Inelastic Material Response Research Articles

IDENTIFICATION OF OPTIMAL REDUCED ORDER HOMOGENIZATION MODELS FOR FAILURE OF HETEROGENEOUS MATERIALS

This manuscript presents a new methodology for the identification of optimal reduced order models for the inelastic and failure response of heterogeneous materials. The proposed methodology employs the eigendeformation-based reduced order homogenization approach. The identification of the optimal reduced order model is posed as an integer optimization problem and the genetic algorithm method is used to evaluate the optimization problem. A second optimization problem is posed to ensure that the errors associated with the optimal reduced order model are minimized through scaling of the failure parameters. The performance and capabilities of the optimal reduced order models identified based on the proposed approach are demonstrated by comparing model predictions with the computational homogenization method with full resolution of the material microstructure. Numerical simulations conducted using unidirectional reinforced matrix microstructures reveal that the reduced order models accurately describe the response characteristics of the composite material for a wide range of loading regimes.

Biaxial ratchetting with novel variations of kinematic hardening

Kinematic hardening and the associated concept of back-stress and its evolution are fundamental constitutive ingredients of classical plasticity theory used to simulate the inelastic material response under stress reversals. Cyclic plasticity addresses such response under a sequence of repeated stress reversals, which results in plastic strain accumulation, called ratchetting. Biaxial ratchetting occurs whenever the material is loaded in two directions although typically the cyclic loading is only in one direction. The realistic description of the material response during cyclic loading depends strongly on the kind of kinematic hardening used. This paper investigates the performance of some existing and novel kinematic hardening rules in the prediction of ratchetting. The multiplicative AF model by Dafalias et al. (2008a,b), which was originally applied to the simulation of uniaxial ratchetting, will be used here to simulate also biaxial ratchetting and will be compared with a model using the concept of a hardening stress threshold. The suggestion of Delobelle et al. (1995) to combine the Armstrong/Frederick and Burlet and Cailletaud (1986) kinematic hardening rules is incorporated in the aforementioned model and used to obtain improved simulations of biaxial ratchetting. After showing a deficiency of the foregoing suggestion which results in the possibility for the back-stress to cross it’s bounding surface and induce a negative plastic modulus, a variation is proposed void of the foregoing deficiency, which is successfully tested in the simulation of multiple biaxial ratchetting experimental results on carbon steel 1026.

A second gradient theoretical framework for hierarchical multiscale modeling of materials

A theoretical framework for the hierarchical multiscale modeling of inelastic response of heterogeneous materials is presented. Within this multiscale framework, the second gradient is used as a nonlocal kinematic link between the response of a material point at the coarse scale and the response of a neighborhood of material points at the fine scale. Kinematic consistency between these scales results in specific requirements for constraints on the fluctuation field. The wryness tensor serves as a second-order measure of strain. The nature of the second-order strain induces anti-symmetry in the first-order stress at the coarse scale. The multiscale internal state variable (ISV) constitutive theory is couched in the coarse scale intermediate configuration, from which an important new concept in scale transitions emerges, namely scale invariance of dissipation. Finally, a strategy for developing meaningful kinematic ISVs and the proper free energy functions and evolution kinetics is presented.

On finite strain micromorphic elastoplasticity

In the micromorphic continuum theory of Eringen, it was proposed that microstructure of materials could be represented in a continuum framework using a micro-deformation tensor governing micro-element deformation, in addition to the deformation gradient governing macro-element deformation. The paper formulates finite strain micromorphic elastoplasticity based on micromorphic continuum mechanics in the sense of Eringen. Multiplicative decomposition into elastic and plastic parts of the deformation gradient and micro-deformation are assumed, and the Clausius–Duhem inequality is formulated in the intermediate configuration B ¯ to analyze what stresses, elastic deformation measures, and plastic deformation rates are used/defined in the constitutive equations. The resulting forms of plastic and internal state variable evolution equations can be viewed as phenomenological at their various scales (i.e., micro-continuum and macro-continuum). The phenomenology of inelastic mechanical material response at the various scales can be different, but for demonstration purposes, J 2 flow plasticity is assumed for each of three levels of plastic evolution equations identified, with different stress, internal state variables, and material parameters. All evolution equations and a semi-implicit time integration scheme are formulated in the intermediate configuration for future coupled Lagrangian finite element implementation. A simpler two-dimensional model for anti-plane shear kinematics is formulated to demonstrate more clearly how such model equations simplify for future finite element implementation.

Cross-section reconstruction during uniaxial loading

The inelastic response of materials to applied uniaxial loading is typically measured using tensile or compressive specimens of an initially circular cross-section. Under deformation, this cross-section may become elliptical due to anisotropic material behaviour. An optical technique for measuring the elliptical deformation of anisotropic, homogeneous cylindrical specimens undergoing uniaxial deformation is presented. It enables the quantification of anisotropic deformation in situ and provides data for material characterization. Three or more silhouette views of a specimen are obtained using multiple cameras or mirrored views. The positions of the edges are computed using a sub-pixel edge detection method, and 3D tangent rays from the camera through these positions are calculated. These bounding tangents are used as the basis for an elliptical fit by least squares at cross-sections along the length of the specimen. Stochastic error estimates are performed by simulation of the experiment. Error estimates, for the experimental set-up used, are also calculated by reconstructing elliptical prisms of precisely measured dimensions. Example reconstructions from specimens of rolled titanium deformed plastically in tension at quasi-static (7 × 10−4 s−1) and high strain rates (3 × 103 s−1) are presented.

On multiscale FE analyses of heterogeneous structures: from homogenization to multigrid solvers

AbstractHeterogeneous structures like composites often need a fine‐scale resolution of micro‐effects which influence the macroscopic overall response. This is of particular relevance in the fully non‐linear range of large strains and inelastic material response of the constituents. Suitable solution methods introduce a multifield scenario of hierarchically superimposed states on different length scales. For big differences of micro‐ and macro‐scales, the argument of scale separation induces the application of homogenization methods. Such types of physical multiscale approaches can be treated by nested multilevel finite element analyses that discretize both the fine‐scale micro‐structure as well as the macroscopic boundary‐value problem. In contrast, small‐scale differences require full resolution of the heterogeneous structure. Effective solution methods for the resulting large‐scale problems with strongly oscillating properties are suitably designed geometric multigrid techniques, which may be considered as numerical multiscale approaches. In both scenarios, a key ingredient is the suitable formulation of scale bridging algorithms that govern the transfer between different scales. The paper outlines new mesh‐bridging techniques in a deformation‐driven context for fully non‐linear response, which exploit in a non‐trivial manner weak constraints on the average deformation in typical finite element patches. The framework is based on an incremental variational structure of finite inelasticity. The proposed new formulations provide variational‐based homogenization algorithms for physical multiscale scenarios and problem‐dependent optimal finite element grid transfers for numerical multiscale scenarios of heterogeneous materials. Copyright © 2007 John Wiley & Sons, Ltd.

Directional response of a reconstituted fine-grained soil—Part I: experimental investigation

This paper discusses the results of a large experimental program designed to investigate in a systematic manner the main features of the incremental response of fine-grained soils. The results are obtained from triaxial stress probing experiments carried out on a French silty clay (Beaucaire Marl). All the tests have been performed on reconstituted specimens, normally consolidated to an initial state which is either isotropic or anisotropic. In the interpretation of the experimental results, extensive use is made of the concept of strain response envelope. The response envelopes obtained for different stress increment magnitudes are remarkably consistent with each other and indicate an inelastic and irreversible material response, i.e. a strong dependence on the stress increment direction, also at relatively small strain levels. A companion paper (Int. J. Numer. Anal. Meth. Geomech., this issue, 2006) assesses the performance of some advanced constitutive models in reproducing the behaviour of reconstituted Beaucaire Marl as observed in this experimental program. Copyright © 2006 John Wiley & Sons, Ltd.

Analysis of material instabilities in inelastic solids by incremental energy minimization and relaxation methods: evolving deformation microstructures in finite plasticity

We propose an approach to the definition and analysis of material instabilities in rate-independent standard dissipative solids at finite strains based on finite-step-sized incremental energy minimization principles. The point of departure is a recently developed constitutive minimization principle for standard dissipative materials that optimizes a generalized incremental work function with respect to the internal variables. In an incremental setting at finite time steps this variational problem defines a quasi-hyperelastic stress potential. The existence of this potential allows to be recast a typical incremental boundary-value problem of quasi-static inelasticity into a principle of minimum incremental energy for standard dissipative solids. Mathematical existence theorems for sufficiently regular minimizers then induce a definition of the material stability of the inelastic material response in terms of the sequentially weakly lower semicontinuity of the incremental variational functional. As a consequence, the incremental material stability of standard dissipative solids may be defined in terms of the quasi-convexity or the rank-one convexity of the incremental stress potential. This global definition includes the classical local Hadamard condition but is more general. Furthermore, the variational setting opens up the possibility to analyze the post-critical development of deformation microstructures in non-stable inelastic materials based on energy relaxation methods. We outline minimization principles of quasi- and rank-one convexifications of incremental non-convex stress potentials for standard dissipative solids. The general concepts are applied to the analysis of evolving deformation microstructures in single-slip plasticity. For this canonical model problem, we outline details of the constitutive variational formulation and develop numerical and semi-analytical solution methods for a first-level rank-one convexification. A set of representative numerical investigations analyze the development of deformation microstructures in the form of rank-one laminates in single slip plasticity for homogeneous macro-deformation modes as well as inhomogeneous macroscopic boundary-value problems. The well-posedness of the relaxed variational formulation is indicated by an independence of typical finite element solutions on the mesh-size.

A material force method for inelastic fracture mechanics

A material force method is proposed for evaluating the energy release rate and work rate of dissipation for fracture in inelastic materials. The inelastic material response is characterized by an internal variable model with an explicitly defined free energy density and dissipation potential. Expressions for the global material and dissipation forces are obtained from a global balance of energy–momentum that incorporates dissipation from inelastic material behavior. It is shown that in the special case of steady-state growth, the global dissipation force equals the work rate of dissipation, and the global material force and J-integral methods are equivalent. For implementation in finite element computations, an equivalent domain expression of the global material force is developed from the weak form of the energy–momentum balance. The method is applied to model problems of cohesive fracture in a remote K-field for viscoelasticity and elastoplasticity. The viscoelastic problem is used to compare various element discretizations in combination with different schemes for computing strain gradients. For the elastoplastic problem, the effects of cohesive and bulk properties on the plastic dissipation are examined using calculations of the global dissipation force.

On the inelastic behavior of crystalline silicon at elevated temperatures

The broad focus of this paper is on developing continuum models of inelastic response of crystalline materials that include the influence of microstructure, material symmetry, and dissipation of energy accompanying the changes in the microstructure. A framework has been developed, built on the idea of natural configurations of a material, that has provisions for explicit treatment of material microstructure (slip planes, dislocations, interfaces, etc.). It is demonstrated that the developed framework is applicable to diverse phenomena such as plasticity due to slip and twinning, and martensitic phase transformations. The developed framework is used to study the inelastic behavior of crystalline silicon at elevated temperatures.

A coupled anisotropic damage model for the inelastic response of composite materials

A coupled incremental damage and plasticity theory for rate-independent and rate-dependent composite materials is introduced here. This allows damage to be path-dependent either on the stress history or thermodynamic force conjugate to damage. This is achieved through the use of an incremental damage tensor. Damage and inelastic deformations are incorporated in the proposed model that is used for the analysis of fiber-reinforced metal–matrix composite materials. The damage is described kinematically in both the elastic and inelastic domains using the fourth-order damage effect tensor which is a function of the second-order damage tensor. A physical interpretation of the second-order damage tensor is given in this work which relates to the microcrack porosity within the unit cell. The inelastic deformation behavior with damage is viewed here within the framework of thermodynamics with internal state variables. Computational aspects of both the rate-independent and rate-dependent models are discussed in this work. The Newton–Rapson iterative scheme is used for the overall laminate system. The constitute equations of both the rate-independent and the rate-dependent plasticity coupled with damage models are additively decomposed into the elastic, inelastic and damage deformations by using the three-step split operator algorithm [J.W. Ju, Internat. J. Solids Struc. 25 (1989) 803–833]. The main framework return maping algorithm [M. Ortiz, C. Simo, Internat. J. Numer. Meth. Eng. 23 (1986) 353–366] is used for the correction of the elasto-plastic and viscoplastic states. However, for the case of the damage model these algorithms are redefined according to the governed damage constitutive relations. In order to test the validity of the model, a series of laminated systems (0 (8s)), (90 (8s)), (0/90) (4s), (−45/45) (2s) are investigated at both room and elevated temperatures of 538°C and 649°C. The results obtained from the special purpose developed computer program, DVP-CALSET (Damage and Viscoplastic Coupled Analysis of Laminate Systems at Elevated Temperatures), are then compared with the available experimental results and other existing theoretical material models obtained from the work of B.S. Majumdar, G.M. Newaz [CR-189095, NASA, 1992] and G.Z. Voyiadjis, A.R. Venson [Internat. J. Damage Mech. 4 (4) (1995) 338–361].

Influence of pore geometry on the effective response of porous media

The generalized method of cells (GMC) is used to study the influence of pore geometry on the effective elastic properties and inelastic response of porous materials. Periodic microstructures with four distinct pore geometries are studied and the results for effective elastic properties are compared with several other available models and experimental results. Predictions for the inelastic response of porous alumina are presented for tensile loading, as a function of pore geometry and pore volume fraction, with the inelastic behavior of the bulk material modeled using a unified visco-plasticity theory. All results are presented for discrete pore shape and discrete porosity. It is shown that pore geometry can have a significant influence on both elastic and inelastic response, that pore geometry can be associated with parameters from other models, and that the generalized method of cells is an efficient, flexible and reliable method of analysis for such problems.

Mechanics of the inelastic behavior of materials. Part II: inelastic response

Abstract This is the second of a two part paper dealing with the inelastic response of materials. Part I (see Rajagopal and Srinivasa, 1998 International Journal of Plasticity 14 , 945–967) dealt with the structure of the constitutive equations for the elastic response of a material with multiple natural configurations. We now focus attention on the evolution of the natural configurations. We introduce two functions-the Helmholtz potential and the rate of dissipation function—representing the rate of conversion of mechanical work into heat. Motivated by, and generalizing the work of Ziegler (1963) in Progress in Solid Mechanics , Vol. 4, North-Holland, Amsterdam/New York; (1983) An Introduction to Thermodynamics, North-Holland, Amsterdam/New York) we then assume that the evolution of the natural configurations occurs in such a way that the rate of dissipation is maximized. This maximization is subject to the constraint that the rate of dissipation is equal to the difference between the rate of mechanical working and the rate of increase of the Helmholtz potential per unit volume. This then allows us to derive the constitutive equations for the stress response and the evolution of the natural configurations from these two scalar functions. Of course, the maximum rate of dissipation criterion that is stated here is only an assumption that holds for a certain class of materials under consideration. Our quest is to see whether such an assumption gives reasonable results. In the process, we hope to gain insight into the nature of such materials. We demonstrate that the resulting constitutive equations allow for response with and without yielding behavior and obtain a generalization of the normality and convexity conditions. We also show that, in the limit of quasistatic deformations, if one considers materials that possess yielding behavior, then the constitutive equations reduce to those corresponding to the strain space formulation of the rate independent theory of plasticity (see e.g. Naghdi (1990) Journal of Applied Mathematics and Physics A345 , 425–458.). Moreover, in this limit, the maximum rate of dissipation criterion, as stated here, is equivalent to the work inequality of Naghdi and Trapp (1975) Quartely Journal of Mechanics and Applied Mathematics 28 , 25–46). The main results together with an illustrative example are presented.

A constitutive frame of elastoplasticity at large strains based on the notion of a plastic metric

The article presents a constitutive framework of large-strain elastoplasticity in both the Lagrangian and the Eulerian geometric setting which takes into account anisotropic material response. In summary, the key ingredients of this framework are: (i) the introduction of a plastic metric which is assumed to describe locally the history-dependent inelastic material response in the sense of an internal variable formulation. (ii) The definition of a convex elastic domain in the space of the local stress-like variable conjugate to the plastic metric, denoted as the plastic force. (iii) An equivalent Lagrangian and Eulerian representation of all constitutive functions as isotropic tensor functions in terms of an extended set of arguments, denoted as anisotropy variables. (iv) The set-up of normality rules for the evolution of the plastic metric and the anisotropy variables, yielding a canonical symmetric form of the elastoplastic tangent moduli. (v) A geometrically exact decomposition of the set of constitutive equations into possibly decoupled volumetric and isochoric contributions. Applications of the constitutive framework are demonstrated by means of several conceptual model problems which cover isotropic, initial anisotropic and induced anisotropic elastoplastic response.

A formulation of finite elastoplasticity based on dual co- and contra-variant eigenvector triads normalized with respect to a plastic metric

The article presents a new formulation of isotropic elastoplasticity at large strains in both the Lagrangian and the Eulerian geometric setting and addresses aspects of its numerical implementation. The key ingredients on the theoretical side are the introduction of a plastic metric for the description of the local history-dependent inelastic material response, the definition of a convex elastic domain in the space of the local stress-like variable conjugate to the plastic metric, denoted as the plastic force, and a fully equivalent Lagrangian and Eulerian representation of all constitutive functions in spectral form for general non-Cartesian coordinate charts in terms of dual co- and contra-variant eigenvector triads which are normalized with respect to the plastic metric. On the numerical side, we propose a stress update algorithm for general non-associative isotropic elasto(visco)plastic response with an arbitrary number of scalar internal variables. The algorithm is based on an exponential map integrator and is recast into a general return mapping scheme, methodically organized with tensorial pre- and postprocessing and a constitutive box in the eigenvalue space. Furthermore, we propose a new perturbation stabilization technique which dramatically enhances the convergence of the general return algorithm. The theoretical and numerical developments are applied to a constitutive model problem with large elastic and large plastic strains: the von Mises-/Tresca-type associative plastic flow in Ogden-type large-strain elastic materials.

Ponding on circular membranes

Nonlinear circular membrane responses under partial and full ponding loads have been solved by the fourth-order Runge-Kutta numerical integration and by finite element simulation with good accuracy. Under partial ponding loads, a discontinuity exists in the curvature of the deformed membrane at the fluid boundary while an inflection point forms inside the ponding. An iterative finite element algorithm has also been developed for solving membrane ponding problems where inelastic material response becomes significant. The finite element simulation procedure was proven to be accurate when validated against test data from membrane forming experiments.

The Effect of Stress-State on the Large Strain Inelastic Deformation Behavior of 304L Stainless Steel

In metals, large strain inelastic deformation processes such as the formation of a preferred crystallographic orientation (crystallographic texture) and strain hardening processes such as the formation and evolution of dislocation substructures depend on stress-state. Much of the current large strain research has focused on texture. Crystallographic texture development and strainhardening processes each contribute to the overall material behavior, and a complete description of large strain inelastic material response should reflect both. An investigation of the large strain behavior of 304L stainless steel (SS 304L) subjected to compression, torsion, and sequences of compression followed by torsion and torsion followed by tension is reported. This paper focuses on the stress-state dependence of strain-hardening processes as well as the relative effect such processes have on the overall material behavior. To characterize these processes, transmission electron microscopy (TEM) as well as magnetization investigations were conducted at different strain levels and under different deformation modes. The γ → α′ martensitic transformation which occurs in this material was found to be related to both the strain level and stress state. Dislocation substructures in the form of Taylor lattices, dense dislocation walls, and microbands were also present. The ramifications of using a thin-walled tubular torsion specimen were also explored.

Viscoplastic flow of planar granular materials

A rate-dependent constitutive model for the viscoplastic flow of granular materials is presented. The kinematics is based on the rate-dependent sliding of layers of granules, which results in overall dilatant shearing of the granular medium. Stress-induced anisotropy in the elastic and inelastic instantaneous material response is incorporated. Typical results for monotonic and cyclic loading are presented for illustration, and are shown to accurately correspond to actual observations. Results from this model for various rate-sensitivity parameters at strain rates are compared with those of the rate-independent model.

Effects of grain size on cyclic strain localization in polycrystalline nickel: Morrison, D.J., Chopra, V. and Jones, J.W. Scr. Metall. Mater. June 1991 25, (6), 1299–1304

Kinematic hardening and the associated concept of back-stress and its evolution are fundamental constitutive ingredients of classical plasticity theory used to simulate the inelastic material response under stress reversals. Cyclic plasticity addresses such response under a sequence of repeated stress reversals, which results in plastic strain accumulation, called ratchetting. Biaxial ratchetting occurs whenever the material is loaded in two directions although typically the cyclic loading is only in one direction. The realistic description of the material response during cyclic loading depends strongly on the kind of kinematic hardening used.This paper investigates the performance of some existing and novel kinematic hardening rules in the prediction of ratchetting. The multiplicative AF model by Dafalias et al., 2008a, Dafalias et al., 2008b, which was originally applied to the simulation of uniaxial ratchetting, will be used here to simulate also biaxial ratchetting and will be compared with a model using the concept of a hardening stress threshold. The suggestion of Delobelle et al. (1995) to combine the Armstrong/Frederick and Burlet and Cailletaud (1986) kinematic hardening rules is incorporated in the aforementioned model and used to obtain improved simulations of biaxial ratchetting. After showing a deficiency of the foregoing suggestion which results in the possibility for the back-stress to cross it’s bounding surface and induce a negative plastic modulus, a variation is proposed void of the foregoing deficiency, which is successfully tested in the simulation of multiple biaxial ratchetting experimental results on carbon steel 1026.

Ductile alloy constitutive response by correlation of iterative finite element simulation with laboratory video images

The accurate description of material response consists of a material constitutive formulation and material parameter values. The reduction of material test specimen load versus displacement data to constitutive parameters is often precluded by inelastic material response and deformation inhomogeneity within the specimen. For ductile engineering alloys, these effects are influenced by specimen geometry and must be uncoupled from specimen geometry to characterize the large strain material response. The accuracy of material parameters at such strains should be demonstrated for subsequent applications to design and analysis. Iterative solution for material constitutive parameters is discussed in the investigation. The use of video processing of laboratory tensile test specimens is combined with successive computational simulations of the specimen responses. The solution for HSLA-80 steel constitutive parameters, in the context of incremental plasticity theory, is presented. The material response is treated as the unknown in the computational simulations. It is iteratively modified to achieve correlation with the laboratory experiments. Two different specimen length-to-diameter aspect ratios are utilized to ensure the geometry independence of the material solution and to facilitate efficient solution. The constitutive iteration sequence illustrates the sensitivity of specimen response to material nonlinearity.