Linear Comparison Composite Research Articles

Soft elastic composites: Microstructure evolution, instabilities and relaxed response by domain formation

This work provides a review of several complementary analytical methods for characterizing the macroscopic response of ‘soft’ elastic composites subjected to large deformations. For this purpose, we first recall the variational definitions of the homogenized response for hyperelastic composites with periodic and random microstructures. We distinguish between the principal solution for the ‘mesoscopic’ stored-energy function, which is based on one-cell-periodic solutions for the periodic microstructures and on the macroscopically uniform solution for the random microstructures, and the ‘relaxed’ or ‘macroscopic’ homogenized stored-energy function, which is obtained after the onset of ‘microscopic’ (periodic on multiple cells) or ‘macroscopic’ (long wavelength) instabilities. The various techniques include variational bounds of the Voigt, and generalized Reuss and Hashin–Shtrikman types, variational estimates based on the use of suitably optimized ‘linear comparison composites,’ and bounds based on the relaxation of the principal solution for the mesoscopic energy by means of a generalized Maxwell procedure consisting in the quasiconvexification of the energy and leading to domain formation. Although the methods are completely general, for simplicity, we review here illustrative results for neo-Hookean laminates and fiber-reinforced elastomers subjected to plane strain loading conditions. Among the main results, it was found that the predicted formation of twins or lamellar domains and associated soft modes of deformation for the laminates is consistent with experimental observations for thermoplastic elastomers with highly ordered lamellar microstructures. In addition, it was found that the predictions for the macroscopic instabilities by means of the relaxation of the mesoscopic energy are generally different (more conservative) and more consistent with the variational definitions of the macroscopic homogenized energy than those based on loss of ellipticity of the incremental elasticity homogenization problem.

An incremental‐secant mean‐field homogenization model enhanced with a non‐associated pressure‐dependent plasticity model

AbstractThis article introduces a, possibly damage‐enhanced, pressure‐dependent‐based incremental‐secant mean‐field homogenization (MFH) scheme for two‐phase composites. The incremental‐secant formulation consists on a fictitious unloading of the material up to a stress‐free state, in which a residual stress is attained in its phases. Then the secant method is performed in order to compute the mean stress fields of each phase. One of the main advantages of this method is the natural isotropicity of the secant tensors that allows defining the linear‐comparison‐composite (LCC). In this work, we show that this isotropic nature is preserved for a non‐associated pressure dependent plastic flow, making possible the direct definition of the LCC. This model is thus able to represent the physics of real polymeric composites. The MFH scheme is then verified by testing its prediction capabilities in several cases, including cyclic and nonproportional loading involving perfectly elastic phases, elasto‐plastic and damage‐enhanced elasto‐plastic phases in random representative volume elements (RVE) of uni‐directional (UD) composites and of composites reinforced with spherical inclusions.

Creep rupture in carbon nanotube-based viscoplastic nanocomposites

While the linear viscoelastic characteristics of polymer composites have been extensively studied, the nonlinear creep with progressive damage leading to the ultimate creep rupture in CNT-based viscoplastic nanocomposites still remains a challenging issue. In this paper, a unified approach is presented to study the whole lifetime of creep process including deformation, damage, and creep rupture. First, based on the concept of secant viscosity, a linear viscoelastic comparison composite is introduced to mimic the nonlinear viscoplastic nanocomposite, with the linear part further formulated in the Laplace transformed domain. Then, both progressive degradation of the interphase and creep damage of the polymer matrix are presented. A joint field-fluctuation method and work-rate equivalence is subsequently utilized to calculate the effective Mises stress and hydrostatic stress of the viscoplastic matrix. The principle of irreversible thermodynamics is then invoked to derive the thermodynamic driving force to characterize the evolution of creep damage and rupture process. After the theory is developed, it is demonstrated that the predicted effective creep strains of multi-walled CNT/polypropylene nanocomposites are calibrated by the experiments spanning over the whole three stages of primary creep, secondary creep, and tertiary creep. Both creep rupture time and rupture strain of the overall nanocomposite are found to increase with CNT volume concentration. It is concluded that population of CNT nanofillers can remarkably enhance the creep rupture resistance of CNT/polymer viscoplastic nanocomposites.

Viscoelastic-viscoplastic polymer composites: Development and evaluation of two very dissimilar mean-field homogenization models

This paper deals with the micromechanical modeling of polymer composites with viscoelastic-viscoplastic (VE-VP) constituents. Two mean-field homogenization (MFH) models based on completely dissimilar theoretical approaches are extended from elasto-viscoplasticity (EVP) to VE-VP and assessed. The first approach is the incremental-secant method. It relies on a fictitious unloading of the composite at the beginning of each time step. Then, a thermoelastic-like Linear Comparison Composite (LCC) is constructed from the computed residual state directly in the time domain. The method provides naturally isotropic per-phase incremental-secant operators for isotropic VE-VP constituents. It takes into account both the first and the second statistical moment estimates of the equivalent stress micro-field. The second approach is the integral affine method. It starts by linearizing the rates of viscoplastic strain and internal variables. The linearized constitutive equations are then recast in a hereditary integral format to which the Laplace-Carson (L-C) transform is applied. A thermoelastic-like LCC is built in the L-C domain, where MFH is carried out. Finally, the composite’s response in the time domain is recovered by numerical inversions of L-C transforms. The method is able to overcome the issue of heterogeneous viscous stresses encountered by time domain MFH models. The two proposed MFH formulations are able to handle non-monotonic, non-proportional and multi-axial loading histories. Their accuracy was assessed against full-field finite element (FE) results for different microstructures and loadings. The computational cost of both methods is negligible compared to FE analyses. Overall, the incremental-secant approach is much simpler mathematically and numerically than the integral affine formulation, its accuracy ranges from acceptable to excellent, and important improvements can be expected in the future by controlling the virtual unloading time increment.

A time-incremental homogenization method for elasto-viscoplastic particulate composites based on a modified secant formulation

Starting from the exact solution of the heterogeneous linear viscoelastic Eshelby ellipsoidal inclusion problem obtained in the time domain for an ellipsoidal inclusion embedded in an isotropic matrix (Berbenni et al., 2015), a direct time-incremental homogenization Mori–Tanaka scheme for two-phase non linear elasto-viscoplastic materials is proposed. The extension to non-linear behavior is based on a “modified secant” formulation to obtain the linear comparison composite (LCC) properties. In contrast with more classic homogenization method based on the first moment of stresses (i.e. classic secant formulation), the present approach is based on the second-order moment of stresses making use of the Hill’s lemma. Hence, a modified secant viscoplastic compliance is considered for the matrix phase. The effective behavior as well as the evolution laws of the averaged strains and stresses per phase are directly solved in the time domain, without the need of Laplace-Carson transforms. The estimates provided by the present homogenization model are compared to full-field calculations from the literature for two-phase non linear particulate composites constituted of an isotropic elasto-viscoplastic matrix phase and isotropic elastic particles, and, for radial and non-radial loadings.

Incremental variational homogenization of elastoplastic composites with isotropic and Armstrong-Frederick type nonlinear kinematic hardening

In order to investigate the behavior of elastoplastic composites exhibiting both isotropic and nonlinear kinematic hardening, we extend the Double Incremental Variational (DIV) formulation of Lucchetta et al. (2019), based on both the incremental variational principles introduced by Lahellec and Suquet (2007) and the formulation proposed by Agoras et al. (2016). However, the Armstrong-Frederick model (Armstrong and Frederick), which is very often used to describe nonlinear kinematic hardening and refers to the framework of non-associated plasticity (Chaboche, 1977), cannot be handled within the framework of generalized standard materials as required by the incremental variational principles on which the DIV formulation relies. That is why we work with an approximation of this model, namely the modified Chaboche model (Chaboche, 1983). As the dissipation potential associated with this model depends on internal variables, we have to extend the incremental variational principles of Lahellec and Suquet to such a situation. Then, we apply twice the variational procedure of Ponte Castañeda (1991), first to linearize the local behavior and then to deal with the intraphase heterogeneity of the thermoelastic Linear Comparison Composite (LCC) induced by the linearisation step. The resulting thermoelastic LCC with per-phase homogeneous properties is homogenized by classical linear schemes. We develop and implement this new incremental variational procedure for two-phase matrix-inclusions composites with an isotropic elastoplastic matrix exhibiting combined isotropic and nonlinear kinematic hardening. For various cyclic loadings, the predictions of the proposed DIV formulation compare favorably with Finite Element simulations based on the Armstrong-Frederick model.

Statistics of the stress, strain-rate and spin fields in viscoplastic polycrystals

This paper provides estimates for the stress, strain-rate and spin field statistics in viscoplastic polycrystals. They are obtained by combining the fully optimized second-order (FOSO) homogenization method with the self-consistent estimate for linear polycrystalline aggregates. The FOSO linearization method allows the consistent estimation of the field statistics in nonlinear composites and polycrystals directly from those in suitably optimized linear comparison composites (LCC). For the LCC, a novel generalized approach is used to extract the field statistics of the stress, the strain-rate and also of the continuum-spin fields (including the second moments). Then, the stress-field statistics are used to extract the plastic-spin statistics, which, together with the continuum-spin statistics, provide consistent estimates for the microstructural-spin statistics. Applications to power-law viscoplastic FCC polycrystals and ice-like HCP polycrystals are considered, and the overall and intragranular field fluctuations are computed and found to increase with nonlinearity exponent and grain anisotropy. In particular, the covariance tensors of the continuum-spin fluctuations within the grains of viscoplastic polycrystals, which are characterized here for the first time by means of nonlinear homogenization methods, are found to be quite significant, even when the corresponding averages of the continuum-spin in the grains are negligible, and thus have a strong influence on the corresponding microstructural-spin fluctuations.

Exact results for weakly nonlinear composites and implications for homogenization methods

Weakly nonlinear composite conductors are characterized by position-dependent dissipation potentials expressible as an additive composition of a quadratic potential and a nonquadratic potential weighted by a small parameter. This additive form carries over to the effective dissipation potential of the composite when expanded to first order in the small parameter. However, the first-order correction of this asymptotic expansion depends only on the zeroth-order values of the local fields, namely, the local fields within the perfectly linear composite conductor. This asymptotic expansion is exploited to derive the exact effective conductivity of a composite cylinder assemblage exhibiting weak nonlinearity of the power-law type (i.e., power law with exponent m=1+δ, such that |δ|≪1), and found to be identical (to first order in δ) to the corresponding asymptotic result for sequentially laminated composites of infinite rank. These exact results are used to assess the capabilities of more general nonlinear homogenization methods making use of the properties of optimally selected linear comparison composites.

A differential homogenization method for estimating the macroscopic response and field statistics of particulate viscoelastic composites

This work provides an analytical method for determining the time-dependent effective constitutive response of linear viscoelastic composites with particulate microstructures by means of a differential-algebraic system (DAS) of equations. By making use of differential equations instead of difference equations and utilizing, as primary variables, the variances of the field fluctuations instead of the properties of fictitious linear comparison composites, the new formulation is found to be more robust and to alleviate some of the issues associated with earlier incremental approaches. Remarkably, the new formulation can be shown to recover the exact results of the correspondence principle for isotropic incompressible composites subjected to proportional loading, as well as for isotropic compressible composites subjected to purely hydrostatic loading. More generally, the proposed formulation is not exact but still delivers reasonably accurate estimates in most cases. Lastly, a generalization of the DAS formulation is provided for nonlinear viscoelastic composites with incompressible phases and also found to be more robust than earlier approaches.

Cohesive-strength homogenisation model of porous and non-porous materials using linear comparison composites and application

An estimation of the strength of composite materials with different strength behaviours of the matrix and inclusion is of great interest in science and engineering disciplines. Linear comparison composite (LCC) is an approach introduced for estimating the macroscopic strength of matrix-inclusion composites. The LCC approach has however not been expanded to model non-porous composites. Therefore, this paper is to fill this gap by developing a cohesive-strength method for modelling frictional composite materials, which can be porous and non-porous, using the LCC approach. The developed cohesive-strength homogenisation model represents the matrix and inclusion as a two-phase composite containing solids and pores. The model is then implemented in a multiscaling model in which porous cohesive-frictional solids intermix with each other at different scale levels classified as micro, meso and macro. The developed model satisfies an upscaling scheme and is suitable for investigating the effects of the microstructure, the composition, and the interface condition of the materials at micro scales on the macroscopic strength of the composites. To further demonstrate the application of the developed cohesive-strength homogenisation model, the cohesive-strength properties of very high strength concrete are determined using instrumented indentation, nonlinear limit analysis and second-order cone programming to obtain material properties at different scale levels.

On the optimality of the variational linear comparison bounds for porous viscoplastic materials

This paper is concerned with the optimality of the variational bounds of the Hashin-Shtrikman type (VHS) for nonlinear composites, first obtained by Ponte Castañeda (1991a) by means of the corresponding HS bounds for suitably optimized linear comparison composites (LCCs). For simplicity, this problem is addressed in the context of porous viscoplastic materials with incompressible, isotropic matrix phase and two-dimensional microstructures, subjected to plane-strain loading conditions. Although special, this case—exhibiting infinite heterogeneity contrast and compressible macroscopic response—is expected to be fully representative of more general three-dimensional porous materials, as well as more general two-phase, well-ordered composites. Thus, it is shown that the VHS bounds, which were originally derived for the class of nonlinear composites with statistically isotropic microstructures, are in fact attained over the larger class of microstructures with anisotropic nonlinear response but isotropic linear response. By appealing to an exact variational representation for the effective potential of finite-rank nonlinear laminates, it is shown that there exist certain values of the applied macroscopic stress for which the finite-rank laminate (closed-cell porous) microstructures attaining the linear HS bounds also attain the nonlinear VHS bound. Explicit results are obtained in the ideally plastic limit for the yield surface of the finite-rank laminates attaining the VHS bound. In particular, the results of the paper highlight the fact that bounds for nonlinear composites are much more sensitive to microstructural details than bounds for linear composites.

A double incremental variational procedure for elastoplastic composites with combined isotropic and linear kinematic hardening

We investigate the nonlinear behavior of elasto-plastic composites with isotropic and linear kinematic hardening. We first rely on the incremental variational principles introduced by Lahellec and Suquet (2007a). We also take advantage of an alternative formulation, recently proposed by Agoras et al. (2016) for visco-plastic composites without hardening, which consists in a double application of the variational procedure of Ponte-Castañeda. We extend in this paper this approach to elasto-plastic composites with combined linear kinematic and isotropic work-hardening. The first application of the variational procedure linearizes the local behavior, including hardening, and leads to a thermo-elastic Linear Comparison Composite (LCC) with a heterogeneous polarization field inside the phases. The second one deals with the heterogeneity of the polarization and results in a new thermo-elastic LCC with a per-phase homogeneous polarization field, which effective behavior can then be estimated by classical linear homogenization schemes. We develop and implement this new incremental variational procedure for composites comprised of linear elastic spherical particles isotropically distributed in an elasto-plastic matrix. The predictions of the model are compared with results available in the literature for cyclic proportional and non-proportional loadings. New results for elasto-plastic composites with combined isotropic and kinematic hardening are also provided. They are in good agreement with the numerical computations we carried out, at both local and macroscopic scales.

Fully optimized second-order homogenization estimates for the macroscopic response and texture evolution of low-symmetry viscoplastic polycrystals

In this paper, we present a finite-strain homogenization model for the macroscopic response of viscoplastic polycrystals deforming by crystallographic slip. The model makes use of the recently developed fully optimized second-order (FOSO) variational homogenization method, together with self-consistent estimates for the instantaneous response of a linear comparison composite (LCC) with optimally selected properties, to generate corresponding estimates for nonlinear viscoplastic polycrystals. The estimates are guaranteed to be exact to second order in the heterogeneity contrast, and to satisfy all known bounds. Unlike earlier second-order methods, the FOSO method has the distinct advantage that the macroscopic behavior and field statistics in the nonlinear composite can be conveniently extracted directly from the corresponding quantities in the LCC. Moreover, consistent homogenization estimates for the average strain-rate and spin fields in the grains are used to derive the evolution equations for the morphological and crystallographic textures of the polycrystals at large deformations. The FOSO method is then used for the first time to investigate the effects of rate sensitivity and grain anisotropy on the macroscopic response and field statistics of untextured (low-symmetry) HCP polycrystals. Comparisons with full-field simulations and earlier homogenization models show that the FOSO method is the most accurate to date. In addition, the FOSO method is used to predict the texture evolution for ice-like HCP polycrystals subjected to finite-strain loading conditions. It is found that the method is able to capture the strong basal texture that is observed experimentally in these materials under compression, leading to strong geometric softening/hardening effects, as well as strong viscous anisotropy in the overall response.

A multi-scale homogenization model for fine-grained porous viscoplastic polycrystals: I – Finite-strain theory

We make use of the recently developed iterated second-order homogenization method to obtain finite-strain constitutive models for the macroscopic response of porous polycrystals consisting of large pores randomly distributed in a fine-grained polycrystalline matrix. The porous polycrystal is modeled as a three-scale composite, where the grains are described by single-crystal viscoplasticity and the pores are assumed to be large compared to the grain size. The method makes use of a linear comparison composite (LCC) with the same substructure as the actual nonlinear composite, but whose local properties are chosen optimally via a suitably designed variational statement. In turn, the effective properties of the resulting three-scale LCC are determined by means of a sequential homogenization procedure, utilizing the self-consistent estimates for the effective behavior of the polycrystalline matrix, and the Willis estimates for the effective behavior of the porous composite. The iterated homogenization procedure allows for a more accurate characterization of the properties of the matrix by means of a finer “discretization” of the properties of the LCC to obtain improved estimates, especially at low porosities, high nonlinearties and high triaxialities. In addition, consistent homogenization estimates for the average strain rate and spin fields in the pores and grains are used to develop evolution laws for the substructural variables, including the porosity, pore shape and orientation, as well as the “crystallographic” and “morphological” textures of the underlying matrix. In Part II of this work has appeared in Song and Ponte Castañeda (2018b), the model will be used to generate estimates for both the instantaneous effective response and the evolution of the microstructure for porous FCC and HCP polycrystals under various loading conditions.

A symmetric fully optimized second‐order method for nonlinear homogenization

AbstractWe consider an alternate formulation of the recently developed ‘fully optimized second‐order’ (FOSO) nonlinear homogenization method, which is based on a stationary variational principle for the macroscopic energy function. In this method, the trial fields are the properties of a suitably designed linear comparison composite (LCC), thus allowing for the estimation of the effective response and field statistics of the nonlinear composite in terms of available estimates for the corresponding quantities in the LCC. The formulation considered in this paper makes use of an alternative choice for the linear comparison composite, leading to homogenization estimates that are more symmetric with respect to Legendre duality than earlier FOSO estimates. The new ‘symmetric’ FOSO method is applied to a class of two‐phase power‐law composites with fibrous microstructures subjected to plane strain loading. The resulting estimates for the effective response and field statistics are found to improve on earlier estimates, and to be in good agreement with full‐field numerical simulations for nonlinear composite cylinder assemblages, as well as with available results for sequentially layered composites.

An incremental-secant mean-field homogenization method with second statistical moments for elasto-visco-plastic composite materials

This paper presents an extension of the recently developed incremental-secant mean-field homogenization (MFH) procedure in the context of elasto-plasticity to elasto-visco-plastic composite materials while accounting for second statistical moments. In the incremental-secant formulation, a virtual elastic unloading is performed at the composite level in order to evaluate the residual stress and strain states in the different phases, from which a secant MFH formulation is applied. When applying the secant MFH process, the linear-comparison-composite (LCC) is built from the piece-wise heterogeneous residual strain-stress state using naturally isotropic secant tensors defined using either first or second statistical moment values. As a result non-proportional and non-radial loading conditions can be considered because of the incremental-secant formulation, and accurate predictions can be obtained as no isotropization step is required. The limitation of the incremental-secant formulation previously developed was the requirement in case of hard inclusions to cancel the residual stress in the matrix phase, resulting from the composite material unloading, to avoid over-stiff predictions. It is shown in this paper that in the case of hard inclusions by defining a proper second statistical moment estimate of the von Mises stress, the residual stress can be kept in the different composite phases. Moreover it is shown that the method can be extended to visco-plastic behaviors without modifying the homogenization process as the incremental-secant formulation only requires the definition of the secant operator of the different phase material models. Finally, it is shown that although it is also possible to define a proper second statistical moment estimate of the von Mises stress in the case of soft inclusions, this does not improve the accuracy as compared to the increment-secant method with first order statistical moment estimates.

A multiscale homogenization model for strength predictions of fully and partially frozen soils

Soil freezing is often used to provide temporary support of soft soils in geotechnical interventions. During the freezing process, the strength properties of the soil–water–ice mixture change from the original properties of the water-saturated soil to the properties of fully frozen soils. In the paper, a multiscale homogenization model for the upscaling of the macroscopic strength of freezing soil based upon information on three individual material phases—the solid particle phase (S), the crystal ice phase (C) and the liquid water phase (L)—is proposed. The homogenization procedure for the partially frozen soil–water–ice composite is based upon an extension of the linear comparison composite (LCC) method for a two-phase matrix–inclusion composite, using a two-step homogenization procedure. In each step, the LCC methodology is implemented by estimating the strength criterion of a two-phase nonlinear matrix–inclusion composite in terms of an optimally chosen linear elastic comparison composite with a similar underlying microstructure. The solid particle phase (S) and the crystal ice phase (C) are assumed to be characterized by two different Drucker–Prager strength criteria, and the liquid water phase (L) is assumed to have zero strength capacity under drained conditions. For the validation of the proposed upscaling strategy, the predicted strength properties for fully and partially frozen fine sands are compared with experimental results, focussing on the investigation of the influence of the porosity and the degree of ice saturation on the predicted failure envelope.

Stationary variational estimates for the effective response and field fluctuations in nonlinear composites

This paper presents a variational method for estimating the effective constitutive response of composite materials with nonlinear constitutive behavior. The method is based on a stationary variational principle for the macroscopic potential in terms of the corresponding potential of a linear comparison composite (LCC) whose properties are the trial fields in the variational principle. When used in combination with estimates for the LCC that are exact to second order in the heterogeneity contrast, the resulting estimates for the nonlinear composite are also guaranteed to be exact to second-order in the contrast. In addition, the new method allows full optimization with respect to the properties of the LCC, leading to estimates that are fully stationary and exhibit no duality gaps. As a result, the effective response and field statistics of the nonlinear composite can be estimated directly from the appropriately optimized linear comparison composite. By way of illustration, the method is applied to a porous, isotropic, power-law material, and the results are found to compare favorably with earlier bounds and estimates. However, the basic ideas of the method are expected to work for broad classes of composites materials, whose effective response can be given appropriate variational representations, including more general elasto-plastic and soft hyperelastic composites and polycrystals.

A methodology for the estimation of the effective yield function of isotropic composites

In this work we derive a general model for N − phase isotropic, incompressible, rate-independent elasto-plastic materials at finite strains. The model is based on the nonlinear homogenization variational (or modified secant) method which makes use of a linear comparison composite (LCC) material to estimate the effective flow stress of the nonlinear composite material. The homogenization approach leads to an optimization problem which needs to be solved numerically for the general case of a N − phase composite. In the special case of a two-phase composite an analytical result is obtained for the effective flow stress of the elasto-plastic composite material. Next, the model is validated by periodic three-dimensional unit cell calculations comprising a large number of spherical inclusions (of various sizes and of two different types) distributed randomly in a matrix phase. We find that the use of the lower Hashin–Shtrikman bound for the LCC gives the best predictions by comparison with the unit cell calculations for both the macroscopic stress-strain response as well as for the average strains in each of the phases. The formulation is subsequently extended to include hardening of the different phases. Interestingly, the model is found to be in excellent agreement even in the case where each of the phases follows a rather different hardening response.

Fully optimized second-order variational estimates for the macroscopic response and field statistics in viscoplastic crystalline composites

A variational method is developed to estimate the macroscopic constitutive response of composite materials consisting of aggregates of viscoplastic single-crystal grains and other inhomogeneities. The method derives from a stationary variational principle for the macroscopic stress potential of the viscoplastic composite in terms of the corresponding potential of a linear comparison composite (LCC), whose viscosities and eigenstrain rates are the trial fields in the variational principle. The resulting estimates for the macroscopic response are guaranteed to be exact to second order in the heterogeneity contrast, and to satisfy known bounds. In addition, unlike earlier ‘second-order’ methods, the new method allows optimization with respect to both the viscosities and eigenstrain rates, leading to estimates that are fully stationary and exhibit no duality gaps. Consequently, the macroscopic response and field statistics of the nonlinear composite can be estimated directly from the suitably optimized LCC, without the need for difficult-to-compute correction terms. The method is applied to a simple example of a porous single crystal, and the results are found to be more accurate than earlier estimates.