Representative Volume Element Level Research Articles

A fully second‐order homogenization formulation for the multi‐scale modeling of heterogeneous materials

AbstractA homogenization‐based multi‐scale model at finite strains is proposed linking the micro‐scale to the macro‐scale level through a second‐gradient constitutive theory. It is suitable for modeling localization and size effects in multi‐phase materials. The macroscopic deformation gradient and second gradient are enforced on a microstructural representative volume element (RVE). The coupling between the scales is defined by means of kinematical insertion and homogenization operators, carefully postulated to ensure kinematical conservation in the scale transition, according to the method of multi‐scale virtual power. From the solution of the micro‐scale equilibrium, the macroscopic stress tensors are derived based on the principle of multi‐scale virtual power. The kinematical constraints at the RVE level are enforced with Lagrange multipliers. A mixed finite element approach is adopted for the numerical solution of the micro and macro second gradient equilibrium problems. It conveniently leads to a formulation where the Lagrange multipliers are the homogenized stresses, and the macroscopic consistent tangent operators for FE2 simulations can be directly derived. The Newton–Raphson scheme is employed to solve the non‐linear systems of equations. The numerical results demonstrate the predictive capability of the proposed model. A systematic analysis of the influence of RVE length and the micro‐constituent size is performed.

Formulation and numerical implementation of a variationally consistent multi-scale model based on second-order computational homogenisation at finite strains for quasi-static problems

A multi-scale model which couples a macro-scale strain-gradient theory with a classical continuum at representative volume element (RVE) level is formulated in this contribution for quasi-static problems. It enables modelling the inelastic response of heterogeneous materials at finite strains incorporating their microstructural size and accounting for non-uniform macroscopic deformation fields within the RVE. The development of the multi-scale model is based on the Method of Multi-Scale Virtual Power from which equilibrium equations and homogenisation relations are derived, ensuring a variationally consistent scale transition. The kinematical constraints at the RVE level are enforced with the Lagrange multiplier method and particularised for direct, periodic and minimal conditions. It is shown that the homogenised stresses can be obtained exclusively from the Lagrange multipliers. The finite element solution of the micro and macro equilibrium problems is presented in detail together with the numerical implementation. An efficient strategy is devised to compute the consistent macroscopic tangents required for the FE2 framework. The Newton–Raphson scheme is employed for the iterative solution of the corresponding non-linear system of equations at both scales. Numerical results show the effectiveness of the numerical strategy. A systematic analysis of the influence of the RVE length, the number and size of the micro-constituents, with different constitutive laws at the micro-scale and subjected to several macroscopic deformation states, is performed. The determination of an RVE that is both representative and descriptive of the macroscopic characteristic length is discussed. The accuracy of the numerical results is assessed against a reference direct numerical simulation.

Tensile and flexural behaviour of a graphene/epoxy composite: experiments and simulation

The tensile and flexural behaviour of a graphene nanoplatelet (GnP) reinforced polymer, Grade M25 GnP / Araldite LY564 is experimentally investigated. This is followed by a multi-scale finite element model to simulate the tensile response as the most critical loading case. The approach is based on the multi-scale method and consists of a unit cell and a representative volume element (RVE). At the unit cell level, the material nanocharacteristics (filler geometry, phase mechanical properties, interfacial properties) are used to calculate the local tensile response. The material architecture is simulated at the RVE level by distributing the locally obtained unit cell mechanical properties, using periodic boundary conditions. A statistical sample was studied and the average mechanical characteristics were compared to the macroscopic measured stress–strain data. Finally, the simulation methodology was validated by comparisons between the effective experimental and numerical results.

Elastic and strength analysis of hybrid 3D woven orthogonal composites: Numerical and analytical modelling

Modern advancement of the 3D woven composites made it possible to adjust the mechanical properties either by laying the fiber tows in a certain angle or by replacing the fibers having nominal mechanical properties with the fiber of higher mechanical properties in a certain proportion, which can be regarded as hybridization of reinforcement material. By adding the fibers with different mechanical properties arises the challenge of estimation of their elastic and strength properties. In the current study, an analytical model is developed based on the volume averaging method to predict the elastic and strength parameters for non-hybrid and hybrid 3D woven composites on the basis of two distinct geometric models. In the later section, the results are verified through numerical simulation on the representative volume element level using finite element simulations. Detailed geometry is constructed for a finite element mosaic model for a weave configuration and the results are verified by comparison with the experimental data. After validating the models successfully, fiber tows in a certain ratio are replaced with other materials to attain the hybridization of 3D woven composite. Finally, the results from the numerical simulation for hybrid and non-hybrid material system are obtained and compared with the analytical results which showed a close agreement among them, making the analytical model a useful tool for estimating the elastic and strength properties for hybrid 3D woven composites.

The method of multiscale virtual power for the derivation of a second order mechanical model

A multi-scale model, based on the concept of Representative Volume Element (RVE), is proposed linking a classical continuum at RVE level to a macro-scale strain-gradient theory. The multi-scale model accounts for the effect of body forces and inertia phenomena occurring at the micro-scale. The Method of Multiscale Virtual Power recently proposed by the authors drives the construction of the model. In this context, the coupling between the macro- and micro-scale kinematical descriptors is defined by means of kinematical insertion and homogenisation operators, carefully postulated to ensure kinematical conservation in the scale transition. Micro-scale equilibrium equations as well as formulae for the homogenised (macro-scale) force- and stress-like quantities are naturally derived from the Principle of Multiscale Virtual Power – a variational extension of the Hill-Mandel Principle that enforces the balance of the virtual powers of both scales. As an additional contribution, further insight into the theory is gained with the enforcement of the RVE kinematical constraints by means of Lagrange multipliers. This approach unveils the reactive nature of homogenised force- and stress-like quantities and allows the characterisation of the homogenised stress-like quantities exclusively in terms of RVE boundary data in a straightforward manner.

3D couple-stress moduli of porous polymeric biomaterials using µCT image stack and FE characterization

The purpose of the current work is to develop a homogeneous, anisotropic couple-stress continuum model as a substitute of the 3D solid phases of porous natural-polymeric biomaterials used for tissue engineering. Consideration of the second gradient of deformation such as in couples stress continuum theories addresses the size dependency that is observed in such porous structures. The equivalent properties of these biomaterials are presently obtained based on the response of different representative volume elements under mixed prescribed boundary conditions comprising both controlled traction and displacement. The elastic mechanical constants of the effective couple-stress continuum are deduced at the representative volume element level by an equivalent strain energy method. We conduct this study computationally using a finite element approach. For this purpose, 3D high-resolution micro-computed tomography (µCT) scans are performed on formerly fabricated specimens. Loadings of representative volume elements include uniaxial extension, biaxial extension, and shear deformation in order to evaluate the first stiffness tensor. Besides, uniaxial twist, biaxial twist, and bending curvature are also imposed in order to obtain an estimation of the second couple stress stiffness tensor. The computed Young's moduli are similar to measurements obtained from uniaxial compression tests performed on circular cylindrical samples.

Finite Element-Based Micromechanical Modeling of Microstructure Morphology in Filler-Reinforced Elastomer

Characteristic properties of elastomers including their fatigue behavior and wear resistance can be tailored by embedding them with filler particles. Along with enhancing the overall properties of the system, filler particles also induce some inelastic effects including the Mullins and the Payne effect. In this paper, computational modeling using finite element methods has been employed to study the mechanical behavior of such systems. A nonlinear model is applied, to predict the macroscopic large deformation behavior, with morphology evolution and deformation occurring at the microscopic level, using the representative volume element (RVE) approach. The approach is based on a micro mechanically motivated hyper-elastic constitutive model, describing the elastomeric matrix within the RVE. In order to simulate the breakdown and reaggregation of filler networks, effects like the change in the glass transition temperature in the vicinity of filler particles, are incorporated in the system. The implementation is designed to be robust, for accommodating large rotations and stretches of the matrix local to, and between, the nanoparticles. The nanocomposite microstructure is reconstructed at the RVE level using a random particle generation algorithm, with the assumption of periodicity. Computational experiments using this methodology enable prediction of the strain softening behavior of filled elastomers, observed experimentally, and to understand the behavior of the interphase between two filler particles and its effect on the characteristic material behavior of filled elastomers.

Thermomechanical Multiscale Constitutive Modeling: Accounting for Microstructural Thermal Effects

In this work we present a thermomechanical multiscale constitutive model for materials with microstructure. In these materials thermal effects at microscale have an impact on the effective macroscopic stress. As a result, it turns out that the homogenized stress depends upon the macroscopic temperature and its gradient. In order to allow this interplay to be thermodynamically valid, we resort to a macroscopic extended thermodynamics whose elements are derived from the microscopic behavior using homogenization concepts. Hence, the thermodynamics implications of this new class of multiscale models are discussed. A variational approach based on the Hill–Mandel Principle of Macro-homogeneity, and which makes use of the volume averaging concept over a local representative volume element (RVE), is employed to derive the thermal and mechanical equilibrium problems at the RVE level and the corresponding homogenization expressions for the effective heat flux and stress. The material behavior at the RVE level is described through standard phenomenological constitutive models. To sum up, the novel contribution of the model presented here is that it allows to include the microscopic temperature fluctuation field, obtained from the multiscale thermal analysis, in the micro-mechanical problem at the RVE level while keeping thermodynamic consistency.

Shakedown analysis of periodic composites with kinematic hardening material model

AbstractLower‐bound limit and shakedown analysis of periodic composites with the consideration of kinematic hardening are carried out on the representative volume element level. In combination with homogenization theory, the homogenized macroscopic admissible loading domains are determined. Furthermore, the strengths of periodic composites by using elastic perfectly plastic, unlimited and linear limited kinematic hardening material models are calculated and compared. (© 2012 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Computational modelling of large deformations in layered-silicate/PET nanocomposites near the glass transition

Layered-silicate nanoparticles offer a cost-effective reinforcement for thermoplastics. Computational modelling has been employed to study large deformations in layered-silicate/poly(ethylene terephthalate) (PET) nanocomposites near the glass transition, as would be experienced during industrial forming processes such as thermoforming or injection stretch blow moulding. Non-linear numerical modelling was applied, to predict the macroscopic large deformation behaviour, with morphology evolution and deformation occurring at the microscopic level, using the representative volume element (RVE) approach. A physically based elasto-viscoplastic constitutive model, describing the behaviour of the PET matrix within the RVE, was numerically implemented into a finite element solver (ABAQUS) using an UMAT subroutine. The implementation was designed to be robust, for accommodating large rotations and stretches of the matrix local to, and between, the nanoparticles. The nanocomposite morphology was reconstructed at the RVE level using a Monte-Carlo-based algorithm that placed straight, high-aspect ratio particles according to the specified orientation and volume fraction, with the assumption of periodicity. Computational experiments using this methodology enabled prediction of the strain-stiffening behaviour of the nanocomposite, observed experimentally, as functions of strain, strain rate, temperature and particle volume fraction. These results revealed the probable origins of the enhanced strain stiffening observed: (a) evolution of the morphology (through particle re-orientation) and (b) early onset of stress-induced pre-crystallization (and hence lock-up of viscous flow), triggered by the presence of particles. The computational model enabled prediction of the effects of process parameters (strain rate, temperature) on evolution of the morphology, and hence on the end-use properties.

On the size of representative volume element for Darcy law in random media

Most studies of effective properties of random heterogeneous materials are based on the assumption of the existence of a representative volume element (RVE), without quantitatively specifying its size L relative to that of the micro-heterogeneity d . In this paper, we study the finite-size scaling trend to RVE of the Darcy law for Stokesian flow in random porous media, without invoking any periodic structure assumptions, but only assuming the microstructure's statistics to be spatially homogeneous and ergodic. By analogy to the existing methodology in thermomechanics of random materials, we first formulate a Hill–Mandel condition for the Darcy flow velocity and pressure gradient fields. This dictates uniform Neumann and Dirichlet boundary conditions, which, with the help of two variational principles, lead to scale-dependent hierarchies on effective (RVE level) permeability. To quantitatively assess the scaling trend towards the RVE, these hierarchies are computed for various porosities of random disc systems, where the disc centres are generated by a planar hard-core Poisson point field. Overall, it turns out that the higher is the density of random discs—or, equivalently, the narrower are the micro-channels in the system—the smaller is the size of RVE pertaining to the Darcy law.

On the scaling from statistical to representative volume element in thermoelasticity of random materials

Under consideration is the finnite-size scaling of effective thermoelastic properties of random microstructures from a Statistical Volume Element (SVE) to a Representative Volume Element (RVE), without invoking any periodic structure assumptions, but only assuming the microstructure's statistics to be spatially homogeneous and ergodic. The SVE is set up on a mesoscale, i.e. any scale finite relative to the microstructural length scale. The Hill condition generalized to thermoelasticity dictates uniform Neumann and Dirichlet boundary conditions, which, with the help of two variational principles, lead to scale dependent hierarchies of mesoscale bounds on effective (RVE level) properties: thermal expansion and stress coefficients, effective stiffness, and specific heats. Due to the presence of a non-quadratic term in the energy formulas, the mesoscale bounds for the thermal expansion are more complicated than those for the stiffness tensor and the heat capacity. To quantitatively assess the scaling trend towards the RVE, the hierarchies are computed for a planar matrix-inclusion composite, with inclusions (of circular disk shape) located at points of a planar, hard-core Poisson point field. Overall, while the RVE is attained exactly on scales infinitely large relative to the microscale, depending on the microstructural parameters, the random fluctuations in the SVE response may become very weak on scales an order of magnitude larger than the microscale, thus already approximating the RVE.

Material spatial randomness: From statistical to representative volume element

The material spatial randomness forces one to re-examine various basic concepts of continuum solid mechanics. In this paper we focus on the Representative Volume Element (RVE) that is commonly taken for granted in most of deterministic as well as in stochastic solid mechanics, although in the latter case it is called the Statistical Volume Element (SVE). The key issue is the scale over which homogenization is being carried out—it is called the mesoscale, separating the microscale (level of microheterogeneities) from the macroscale (level of RVE). As the mesoscale grows, the SVE tends to become the RVE. This occurs in terms of two hierarchies of bounds stemming from Dirichlet and Neumann boundary value problems on the mesoscale, respectively. Since generally there is no periodicity in real random media, the RVE can only be approached approximately on finite scales. We review results on this subject in the settings of linear elasticity, finite elasticity, plasticity, viscoelasticity, thermoelasticity, and permeability.