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Accepted Manuscript A micromechanics approach for effective elastic properties of nano-composites with energetic surfaces/interfaces Yao Koutsawa, Sonnou Tiem, Wenbin Yu, Frédéric Addiego, Gaetano Guinta PII: DOI: Reference:
S0263-8223(16)30983-7 http://dx.doi.org/10.1016/j.compstruct.2016.09.066 COST 7792
To appear in:
Composite Structures
Received Date: Accepted Date:
20 June 2016 24 September 2016
Please cite this article as: Koutsawa, Y., Tiem, S., Yu, W., Addiego, F., Guinta, G., A micromechanics approach for effective elastic properties of nano-composites with energetic surfaces/interfaces, Composite Structures (2016), doi: http://dx.doi.org/10.1016/j.compstruct.2016.09.066
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A micromechanics approach for effective elastic properties of nano-composites with energetic surfaces/interfaces Yao Koutsawaa,b,∗, Sonnou Tiemb , Wenbin Yuc , Frédéric Addiegoa , Gaetano Guintaa a Materials
Research and Technology Department, Luxembourg Institute of Science and Technology, 41, rue du Brill, L-4422 Belvaux, Luxembourg b Equipe de Recherche en Science de l’Ingénieur, Ecole Nationale Supérieure d’Ingénieurs, Université de Lomé, BP. 1515, Lomé, Togo c Purdue University, West Lafayette, IN 47907-2045, USA
Abstract The mechanics of structure genome is generalized to model nanocomposites taking into account the surfaces/interfaces stress effect at nano-scale. This full field micromechanics approach is applied to predict the effective properties of composites containing nano-inhomogeneities. Examples of binary composite materials, fiber reinforced composite materials and particle reinforced composite materials are used to demonstrate the robustness and accuracy of this micromechanics theory with surfaces/interfaces effects. The size-dependency of the overall elastic moduli shows the importance of energetic surfaces/interfaces in modeling the mechanical behavior of nano-scale composite materials. The proposed micromechanics approach is versatile enough to be applied for estimating the effective elastic properties of many nano-composite materials. Keywords: Micromechanics, Interface elasticity, Nano-inhomogeneities, Elastic properties
1. Introduction Many research investigations are devoted to nano-scale science and developments of nano-composites due to the advances in nanotechnology. Nano-composites/nano-materials 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. Nano-materials can be generally classified into two groups. In the first group (called nano-structured material), the characteristic length of the microstructure, such as the grain size of a polycrystal material, is in the nano-meter range. In the second group (called nano-sized structural element), at least one of the overall dimensions of a structural element is in the nano-meter range. Therefore, this may include nano-particles, nano-films, nano-wires [1, 2]. In the present study, nano-composites are defined as either bulk materials that consist of a matrix containing inhomogeneities with at least one dimension within 1 to 100 nm, or a nano-scale structure with inhomogeneities. Obviously, the latter case involves nano-scale inhomogeneities since these inhomogeneities should be about one order smaller ∗ Corresponding
author. Tel.: +352 275 88 87 48 79, Fax: +352 275 885. Email addresses: [email protected] (Yao Koutsawa ), [email protected] (Wenbin Yu)
Preprint submitted to Composite Structures
September 24, 2016
than the smallest dimension of the structure itself. The size-dependency in nano-composite materials is well known and has been investigated in terms of surface/interface energies, stresses and strains [2–8]. The classical micromechanics approaches for conventional composite materials neglect 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 nano-meters, where one contends with a significant surface-to-volume ratio. Hence, due to the large ratio of surface area to volume in nano-sized objects, the behavior of surfaces and interfaces becomes a prominent factor controlling the nano-mechanical properties of nano-structured materials. The surfaces of solid bodies and the interfaces in composite materials generally exhibit properties different from those of the bulk materials. Such differences are typically caused by processes such as surface oxidation, ageing, coating, atomic rearrangement or the termination of atomic bonds. 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 [2, 4]. Numerous micromechanics models have been proposed to investigate the role of surface/interface elasticity [9] and the size-dependence on the elastic response of nano-composite materials, see [5, 6, 10–15] and references cited therein. Based on the variational asymptotic method of unit cell homogenization (VAMUCH) [16–22], the mechanics of structure genome (MSG) has been recently discovered to provide a unified theory for constitutive modeling of heterogeneous materials and structures [23–26]. MSG unifies micromechanics and structural mechanics to provide a single approach to model all types of composite structures. MSG is based on the the concept of Structure Genome (SG), which is defined as the smallest mathematical building block of the material. MSG minimizes the loss of information during homogenization of anisotropic, heterogeneous bodies by the variational asymptotic method (VAM) [27] which is applicable to any solid mechanics problem admitting a variational structure where one or more relatively small parameters are involved. The “smallness” of these parameters is exploited by using an asymptotic expansion structure of the functional of the problem (and not of the unknown field quantities as done in conventional asymptotic methods). VAM combines the advantages of both variational and asymptotic methods so that the models constructed using VAM can be directly implemented using the well established finite element method (FEM). Motivated by the interface/surface excess energy effects and other non-classical behavior of composite materials at the nano-scale, the objective of this contribution is to extend MSG for micromechanics problems where the microstructure possesses interface/surfaces. In compar2
ison to existing finite element (FE) based micromechanics approaches such as representative volume element (RVE) [28] analysis and the mathematical homogenization theories (MHT) [29], as far as a heterogeneous material featuring a 3D RVE with periodic boundary conditions is concerned, RVE analysis, MHT and MSG will provide exactly the same results for both effective properties and local fields. As far as efficiency is concerned, computing the complete stiffness matrix, RVE analysis requires solving the six static problems because the coefficient matrix of the linear system is affected by the coupled equation constraints used to apply the periodic boundary conditions. MHT and MSG can be implemented using the finite element method so that the linear system will be factorized once and solve for six load steps. Theoretically speaking, MHT and MSG could be six times more efficient than RVE analysis. However, such equivalence does not exist for situations when periodic boundary conditions cannot be applied. MHT is not applicable for general aperiodic materials. RVE analysis and MSG can still use appropriate boundary conditions but the results could differ from each other. For materials featuring lower-dimensional heterogeneities such as binary composites or unidirectional continuous fiber reinforced composites, RVE analysis and MHT can only obtain properties and local fields with the same dimensionality as that of the RVE, while MSG can still obtain the complete set of 3D properties and local fields out of a 1D or 2D analysis. The main reason is that numerical implementations of MHT and RVE are based on a weak form converted from the strong form of a boundary value problem while MSG directly solve a variational statement based on energetic considerations. Last but not least, MSG has the capability to directly construct models for beams/plates/shells based on the same principle of minimum information loss which is different from the RVE analysis and MHT although it is possible to modify RVE analysis and MHT to construct models for beams/plates/shells. For example MSG can be used to construct beam models with accuracy comparable to 3D FEA for slender structures [25] and can provide a general-purpose solution for the free-edge stress problem of composite laminates [26]. The paper is organized as follows. Section 2 presents an overview of the interface/surface excess energy effect in nano-scale heterogeneous solids. Section 3 gives the theoretical formulation of MSG with interafce/surface effects. Section 4 presents some numerical examples to show the accuracy of the predictions of the proposed size-dependent MSG based micromechanics approach. Some conclusions are given in section 5. The results presented in this study are obtained using GetFEM++ [30] as a FE library and Gmsh [31] as a pre-processing tool.
2. Interface/Surface excess energy in nano-scale composite materials: problem statement We consider a simple unit cell of a two-phase composite, see Fig. 1. The inclusion occupies a finite region of arbitrary shape, bounded by a smooth surface, S, with outward unit normal, n. The interface between the
3
matrix and the inclusion is assumed coherent, i.e., JuK = u(S+ ) − u(S− ) = 0,
(1)
where u is the displacement field, S+ and S− mean approaching the interface from the matrix side (+) or the inclusion side (-), respectively, see Fig. 1.
Figure 1: Unit cell of a 2D two-phase composite material with energetic interface/surface.
Our objective is to find the effective elastic properties of the composite material using MSG in the case where the inclusion is very small so that interfacial stress becomes non-negligible due to the high surface-tovolume ratio. This elasticity problem for a conventional size inclusion has been solved in [16]. In what follows, the dot symbol “·” denotes the dot product, the colon symbol “:” denotes the double dot product and the symbol “>” denotes tensor transposition. The interfacial excess energy density is given as, see [3], 1 Γ = Γ(0) + Γ(1) : s + s : Γ(2) : s , 2
(2)
where Γ(0) , Γ(1) and Γ(2) are the intrinsic interfacial elastic properties that can be computed once the material system is defined, see [3, 4]. The interfacial or surface strain, s , is defined for the classical bulk strain, , as s = P · · P = P : ,
(3)
where the second order tensor P = I − n ⊗ n (i.e. Pi j = δi j − ni n j with δi j being the Kronecker delta) and the fourth order tensor P = P⊗P (i.e. Pi jkl = Pik Pl j ). The interfacial or surface stress, σs , is given by the Shuttleworth equation, see [32], as σs =
∂Γ = Γ(1) + Γ(2) : s . ∂ s
(4)
For isotropic linear elastic interface/surface, Γ(1) and Γ(2) are defined as (see [33]) (1)
Γi j = τs Pi j
! (2) and Γi jkl = λs Pi j Plk + µs Pik Pjl + Pil Pjk ,
(5)
where τs is the strain-independent surface/interfacial tension, λs and µs are the Lamé’s constants of the interface/surface. This paper deals with the elastic properties so the influence of the surface tension, τs , on the 4
effective mechanical behavior is not investigated. Hence, τs is set to zero in what follows and Eq. (4) becomes σs =
∂Γ = Γ(2) : s . ∂ s
(6)
3. Theoretical formulation of MSG with interface/surface excess energy In this section, we will first provide a brief introduction to MSG and its application to micromechanics. MSG provides a unified theory to construct macroscopic structural models for capturing both anisotropy and heterogeneity of composites at the microscopic scale. The macroscopic structural model could be a beam model (e.g. Euler-Bernoulli model, Timoshenko model, or Vlasov model), a plate/shell model (e.g. KirchhoffLove model or Reissner-Mindlin model), or a 3D model (e.g. Cauchy continuum model). The focus of this work is to predict the effective elastic properties for Cauchy continuum model. In other words, we plan to construct micromechanics models using MSG. According to [24], structure genome (SG) is defined as the smallest mathematical building block of the structure to emphasize the fact that it contains all the constitutive information needed for a structure in the same fashion as the genome contains all the genetic information for an organism’s growth and development. Particularly to construct micromechanics models, we need to define the corresponding SG for a 3D heterogeneous body. As shown in Fig. 2, analyses of 3D heterogeneous structures can be approximated by a 3D macroscopic structural analysis with the material properties provided by a constitutive modeling of a SG. For 3D structures, SG serves a similar role as representative volume element (RVE) in micromechanics. However, they are different in many aspects. For example, for a structure made of composites featuring 1D heterogeneity (e.g. binary composites made of two alternating layers, Fig. 2a), the SG will be a straight line with two segments denoting corresponding phases. One can mathematically repeat this line in-plane to build the two layers of the binary composite, and then repeat the binary composite out of plane to build the entire structure. Another possible application is to model a laminate as an equivalent homogeneous solid. The transverse normal line is the 1D SG for the laminate. The constitutive modeling over the 1D SG can compute the complete set of 3D properties. For a structure made of composites featuring 2D heterogeneity (e.g. continuous unidirectional fiber reinforced composites, Fig. 2b), the SG will be 2D. Although 2D RVEs are also used in micromechanics, only in-plane properties can be obtained from common RVE-based models. If the complete set of properties are needed for 3D structural analysis, a 3D RVE is usually required [28], while a 2D domain is sufficient if it is modeled using SG-based models (Fig. 2b). For a structure made of composites featuring 3D heterogeneity (e.g. particle reinforced composites, Fig. 2c), the SG will be a 3D volume. Although 3D SG for 3D structures represents the most similar case to RVE, boundary conditions in terms of displacements and tractions indispensable in RVE-based models are not needed for SG-based models. We consider a solid continuum that occupies a region Ω of the three-dimensional Euclidean space in its reference configuration. In view of the fact that the size of SG is much smaller than the overall size of the 5
Actual problem
b) 2D SG a) 1D SG
+
C) 3D SG
SG-based Representa5on
3D macroscopic structural analysis
Figure 2: Analysis of 3D heterogeneous structures approximated by a constitutive modeling over SG and a corresponding 3D macroscopic structural analysis.
macroscopic structure, we introduce a set of micro-coordinates yi = xi /ε with ε being a small parameter to describe the SG. This basically enables a zoom-in view of the SG at the size similar as the macroscopic structure. In multi-scale analysis nomenclature, xi is called the slow field and yi is called the fast or rapid field, see [29]. Let us denote the SG as Ωµ . The domains Ω and Ωµ are referred to as the macro- and micro-scale, respectively. The first step of MSG is to express the kinematics of the original model using the kinematic variables of the macroscopic model. We can express the micro-scale displacement field uµ over Ωµ as uµ (x, y) = u(x) + εξ (x, y),
(7)
where x = (x1 , x2 , x3 ), y = (y1 , y2 , y3 ), u(x) is the displacement of the corresponding point x of the macroscale, and ξ (x, y) is defined as the displacement fluctuation field of the SG which is a periodic for periodic microstructures. Eq. (7) can be considered as a change of variable. To render the mapping unique, we need to define the macroscopic displacement as 1 u(x) = |Ωµ |
Z
Ωµ
uµ (x; y)dΩµ ,
(8)
where |Ωµ | is the volume of Ωµ . This definition result in the following constraints on the fluctuation field: Z
Ωµ
ξ (x, y)dΩµ = 0,
(9)
The general MSG theory, see [24], can be specified for the micromechanics analysis of elastic composite materials with energetic surfaces/interfaces as the minimization of the following functional Z Z 1 s (2) s : C : dΩµ + : Γ : dS , Π= 2|Ωµ | Ωµ S
(10)
subject to the constraints Eq. (9) and possible periodicity requirement of ξ (x, y) for periodic materials. In Eq. (10), C is the fourth order elastic properties tensor, is the strain field in the SG and s = P · · P = P : 6
is the surface/interface strain. The expression of is given as def
≡
i 1 h! i 1h i εh i > > 1 h! , + εˆ ∇uµ + ∇uµ = ∇µ ξ + ∇µ ξ + (∇ξ )> + ∇ξ = ¯ + ˜ (∇u)> + ∇u + 2 2 2 2
(11)
with ∇ denoting the gradient with respect to the macro-scale coordinates, ∇µ denoting the gradient with respect to the micro-scale coordinates, ¯ is the macro-scale strain at the point x and ˜ is the fluctuation strain field in h i > . It the SG, and the last term, ε ˆ = ε (∇ξ ) + ∇ξ /2 is a high-order term and can be omitted so that ≈ ¯ + ˜
is noted that the constraints Eq. (9) do not influence the minimum values of the functional, Π, but help uniquely
determine the fluctuation functions, ξ . In practice, one can constrain the fluctuation functions at an arbitrary node to be zero and later use these constraints to recover the unique fluctuation functions. Here we consider periodic microstructures. It is fine to use penalty function method to introduce the periodic boundary conditions on ξ . However, this method introduces additional approximation and the robustness of the solution depends on the choice of large penalty number. Here, we choose to make the nodes on the positive boundary surface (i.e. yi = li /2) slave to the nodes on the opposite negative boundary surface (i.e. yi = −li /2), li being the dimension of the SG in the i−th direction. By assembling all the independent active degrees of freedom, one can implicitly and exactly incorporate the periodic boundary conditions on ξ . In this way, one also reduces the total number of unknowns in the linear system, see [16]. Using matrix notation, the functional, Π, can be written as Z Z 1 > > s (¯ + ˜ ) C(¯ + ˜ )dΩµ + (¯ + ˜ ) C (¯ + ˜ )dS , Π= 2|Ωµ | Ωµ S
(12)
where C is the matrix representation of the fourth order tensor, C, and Cs is the matrix representation of the fourth order tensor, P> : Γ(2) : P. The minimization of the functional in Eq. (10) can be found analytically for simple and restricted cases. To deal with a more general microstructure, numerical techniques such as the finite element method should be used to solve this variational statement. The fluctuation functions, ξ (x, y), are approximated as
ξ (x, y) = ∑ Ni (y)qi (x),
(13)
i
where Ni (y) are the ordinary finite element shape functions and qi (x) are the nodal values of the fluctuation can be interpolated as functions. Making use of the finite element discretisation (13), ˜ ˜ = Bq,
(14)
where B is the classical FEM B-matrix containing some gradient operators and q is the vector of nodal values of the unknown fluctuation functions, ξ (x, y). Substituting Eqs. (13) and (14) into Eq. (12), one obtains a discretized form of the functional, Π, as Π=
1 > q Kqq q + 2q> Kq ¯ + ¯ > K ¯ , 2|Ωµ |
(15) 7
where: Kqq =
Z
B CBdΩµ +
Kq =
Z
B> CdΩµ +
K =
Z
CdΩµ +
Ωµ
Ωµ
Ωµ
>
Z
Z
Z
B> Cs BdS,
(16)
S
B> Cs dS,
(17)
S
Cs dS.
(18)
S
Minimizing Π in Eq. (15) with respect to q, one obtains the following linear system ¯ Kqq q = −Kq ,
(19)
to solve for the unknown, q. The solution of Eq. (19) can be written symbolically as ¯ q = q0 ,
(20)
where q0 is the solution of the linear system Kqq q0 = −Kq .
(21)
Here, the linear system Eq. (21) is solved using the multifrontal massively parallel sparse direct solver (MUMPS) [34, 35], within the GetFEM++ open source FE library [30]. Substituting Eq. (20) in Eq. (15), one obtains Π=
1 ¯ 1 ¯ = ¯ > C ¯ ¯ > q> , K + K 0 q 2|Ωµ | 2
(22)
¯ is the effective elastic properties matrix of the composite containing energetic interfaces/surfaces. where C Having obtained the effective elastic properties, one can use these properties to carry out the macroscopic analysis of the complete structure to predict the global mechanical behavior of the engineering system integrating this composite material with energetic surfaces/interfaces. If the point-wise distribution of the elastic fields (displacement, strain, stresses, etc.) within the microstructure is needed, one has to uniquely determine the fluctuation functions, ξ (x, y), using the constraints in Eq. (9) and follow the recovery procedure explained in [16, 18].
4. Numerical examples and discussion MSG, applied to micromechanics modeling, provides a unified modeling approach for 1D (e.g. binary composites), 2D (e.g. fibers reinforced composites) and 3D (e.g. particles reinforced composites) SGs. In what follows, we present some numerical results to demonstrate the prediction capabilities of the present micromechanics approach with energetic surfaces/interfaces.
8
4.1. Comparison with results available in the literature for infinite cylindrical fibers Solution for nano-composites containing infinite nano-cylindrical fibers has been obtained by solving a number of different two-dimensional problems in [13] and very recently by the method of conditional moments (MCM) in [15]. In this section, the effective elastic properties obtained using MSG are compared with those reported in [13] and [15] who considered aluminum matrix with bulk modulus κ = 75.2 GPa and Poisson’s ratio ν = 0.3 containing cylindrical cavities. The two sets of the free surface properties used in [13] and [15] are those reported in [36]. The surface A associated with [1 0 0] orientation is characterized by λs = 3.4893 N/m and µs = −6.2178 N/m and the surface B with [1 1 1] orientation is characterized by λs = 12.6838 N/m and µs = −0.3755 N/m. For this example, a 2D SG with hexagonal packing (see Fig. 8) is used to compute the 3D size-dependent effective elastic properties. Figures 3 and 4 show the variations of the plane bulk modulus κ¯ and the effective elastic moduli C¯33 , C¯13 , C¯44 and C¯66 with the volume fraction of the cylindrical cavities for a radius of 5 nm. The effective elastic properties including surface effects are normalized by the corresponding classical effective properties neglecting those effects (and identified by the subscript c) in order to emphasis the surface effects. In Figs. 3 and 4, A and B refer to the surface A and B, respectively; MSG, MCM and CHEN refer to the results obtained by the present model, the MCM in [15] and the models in [13], respectively. Figs. 3 and 4 show that the results of the MSG approach and those obtained by the CHEN approach [13] are very similar for all volume fractions of the cylindrical cavities and for both sets of surface properties except for the elastic modulus C¯66 . This exception for the elastic modulus C¯66 can be explained by the fact that, in the CHEN approach, κ¯ , C¯33 , C¯13 and C¯44 are computed using the neutral inhomogeneity approach which is mathematically equivalent to that of constructions of composite cylinder assemblages [37], while C¯66 is computed by the Generalized Self-Consistent Method (GSCM) [38]. It is seen that the results obtained by the three different micromechanics approaches are very
(b) Elastic modulus C¯33
(a) Plane bulk modulus
Figure 3: Normalized bulk and C¯33 moduli as function of the volume fraction of cylindrical cavities, R = 5 nm.
9
similar for all volume fractions of the cylindrical cavities less than 30% and for both sets of surface properties. For the volume fractions of cavities higher than 30%, the values of the normalized moduli for the surfaces A and B differ by a larger amount. The MCM predictions show larger influence of porosity than those of MSG and CHEN. For all the cases, the effective properties obtained by the MSG micromechanics approach are bounded by those predicted by the MCM and CHEN micromechanics approaches.
(a) Elastic modulus C¯13
(b) Elastic modulus C¯44
(c) Elastic modulus C¯66 Figure 4: Normalized C¯13 , C¯44 and C¯66 moduli as function of the volume fraction of cylindrical cavities, R = 5 nm.
4.2. Porous aluminium with spherical voids In this example, we consider an aluminum matrix containing spherical nano-voids. The material and surface properties are those used in the example of the Section 4.1. The effective elastic properties of the voided aluminium are computed with MSG using a 3D face-centered cubic (FCC) ordered SG, see Fig 5. Obviously, for the FCC SG, the effective properties are not isotropic (i.e. the material is characterized by 2 material constants: Young modulus, E and Poisson ratio, ν ) but have a cubic symmetry (i.e. the material is characterized 10
by 3 material constants: Young modulus, E, Poisson ratio, ν and shear modulus, µ 6= E/(2 + 2ν )). Hence in our results, we define the bulk modulus, κ , for the FCC material as κ = E µ /(9µ − 3E).
Y Z
X
Figure 5: Face-centered cubic structure genome of the porous aluminium material.
The normalized shear (µ¯ /µ¯ c ) and bulk (κ¯ /κ¯ c ) moduli of the porous aluminium material for different surface properties as a function of the void radius are plotted in Fig. 6, where µ¯ c and κ¯ c represent the classical results without surface effect. The volume fraction, f , of the voids is equal to 30%. It is seen from Fig. 6 that µ¯ /µ¯ c and
κ¯ /κ¯ c increases (decreases) with an increase of the void size depending on the considered surface. In general, the surface effect is more pronounced for surface A than for surface B and becomes negligible for a void radius larger than 50 nm. The variations of the normalized shear (µ¯ /µ¯ c ) and bulk (κ¯ /κ¯ c ) moduli with the void volume fraction for a void radius of 5 nm are shown in Fig. 7. These results are in good agreement with those reported in [5] for isotropic and random distribution of spherical voids.
(a) Shear modulus
(b) Bulk modulus
Figure 6: Effective normalized shear and bulk moduli as function of void radius, f = 30%.
11
(a) Shear modulus
(b) Bulk modulus
Figure 7: Effective normalized shear and bulk moduli as function of void volume fraction, R = 5 nm.
4.3. Long fibers reinforced composite material In this example, we consider a long fiber reinforced composite material. The matrix (M) and inclusion (I) M = 118 GPa, CM = 54 GPa are elastic materials with a cubic symmetry. In the numerical results, we use C11 12 M = 59 GPa for the matrix and CI = 83 GPa, CI = 45 GPa and CI = 40 GPa for the inclusion. The and C44 11 12 144
interfacial elasticity tensor is assumed to be isotropic and identical to that of the surface A [1 0 0] used in Section 4.1 (i.e. λs = 3.4893 N/m and µs = −6.2178 N/m). A 2D SG with hexagonal packing (see Fig. 8) is used to compute the 3D size-dependent effective elastic properties of this composite material.
Y Z
X
Figure 8: Hexagonal structure genome of the long fiber reinforced composite material.
The normalized effective Young moduli, shear moduli and Poisson ratios of the composite as a function of the inclusion radius are plotted in Fig. 9. The volume fraction, f , of the inclusions is set to 30%. It is seen from Figs. 9(a) and (b) that the effective Young and shear moduli of this fiber reinforced composite material increase with an increase in the inclusion size. The size effect is more pronounced for inclusion sizes below 10 nm. Above 50 nm, the surface effect is negligible and one recovers the classical effective properties. Fig. 9(c) shows that the effective Poisson ratios , ν¯ 23 and ν¯ 13 , decrease with an increase in the inclusion size whereas the 12
Poisson ratio, ν¯ 12 , increases with an increase in the inclusion size. Again, the size effect is more pronounced for inclusion sizes below 10 nm and negligible above 50 nm.
(a) Young moduli
(b) Shear moduli
(c) Poisson ratios Figure 9: Effective normalized Young moduli, shear moduli and Poisson ratios as function of fiber radius, f = 30%.
The normalized effective Young moduli, shear moduli and Poisson ratios of the composite as a function of the inclusion volume fraction are plotted in Fig. 10 for a fiber radius R = 1 nm. It is seen that the effective Young and shear moduli as well as the Poisson ratio, ν¯ 12 , are less that those of the traditional composite while the Poisson ratios, ν¯ 23 and ν¯ 13 , are bigger than those of the classical composite without the size effect. 4.4. Binary composite material The last example considers a binary composite material (i.e. periodic two-phase layered composite material). The two phases are isotropic elastic materials. We use E1 = 51.818 GPa and ν1 = 0.36 for the phase 1 and E2 = 87.579 GPa and ν2 = 0.33 for the phase 2. The interfacial elastic properties are those of the surface A [1 0 0] (i.e. λs = 3.4893 N/m and µs = −6.2178 N/m) given in Section 4.1. For this binary composite material, a 1D SG is used to compute the 3D size-dependent effective elastic properties. 13
(a) Young moduli
(b) Shear moduli
(c) Poisson ratios Figure 10: Effective normalized Young moduli, shear moduli and Poisson ratios as function of fiber volume fraction, R = 1 nm.
14
The normalized effective Young moduli, shear moduli and Poisson ratios of the composite as a function of the phase 1 thickness are plotted in Fig. 11. The volume fraction, f , of the phase 1 is equale to 30%. It is seen from Figs. 11(a) and (b) that the effective Young moduli and the shear modulus µ¯ 23 of this binary composite material increase with an increase of the phase 1 size. The size effect is more pronounced for inclusion sizes below 10 nm. Above 50 nm, the surface effect is negligible and one recovers the classical effective properties. The effect of the interfacial stress on the shear moduli, µ¯ 13 and µ¯ 12 is negligible. Figure 11(c) shows that the effective Poisson ratios decrease with an increase of the phase 1 size. The size effect is more pronounced for the thicknesses below 10 nm and negligible above 50 nm.
(a) Young moduli
(b) Shear moduli
(c) Poisson ratios Figure 11: Effective normalized Young moduli, shear moduli and Poisson ratios as function of phase 1 thickness, f = 30%.
The normalized effective Young moduli, shear moduli and Poisson ratios of the composite as a function of the phase 1 volume fraction are plotted in Fig. 12 for a thickness h = 1 nm. It is seen that the effective Young and shear moduli are lower that those of the traditional composite while the Poisson ratios are higher than those of the classical composite without the size effect. The effect of the interfacial stress on the shear moduli, µ¯ 13 15
and µ¯ 12 is negligible. The Poisson ratios are higher than those predicted for the classical composite without the size effect.
(a) Young moduli
(b) Shear moduli
(c) Poisson ratios Figure 12: Effective normalized Young moduli, shear moduli and Poisson ratios as function of phase 1 volume fraction, h = 1 nm.
5. Conclusion The interface/surface excess energy is introduced in the mechanics of structure genome approach to study composite materials containing nano-inhomogeneities within the framework of linear elasticity. To this end, the coherent interface model has been adopted, leading to additional stiffness matrices. The proposed approach has been used to predict the effective elastic moduli of composite materials with 3D (spherical nano-porous material), 2D (long nano-fibers reinforced composite material) and 1D (binary composite material) structure genomes. As expected, the numerical results show that the effective elastic properties predicted by this full field MSG based micromechanics method depend not only on the material properties and volume fractions, but also on the surface-to-volume ratio. The versatility and robustness of the MSG approach allows us to apply 16
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20
S0263-8223(16)30983-7 http://dx.doi.org/10.1016/j.compstruct.2016.09.066 COST 7792
To appear in:
Composite Structures
Received Date: Accepted Date:
20 June 2016 24 September 2016
Please cite this article as: Koutsawa, Y., Tiem, S., Yu, W., Addiego, F., Guinta, G., A micromechanics approach for effective elastic properties of nano-composites with energetic surfaces/interfaces, Composite Structures (2016), doi: http://dx.doi.org/10.1016/j.compstruct.2016.09.066
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A micromechanics approach for effective elastic properties of nano-composites with energetic surfaces/interfaces Yao Koutsawaa,b,∗, Sonnou Tiemb , Wenbin Yuc , Frédéric Addiegoa , Gaetano Guintaa a Materials
Research and Technology Department, Luxembourg Institute of Science and Technology, 41, rue du Brill, L-4422 Belvaux, Luxembourg b Equipe de Recherche en Science de l’Ingénieur, Ecole Nationale Supérieure d’Ingénieurs, Université de Lomé, BP. 1515, Lomé, Togo c Purdue University, West Lafayette, IN 47907-2045, USA
Abstract The mechanics of structure genome is generalized to model nanocomposites taking into account the surfaces/interfaces stress effect at nano-scale. This full field micromechanics approach is applied to predict the effective properties of composites containing nano-inhomogeneities. Examples of binary composite materials, fiber reinforced composite materials and particle reinforced composite materials are used to demonstrate the robustness and accuracy of this micromechanics theory with surfaces/interfaces effects. The size-dependency of the overall elastic moduli shows the importance of energetic surfaces/interfaces in modeling the mechanical behavior of nano-scale composite materials. The proposed micromechanics approach is versatile enough to be applied for estimating the effective elastic properties of many nano-composite materials. Keywords: Micromechanics, Interface elasticity, Nano-inhomogeneities, Elastic properties
1. Introduction Many research investigations are devoted to nano-scale science and developments of nano-composites due to the advances in nanotechnology. Nano-composites/nano-materials 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. Nano-materials can be generally classified into two groups. In the first group (called nano-structured material), the characteristic length of the microstructure, such as the grain size of a polycrystal material, is in the nano-meter range. In the second group (called nano-sized structural element), at least one of the overall dimensions of a structural element is in the nano-meter range. Therefore, this may include nano-particles, nano-films, nano-wires [1, 2]. In the present study, nano-composites are defined as either bulk materials that consist of a matrix containing inhomogeneities with at least one dimension within 1 to 100 nm, or a nano-scale structure with inhomogeneities. Obviously, the latter case involves nano-scale inhomogeneities since these inhomogeneities should be about one order smaller ∗ Corresponding
author. Tel.: +352 275 88 87 48 79, Fax: +352 275 885. Email addresses: [email protected] (Yao Koutsawa ), [email protected] (Wenbin Yu)
Preprint submitted to Composite Structures
September 24, 2016
than the smallest dimension of the structure itself. The size-dependency in nano-composite materials is well known and has been investigated in terms of surface/interface energies, stresses and strains [2–8]. The classical micromechanics approaches for conventional composite materials neglect 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 nano-meters, where one contends with a significant surface-to-volume ratio. Hence, due to the large ratio of surface area to volume in nano-sized objects, the behavior of surfaces and interfaces becomes a prominent factor controlling the nano-mechanical properties of nano-structured materials. The surfaces of solid bodies and the interfaces in composite materials generally exhibit properties different from those of the bulk materials. Such differences are typically caused by processes such as surface oxidation, ageing, coating, atomic rearrangement or the termination of atomic bonds. 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 [2, 4]. Numerous micromechanics models have been proposed to investigate the role of surface/interface elasticity [9] and the size-dependence on the elastic response of nano-composite materials, see [5, 6, 10–15] and references cited therein. Based on the variational asymptotic method of unit cell homogenization (VAMUCH) [16–22], the mechanics of structure genome (MSG) has been recently discovered to provide a unified theory for constitutive modeling of heterogeneous materials and structures [23–26]. MSG unifies micromechanics and structural mechanics to provide a single approach to model all types of composite structures. MSG is based on the the concept of Structure Genome (SG), which is defined as the smallest mathematical building block of the material. MSG minimizes the loss of information during homogenization of anisotropic, heterogeneous bodies by the variational asymptotic method (VAM) [27] which is applicable to any solid mechanics problem admitting a variational structure where one or more relatively small parameters are involved. The “smallness” of these parameters is exploited by using an asymptotic expansion structure of the functional of the problem (and not of the unknown field quantities as done in conventional asymptotic methods). VAM combines the advantages of both variational and asymptotic methods so that the models constructed using VAM can be directly implemented using the well established finite element method (FEM). Motivated by the interface/surface excess energy effects and other non-classical behavior of composite materials at the nano-scale, the objective of this contribution is to extend MSG for micromechanics problems where the microstructure possesses interface/surfaces. In compar2
ison to existing finite element (FE) based micromechanics approaches such as representative volume element (RVE) [28] analysis and the mathematical homogenization theories (MHT) [29], as far as a heterogeneous material featuring a 3D RVE with periodic boundary conditions is concerned, RVE analysis, MHT and MSG will provide exactly the same results for both effective properties and local fields. As far as efficiency is concerned, computing the complete stiffness matrix, RVE analysis requires solving the six static problems because the coefficient matrix of the linear system is affected by the coupled equation constraints used to apply the periodic boundary conditions. MHT and MSG can be implemented using the finite element method so that the linear system will be factorized once and solve for six load steps. Theoretically speaking, MHT and MSG could be six times more efficient than RVE analysis. However, such equivalence does not exist for situations when periodic boundary conditions cannot be applied. MHT is not applicable for general aperiodic materials. RVE analysis and MSG can still use appropriate boundary conditions but the results could differ from each other. For materials featuring lower-dimensional heterogeneities such as binary composites or unidirectional continuous fiber reinforced composites, RVE analysis and MHT can only obtain properties and local fields with the same dimensionality as that of the RVE, while MSG can still obtain the complete set of 3D properties and local fields out of a 1D or 2D analysis. The main reason is that numerical implementations of MHT and RVE are based on a weak form converted from the strong form of a boundary value problem while MSG directly solve a variational statement based on energetic considerations. Last but not least, MSG has the capability to directly construct models for beams/plates/shells based on the same principle of minimum information loss which is different from the RVE analysis and MHT although it is possible to modify RVE analysis and MHT to construct models for beams/plates/shells. For example MSG can be used to construct beam models with accuracy comparable to 3D FEA for slender structures [25] and can provide a general-purpose solution for the free-edge stress problem of composite laminates [26]. The paper is organized as follows. Section 2 presents an overview of the interface/surface excess energy effect in nano-scale heterogeneous solids. Section 3 gives the theoretical formulation of MSG with interafce/surface effects. Section 4 presents some numerical examples to show the accuracy of the predictions of the proposed size-dependent MSG based micromechanics approach. Some conclusions are given in section 5. The results presented in this study are obtained using GetFEM++ [30] as a FE library and Gmsh [31] as a pre-processing tool.
2. Interface/Surface excess energy in nano-scale composite materials: problem statement We consider a simple unit cell of a two-phase composite, see Fig. 1. The inclusion occupies a finite region of arbitrary shape, bounded by a smooth surface, S, with outward unit normal, n. The interface between the
3
matrix and the inclusion is assumed coherent, i.e., JuK = u(S+ ) − u(S− ) = 0,
(1)
where u is the displacement field, S+ and S− mean approaching the interface from the matrix side (+) or the inclusion side (-), respectively, see Fig. 1.
Figure 1: Unit cell of a 2D two-phase composite material with energetic interface/surface.
Our objective is to find the effective elastic properties of the composite material using MSG in the case where the inclusion is very small so that interfacial stress becomes non-negligible due to the high surface-tovolume ratio. This elasticity problem for a conventional size inclusion has been solved in [16]. In what follows, the dot symbol “·” denotes the dot product, the colon symbol “:” denotes the double dot product and the symbol “>” denotes tensor transposition. The interfacial excess energy density is given as, see [3], 1 Γ = Γ(0) + Γ(1) : s + s : Γ(2) : s , 2
(2)
where Γ(0) , Γ(1) and Γ(2) are the intrinsic interfacial elastic properties that can be computed once the material system is defined, see [3, 4]. The interfacial or surface strain, s , is defined for the classical bulk strain, , as s = P · · P = P : ,
(3)
where the second order tensor P = I − n ⊗ n (i.e. Pi j = δi j − ni n j with δi j being the Kronecker delta) and the fourth order tensor P = P⊗P (i.e. Pi jkl = Pik Pl j ). The interfacial or surface stress, σs , is given by the Shuttleworth equation, see [32], as σs =
∂Γ = Γ(1) + Γ(2) : s . ∂ s
(4)
For isotropic linear elastic interface/surface, Γ(1) and Γ(2) are defined as (see [33]) (1)
Γi j = τs Pi j
! (2) and Γi jkl = λs Pi j Plk + µs Pik Pjl + Pil Pjk ,
(5)
where τs is the strain-independent surface/interfacial tension, λs and µs are the Lamé’s constants of the interface/surface. This paper deals with the elastic properties so the influence of the surface tension, τs , on the 4
effective mechanical behavior is not investigated. Hence, τs is set to zero in what follows and Eq. (4) becomes σs =
∂Γ = Γ(2) : s . ∂ s
(6)
3. Theoretical formulation of MSG with interface/surface excess energy In this section, we will first provide a brief introduction to MSG and its application to micromechanics. MSG provides a unified theory to construct macroscopic structural models for capturing both anisotropy and heterogeneity of composites at the microscopic scale. The macroscopic structural model could be a beam model (e.g. Euler-Bernoulli model, Timoshenko model, or Vlasov model), a plate/shell model (e.g. KirchhoffLove model or Reissner-Mindlin model), or a 3D model (e.g. Cauchy continuum model). The focus of this work is to predict the effective elastic properties for Cauchy continuum model. In other words, we plan to construct micromechanics models using MSG. According to [24], structure genome (SG) is defined as the smallest mathematical building block of the structure to emphasize the fact that it contains all the constitutive information needed for a structure in the same fashion as the genome contains all the genetic information for an organism’s growth and development. Particularly to construct micromechanics models, we need to define the corresponding SG for a 3D heterogeneous body. As shown in Fig. 2, analyses of 3D heterogeneous structures can be approximated by a 3D macroscopic structural analysis with the material properties provided by a constitutive modeling of a SG. For 3D structures, SG serves a similar role as representative volume element (RVE) in micromechanics. However, they are different in many aspects. For example, for a structure made of composites featuring 1D heterogeneity (e.g. binary composites made of two alternating layers, Fig. 2a), the SG will be a straight line with two segments denoting corresponding phases. One can mathematically repeat this line in-plane to build the two layers of the binary composite, and then repeat the binary composite out of plane to build the entire structure. Another possible application is to model a laminate as an equivalent homogeneous solid. The transverse normal line is the 1D SG for the laminate. The constitutive modeling over the 1D SG can compute the complete set of 3D properties. For a structure made of composites featuring 2D heterogeneity (e.g. continuous unidirectional fiber reinforced composites, Fig. 2b), the SG will be 2D. Although 2D RVEs are also used in micromechanics, only in-plane properties can be obtained from common RVE-based models. If the complete set of properties are needed for 3D structural analysis, a 3D RVE is usually required [28], while a 2D domain is sufficient if it is modeled using SG-based models (Fig. 2b). For a structure made of composites featuring 3D heterogeneity (e.g. particle reinforced composites, Fig. 2c), the SG will be a 3D volume. Although 3D SG for 3D structures represents the most similar case to RVE, boundary conditions in terms of displacements and tractions indispensable in RVE-based models are not needed for SG-based models. We consider a solid continuum that occupies a region Ω of the three-dimensional Euclidean space in its reference configuration. In view of the fact that the size of SG is much smaller than the overall size of the 5
Actual problem
b) 2D SG a) 1D SG
+
C) 3D SG
SG-based Representa5on
3D macroscopic structural analysis
Figure 2: Analysis of 3D heterogeneous structures approximated by a constitutive modeling over SG and a corresponding 3D macroscopic structural analysis.
macroscopic structure, we introduce a set of micro-coordinates yi = xi /ε with ε being a small parameter to describe the SG. This basically enables a zoom-in view of the SG at the size similar as the macroscopic structure. In multi-scale analysis nomenclature, xi is called the slow field and yi is called the fast or rapid field, see [29]. Let us denote the SG as Ωµ . The domains Ω and Ωµ are referred to as the macro- and micro-scale, respectively. The first step of MSG is to express the kinematics of the original model using the kinematic variables of the macroscopic model. We can express the micro-scale displacement field uµ over Ωµ as uµ (x, y) = u(x) + εξ (x, y),
(7)
where x = (x1 , x2 , x3 ), y = (y1 , y2 , y3 ), u(x) is the displacement of the corresponding point x of the macroscale, and ξ (x, y) is defined as the displacement fluctuation field of the SG which is a periodic for periodic microstructures. Eq. (7) can be considered as a change of variable. To render the mapping unique, we need to define the macroscopic displacement as 1 u(x) = |Ωµ |
Z
Ωµ
uµ (x; y)dΩµ ,
(8)
where |Ωµ | is the volume of Ωµ . This definition result in the following constraints on the fluctuation field: Z
Ωµ
ξ (x, y)dΩµ = 0,
(9)
The general MSG theory, see [24], can be specified for the micromechanics analysis of elastic composite materials with energetic surfaces/interfaces as the minimization of the following functional Z Z 1 s (2) s : C : dΩµ + : Γ : dS , Π= 2|Ωµ | Ωµ S
(10)
subject to the constraints Eq. (9) and possible periodicity requirement of ξ (x, y) for periodic materials. In Eq. (10), C is the fourth order elastic properties tensor, is the strain field in the SG and s = P · · P = P : 6
is the surface/interface strain. The expression of is given as def
≡
i 1 h! i 1h i εh i > > 1 h! , + εˆ ∇uµ + ∇uµ = ∇µ ξ + ∇µ ξ + (∇ξ )> + ∇ξ = ¯ + ˜ (∇u)> + ∇u + 2 2 2 2
(11)
with ∇ denoting the gradient with respect to the macro-scale coordinates, ∇µ denoting the gradient with respect to the micro-scale coordinates, ¯ is the macro-scale strain at the point x and ˜ is the fluctuation strain field in h i > . It the SG, and the last term, ε ˆ = ε (∇ξ ) + ∇ξ /2 is a high-order term and can be omitted so that ≈ ¯ + ˜
is noted that the constraints Eq. (9) do not influence the minimum values of the functional, Π, but help uniquely
determine the fluctuation functions, ξ . In practice, one can constrain the fluctuation functions at an arbitrary node to be zero and later use these constraints to recover the unique fluctuation functions. Here we consider periodic microstructures. It is fine to use penalty function method to introduce the periodic boundary conditions on ξ . However, this method introduces additional approximation and the robustness of the solution depends on the choice of large penalty number. Here, we choose to make the nodes on the positive boundary surface (i.e. yi = li /2) slave to the nodes on the opposite negative boundary surface (i.e. yi = −li /2), li being the dimension of the SG in the i−th direction. By assembling all the independent active degrees of freedom, one can implicitly and exactly incorporate the periodic boundary conditions on ξ . In this way, one also reduces the total number of unknowns in the linear system, see [16]. Using matrix notation, the functional, Π, can be written as Z Z 1 > > s (¯ + ˜ ) C(¯ + ˜ )dΩµ + (¯ + ˜ ) C (¯ + ˜ )dS , Π= 2|Ωµ | Ωµ S
(12)
where C is the matrix representation of the fourth order tensor, C, and Cs is the matrix representation of the fourth order tensor, P> : Γ(2) : P. The minimization of the functional in Eq. (10) can be found analytically for simple and restricted cases. To deal with a more general microstructure, numerical techniques such as the finite element method should be used to solve this variational statement. The fluctuation functions, ξ (x, y), are approximated as
ξ (x, y) = ∑ Ni (y)qi (x),
(13)
i
where Ni (y) are the ordinary finite element shape functions and qi (x) are the nodal values of the fluctuation can be interpolated as functions. Making use of the finite element discretisation (13), ˜ ˜ = Bq,
(14)
where B is the classical FEM B-matrix containing some gradient operators and q is the vector of nodal values of the unknown fluctuation functions, ξ (x, y). Substituting Eqs. (13) and (14) into Eq. (12), one obtains a discretized form of the functional, Π, as Π=
1 > q Kqq q + 2q> Kq ¯ + ¯ > K ¯ , 2|Ωµ |
(15) 7
where: Kqq =
Z
B CBdΩµ +
Kq =
Z
B> CdΩµ +
K =
Z
CdΩµ +
Ωµ
Ωµ
Ωµ
>
Z
Z
Z
B> Cs BdS,
(16)
S
B> Cs dS,
(17)
S
Cs dS.
(18)
S
Minimizing Π in Eq. (15) with respect to q, one obtains the following linear system ¯ Kqq q = −Kq ,
(19)
to solve for the unknown, q. The solution of Eq. (19) can be written symbolically as ¯ q = q0 ,
(20)
where q0 is the solution of the linear system Kqq q0 = −Kq .
(21)
Here, the linear system Eq. (21) is solved using the multifrontal massively parallel sparse direct solver (MUMPS) [34, 35], within the GetFEM++ open source FE library [30]. Substituting Eq. (20) in Eq. (15), one obtains Π=
1 ¯ 1 ¯ = ¯ > C ¯ ¯ > q> , K + K 0 q 2|Ωµ | 2
(22)
¯ is the effective elastic properties matrix of the composite containing energetic interfaces/surfaces. where C Having obtained the effective elastic properties, one can use these properties to carry out the macroscopic analysis of the complete structure to predict the global mechanical behavior of the engineering system integrating this composite material with energetic surfaces/interfaces. If the point-wise distribution of the elastic fields (displacement, strain, stresses, etc.) within the microstructure is needed, one has to uniquely determine the fluctuation functions, ξ (x, y), using the constraints in Eq. (9) and follow the recovery procedure explained in [16, 18].
4. Numerical examples and discussion MSG, applied to micromechanics modeling, provides a unified modeling approach for 1D (e.g. binary composites), 2D (e.g. fibers reinforced composites) and 3D (e.g. particles reinforced composites) SGs. In what follows, we present some numerical results to demonstrate the prediction capabilities of the present micromechanics approach with energetic surfaces/interfaces.
8
4.1. Comparison with results available in the literature for infinite cylindrical fibers Solution for nano-composites containing infinite nano-cylindrical fibers has been obtained by solving a number of different two-dimensional problems in [13] and very recently by the method of conditional moments (MCM) in [15]. In this section, the effective elastic properties obtained using MSG are compared with those reported in [13] and [15] who considered aluminum matrix with bulk modulus κ = 75.2 GPa and Poisson’s ratio ν = 0.3 containing cylindrical cavities. The two sets of the free surface properties used in [13] and [15] are those reported in [36]. The surface A associated with [1 0 0] orientation is characterized by λs = 3.4893 N/m and µs = −6.2178 N/m and the surface B with [1 1 1] orientation is characterized by λs = 12.6838 N/m and µs = −0.3755 N/m. For this example, a 2D SG with hexagonal packing (see Fig. 8) is used to compute the 3D size-dependent effective elastic properties. Figures 3 and 4 show the variations of the plane bulk modulus κ¯ and the effective elastic moduli C¯33 , C¯13 , C¯44 and C¯66 with the volume fraction of the cylindrical cavities for a radius of 5 nm. The effective elastic properties including surface effects are normalized by the corresponding classical effective properties neglecting those effects (and identified by the subscript c) in order to emphasis the surface effects. In Figs. 3 and 4, A and B refer to the surface A and B, respectively; MSG, MCM and CHEN refer to the results obtained by the present model, the MCM in [15] and the models in [13], respectively. Figs. 3 and 4 show that the results of the MSG approach and those obtained by the CHEN approach [13] are very similar for all volume fractions of the cylindrical cavities and for both sets of surface properties except for the elastic modulus C¯66 . This exception for the elastic modulus C¯66 can be explained by the fact that, in the CHEN approach, κ¯ , C¯33 , C¯13 and C¯44 are computed using the neutral inhomogeneity approach which is mathematically equivalent to that of constructions of composite cylinder assemblages [37], while C¯66 is computed by the Generalized Self-Consistent Method (GSCM) [38]. It is seen that the results obtained by the three different micromechanics approaches are very
(b) Elastic modulus C¯33
(a) Plane bulk modulus
Figure 3: Normalized bulk and C¯33 moduli as function of the volume fraction of cylindrical cavities, R = 5 nm.
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similar for all volume fractions of the cylindrical cavities less than 30% and for both sets of surface properties. For the volume fractions of cavities higher than 30%, the values of the normalized moduli for the surfaces A and B differ by a larger amount. The MCM predictions show larger influence of porosity than those of MSG and CHEN. For all the cases, the effective properties obtained by the MSG micromechanics approach are bounded by those predicted by the MCM and CHEN micromechanics approaches.
(a) Elastic modulus C¯13
(b) Elastic modulus C¯44
(c) Elastic modulus C¯66 Figure 4: Normalized C¯13 , C¯44 and C¯66 moduli as function of the volume fraction of cylindrical cavities, R = 5 nm.
4.2. Porous aluminium with spherical voids In this example, we consider an aluminum matrix containing spherical nano-voids. The material and surface properties are those used in the example of the Section 4.1. The effective elastic properties of the voided aluminium are computed with MSG using a 3D face-centered cubic (FCC) ordered SG, see Fig 5. Obviously, for the FCC SG, the effective properties are not isotropic (i.e. the material is characterized by 2 material constants: Young modulus, E and Poisson ratio, ν ) but have a cubic symmetry (i.e. the material is characterized 10
by 3 material constants: Young modulus, E, Poisson ratio, ν and shear modulus, µ 6= E/(2 + 2ν )). Hence in our results, we define the bulk modulus, κ , for the FCC material as κ = E µ /(9µ − 3E).
Y Z
X
Figure 5: Face-centered cubic structure genome of the porous aluminium material.
The normalized shear (µ¯ /µ¯ c ) and bulk (κ¯ /κ¯ c ) moduli of the porous aluminium material for different surface properties as a function of the void radius are plotted in Fig. 6, where µ¯ c and κ¯ c represent the classical results without surface effect. The volume fraction, f , of the voids is equal to 30%. It is seen from Fig. 6 that µ¯ /µ¯ c and
κ¯ /κ¯ c increases (decreases) with an increase of the void size depending on the considered surface. In general, the surface effect is more pronounced for surface A than for surface B and becomes negligible for a void radius larger than 50 nm. The variations of the normalized shear (µ¯ /µ¯ c ) and bulk (κ¯ /κ¯ c ) moduli with the void volume fraction for a void radius of 5 nm are shown in Fig. 7. These results are in good agreement with those reported in [5] for isotropic and random distribution of spherical voids.
(a) Shear modulus
(b) Bulk modulus
Figure 6: Effective normalized shear and bulk moduli as function of void radius, f = 30%.
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(a) Shear modulus
(b) Bulk modulus
Figure 7: Effective normalized shear and bulk moduli as function of void volume fraction, R = 5 nm.
4.3. Long fibers reinforced composite material In this example, we consider a long fiber reinforced composite material. The matrix (M) and inclusion (I) M = 118 GPa, CM = 54 GPa are elastic materials with a cubic symmetry. In the numerical results, we use C11 12 M = 59 GPa for the matrix and CI = 83 GPa, CI = 45 GPa and CI = 40 GPa for the inclusion. The and C44 11 12 144
interfacial elasticity tensor is assumed to be isotropic and identical to that of the surface A [1 0 0] used in Section 4.1 (i.e. λs = 3.4893 N/m and µs = −6.2178 N/m). A 2D SG with hexagonal packing (see Fig. 8) is used to compute the 3D size-dependent effective elastic properties of this composite material.
Y Z
X
Figure 8: Hexagonal structure genome of the long fiber reinforced composite material.
The normalized effective Young moduli, shear moduli and Poisson ratios of the composite as a function of the inclusion radius are plotted in Fig. 9. The volume fraction, f , of the inclusions is set to 30%. It is seen from Figs. 9(a) and (b) that the effective Young and shear moduli of this fiber reinforced composite material increase with an increase in the inclusion size. The size effect is more pronounced for inclusion sizes below 10 nm. Above 50 nm, the surface effect is negligible and one recovers the classical effective properties. Fig. 9(c) shows that the effective Poisson ratios , ν¯ 23 and ν¯ 13 , decrease with an increase in the inclusion size whereas the 12
Poisson ratio, ν¯ 12 , increases with an increase in the inclusion size. Again, the size effect is more pronounced for inclusion sizes below 10 nm and negligible above 50 nm.
(a) Young moduli
(b) Shear moduli
(c) Poisson ratios Figure 9: Effective normalized Young moduli, shear moduli and Poisson ratios as function of fiber radius, f = 30%.
The normalized effective Young moduli, shear moduli and Poisson ratios of the composite as a function of the inclusion volume fraction are plotted in Fig. 10 for a fiber radius R = 1 nm. It is seen that the effective Young and shear moduli as well as the Poisson ratio, ν¯ 12 , are less that those of the traditional composite while the Poisson ratios, ν¯ 23 and ν¯ 13 , are bigger than those of the classical composite without the size effect. 4.4. Binary composite material The last example considers a binary composite material (i.e. periodic two-phase layered composite material). The two phases are isotropic elastic materials. We use E1 = 51.818 GPa and ν1 = 0.36 for the phase 1 and E2 = 87.579 GPa and ν2 = 0.33 for the phase 2. The interfacial elastic properties are those of the surface A [1 0 0] (i.e. λs = 3.4893 N/m and µs = −6.2178 N/m) given in Section 4.1. For this binary composite material, a 1D SG is used to compute the 3D size-dependent effective elastic properties. 13
(a) Young moduli
(b) Shear moduli
(c) Poisson ratios Figure 10: Effective normalized Young moduli, shear moduli and Poisson ratios as function of fiber volume fraction, R = 1 nm.
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The normalized effective Young moduli, shear moduli and Poisson ratios of the composite as a function of the phase 1 thickness are plotted in Fig. 11. The volume fraction, f , of the phase 1 is equale to 30%. It is seen from Figs. 11(a) and (b) that the effective Young moduli and the shear modulus µ¯ 23 of this binary composite material increase with an increase of the phase 1 size. The size effect is more pronounced for inclusion sizes below 10 nm. Above 50 nm, the surface effect is negligible and one recovers the classical effective properties. The effect of the interfacial stress on the shear moduli, µ¯ 13 and µ¯ 12 is negligible. Figure 11(c) shows that the effective Poisson ratios decrease with an increase of the phase 1 size. The size effect is more pronounced for the thicknesses below 10 nm and negligible above 50 nm.
(a) Young moduli
(b) Shear moduli
(c) Poisson ratios Figure 11: Effective normalized Young moduli, shear moduli and Poisson ratios as function of phase 1 thickness, f = 30%.
The normalized effective Young moduli, shear moduli and Poisson ratios of the composite as a function of the phase 1 volume fraction are plotted in Fig. 12 for a thickness h = 1 nm. It is seen that the effective Young and shear moduli are lower that those of the traditional composite while the Poisson ratios are higher than those of the classical composite without the size effect. The effect of the interfacial stress on the shear moduli, µ¯ 13 15
and µ¯ 12 is negligible. The Poisson ratios are higher than those predicted for the classical composite without the size effect.
(a) Young moduli
(b) Shear moduli
(c) Poisson ratios Figure 12: Effective normalized Young moduli, shear moduli and Poisson ratios as function of phase 1 volume fraction, h = 1 nm.
5. Conclusion The interface/surface excess energy is introduced in the mechanics of structure genome approach to study composite materials containing nano-inhomogeneities within the framework of linear elasticity. To this end, the coherent interface model has been adopted, leading to additional stiffness matrices. The proposed approach has been used to predict the effective elastic moduli of composite materials with 3D (spherical nano-porous material), 2D (long nano-fibers reinforced composite material) and 1D (binary composite material) structure genomes. As expected, the numerical results show that the effective elastic properties predicted by this full field MSG based micromechanics method depend not only on the material properties and volume fractions, but also on the surface-to-volume ratio. The versatility and robustness of the MSG approach allows us to apply 16
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