Computational multiscale modelling of heterogeneous material layers

A computational homogenization procedure for a material layer that possesses an underlying heterogeneous microstructure is introduced within the framework of finite deformations. The macroscopic material properties of the material layer are obtained from multiscale considerations. At the macro level, the layer is resolved as a cohesive interface situated within a continuum, and its underlying microstructure along the interface is treated as a continuous representative volume element of given height. The scales are linked via homogenization with customized hybrid boundary conditions on this representative volume element, which account for the deformation modes along the interface. A nested numerical solution scheme is adopted to link the macro and micro scales. Numerical examples successfully display the capability of the proposed approach to solve macroscopic boundary value problems with an evaluation of the constitutive properties of the material layer based on its micro-constitution.     

A two‐scale failure model for heterogeneous materials: numerical implementation based on the finite element method

In the first part of this contribution, a brief theoretical revision of the mechanical and variational foundations of a Failure‐Oriented Multiscale Formulation devised for modeling failure in heterogeneous materials is described. The proposed model considers two well separated physical length scales, namely: (i) the macroscale where nucleation and evolution of a cohesive surface is considered as a medium to characterize the degradation phenomenon occurring at the lower length scale, and (ii) the microscale where some mechanical processes that lead to the material failure are taking place, such as strain localization, damage, shear band formation, and so on. These processes are modeled using the concept of Representative Volume Element (RVE). On the macroscale, the traction separation response, characterizing the mechanical behavior of the cohesive interface, is a result of the failure processes simulated in the microscale. The traction separation response is obtained by a particular homogenization technique applied on specific RVE sub‐domains. Standard, as well as, Non‐Standard boundary conditions are consistently derived in order to preserve objectivity of the homogenized response with respect to the micro‐cell size. In the second part of the paper, and as an original contribution, the detailed numerical implementation of the two‐scale model based on the finite element method is presented. Special attention is devoted to the topics, which are distinctive of the Failure‐Oriented Multiscale Formulation, such as: (i) the finite element technologies adopted in each scale along with their corresponding algorithmic expressions, (ii) the generalized treatment given to the kinematical boundary conditions in the RVE, and (iii) how these kinematical restrictions affect the capturing of macroscopic material instability modes and the posterior evolution of failure at the RVE level. Finally, a set of numerical simulations is performed in order to show the potentialities of the proposed methodology, as well as, to compare and validate the numerical solutions furnished by the two‐scale model with respect to a direct numerical simulation approach. Copyright © 2013 John Wiley & Sons, Ltd.     

Computational homogenization for heat conduction in heterogeneous solids

In this paper, a multi‐scale analysis method for heat transfer in heterogeneous solids is presented. The principles of the method rely on a two‐scale computational homogenization approach which is applied successfully for the stress analysis of multi‐phase solids under purely mechanical loading. The present paper extends this methodology to heat conduction problems. The flexibility of the method permits one to take into account local microstructural heterogeneities and thermal anisotropy, including non‐linearities which might arise at some stage of the thermal loading history. The resulting complex microstructural response is transferred back to the macro level in a consistent manner. A proper macro to micro transition is established in terms of the applied boundary conditions and likewise a micro to macro transition is formulated in the form of consistent averaging relations. Imposition of boundary conditions and extraction of macroscopic quantities are elaborated in detail. A nested finite element solution procedure is outlined, and the effectiveness of the approach is demonstrated by some illustrative example problems. Copyright © 2007 John Wiley & Sons, Ltd.     

Multi‐scale modeling of moving interface problems with flux and field jumps: Application to oxidative degradation of ceramic matrix composites

Problems involving reaction and species diffusion involve field and flux jumps at a moving reaction front. In multi‐scale problems such as carbon fiber composite oxidation, these effects need to be tracked at the microscopic scale of individual carbon fibers. A multi‐scale model is derived in this paper for predicting species distribution in such problems using a fully coupled multi‐scale homogenization approach. The homogenized fluxes from the micro‐scale are derived using Hill's macro‐homogeneity condition accounting for both flux jumps and species density field jumps at the reacting interface in the micro‐scale unit cell. At the macro‐scale, the competition between the transport of reacting species (oxygen) and the reaction product (carbon dioxide) is modeled using homogenized mass conservation equations. The moving reaction front in carbon fibers at the micro‐scale is tracked using level set method and an adaptive meshing strategy. The macroscopic weight loss of the composite when exposed to oxygen is simulated as a function of time using a coupled finite element methodology at various locations in a validated macroscopic model. Copyright © 2010 John Wiley & Sons, Ltd.     

Micro and macro contact mechanics for interacting asperities

Contact of rough surfaces at micro and macro scales is studied in this paper. The asperities at micro scale are characterised by small radius of curvature whereas the waviness is characterised by large radius of curvature. When two rough surfaces come in contact, on the micro scale, of asperities contacts in a very small area leave large gaps between the surfaces; whereas on the macro scale the surfaces conform to each other under the application of load without gaps. Contact at micro scale is modelled by superposition of Hertzian stress fields of individual asperity contacts and the waviness at macro scale is modelled as a mixed boundary problem of rough punch indentation where displacements of uneven profile are prescribed along the region of contact. In both the cases for simplification the roughness is assigned to one surface making the other surface perfectly flat an assumption often made in contact mechanics of rough bodies. The motivation for modelling the asperities at micro scales comes from the preliminary results obtained from photoelastic experiments. Numerical results are presented based on the analytical results available for Hertzian contacts. The motivation for modelling the asperities at macro scales comes from the results available in literature for flat contacts from solving mixed boundary elasticity problems. A condition of full stick is assumed along the contact which is a common assumption made for rough contacts. The numerical results are presented for both the cases of rough contact at micro and macro scales.     

Consistent micro–macro transitions at large objective strains in curvilinear convective coordinates

This paper presents a computational homogenization scheme that is of particular interest for problems formulated in curvilinear coordinates. The main goal of this contribution is to generalize the computational homogenization scheme to a formulation of micro–macro transitions in curvilinear convective coordinates, where different physical spaces are considered at the homogenized macro‐continuum and at the locally attached representative micro‐structures. The deformation and the coordinate system of the micro‐structure are assumed to be coupled with the local deformation and the local coordinate system at a corresponding point of the macro‐continuum. For the consistent formulation of micro–macro transitions, the operations scale‐up and scale‐down are introduced, considering the rotated representation of tensor variables at the different physical reference frames of micro‐ and macro‐structure. The second goal of this paper is to use objective strain measures like the Green–Lagrange strain tensor for the solution of boundary value problems on the micro‐ and macro‐scale by providing the required transformations for the work‐conjugate stress, strain and tangent tensors into variables admissible for the considered micro–macro transitions and satisfying the averaging theorem. Copyright © 2007 John Wiley & Sons, Ltd.     

Analysis of micro fracture in human Haversian cortical bone under transverse tension using extended physical imaging

We propose a procedure to investigate local stress intensity factors at the scale of the osteons in human Haversian cortical bone. The method combines a specific experimental setting for a three‐point bending millimetric specimen and a numerical method using the eXtended Finite Element Method (X‐FEM). The interface between the experimental setting and the numerical method is ensured through an imaging technique that analyses the light microscopy observations to import the geometrical heterogeneity of the Haversian microstructures, the boundary conditions and appearing crack discontinuities into the numerical model. The local mechanical elastic Young's moduli are measured by nano‐indentation, and the Poisson ratios are determined by an imaging technique of the stress–strain fields. The model is able to access three scales of measurement: the macro scale of the material level (mm), the micro scale inside the Haversian material for stress–strain fields (10–100µm), and the sub‐micro scale for the crack opening profiles (1–10µm ) and fracture parameters (stress intensity factors). The model is applied to several patients at different aging stages. Copyright © 2009 John Wiley & Sons, Ltd.     

Multi‐scale modeling of surface effects in nano‐materials with temperature‐related Cauchy‐Born hypothesis via the modified boundary Cauchy‐Born model

In nano‐structures, the influence of surface effects on the properties of material is highly important because the ratio of surface to volume at the nano‐scale level is much higher than that of the macro‐scale level. In this paper, a novel temperature‐dependent multi‐scale model is presented based on the modified boundary Cauchy‐Born (MBCB) technique to model the surface, edge, and corner effects in nano‐scale materials. The Lagrangian finite element formulation is incorporated into the heat transfer analysis to develop the thermo‐mechanical finite element model. The temperature‐related Cauchy‐Born hypothesis is implemented by using the Helmholtz free energy to evaluate the temperature effect in the atomistic level. The thermo‐mechanical multi‐scale model is applied to determine the temperature related characteristics at the nano‐scale level. The first and second derivatives of free energy density are computed using the first Piola‐Kirchhoff stress and tangential stiffness tensor at the macro‐scale level. The concept of MBCB is introduced to capture the surface, edge, and corner effects. The salient point of MBCB model is the definition of radial quadrature used at the surface, edge, and corner elements as an indicator of material behavior. The characteristics of quadrature are derived by interpolating the data from the atomic level laid in a circular support around the quadrature in a least‐square approach. Finally, numerical examples are modeled using the proposed computational algorithm, and the results are compared with the fully atomistic model to illustrate the performance of MBCB multi‐scale model in the thermo‐mechanical analysis of metallic nano‐scale devices. Copyright © 2013 John Wiley & Sons, Ltd.     

Mesoscopic mechanisms in fatigue crack initiation in an aluminium alloy

The mechanistic aspects of process of initiation of a mode‐I fatigue crack in an aluminium alloy (AA 2219‐T87) are studied in detail, both computationally as well as experimentally. Simulations are carried out under plane strain conditions with fatigue process zone modelled as stress‐state–dependent cohesive elements along the expected mode‐I failure path. An irreversible damage parameter that accounts for the progressive microstructural damage due to fatigue is employed to degrade cohesive properties. The simulations predict the location of initiation of the fatigue crack to be subsurface where the triaxiality and the opening tensile stresses are higher in comparison with that at the notch surface. Examination of the fracture surface profile of fracture test specimens near notch tip reveals a few types of regions and existence of a mesoscopic length scale that is the distance of the location of highest roughness from the notch root. A discussion is developed on the physical significance of the experimentally observed length scale.     

Computational Modeling of Columnar to Equiaxed Transition in Alloy Solidification

The columnar to equiaxed transition (CET) provides a challenging simulation goal for computational models of alloy solidification, in addition to being an important technological feature of many casting processes. CET thus provides an industrially relevant test‐case for those developing numerical models across a range of scales. Whether or not CET occurs depends on numerous experimental parameters such as cooling rate, speed of columnar growth, thermal gradient in the liquid, and level of grain refiner in the alloy. Information on columnar and equiaxed grain structure, and the transition between the two, is very useful for foundry engineers, at the macroscopic scale of the casting. The detailed microstructure within each grain is determined by typically dendritic growth and local transport of solute and heat. This paper presents a review of recent progress on modeling CET at multiple length scales. It is evident that, whilst micro‐models can provide simulations of physical phenomena, such as the evolution of dendrite morphology, at scales 10^?3 to 10^?5 m, finite computational resources preclude this resolution over the length scale of castings which is in the 10^?2–10⁰ m range. Instead, reasonably accurate models of CET formation in castings can be achieved by meso‐scale modeling featuring 10^?3–10^?2 m phenomena. Such meso‐scale models make use of analytical expressions to simulate dendrite growth in undercooled melts. Recent progress in modeling of CET, at both macro/meso‐ and micro‐scales is reviewed, and computational challenges yet to be met are summarized.     

Combined continuum damage‐embedded discontinuity model for explicit dynamic fracture analyses of quasi‐brittle materials

In this paper, a novel constitutive model combining continuum damage with embedded discontinuity is developed for explicit dynamic analyses of quasi‐brittle failure phenomena. The model is capable of describing the rate‐dependent behavior in dynamics and the three phases in failure of quasi‐brittle materials. The first phase is always linear elastic, followed by the second phase corresponding to fracture‐process zone creation, represented with rate‐dependent continuum damage with isotropic hardening formulated by utilizing consistency approach. The third and final phase, involving nonlinear softening, is formulated by using an embedded displacement discontinuity model with constant displacement jumps both in normal and tangential directions. The proposed model is capable of describing the rate‐dependent ductile to brittle transition typical of cohesive materials (e.g., rocks and ice). The model is implemented in the finite element setting by using the CST elements. The displacement jump vector is solved for implicitly at the local (finite element) level along with a viscoplastic return mapping algorithm, whereas the global equations of motion are solved with explicit time‐stepping scheme. The model performance is illustrated by several numerical simulations, including both material point and structural tests. The final validation example concerns the dynamic Brazilian disc test on rock material under plane stress assumption. Copyright © 2014 John Wiley & Sons, Ltd.     

Adaptive mesh refinement and coarsening for cohesive zone modeling of dynamic fracture

Adaptive mesh refinement and coarsening schemes are proposed for efficient computational simulation of dynamic cohesive fracture. The adaptive mesh refinement consists of a sequence of edge‐split operators, whereas the adaptive mesh coarsening is based on a sequence of vertex‐removal (or edge‐collapse) operators. Nodal perturbation and edge‐swap operators are also employed around the crack tip region to improve crack geometry representation, and cohesive surface elements are adaptively inserted whenever and wherever they are needed by means of an extrinsic cohesive zone model approach. Such adaptive mesh modification events are maintained in conjunction with a topological data structure (TopS). The so‐called PPR potential‐based cohesive model (J. Mech. Phys. Solids 2009; 57 :891–908) is utilized for the constitutive relationship of the cohesive zone model. The examples investigated include mode I fracture, mixed‐mode fracture and crack branching problems. The computational results using mesh adaptivity (refinement and coarsening) are consistent with the results using uniform mesh refinement. The present approach significantly reduces computational cost while exhibiting a multiscale effect that captures both global macro‐crack and local micro‐cracks. Copyright © 2012 John Wiley & Sons, Ltd.     

A new multi‐scale scheme for modeling heterogeneous incompressible hyperelastic materials

A computational homogenization scheme is developed to model heterogeneous hyperelastic materials undergoing large deformations. The homogenization scheme is based on a so‐called computational continua formulation in which the macro‐scale model is assumed to consist of disjoint unit cells. This formulation adds no higher‐order boundary conditions and extra degrees of freedom to the problem. A computational procedure is presented to calculate the macroscopic quantities from the solution of the representative volume element boundary value problem. The proposed homogenization scheme is verified against a direct numerical simulation. It is also shown that the computational cost of the proposed model is lower than that of standard homogenization schemes. Copyright © 2015 John Wiley & Sons, Ltd.     

Cracks and Plasticity

The paper gives a review of the state‐of‐the‐art of modeling deformation, damage, and fracture of ductile metals on various length scales, namely numerical simulation of micro‐void nucleation and growth, porosity‐based constitutive equations, the cohesive zone model and the macroscopic fracture‐mechanics approach. Links between the various models are outlined and their respective favorable and preferential applications are discussed. The potential of a hierarchical modeling approach from the micro‐ to the macroscale is emphasized.     

Wavelet-based homogenization of unidirectional multiscale composites

The main aim is to present a homogenization algorithm for the multiscale heterogeneous (composite) materials, which is based on the wavelet representation of material properties and the relevant multiscale reduction. It is shown that classical homogenization method used before for two-scale composites (with micro and macro scales) is a special case of general multiresolutional strategy, where a single scale parameter tends to 0. The approach presented is applied to unidirectional wavelet-based homogenization of linear elasticity heterogeneous problem and to wave propagation, which may be applied in conjunction with various discrete numerical methods for efficient modeling of heterogeneous solids, fluids and multiphase media.     

A Rate Independent Cohesive Zone Model for Modeling Failure in Quasi-Brittle Materials

Computational modeling of failure in quasi-brittle materials at various length scales is important. In this work we present a rate independent cohesive zone model for modeling failure in quasi-brittle materials. The proposed model can simulate cracking, slipping, and crushing of planes through a traction-separation law. A single surface hyperbolic failure criterion, which naturally comes as a direct extension of Coulomb friction criterion with cut-off in tension and cap-off in compression, has been developed. A Euler backward integration scheme together with a global-local Newton solver compatible with a substepping strategy has been used in numerical computations. The proposed model is then used for modeling of shear wall panels. The numerical results obtained are validated by comparing them with experimental results available in literatures.     

A continuous–discontinuous approach to simulate physical degradation processes in porous media

A macroscopic framework for the simulation of physical degradation processes in quasi‐brittle porous materials is proposed. The framework employs the partition of unity (PU) concept and introduces a cohesive zone model, capturing the entire failure process starting from the growth and coalescence of micro‐defects until the formation of macro‐cracks. The framework incorporates the interaction between the failure process and the heat and mass transfer in the porous medium. As an example, physical degradation of an outside render is studied. The analysis illustrates that both material and interface failure can be investigated with this formulation. Depending on the boundary conditions, either one dominant crack or a network of small cracks is formed. Copyright © 2010 John Wiley & Sons, Ltd.     

Cohesive modeling of dynamic fracture in functionally graded materials

The dynamic fracture of functionally graded materials (FGMs) is modeled using an explicit cohesive volumetric finite element scheme that incorporates spatially varying constitutive and failure properties. The cohesive element response is described by a rate-independent bilinear cohesive failure model between the cohesive traction acting along the cohesive zone and the associated crack opening displacement. A detailed convergence analysis is conducted to quantify the effect of the material gradient on the ability of the numerical scheme to capture elastodynamic wave propagation. To validate the numerical scheme, we simulate dynamic fracture experiments performed on model FGM compact tension specimens made of a polyester resin with varying amounts of plasticizer. The cohesive finite element scheme is then used in a parametric study of mode I dynamic failure of a Ti/TiB FGM, with special emphasis on the effect of the material gradient on the initiation, propagation and arrest of the crack.     

Time‐domain hybrid formulations for wave simulations in three‐dimensional PML‐truncated heterogeneous media

We are concerned with the numerical simulation of wave motion in arbitrarily heterogeneous, elastic, perfectly‐matched‐layer‐(PML)‐truncated media. We extend in three dimensions a recently developed two‐dimensional formulation, by treating the PML via an unsplit‐field, but mixed‐field, displacement‐stress formulation, which is then coupled to a standard displacement‐only formulation for the interior domain, thus leading to a computationally cost‐efficient hybrid scheme. The hybrid treatment leads to, at most, third‐order in time semi‐discrete forms. The formulation is flexible enough to accommodate the standard PML, as well as the multi‐axial PML. We discuss several time‐marching schemes, which can be used à la carte, depending on the application: (a) an extended Newmark scheme for third‐order in time, either unsymmetric or fully symmetric semi‐discrete forms; (b) a standard implicit Newmark for the second‐order, unsymmetric semi‐discrete forms; and (c) an explicit Runge–Kutta scheme for a first‐order in time unsymmetric system. The latter is well‐suited for large‐scale problems on parallel architectures, while the second‐order treatment is particularly attractive for ready incorporation in existing codes written originally for finite domains. We compare the schemes and report numerical results demonstrating stability and efficacy of the proposed formulations. Copyright © 2014 John Wiley & Sons, Ltd.