Macroscopic Constitutive Laws Research Articles

Variational upscaling for modeling state of strain-dependent behavior and stress-induced crystallization in rubber-like materials

The purpose of this paper is to present a general upscaling strategy for deriving macroscopic constitutive laws for rubber-like materials from the knowledge of the network distribution and a mechanical description of the individual chains and of their free energy. The microscopic configuration is described by the position of the cross-links and is not obtained by an affine assumption but by minimizing the corresponding free energy on stochastic large representative volume elements with adequate boundary conditions. This general framework is then approximated by using a microsphere (directional) description of the network. It is presented in a global setting and is extended in order to handle situations with tube-like constraints and stress-induced crystallization.

Viscous phase-field modeling for chemo-mechanical microstructural evolution: application to geomaterials and pressure solution

The microstructural geometry of materials has a significant influence on their macroscopic response, especially when the process is essentially microscopic as for chemo-mechanical processes. In this work, we are pursuing the idea that interface-tracking models like phase-field modeling can capture the microstructural dynamics and enrich macroscopic constitutive laws for geomaterials. We are introducing a phase-field theory for chemo-mechanical processes derived from fully-dissipative thermodynamics, enabling the inclusion of dissipative effects such as the evolution of the microstructural curvature. The resulting microstructural viscosity introduces rate-dependency through the associated relaxation time. To emphasize the influence of irregular microstructural geometries, we study numerically the chemo-mechanical response of digitalized geomaterials at the grain scale under pressure solution creep. We show that tracking the grain boundary evolution allows for capturing the microscopic transient nature of pressure solution. Our numerical results are discussed against experimental results from the literature and the properties of macroscopic creep laws are shown to be linked to the evolution of the microstructure.

A multiscale multi-permeability poroplasticity model linked by recursive homogenizations and deep learning

Many geological materials, such as shale, mudstone, carbonate rock, limestone and rock salt are multi-porosity porous media in which pores of different scales may co-exist in the host matrix. When fractures propagate in these multi-porosity materials, these pores may enlarge and coalesce and therefore change the magnitude and the principal directions of the effective permeability tensors. The pore-fluid inside the cracks and the pores of host matrix may interact and exchange fluid mass, but the difference in hydraulic properties of these pores often means that a single homogenized effective permeability tensor field is insufficient to characterize the evolving hydraulic properties of these materials at smaller time scale. Furthermore, the complexity of the hydro-mechanical coupling process and the induced mechanical and hydraulic anisotropy originated from the micro-fracture and plasticity at grain scale also makes it difficult to propose, implement and validate separated macroscopic constitutive laws for numerical simulations. This article presents a hybrid data-driven method designed to capture the multiscale hydro-mechanical coupling effect of porous media with pores of various different sizes. At each scale, data-driven models generated from supervised machine learning are hybridized with classical constitutive laws in a directed graph that represents the numerical models. By using sub-scale simulations to generate database to train material models, an offline homogenization procedure is used to replace the up-scaling procedure to generate cohesive laws for localized physical discontinuities at both grain and specimen scales. Through a proper homogenization procedure that preserves spatial length scales, the proposed method enables field-scale simulations to gather insights from meso-scale and grain-scale micro-structural attributes. This method is proven to be much more computationally efficient than the classical DEM–FEM or FEM2 approach while at the same time more robust and flexible than the classical surrogate modeling approach. Due to the usage of bridging-scale technique, the proposed model may provide multiple opportunities to incorporate different types of simulations and experimental data across different length scales for machine learning. Numerical issues will also be discussed.

Numerical homogenization of elastic and thermal material properties for metal matrix composites (MMC)

A two-scale material modeling approach is adopted in order to determine macroscopic thermal and elastic constitutive laws and the respective parameters for metal matrix composite (MMC). Since the common homogenization framework violates the thermodynamical consistency for non-constant temperature fields, i.e., the dissipation is not conserved through the scale transition, the respective error is calculated numerically in order to prove the applicability of the homogenization method. The thermomechanical homogenization is applied to compute the macroscopic mass density, thermal expansion, elasticity, heat capacity and thermal conductivity for two specific MMCs, i.e., aluminum alloy Al2024 reinforced with 17 or 30 % silicon carbide particles. The temperature dependency of the material properties has been considered in the range from 0 to $$500{\,}^\circ \mathrm {C}$$ , the melting temperature of the alloy. The numerically determined material properties are validated with experimental data from the literature as far as possible.

Nonlocal or gradient elasticity macroscopic models: A question of concentrated or distributed microstructure

One dimensional discrete systems, such as axial lattices, may be investigated by using some enriched continuum models. In this paper, strain gradient models (also called gradient elasticity) and stress gradient models (also called nonlocal elasticity) are both shown to be supported by some microstructured physical configurations. Starting from the difference equations associated with each discrete system, a continualization approach is applied to the governing difference equations. Alternatively, one may use energy considerations to derive these higher-order continua. Stress gradient models are built from concentrated microstructure (with direct neighboring interaction) whereas strain gradient models are associated to some distributed microstructure (also with direct neighboring interaction). Each model leads to opposite effect, namely the softening effect of the small scale terms for the stress gradient model (built from concentrated microstructure), and the stiffening effect of the small scale terms for the strain gradient model (built from distributed microstructure) with respect to the asymptotic local model. We also discuss the link between lattice equations, finite difference formulation or finite element formulation of the continuous local problem. The paper concludes that the local neighboring interaction at the discrete scale may transmit some higher-order effects at the macroscopic scale. Hence, the higher-order nature of the macroscopic constitutive laws may not necessarily be seen as the consequence of nonlocal interaction at the lattice scale.

The heterogeneous multiscale method as a framework for coupling the finite element method and molecular dynamics

AbstractHierarchical two‐scale methods are computationally very powerful as there is no direct coupling between the macro‐ and microscale. Such schemes develop first a microscale model under macroscopic constraints, then the macroscopic constitutive laws are found by averaging over the microscale. The heterogeneous multiscale method (HMM) is a general top‐down approach for the design of multiscale algorithms. While this method is mainly used for concurrent coupling schemes in the literature, the proposed methodology also applies to a hierarchical coupling. This contribution discusses a hierarchical two‐scale setting based on the heterogeneous multi‐scale method for quasi‐static problems: the macroscale is treated by continuum mechanics and the finite element method and the microscale is treated by statistical mechanics and molecular dynamics. Our investigation focuses on an optimised coupling of solvers on the macro‐ and microscale which yields a significant decrease in computational time with no associated loss in accuracy. In particular, the number of time steps used for the molecular dynamics simulation is adjusted at each iteration of the macroscopic solver. A numerical example demonstrates the performance of the model. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Elastic Properties of Swelling Clay Particles at Finite Temperature upon Hydration

The knowledge of mechanical properties of swelling clays on the scale of the clay layer is of crucial importance to build predictive hydromechanical macroscopic constitutive laws in a bottom-up approach. In this work, we computed the elastic properties of particles of a swelling clay (i.e., montmorillonite) by molecular dynamics simulations. Because of the softness of the material in the direction orthogonal to the clay layer with increasing water content, the computation of the whole stiffness tensor was not trivial: we needed to implement the elastic bath method,1 which allows us to attenuate thermal strains and compute the stiffness via strain fluctuations in isothermal–isobaric simulations. We investigated the effect of water content, temperature, and the interlayer cation on the elastic properties of swelling clays. In particular, we showed that the out-of-plane stiffness coefficients computed at 300 K differed significantly from the same coefficients computed at 0 K. The out-of-plane coefficients were very sensitive to both temperature and water content. In contrast, the dependence of the in-plane coefficients to temperature was slight. Moreover, the decrease of the in-plane coefficients with water content could be entirely explained by the change of geometry of the system due to the swelling of the interlayer space. The interlayer cation impacted the elastic properties of montmorillonite only in the driest states.

A Modified Effective Stress Principle for Unsaturated Swelling Clays Derived from Microstructure

A three‐scale model is proposed to describe electro‐chemo‐mechanical couplings in unsaturated swelling clays characterized by two porosity levels and three separate length scales. The nanoscale portrait consists of charged clay particles separated by a nanoporous network saturated by a binary monovalent aqueous electrolyte solution. Local ion distribution and electric potential are governed by the Poisson–Boltzmann problem. At the microscale, the system is represented by swollen clay clusters separated from each other by a network of micropores filled with a mixture of bulk water and air. Under mechanical equilibrium characterized by the competition between disjoining forces of electrochemical nature and capillary attraction effects, we derived a novel form of the effective stress principle in the asymptotic limit of scale separation. Such form includes a new macroscopic equivalent pore pressure weighted by a two‐scale effective Bishop coefficient that incorporates the effects of the adsorbed water at the secondary nanopore level in addition to the contributions of the water–air interfaces at the micropore level. Within the thermodynamic context for constructing macroscopic constitutive laws based on stress–strain variables, the three‐scale model leads to a set of three work‐conjugated state variables. In addition to the contact stress between particles and the new effective Bishop‐type component, the novel form includes salinity as an additional stress‐conjugated variable and a three‐scale version of the electrochemical swelling stress. The potential of the multiscale approach in capturing the complex features of unsaturated expansive clays is illustrated by numerical reconstruction of the effective coefficients in a simplified isotropic microstructure.

Reliability analysis and micromechanics: A coupled approach for composite failure prediction

This work aims at associating two classical approaches for the design of composite materials: first, reliability methods that allow to account for the various uncertainties involved in the composite materials behaviour and lead to a rational estimation of their reliability level; on the other hand, micromechanics that derive macroscopic constitutive laws from micromechanical features. Such approach relies on the introduction of variabilities defined at the microscale and on the investigation of their consequences on the material macroscopic response through an homogenization scheme. Precisely, we propose here a systematic treatment of variability which involves a strong link between micro- and macroscales and provides a more exhaustive analysis of the influence of uncertainties. The paper intends to explain the main steps of such coupling and demonstrate its interests for material engineering, especially for constitutive modelling and composite materials optimization. An application case is developed throughout on the failure of unidirectional carbon fibre-reinforced composites with a comparative analysis between experimental data and simulation results.

Mesoscale Approach to Modeling Concrete Subjected to Thermomechanical Loading

Concrete subjected to combined compressive stresses and temperature loading exhibits compressive strains, which are considerably greater than for concrete subjected to compressive stresses alone. This phenomenon is called transient thermal creep or load induced thermal strain and is usually modeled by macroscopic phenomenological constitutive laws which have only limited predictive capabilities. In the present study a mesoscale modeling approach is proposed in which the macroscopically observed transient thermal creep results from the mismatch of thermal expansions of the mesoscale constituents. The mesostructure of concrete is idealized as a two-dimensional three-phase material consisting of aggregates, matrix, and interfacial transition zones. The nonlinear material response of the phases is described by a plasticity interface model. The mesoscale approach was applied to analyze compressed concrete specimens subjected to uniform temperature histories and the analysis results were compared to experimental results reported in the literature.

Homogenized mechanical behavior of composite micro-structures including micro-cracking and contact evolution

The aim of this paper is to study the effects of micro-cracking on the homogenized constitutive properties of elastic composite materials. To this end a novel micro-mechanical approach based on homogenization techniques and fracture mechanics concepts, is proposed and an original J-integral formulation is established for composite micro-structures. Accurate non-linear macroscopic constitutive laws are developed for a uniaxial and a shear macro-strain path by taking into account changes in micro-structural configuration owing to crack growth and crack face contact. Numerical results, carried out by coupling a finite element formulation and an interface model, are applied to a porous composite with edge cracks and a debonded short fiber-reinforced composite. The composite micro-structure is controlled by the macroscopic strain and the micro-to-macro transition, settled in a variational formulation, is obtained for three types of boundary conditions, i.e. linear displacements, uniform tractions and periodic fluctuations and anti-periodic tractions. The accuracy of the determined macroscopic constitutive properties to represent the failure characteristics of locally periodic defected composites is also investigated in terms of energy release rate predictions, by comparisons between a direct analysis and homogenization approaches. Results highlight the dependence of the macroscopic constitutive law for a micro-structure with evolving defects on both the macro-strain path and the type of boundary conditions and the capability of the proposed model to provide a failure model for a composite material undergoing micro-cracking and contact.

Influence of micro-cracking and contact on the effective properties of composite materials

In the present work a novel micro-mechanical approach to analyze the influence of micro-crack evolution and contact on the effective properties of elastic composite materials is proposed, based on homogenization techniques, interface models and fracture mechanics concepts. By means of the finite element method, enhanced non-linear macroscopic constitutive laws are developed by taking into account changes in micro-structural configuration associated with the growth of micro-cracks and with contact between crack faces. Numerical simulations are carried out for the cases of a porous composite with edge cracks and of a debonded fibre reinforced composite, loaded along extension/compression uniaxial macro-strain paths. Micro-crack propagation is modelled by using an original methodology based on the J-integral technique in conjunction with an interface model taking into account the unilateral contact of crack faces. In the context of a micro-to-macro transition obtained by controlling the macro-deformation of the micro-structure, the effects of adopting three types of boundary conditions on the macroscopic constitutive law, namely linear deformation, uniform tractions and periodic deformations and anti-periodic tractions, are studied. As a consequence, the proposed method can be applied to a large class of problems including periodic, locally periodic and irregular composite materials. Micro-crack and contact evolution result in a progressive loss of stiffness and can lead to failure for homogeneous macro-deformations associated with unstable crack propagation.

Structural Damage Analysis of Masonry Walls using Computational Homogenization

This contribution deals with the application of computational homogenization techniques for structural masonry computations, as an alternative to the formulation of complex closed-form macroscopic constitutive laws. The complexity of modeling masonry material stems from the anisotropy evolution and localization induced by mesostructural damage. This phenomenon appears with preferential damage orientations, which are intimately related to the initial periodic structure of the material. The upscaling procedure used here relies on the formulation of mesoscopic constitutive laws at the level of the individual brick and mortar materials. A mesostructural unit cell with its corresponding periodicity requirements is used to deduce the average response of the masonry material through a scale transition. At the macroscopic scale, this averaged material response is used in the frame of an enhanced continuum approach with embedded localization bands, the widths of which are directly deduced from the initial periodicity of the material. The results obtained by the framework are illustrated and discussed by means of a structural computation example, which involves a complex cracking evolution together with fully anisotropic damage development.

Parameter identification in constitutive models via optimization with a posteriori error control

AbstractIn this paper we outline a computational technique for the calibration of macroscopic constitutive laws with automatic error control. In the most general situation the state variables of the constitutive law, as well as the material parameters, are spatially non‐homogeneous. The experimental observations are given in space–time. Based on an appropriate dual problem, we compute a posteriori the discretization error contributions from approximations of the parameter, state and costate fields in space–time for an arbitrarily chosen goal‐oriented error measure of engineering significance. Such a measure can be used in an adaptive strategy (not discussed in this paper) to meet a predefined error tolerance. An important observation is that the Jacobian matrix associated with the resulting Newton method is used (in principle) in solving the dual problem. Rather than treating the Jacobian in a monolithic fashion, we utilize a sequential solution strategy, whereby the FE‐topology of the discretized state problem is used repeatedly. Moreover, the proposed solution strategy lends itself naturally to the computation of first and second order sensitivities, which are obtained with little extra computational effort. Numerical results are given for the prototype model of confined aquifer flow with spatially non‐homogeneous permeability. The efficiency of the optimization strategy and the effectivity of the error computation are assessed. Copyright © 2005 John Wiley & Sons, Ltd.

High-Fidelity Micromechanical Modeling of Continuously Reinforced Elastic Multiphase Materials Undergoing Finite Deformations

The purpose of this communication is twofold. First, we demonstrate the predictive capability of a recently extended micromechanics model known as the high-fidelity generalized method of cells, originally developed for unidirectionally reinforced periodic multiphase materials characterized by elastic or elasto-plastic phases undergoing infinitesimal deformation. The recent extension incorporates finite-deformation capabilities to enable modeling of heterogeneous materials such as fiber-reinforced rubbers or certain types of biological tissues characterized by potential-based, nonlinear elastic phases. The model's capability to accurately estimate both the homogenized nonlinear elastic response and the local stress fields in the individual phases is demonstrated by comparison with an exact elasticity solution for a porous composite with four different types of the matrix material under axisyrmnetric loading, and a finite-element analysis of a repeating unit cell representative of a unidirectionally reinforced periodic composite subjected to transverse loading. Second, we demonstrate the micromechanics model's utility as a subroutine in a structural analysis setting by implementing it into a specialized lamination theory framework in the absence of bending. Examples of the nonlinear response of families of [±θ]s lay-ups under biaxial inplane loading are provided, demonstrating how the developed model can be used either to validate or to construct macroscopic constitutive laws for materials, such as certain biological tissues, characterized by multi-directional reinforcement.

Structural Characterization of Modern Highway Overcrossings—Case Study

This paper revisits the mechanical characterization of the main substructure elements of highway overcrossings in an effort to build a more balanced methodology to compute the earthquake response of seismically protected bridges accounting for the effects of soil–structure interaction. Various macroscopic constitutive laws that are well known in literature are summarized and synthesized in a way that can be incorporated in reduced-order stick models used in practice. Here, we discuss the required sophistication level of the proposed macroscopic constitutive laws in order to capture the leading mechanical behavior of each substructural element without adding unnecessary complications when incorporated in a stick model idealization of the bridge.

Seismic Response Analysis of a Highway Overcrossing Equipped with Elastomeric Bearings and Fluid Dampers

This paper presents a case study on the seismic response of a recently constructed freeway overcrossing located in southern California, which is equipped with elastomeric bearings and fluid dampers at its end abutments. The analysis employs the substructure method and a reduced order stick model. The macroscopic constitutive laws of the springs and dashpots that approximate the mechanical behavior of approach embankments, pile foundations, center bent, abutments, elastomeric bearings, and fluid dampers are established in the companion paper. The paper presents an in-depth analysis of the seismic response of the bridge equipped with the response modification devices accounting for the effects of soil–structure interaction. The various response quantities presented are compared with the corresponding response quantities of a hypothetical bridge with integral abutments. Advantages and challenges in the two design configurations are identified and discussed.

A comparison of damage models formulated on different material scales

This contribution aims at characteristic damage mechanisms of materials at two levels of observation, (a) at the macro-scale in the sense of classical continuum damage mechanics, and (b) at the meso-scale utilizing the so-called microplane concept. The constitutive formulations are embedded in a thermodynamically consistent framework. The microplane formulation is based on a volumetric–deviatoric split (V–D), which is motivated from a macroscopic viewpoint. The main advantage of this formulation is that the material behavior at a material point is characterized by constitutive laws formulated on the individual microplanes, allowing to describe anisotropic material behavior in a very natural and simple way. One particular advantage of the present version of a microplane formulation is also its close relation to macroscopic models which enables the interpretation and identification of the microplane constitutive laws in terms of well-known macroscopic constitutive laws. The appropriate choice of the microplane loading functions is illustrated by a comparison of the microplane damage formulation with a macroscopic two-parameter damage model.

Intermittent dislocation flow in viscoplastic deformation.

The viscoplastic deformation (creep) of crystalline materials under constant stress involves the motion of a large number of interacting dislocations. Analytical methods and sophisticated 'dislocation dynamics' simulations have proved very effective in the study of dislocation patterning, and have led to macroscopic constitutive laws of plastic deformation. Yet, a statistical analysis of the dynamics of an assembly of interacting dislocations has not hitherto been performed. Here we report acoustic emission measurements on stressed ice single crystals, the results of which indicate that dislocations move in a scale-free intermittent fashion. This result is confirmed by numerical simulations of a model of interacting dislocations that successfully reproduces the main features of the experiment. We find that dislocations generate a slowly evolving configuration landscape which coexists with rapid collective rearrangements. These rearrangements involve a comparatively small fraction of the dislocations and lead to an intermittent behaviour of the net plastic response. This basic dynamical picture appears to be a generic feature in the deformation of many other materials. Moreover, it should provide a framework for discussing fundamental aspects of plasticity that goes beyond standard mean-field approaches that see plastic deformation as a smooth laminar flow.

Viscoelasticity of granular materials

In a former work (V.T. Granik and M. Ferrari (1993), Microstructural mechanics of granular media, Mech. Mater. 15, 301–322) a micromechanical theory for elastic granular media was deduced on the basis of the identification of the constituent grains with the nodes of a Bravais lattice. The transition from the discrete structure to the continuum level was achieved through assumptions on the kinematical fields and through a variational formulation establishing the relationship between microstresses and macrostresses. In the present paper we extend this theory to the linear viscoelasticity domain. We formulate the general linear viscoelastic relations in terms of microstresses and microdeformations, deducing the boundary value problems for the microstresses and for macrostresses. Then we focus our attention on the links between constitutive equations governing the microscopic level (formulated following the general methods of linear viscoelasticity) and their macroscopic counterparts. Thus we directly relate microstructural information to the macroscopic constitutive laws. deducing classical macroscopic properties in terms of structural parameters. Finally the dynamical problem of plane waves propagation in a semi-infinite granular medium is analyzed and the influence that the arrangement of the particles exerts on the pulse propagation is discussed.