Effective Stress Strain Response Research Articles

A multi-scale computational model for the passive mechanical behavior of right ventricular myocardium

We have previously demonstrated the importance of myofiber–collagen mechanical interactions in modeling the passive mechanical behavior of right ventricle free wall (RVFW) myocardium. To gain deeper insights into these coupling mechanisms, we developed a high-fidelity, micro-anatomically realistic 3D finite element model of right ventricle free wall (RVFW) myocardium by combining high-resolution imaging and supercomputer-based simulations. We first developed a representative tissue element (RTE) model at the sub-tissue scale by specializing the hyperelastic anisotropic structurally-based constitutive relations for myofibers and ECM collagen, and equi-biaxial and non-equibiaxial loading conditions were simulated using the open-source software FEniCS to compute the effective stress–strain response of the RTE. To estimate the model parameters of the RTE model, we first fitted a ’top-down’ biaxial stress–strain behavior with our previous structurally based (tissue-scale) model, informed by the measured myofiber and collagen fiber composition and orientation distributions. Next, we employed a multi-scale approach to determine the tissue-level (5 x 5 x 0.7 mm specimen size) RVFW biaxial behavior via ’bottom-up’ homogenization of the fitted RTE model, recapitulating the histologically measured myofiber and collagen orientation to the biaxial mechanical data. Our homogenization approach successfully reproduced the tissue-level mechanical behavior of our previous studies in all biaxial deformation modes, suggesting that the 3D micro-anatomical arrangement of myofibers and ECM collagen is indeed a primary mechanism driving myofiber–collagen interactions.

Refined nonlinear micromechanical models using artificial neural networks for multiscale analysis of laminated composites subject to low-velocity impact

The parametric high fidelity generalized method of cells (PHFGMC) is an advanced micromechanical method that can be used for the nonlinear and failure analysis of several composite materials. The computational effort required for studying the nonlinear and damage multiaxial behavior is relatively small, depending on the size of the discretized repeating unit cell (RUC). However, it is computationally challenging, if not impossible, to integrate refined nonlinear micromechanical models within a multiscale analysis of composite structures. This is due to the thousands or more RUC models required at the integration points within a multiscale finite-element (FE) model of laminated structures. To that end, we propose a new artificial neural network (ANN) based micromechanical modeling framework, termed ANN-PHFGMC, for exploring the nonlinear behavior of fiber-reinforced polymeric (FRP) materials. Pre-simulated mechanical stress–strain responses and behaviors are determined using the PHFGMC to generate a multiaxial training database for the ANN micromodel. The simulated training data is founded on the PHFGMC-RUC results based on a hexagonal RUC. The PHFGMC effective stress–strain responses for different applied multiaxial strain paths are divided into two sets of data; one for the training and the other for verifying the trained ANN-PHFGMC model. The resulting trained ANN-PHFGMC is accurate, with less than 5% error in the verified predictions. The ANN-PHFGMC can be used as a stand-alone or embedded as a surrogate proxy model within a multiscale analysis of composite structures. Next, the ANN-PHFGMC model is integrated within a commercial explicit FE code for low-velocity impact (LVI) analysis of laminated composite plates. Multiscale LVI analyses are performed for two composite plates with different layups. Further, results are compared to experimental data to demonstrate the new model's ability to integrate refined nonlinear micromechanical models within a multiscale analysis.

ON TWINNING-MEDIATED VOID GROWTH IN HEXAGONAL CRYSTALS

Aspects of plastic anisotropy in damage accumulation are considered for a class of hexagonal crystals that deform by combined slip and twinning. Focus is placed on crystallographic aspects that are currently absent from constitutive formulations of ductile damage. To this end, three-dimensional finite-element calculations are carried out using a cubic unit cell containing a single void embedded in a crystal matrix. Plastic flow in the latter is described using crystal plasticity with parameters representative of single crystal pure magnesium. The effect of void oblateness is analyzed in some detail, as voids often form as blunted microcracks in Mg alloys. The analyses reveal two aspects peculiar to twinning-mediated void growth: (1) insensitivity of the effective stress-strain response to void oblateless and (2) a plastic auxetic effect. Both aspects manifest under certain circumstances. Some implications in terms of incorporating the uncovered crystallographic aspects in coupled plasticity-damage formulations of anisotropic materials are discussed.

A multi-scale, multi-physics modeling framework to predict spatial variation of properties in additive-manufactured metals

The microstructure of additively manufactured (AM) metals has been shown to be heterogeneous and spatially non-uniform when compared to conventionally manufactured metals. Consequently, the effective mechanical properties of AM-metal parts are expected to vary both within and among builds. Here, we present a framework for simulating process–(micro)structure–property relationships of AM metals produced via direct laser deposition (DLD). The framework predicts grain nucleation and competitive growth as a function of thermal history for a multi-pass, multi-layer DLD process. The resulting three-dimensional microstructure is automatically sub-sampled to perform virtual mechanical testing throughout the build domain using a parallelized elasto-viscoplastic fast Fourier transform code, accounting for grain-boundary strengthening. The effective stress–strain response of each sub-sampled volume is automatically analyzed to extract effective mechanical properties, which are used to generate property maps showing the spatial variability of effective mechanical properties throughout the simulated build volume. As a demonstration, the framework is applied to different DLD stainless steel 316L build volumes having different process-induced microstructures. The multi-physics framework and property maps could provide a path toward qualification of AM-metal parts.

Materials knowledge system for nonlinear composites

In this contribution, we present a new Materials Knowledge System framework for microstructure-sensitive predictions of effective stress–strain responses in composite materials. The model is developed for composites with a wide range of combinations of strain hardening laws and topologies of the constituents. The theoretical foundation of the model is inspired by statistical continuum theories, leveraged by mean-field approximation of self-consistent models, and calibrated to data obtained from micromechanical finite element simulations. The model also relies on newly formulated data-driven linkages between micromechanical responses (phase-average strain rates and effective strength) and microstructure as well as strength contrast of the constituents. The paper describes in detail the theoretical development of the model, its implementation into an efficient computational plasticity framework, calibration of the linkages, and demonstration of the model predictions on two-phase composites with isotropic constituents exhibiting linear and power-law strain hardening laws. It is shown that the model reproduces finite element results reasonably well with significant savings of the computational cost.

Prediction of Stress–Strain Curves of Metastable Austenitic Stainless Steel Considering Deformation-Induced Martensitic Transformation

Transformation-induced plasticity (TRIP) effect is the outstanding mechanism of austenitic stainless steel. It plays an important role in increasing formability of the steel due to higher local strain hardening during deformation. In order to better understand forming behavior of this steel grade, the strain-induced martensitic transformation of the 304 stainless steel was investigated. Uniaxial tensile tests were performed at different temperatures for the steel up to varying strain levels. Stress–strain curves and work hardening rates with typical TRIP effect characteristics were obtained. Metallographic observations in combination with X-ray diffraction method were employed for determining microstructure evolution. Higher volume fraction of martensite was found by increasing deformation level and decreasing forming temperature. Subsequently, micromechanics models based on the Mecking–Kocks approach and Gladman-type mixture law were applied to predict amount of transformed martensite and overall flow stress curves. Hereby, individual constituents of the steel and their developments were taken into account. Additionally, finite element (FE) simulations of two representative volume element (RVE) models were conducted, in which effective stress–strain responses and local stress and strain distributions in the microstructures were described under consideration of the TRIP effect. It was found that flow stress curves calculated by the mixture law and RVE simulations fairly agreed with the experimental results. The RVE models with different morphologies of martensite provided similar effective stress–strain behavior, but unlike local stress and strain distributions, which could in turn affect the strain-induced martensitic transformation.

Numerical assessment of an anisotropic porous metal plasticity model

The objective of this paper is to perform numerical assessment of a micromechanical model of porous metal plasticity developed previously by the authors. First, upper bound estimates for the yield loci are computed using homogenization and limit analysis of a spheroidal representative volume element containing a confocal spheroidal void, neglecting elasticity. Unlike in the development of the analytical model, the computational limit analysis is performed without recourse to approximations so that the obtained yield loci are rigorous upper bounds for the true criterion. Next, the model’s macroscopic dilatancy at incipient plastic flow is compared against that of the numerical limit analysis approach. Finally, finite-element calculations, with elasticity included, are presented for transversely isotropic porous unit-cells loaded axisymmetrically. The effective stress–strain response as well as evolution of the unit-cell porosity and void aspect ratio are compared with theoretical predictions.

Multi‐scale computational homogenization–localization for propagating discontinuities using X‐FEM

SummaryThe aim of this paper is to propose a continuous–discontinuous computational homogenization–localization framework to upscale microscale localization toward the onset and propagation of a cohesive discontinuity at the macroscale. The major novelty of this contribution is the development of a fully coupled micro–macro solution strategy, where the solution procedure for the macroscopic domain is based on the extended finite element method. The proposed approach departs from classical computational homogenization, which allows to derive the effective stress–strain response before the onset of localization. Upon strain localization, the microscale is characterized by a strain localization band where damage grows and by two adjacent unloading bulk regions at each side of the localization zone. The microscale localization band is lumped into a macroscopic cohesive crack, accommodated through discontinuity enriched macroscale kinematics. The governing response of the continuum with a discontinuity is obtained numerically based on proper scale transition relations in terms of the traction–separation law and the stress–strain description of the continuous surrounding material at both sides of the discontinuity. The potential of the method is demonstrated with a numerical example, which illustrates the onset and propagation of a macroscale cohesive crack emerging from microstructural damage within the underlying microstructure. Copyright © 2015 John Wiley & Sons, Ltd.

Microstructural pattern formation in finite-deformation single-slip crystal plasticity under cyclic loading: Relaxation vs. gradient plasticity

We investigate microstructure formation and evolution during cyclic loading in rate-dependent crystal plasticity at finite strains. The non-quasiconvex free energy density in multiplicative single-slip crystal plasticity leads to fine-scale microstructure whose characteristics and resulting effective stress–strain response are studied by two independent approaches: (i) using an incremental formulation based on variational constitutive updates we approximate the quasiconvex hull by lamination, i.e. by constructing an energy-minimizing first-order laminate microstructure, and (ii) a strain-gradient plasticity model applied to a representative unit cell whose effective properties are obtained from homogenization. In the lamination model, three different formulations for updating the accumulated plastic strains are compared and discussed with a specific focus on identifying a suitable description to account for hardening due to changes of the laminate volume fractions. The gradient-plasticity model also predicts a first-order laminate microstructure to form at a comparable stress level upon microstructure initiation. However, the energy associated with the dislocation network is shown to affect the microstructure evolution, leading to considerably higher strain levels at laminate initiation and a stress overshoot. In both models, cyclic loading leads to a degeneration of the stress–strain hysteresis which ultimately experiences elastic shakedown. The amount of work hardening significantly depends on how fast the degeneration occurs. To allow for a comparison, we consider cyclic loading after pre-deformation in the gradient model which delays the degeneration of the stress–strain hysteresis. For low hardening, the two models predict differences in the stress–strain hysteresis, mainly owing to laminate migration in the gradient-plasticity model. As work hardening increases, this phenomenon is restricted and the agreement of the effective stress–strain response between the two models is excellent. Accounting for the energy stored in the domain walls leads to a delayed lamination which is in agreement with the gradient plasticity model.

Can equations of equilibrium predict all physical equilibria? A case study from Field Dislocation Mechanics

Numerical solutions of a one-dimensional model of screw dislocation walls (twist boundaries) are explored. The model is an exact reduction of the three-dimensional system of partial differential equations of Field Dislocation Mechanics. It shares features of both Ginzburg–Landau (GL)-type gradient flow equations and hyperbolic conservation laws, but is qualitatively different from both. We demonstrate such similarities and differences in an effort to understand the equation through simulation. A primary result is the existence of spatially non-periodic, extremely slowly evolving (quasi-equilibrium) cell-wall dislocation microstructures practically indistinguishable from equilibria, which however cannot be solutions to the equilibrium equations of the model, a feature shared with certain types of GL equations. However, we show that the class of quasi-equilibria comprising a spatially non-periodic microstructure consisting of fronts is larger than that of the GL equations associated with the energy of the model. In addition, under applied strain-controlled loading, a single dislocation wall is shown to be capable of moving as a localized entity, as expected in a physical model of dislocation dynamics, in contrast to the associated GL equations. The collective evolution of the quasi-equilibrium cell-wall microstructure exhibits a yielding-type behavior as bulk plasticity ensues, and the effective stress–strain response under loading is found to be rate-dependent. The numerical scheme employed is non-conventional, since wave-type behavior has to be accounted for, and interesting features of two different schemes are discussed. Interestingly, a stable scheme conjectured by us to produce a non-physical result in the present context nevertheless suggests a modified continuum model that appears to incorporate apparent intermittency.

Atomistic insights into dislocation-based mechanisms of void growth and coalescence

One of the low-temperature failure mechanisms in ductile metallic alloys is the growth of voids and their coalescence. In the present work we attempt to obtain atomistic insights into the mechanisms underpinning cavitation in a representative metal, namely Aluminum. Often the pre-existing voids in metallic alloys such as Al have complex shapes (e.g. corrosion pits) and the defromation/damage mechanisms exhibit a rich size-dependent behavior across various material length scales. We focus on these two issues in this paper through large-scale calculations on specimens of sizes ranging from 18 thousand to 1.08 million atoms. In addition to the elucidation of the dislocation propagation based void growth mechanism we highlight the observed length scale effect reflected in the effective stress–strain response, stress triaxiality and void fraction evolution. Furthermore, as expected, the conventionally used Gurson's model fails to capture the observed size-effects calling for a mechanistic modification that incorporates the mechanisms observed in our (and other researchers') simulation. Finally, in our multi-void simulations, we find that, the splitting of a big void into a distribution of small ones increases the load-carrying capacity of specimens. However, no obvious dependence of the void fraction evolution on void coalescence is observed.

Void growth and coalescence in anisotropic plastic solids

Large strain finite element calculations of unit cells subjected to triaxial axisymmetric loadings are presented for plastically orthotropic materials containing a periodic distribution of aligned spheroidal voids. The spatial distribution of voids and the plastic flow properties of the matrix are assumed to respect transverse isotropy about the axis of symmetry of the imposed loading so that a two-dimensional axisymmetric analysis is adequate. The parameters varied pertain to load triaxiality, matrix anisotropy, initial porosity and initial void shape so as to include the limiting case of penny-shaped cracks. Attention is focussed on comparing the individual and coupled effects of void shape and material anisotropy on the effective stress–strain response and on the evolution of microstructural variables. In addition, the effect of matrix anisotropy on the mode of plastic flow localization is discussed. From the results, two distinct regimes of behavior are identified: (i) at high triaxialities, the effect of material anisotropy is found to be persistent, unlike that of initial void shape and (ii) at moderate triaxialities the influence of void shape is found to depend strongly on matrix anisotropy. The findings are interpreted in light of recent, microscopically informed models of porous metal plasticity. Conversely, observations are made in relation to the relevance of these results in the development and calibration of a broader set of continuum damage mechanics models.

Characterising material and process variation effects on springback robustness for a semi-cylindrical sheet metal forming process

Variation in the incoming sheet material and fluctuations in the press setup is unavoidable in many stamping plants. The effect of these variations can have a large influence on the quality of the final stamping, in particular, unpredictable springback of the sheet when the tooling is removed. While stochastic simulation techniques have been developed to simulate this problem, there has been little research that connects the influence of the noise sources to springback. This paper characterises the effect of material and process variation on the robustness of springback for a semi-cylindrical channel forming operation, which shares a similar cross-section profile as many automotive structural components. The study was conducted using the specialised sheet metal forming package AutoForm™ Sigma, for which a series of stochastic simulations were performed with each of the noise sources incrementally introduced. The effective stress and effective strain scatter in a critical location of the part was examined and a response window, which indicates the respective process robustness, was defined. The incremental introduction of the noise sources allows the change in size of the stress–strain response window to be tracked. The results showed that changes to process variation parameters, such as BHP and friction coefficient, directly affect the strain component of the stress–strain response window by altering the magnitude of external work applied to forming system. Material variation, on the other hand, directly affected the stress component of the response window. A relationship between the effective stress–strain response window and the variation in springback was also established.

Elastic-damage Modeling for Particulate Composites Considering Cumulative Damage

A micromechanics-based elastic-damage model considering cumulative damage is proposed to predict the effective stress—strain response and damage evolution in particulate composites. A cumulative damage model combining the Weibull probability distribution function (Weibull, 1951) and the step-stress concept (ReliaSoft, 2003) is incorporated into micromechanical formulations (Ju and Chen, 1994a,b) for more realistic prediction of evolutionary interfacial debonding. A multistep damage process is introduced to model accumulated damage induced by the interfacial debonding. A parametric study of Weibull damage parameters is carried out to examine the sensitivity of the parameters to the proposed model. Finally, present predictions are compared with other numerical predictions and experimental data to assess the predictive capability of the proposed model.

A novel grain cluster-based homogenization scheme

An efficient homogenization scheme, termed the relaxed grain cluster (RGC), for elasto-plastic deformations of polycrystals is presented. The scheme is based on a generalization of the grain cluster concept. A volume element consisting of eight (= 2 × 2 × 2) hexahedral grains is considered. The kinematics of the RGC scheme is formulated within a finite deformation framework, where the relaxation of the local deformation gradient of each individual grain is connected to the overall deformation gradient by the, so-called, interface relaxation vectors. The set of relaxation vectors is determined by the minimization of the constitutive energy (or work) density of the overall cluster. An additional energy density associated with the mismatch at the grain boundaries due to relaxations is incorporated as a penalty term into the energy minimization formulation. Effectively, this penalty term represents the kinematical condition of deformation compatibility at the grain boundaries.Simulations have been performed for a dual-phase grain cluster loaded in uniaxial tension. The results of the simulations are presented and discussed in terms of the effective stress–strain response and the overall deformation anisotropy as functions of the penalty energy parameters. In addition, the prediction of the RGC scheme is compared with predictions using other averaging schemes, as well as to the result of direct finite element (FE) simulation. The comparison indicates that the present RGC scheme is able to approximate FE simulation results of relatively fine discretization at about three orders of magnitude lower computational cost.

Fluid coupled-DEM modelling of undrained behavior of granular media

A 2D coupled continuum-discrete model is utilized to simulate the undrained behavior of saturated granular media subjected to monotonic biaxial shearing. The fluid phase is considered as continuum, and its flow is modeled using equation of continuity and according to Darcy Law. Discrete element method is employed to model the solid phase discrete particles. Fluid–particle interactions are taken into account based on semi-empirical relations. Loose and dense samples are prepared and biaxially sheared in drained and undrained conditions. The undrained condition is simulated using the constant volume method as well as the present coupled method. The macroscopic behavior, including stress–strain response, stress path and volumetric change, of the assemblies is studies. The results of the undrained responses comply fully with each other in the constant volume and coupling methods; they also show good compatibility with their drained behavior. Bulk modulus of the pore fluid has a substantial effect on pore pressure generation, with negligible effect on the effective stress–strain response of the assemblies.

A pattern-based method for bounding the effective response of a nonlinear composite

This paper deals with the prediction of the effective properties of nonlinear composites. Rather than bounding the effective energy, this work aims at bounding directly the effective stress–strain response, by extending a method originally introduced by Milton and Serkov (J. Mech. Phys. Solids 48 (2000) 1295) and recently refined by Talbot and Willis (Proc. Roy. Soc. 460 (2004) 2705). In this paper, bounding the effective response is achieved by introducing a linear comparison composite with the same micro-geometry as the given nonlinear composite, as Ponte Castañeda (J. Mech. Phys. Solids 39 (1991) 45) did for the energy. It is found that any lower bound for the energy of the linear comparison composite generates a corresponding bound for the stress–strain response of the nonlinear composite. A selection of examples is presented to illustrate the method and compare the bounds obtained with existing results.

The Effect of Perfusion on Soft Tissue Mechanical Properties: A Computational Model

Some simple finite element models were constructed to investigate the magnitude and character of changes in mechanical properties of very soft tissues due to the loss of perfusion. Changes in the apparent stress-strain curve were used to characterise the effect of simulated blood perfusion pressure on the engineering stress-strain curve. The results indicated that the blood to tissue volume ratio and the perfusion pressure have the strongest effect on the effective stress-strain response of a representative tissue cell. Tissue viscoelasticity increased the sensitivity of the system to perfusion pressure changes.

Simulated Micromechanical Models Using Artificial Neural Networks

A new method, termed simulated micromechanical models using artificial neural networks (MMANN), is proposed to generate micromechanical material models for nonlinear and damage behavior of heterogeneous materials. Artificial neural networks (ANN) are trained with results from detailed nonlinear finite-element (FE) analyses of a repeating unit cell (UC), with and without induced damage, e.g., voids or cracks between the fiber and matrix phases. The FE simulations are used to form the effective stress-strain response for a unit cell with different geometry and damage parameters. The FE analyses are performed for a relatively small number of applied strain paths and damage parameters. It is shown that MMANN material models of this type exhibit many interesting features, including different tension and compression response, that are usually difficult to model by conventional micromechanical approaches. MMANN material models can be easily applied in a displacement-based FE for nonlinear analysis of composite structures. Application examples are shown where micromodels are generated to represent the homogenized nonlinear multiaxial response of a unidirectional composite with and without damage. In the case of analysis with damage growth, thermodynamics with irreversible processes (TIP) is used to derive the response of an equivalent homogenized damage medium with evolution equations for damage. The proposed damage formulation incorporates the generalizations generated by the MMANN method for stresses and other possible responses from analysis results of unit cells with fixed levels of damage.

Mean field estimates of the response of fiber composites with nonlinear interface

This paper presents a theory of the effective response of composites containing general, nonlinearly separating inclusion–matrix interfaces. The direct method of composite materials theory is utilized to pass from local nonlinear behavior of a solitary inclusion problem to nonlinear aggregate response. Interaction effects at finite volume concentration are captured in the representative solitary problem by employing the Mori–Tanaka mean field estimate. The resulting model falls within the conceptual framework of continuum damage mechanics in that nonlinear effective response depends on internal variables that are governed by local evolution equations. The damage variables turn out to be the expansion coefficients arising in an eigenfunction representation of the displacement jump at a representative inclusion–matrix interface. Interfaces are generally modeled according to a nonlinear force-separation law that allows for both normal and shear decohesion (X-.P. Xu, A. Needleman, Modell. Simul. Mater. Sci. Eng. 1 (1993) 111). Detailed calculations of effective stress–strain response are carried out for the case of transverse shear and plane dilatation of unidirectional fiber composites at various values of concentration and interface constitutive constants. The effects of the various parameters on bifurcation of equilibrium separation in the solitary inclusion problem and on overall composite stability are demonstrated as well.