellipsoidal inhomogeneities and anisotropic interfacial elastic properties. The interfacial excess anisotropic elastic properties are taken from Dingreville and Qu (2)

(2)

(2)

(2008). The non zero components of Γ(2) (J/m2 ) are Γ1111 = −10.679, Γ2222 = −10.510, Γ1212 = (2)

−2.489, Γ1122 = −14.908. The non zero components of H(nm) are H111 = −3 × 10−4 , H122 = 2 × 10−4 , H211 = 0.046, H222 = 0.057, H311 = 0.35 and H322 = 0.618. The non zero components (2)

(2)

(2)

of Λ(2) (10−11 nm/Pa) are Λ11 = 0.494, Λ22 = −0.185 and Λ33 = 0.121. The stiffness matrix of the 12

interphase associated to these interfacial elastic properties is computed using Eqs. (25), (26) and (27). In the present case, the matrix is copper (Cu) which bulk and shear moduli are 138.53 GPa and 21.65 GPa respectively. The effective properties of this ellipsoidal nano-voided composite with a = 5 nm, b = 3a, c = 5a and ϕ1 = 30% are presented in table 3. The effective material is orthotropic. E1eff

E2eff

E3eff

µeff 23

µeff 13

µeff 12

eff ν23

eff ν13

eff ν12

20.64.99

38.83

40.23

13.74

9.65

9.53

0.39

0.22

0.21

Table 3: Effective properties with anisotropic interface effect (GPa).

The numerical results presented above show the capacity of the present models to efficiently handle the nano-inhomogeneity Eshelby’s problem by taking into account the atomistic level informations. In contrast to the previous models which have been devoted to this problem, the present modeling approach is able to tackle any material/interface anisotropy and a general ellipsoidal inhomogeneity shape. It is shown from this modeling approach that a nano-particles-reinforced composite can exhibit locally negative stiffness behavior. This observation is very interesting since it may lead to new avenues in materials design strategies. Many potential applications can be made: damping, piezoelectricity, low-k materials, magnetostriction of magnetic materials etc.

5. Conclusion This paper has introduced an interphase model that bridges the gap between discrete systems (atomic level interactions) and continuum mechanics. The atomistic informations gotten through molecular static (MS) simulations are put in micromechanical-based multi-coatings scheme to determine the effective properties of composite materials containing nano-inhomogeneities. Contrary to the previous modeling schemes that considers interface approach, the present modeling schemes deal with the more general case of anisotropic behaviors and ellipsoidal inhomogeneities. Numerical results for solids with spherical nano-voids show that with an appropriate value of the interphase’s thickness, the present modeling schemes reproduce the results in the paper of Duan et al. (2005a). Other results show the capability of this approach to handle an anisotropic material/interface behavior and ellipsoidal nano-inhomogeneities.

Appendix A. “T” stress decomposition Consider an inhomogeneous, linearly elastic solid with strain energy density per unit undeformed volume defined by 1 w = w0 + τij εij + Cijkl εij εkl , 2

(A.1)

13

where εij is the Lagrangian strain tensor. The corresponding second Piola-Kirchhoff stress tensor is thus given by σij =

∂w = τij + Cijkl εij . ∂εij

(A.2)

Equivalently, (A.2) can be written as s s σαβ = ταβ + Cαβκλ εsκλ + Cαβ3k εtk ,

σjt = τjt + C3jκλ εsκλ + C3j3k εtk ,

(A.3)

where the summation convention is implied, and the lower case Roman subscripts go from 1 to 3 and the lower case Greek subscripts go from 1 to 2, and s s = ταβ , τjt = τ3j . = σαβ , εsαβ = εαβ , σjt = σ3j , εtα = 2εα3 , εt3 = ε33 , ταβ σαβ

(A.4)

Assuming that the second order, C3k3j , is invertible, the second part of Eq. (A.3) can be rewritten as εtk = −Mkj τjt + Mjk σjt − γkαβ εsαβ ,

(A.5)

−1 Mkj = C3k3j ,

(A.6)

where γkαβ = Mkj C3kαβ .

Substituting (A.5) into the first of (A.3) yields s s s εsκλ + γjαβ σjt , + Cαβκλ = τˆαβ σαβ

(A.7)

s s τˆαβ = ταβ − τjt γjαβ ,

(A.8)

where s Cαβκλ = Cαβκλ − Cαβ3j γjκλ .

Using tensorial notation, Eqs. (A.1), (A.5) and (A.7) can be written, respectively, as 1 1 1 w = w0 − τ t · M · τ t + τˆ s : εs + εs : Cs : εs + σt · M · σt , 2 2 2

(A.9)

εt = −M · τ t + M · σ t − γ : εs ,

(A.10)

σs = τˆ s + Cs : εs + γ · σ t .

(A.11)

In addition, if the material is isotropic, that is Cijkl = λδij δkl + µ (δik δjl + δil δjk ) , where λ and µ are the Lamé constants. The other quantities, in this special case, are such as  1 λ+µ  δ3k δ3j + δkj ,  C3k3j = (λ + µ) δ3k δ3j + µδkj , Mkj = − µ (λ + 2µ) µ λ 2λµ   Cs δαβ δκλ + µ (δακ δβλ + δαλ δβκ ) , γiαβ = δ3i δαβ . αβκλ = λ + 2µ λ + 2µ 14

(A.12)

(A.13)

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