Foundational aspects of a multiscale modeling framework for composite materials
 Somnath Ghosh^{1}Email author
DOI: 10.1186/s401920150036x
© Ghosh. 2015
Received: 27 November 2014
Accepted: 9 April 2015
Published: 11 June 2015
Abstract
The objective of this paper is to provide an integrated computational materials science and engineering or ICMSE perspective on various aspects governing multiscale analysis of composite materials. These include microstructural characterization, micromechanical analysis of microstructural regions, and bridging length scales through hierarchical modeling. The paper discusses methods of identifying representative volume elements or RVEs in the material microstructure using both morphology and micromechanicsbased methods. For microstructures with nonuniform distributions, a statistically equivalent RVE or SERVE is identified for developing homogenized properties under undamaged and damaging conditions. A particularly novel development is the introduction of SERVE boundary conditions based on the statistical distribution of heterogeneities in the domain exterior to the SERVE. A micromechanical model of the SERVE incorporating explicit damage mechanisms like interfacial debonding, and fiber and matrix damage is developed for crack propagation. Finally, a microstructural homogenizationbased continuum damage mechanics (HCDM) model is developed that accounts for the microstructural distributions as well as the evolution of damage. The HCDM modelbased simulations are able to provide both macroscopic and microscopic information on evolving damage and failure.
Keywords
Statistically equivalent RVE Statistical distributionbased SERVE boundary conditions Cohesive zone models Homogenizationbased continuum damage mechanics (HCDM)Background
Multiscale modeling has become a familiar theme, integral to the modeling of heterogeneous materials, such as composites. The ability of powerful computational methods to resolve material behavior at different scales and communicate across them is fostering unprecedented advances in multiscale modeling. These models provide indepth understanding of material deformation and failure that can revolutionize integrated structurematerial design. It is prudent to use the notion of multiple spatial scales in the analysis of composite materials and structures due to the inherent existence of various scales. Conventional methods of analysis have used effective properties obtained from homogenization of response at microscopic length scales. A number of analytical models have evolved within the framework of small deformation linear elasticity theory to predict homogenized macroscale constitutive response of heterogeneous materials, accounting for the characteristics of microstructural behavior. The underlying principle of these models is the HillMandel condition of homogeneity, which states that for large differences in microscopic and macroscopic length scales, the volumeaveraged strain energy is obtained as the product of the volumeaveraged stresses and strains in the representative volume element (RVE). Cogent reviews of various homogenization models are presented in [1]. Notable among the various estimates and bounds on the elastic properties are the variational approach using extremum principles [2,3], selfconsistent model [4,5], etc. These analytical models however do not provide adequate resolution to capture the fluctuations in microstructural variables that can have significant effects on properties.
The use of computational micromechanical methods like the finite element method, boundary element method, spring lattice models, etc. has become increasingly popular for accurate prediction of stresses, strains, and other evolving variables in composite materials. Within the framework of computational multispatial scale analyses, two categories of methods have emerged. The first group, known as ‘hierarchical models,’ entails bottomup coupling for transfer of information from lower to higher scales [612]. Homogenization theory is based on complete scale separation with implicit assumptions of uniformity of macroscopic variables. Uncoupling of governing equations at different scales is often achieved through incorporation of periodicity boundary conditions on the microscopic RVEs, implying periodic repetition of a local microstructural region. The models can simultaneously predict evolution of macroscale variables using homogenized material properties and microscale variables in the periodic microstructural RVE as a postprocessor to the macroscopic analysis module. A subset of the hierarchical models has been branded as the ‘FE ^{2} multiscale methods’ in [13], where micromechanical RVE models are solved in every increment to obtain homogenized properties for macroscopic analysis. However, this method can be very expensive as it entails solving the RVE micromechanical problem for every element integration point in the computational domain. To overcome the limitations of prohibitive computational overhead, macroscopic constitutive laws of elastic damage and elasticplastic damage have been developed in [11,14] from homogenization of RVE response at microscopic scales. The constitutive models represent the effect of morphological features and evolving microstructural mechanisms through evolving, anisotropic homogenized parameters. These reducedorder constitutive models are significantly more efficient than the FE ^{2}type models since they have limited information on microstructural morphology and do not have to solve the RVE problem in every step.
The second category of concurrent multiscale modeling methods has been proposed for problems of heterogeneous materials involving high solution gradients in [1522]. Concurrent multiscale models differentiate between regions that require differential resolutions and invoke twoway (both bottomup and topdown) coupling of scales. They introduce a platform for coherent, coupled analysis through substructuring of the computational domain into (a) regions of macroscopic analysis using homogenized material properties and (b) embedded local regions of detailed micromechanical modeling. Macroscopic analysis with homogenized constitutive models in regions of low deformation or stress gradients enhances the efficiency of the computational analysis due to reducedorder constitutive representation. Topdown localization, on the other hand, requires cascading down and embedding critical regions of localized damage or instability with explicit representation of the microstructure and micromechanisms. The computational model concurrently performs micromechanical analysis in these regions with direct interfaces to the surrounding homogenized region of macroscopic analysis [2022]. In other approaches in [23,24], higherorder gradients have been introduced to regularize the material model. The present paper will focus on hierarchical models only and not discuss concurrent multiscale models further.
Multiscale modeling of composites, especially for structures and materials in extreme environments such as failure and fatigue loading conditions, requires detailed scalespecific models that incorporate the underpinnings of the microstructure on material behavior. A holistic approach requires both characterization and modeling at each relevant scale and consequently establishes bridges between them. Emergent thrusts in integrated computational materials science and engineering or ICMSE and virtual materials systems are fostering unprecedented advances, integrating microstructure representations, constitutive descriptions, computational algorithms, and experimental methods. The objective of this paper is to provide an ICMSE perspective on different aspects governing multiscale analysis of composite materials. These include microstructural characterization, micromechanical analysis of microstructural regions, and bridging length scales involving bottomup or hierarchical modeling. The paper begins with a discussion of different methods of identifying RVEs in the material microstructure using both morphology and micromechanicsbased methods. For microstructures with nonuniform distributions, a statistically equivalent RVE or SERVE is identified for developing homogenized properties under undamaged and damaging conditions. A particularly novel development is the introduction of SERVE boundary conditions based on the statistical distribution of heterogeneities in the domain exterior to the SERVE in the ‘A novel boundary condition for defining SERVE based on statistical distribution of heterogeneities’ section. Section ‘Micromechanical model of SERVE undergoing damage and failure’ develops a micromechanical model of the SERVE incorporating explicit damage mechanisms like interfacial debonding and fiber and matrix damage. Finally, a microstructural homogenizationbased continuum damage model (HCDM) is developed in the ‘Homogenizationbased continuum damage mechanics model’ section that accounts for the microstructural distributions as well as the evolution of damage. The HCDM model corresponds to diffused damage in the macrostructure and is not valid in regions of severe macroscopic localization. In such cases, a concurrent multiscale model as developed by the authors in [21,22] is desirable. The author has developed 3D HCDM model in [2527] in a principal damage coordinate system (PDCS), which evolves with the load history. These are essential steps in developing rigorous multiscale models of damage and failure in heterogeneous materials.
Identification of the representative volume element or RVE for homogenization
 1.
Effective material properties, e.g., stressstrain behavior in the SERVE should be equivalent to the properties of the entire microstructure, at least locally.
 2.
Distribution functions of parameters reflecting the local morphology, like local volume fraction, neighbor distance, or radial distributions in the SERVE should be equivalent to those for the overall microstructure.
 3.
The SERVE should be independent of location in the local microstructure, as well as of the applied loading.
The second characteristic is driven by the postulate that response functions characterizing a material behavior should have a strong dependence on morphological parameters of the microstructure. Assuming that the SERVE corresponds to a converged property or response function, the set of distribution functions should depend on what set of morphological parameters, e.g., local/overall volume fraction, nearest neighbor distance, shape, etc., control that response function. The necessary set should be at least the parameters that most strongly affect that response function, while the sufficient set should consist of additional parameters that will have smaller effects on the behavior.
Figure 1 shows a 100 µm × 79.09 µm optical micrograph of a steel fiberreinforced epoxy matrix composite. The matrix material is an epoxy with Young’s modulus: E _{ m }=4.6 GPa and Poisson ratio: ν _{ m }=0.4. The steel fiber material has Young’s modulus: E _{ f }=210 GPa and Poisson ratio: ν _{ f }=0.3. All fibers are aligned perpendicular to the plane of the paper and have circular cross sections with a radius of 1.75 µm. Figure 1b shows a computergenerated image of the optical micrograph that is tessellated into a network of Voronoi cells [34]. The circular region is used to identify N inclusions belonging to a SERVE. While satisfying different criteria may lead to nonuniqueness, it is possible to postulate the SERVE as the microregion that will satisfy all of the above requirements. Arriving at the optimal SERVE size is important to prevent risking erroneous estimation of effective properties with smaller RVEs or requiring huge computational resources with larger RVEs. Two metrics are discussed here for identification of the SERVE in undamaged and damaged composite microstructures.
Convergence of homogenized tangent stiffness tensor
Size (in µm), number of fibers, and area fraction of RVEs
Size  15  22  25  29  35  52  63  Micrograph 

# fibers  10  20  25  35  50  100  150  264 
AF (%)  31.5  32.1  32.2  31.5  33.1  32.3  31.3  32.3 
Estimation of the SERVE for microstructures with evolving damage is a more extensive exercise [33,34]. Geometric parameters play a lesser role since the evolution of stresses and strains is affected by the distribution of evolving damage as well. Even when the microstructure is geometrically uniform, initiation and progression of damage can result in a SERVE that is considerably larger than a unit cell. Analysis of the SERVE for heterogeneous microstructures undergoing interfacial debonding has been discussed in [33,34] using approaches similar to those used for the undamaged material. Initiation and progression of damage in the microstructure require the consideration of an evolving SERVE. Convergence of the degrading homogenized stiffness tensor \(\left [E^{H}_{\textit {ijkl}}\right ]\) is taken as an indicator of the region of influence and hence is a metric for estimating the SERVE. Instead of the tangent stiffness tensor, \(\left [E^{H}_{\textit {ijkl}}\right ]\) is represented as the linear unloading stiffness tensor from the point of loading in the macroscopic stressstrain plot.
Convergence of statistical functions of microstructural variables
I _{ k }(r) is the number of additional inclusion centers that lie inside a circle of radius r about an arbitrarily chosen inclusion. K(r) is a secondorder intensity function, defined in [36] as the number of additional inclusions that lie within a distance r of an inclusion center and divided by the number density N/A of inclusions.
A declining value of M(r) indicates reduced correlation between elements of the microstructure. It is therefore a good metric for the estimation of SERVE or the region of influence for a nonuniform microstructure. Steps to evaluate the SERVE size from M(r) have been described in [32,34]. For a random distribution, the pair distribution function g(r) approaches unity at large radial distances r. The radius of convergence r _{0} is identified from the g(r) plot for which g(r)≈1 for r>r _{0}. Appropriate microstructural variable fields associated with each inclusion are assigned as a ‘mark’, e.g., principal stresses and strains, Von Mises stress, etc. M(r)=1 corresponds to an uncorrelated random distribution of circular heterogeneities having identical marks. For nonuniform microstructures, M(r) stabilizes to nearunit values at a radius of convergence r _{ p }, such that for r>r _{ p }, M(r)≈1 and the local morphology ceases to have significant influence on the state variables. The radius r _{ p } corresponds to the lengthscale of correlation between the physical behavior and microstructural morphology. It provides an estimate for SERVE size.
Marked correlation function with geometric parameterbased marks
In summary, similar sizes of the SERVE are predicted by these alternate methods for given response functions. The successful use of the geometrybased indicators for problems without significant microstructural evolution, point to the fact that the morphological parameters strongly affect the response functions considered in the estimation of the SERVE. In these cases, the SERVE can be estimated without having to solve the entire micromechanics problem multiple times.
Marked correlation function with damage variables
A novel boundary condition for defining SERVE based on statistical distribution of heterogeneities
 1.
Affine displacement boundary conditions, \({u_{i}^{0}} = \epsilon _{\textit {ij}}^{0} x_{j}\) on ∂ Ω, where \(\epsilon _{\textit {ij}}^{0}\) is a constant applied farfield strain and x _{ j } are the boundary positions, measured from the geometrical centroid of the RVE.
 2.
Uniform traction boundary condition \(T_{i} = \sigma _{\textit {ij}}^{0} n_{j}\) on ∂ Ω, where \(\sigma _{\textit {ij}}^{0}\) is the constant applied stress, n _{ j } is the unit normal to the boundary of the RVE, and T _{ i } is the applied traction on the RVE boundary.
 3.
Periodic boundary conditions \({u_{i}^{p}}=\epsilon _{\textit {ij}}^{0} x_{j} + u_{i}^{pd}\) on ∂ Ω, with a periodic additional displacement \(u_{i}^{pd}\) which are equal on opposite faces of the RVE.
Typically, the uniform traction case gives the upper bound (Reuss bound), and the uniform strain condition provides the lower bound (Voigt bounds). These are not accurate for heterogeneous materials with nonuniform distributions. A drawback of the periodic boundary condition is that it automatically repeats the microstructure and associated deformation and damage patterns in the domain exterior to the RVE. This may not be accurate for nonuniform microstructures in general. Application of the above boundary conditions can result in an overestimation of the RVE region due to convergence requirements. Accurate boundary conditions play an important role in determining the SERVE size. In an attempt to redefine the SERVE with respect to both the volume and boundary conditions, this section introduces a novel boundary condition required to meet the criteria described in the ‘Identification of the representative volume element or RVE for homogenization’ section. Optimally, the SERVE should (a) encompass the region required to represent essential deformation mechanisms and (b) represent boundary conditions that reflect the effect of the region exterior to the SERVE domain.
Displacement boundary conditions are imposed on the SERVE as an augmentation of the affine displacements through the expression \(u_{i}={u_{i}^{0}}+u_{i}^{*}\). Here, the augmented displacement field \(u_{i}^{*}\) is derived by accounting for the microstructural distribution of the domain exterior to the RVE through a Green’s functionbased approach. While this approach of describing the SERVE boundary conditions is applicable to general threedimensional problems with inclusions of arbitrary shapes, the present study is restricted to unidirectional cylindrical fibers. For the sake of simplicity, all the fibers are assumed to have identical elastic properties, specifically stiffness components \(C_{\textit {ijkl}}^{F}\) and same crosssectional radius a. The fibers are however dispersed randomly in the matrix as seen from the microstructure in Figure 8a.
In Equation (14), the thirdorder tensor L _{ pql } relates the disturbance displacement to the eigenstrains \(\epsilon _{\textit {kl}}^{\Lambda }\). The tensor L _{ pql } is obtained from the Green’s function solution of the interacting fibers.
Micromechanical model of SERVE undergoing damage and failure
Micromechanical analyses of the identified composite microstructural SERVE are necessary ingredients for the development of a homogenizationbased continuum damage mechanics or HCDM model. This section briefly discusses the computational models that describe the deformation and damage response of fiberreinforced matrix composites exhibiting fiber and matrix cracking as well as interfacial debonding.
Constitutive model for fiber and matrix in the composite microstructure
κ is a constant parameter and l _{ c } is a characteristic length that is of the same order of magnitude as the maximum size of inhomogeneities [46]. In these simulations, a value l _{ c }=8 µm is taken.
Cohesive zone model for interfacial debonding
A 3D micromechanical model for composite microstructures undergoing fibermatrix interfacial debonding has been developed in [47]. In this model, the fibermatrix interface behavior in the normal and tangential directions is described by a nonlinear 3D cohesive zone model with bilinear tractiondisplacement relations. The interface is represented by a set of cohesive springs of infinitesimal length that are attached to the fiber and the matrix at opposite ends. With increasing displacement the traction across the interface reaches a maximum value, then decreases with further displacement increase, and finally vanishes indicating failure of the spring. Ortiz and coworkers [48,49] have developed irreversible cohesive laws for the unloading path after the interfacial softening.
Micromechanical analysis of a SERVE undergoing interfacial debonding and fiber/matrix damage
Fiber and matrix properties and interface (cohesive zone) material properties
Medium  E ^{ 0 }  υ  ρ  ε ^{ D }  ε ^{ 0 }  S ^{ ∗ }  σ _{ max }  δ _{ e }  σ _{ max } / δ _{ c } 

(GPa)  (kg/m ^{ 3 } )  (MPa)  (µs)  (N/m)  
Fiber (f)  64  0.26  2,230  0.0164  0.001  7  
Matrix (m)  2.5  0.40  1,170  1.50  0.001  7  
Interface (CZM)  4.0  5  20,000 
Homogenizationbased continuum damage mechanics model
where −T corresponds to the transpose of the inverse of the fourthorder M tensor. With the choice of an appropriate order of the damage tensor and the assumption of a function for M _{ ijkl }, Equation (30) can be used to formulate a damage evolution model using micromechanics and homogenization. The HCDM model assumes a diffused damage state and correspondingly a positive semidefinite stiffness matrix, even though the microstructural SERVE from which it is developed has regions of softening and damage localization.
P _{ ijkl } is a fourthorder symmetric negative definite tensor that corresponds to the direction of the rate of stiffness degradation tensor \(\dot {C}_{\textit {ijkl}}\). P _{ ijkl } is expressed as a function of strain e _{ ij }, α is a scaling parameter, and κ is a function of W _{ d }.
HCDM model in the principal damage coordinate system
Numerical examples in [11] have shown that material symmetry is considerably affected by damage evolution in composite microstructures. Different load paths will yield different damage profiles in the microstructure, and this will alter the initial material symmetry in \(C_{\textit {ijkl}}^{o}\) in different ways. In a fixed coordinate system, an RVE exhibiting orthotropy in \(C_{\textit {ijkl}}^{o}\) can exhibit general anisotropy with evolving damage under multiaxial loading. In the fixed coordinate system, the anisotropic C _{ ijkl } will couple normal and shear strain components in the elastic energy expression. However, when the strains are represented in a coordinate system that corresponds to the principal damage axes, the coupling terms in the stiffness C _{ ijkl } reduce to near vanishing values and the initial symmetry properties are retained. Hence, the homogenized stiffness matrix has been represented in the PDCS in [2527]. Determination of the continuously evolving principal damage coordinate system requires the determination of the secondorder damage tensor D _{ ij } and subsequent evaluation of its eigenvectors at each step of the incremental loading process. A transformation matrix [Q]^{ D } is formed from the eigenvectors of D _{ ij }, which leads to the rotation of the global coordinate system to the principal damage coordinate system.
An issue with the HCDM formulation in [2527] is that the coefficients P _{ ijkl } have been derived in terms of the applied strain to the RVE. This makes them dependent on the external load, which leads to a loaddependent (rather than material statedependent) damage relation. To overcome this, a new formulation is proposed in this work.
Damage evolution laws in the PDCS
Evaluation of P ijkl′ from micromechanical analysis of the RVE
Macromicro analysis of a composite structure with the HCDM model

Unidirectional 3D uniform composite microstructure with fibers arranged in a rectangular array. The RVE is a unit cell containing a single cylindrical fiber of volume fraction 20%.

Unidirectional 3D uniform composite microstructure with fibers arranged in a rectangular array. The RVE is a unit cell containing a single fiber of elliptical cross section. Volume fraction is 20%, and the aspect ratio of the elliptical cross section is \(\frac {a}{b}=2\).

Unidirectional 3D composite microstructure with hexagonal arrangement of fibers. The RVE contains cylindrical fibers with a fiber volume fraction is 20%.

Crossply 3D composite microstructure with its RVE consisting of two cylindrical fibers oriented at 90° with respect to each other. The fiber volume fraction in the RVE is 20%.
The simulations involve a deformable composite projectile impactor colliding with a rigid surface. The impactors, consisting of one of the four microstructural architectures, is macroscopically cylindrical in shape with radius of 3.2 mm and length of 32.4 mm. It moves with an initial velocity of 10 m/s. The density of epoxy matrix is assumed to be 750 kg/m ^{3} while that of steel fiber is 7,800 kg/m ^{3} so that, for fiber volume fraction of 20%, the density of the aggregate is 2,160 kg/m ^{3}. The rigid surface and the projectile are modeled using continuum elements, and contact between projectile and rigid surface is assumed to be frictionless.
Conclusions
This paper provides an ICMSE perspective on some of the aspects, governing multiscale modeling of composite materials. While a rigorous ICMSE framework encompasses several ingredients, only a partial list is dealt with in this paper. The major items discussed include microstructure characterization and statistically equivalent representative volume elements or SERVE identification, micromechanical analyses of the SERVE, and development of homogenized models by hierarchical modeling.
Alternative methods of identifying SERVEs using both morphology and micromechanicsbased methods are proposed for undamaged and damaging conditions. Similar sizes of the SERVE are predicted by these alternate methods, suggesting that the different indicators are able to predict similar influence of microstructural elements on one another. An important contribution to the identification of the SERVE is the development of boundary conditions based on the statistical distribution of heterogeneities in the domain exterior to the SERVE. In this formulation, the twopoint correlation function is combined with the Green’s function approach to derive accurate displacement boundary conditions on the SERVE. The SERVE continuously evolves in size with the evolution of microscopic damage. It is expected that the imposition of accurate boundary conditions will reduce the size of the SERVE that is conventionally needed with periodic, uniform traction, or displacement boundary conditions.
A micromechanical analysis model of the composite microstructural SERVE, undergoing damage evolution in the form of fiber and matrix cracking and interfacial debonding, is developed. Interfacial debonding is modeled using a bilinear cohesive zone model, while nonlocal continuum damage mechanics models are used to represent fiber and matrix damage. Threedimensional microstructures are simulated for crack propagation with this model.
Finally, an accurate and computationally efficient 3D HCDM is developed for composites undergoing interfacial debonding. The HCDM model represents orthotropic behavior in the PDCS and uses a fourthorder damage tensor which characterizes the stiffness as an internal variable. The model is found to accurately predict damage behavior for a wide range of proportional and nonproportional loading. The HCDM modelbased simulations are able to provide both macroscopic and microscopic information on evolving damage and failure. The capability of macroscopic damage predictions in structures with explicit reference to the microstructural composition is largely lacking in the literature. The overall model framework presented in this study can be used in material design to enhance structural performance and life.
Declarations
Acknowledgements
This author acknowledges the contributions of his research group members Dr. D. Kubair, Ms. Z. Li, and Mr. S. Baby to the paper. This work has been partially supported through a grant No. FA95501210445 to the Center of Excellence on Integrated Materials Modeling (CEIMM) at Johns Hopkins University awarded by the AFOSR/RSL Computational Mathematics Program (Manager Dr. F. Fahroo) and AFRL/RX (Monitors Drs. C. Woodward and C. Przybyla). It is also partially sponsored by the Center for Materials in Extreme Dynamic Environments (CMEDE) of the Army Research Laboratory and was accomplished under Cooperative Agreement Number W911NF1220022. These sponsorships are gratefully acknowledged.
Authors’ Affiliations
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