Finite Deformation Framework Research Articles

Numerical modelling of transformation-induced damage and plasticity in metals

In multiphase carbon steels, irreversible martensitic transformations from a parent austenitic phase are often accompanied by plastic deformations in the surrounding phases and damage growth in the martensitic product phase. In the present communication the interactions between these complex processes are studied through three-dimensional numerical analyses, where the response of the austenitic phase is simulated using a phase-changing damage model and the response of the surrounding matrix is mimicked with a plasticity model. A numerical update algorithm is provided for the phase-changing damage model, which is based on a fully implicit backward Euler scheme formulated within the framework of finite deformations. The consistent tangent operator for the model is computed using a numerical differentiation technique. Special attention is given to the robust and accurate treatment of the various constraints related to the volume fractions and damaged volume fractions of the transformation parent and product phases. The numerical analyses of a multiphase carbon steel microstructure illustrate the effect of the transformation, plasticity and damage processes on the overall stress–strain response and on the transformation and damage evolutions of the sample for various austenite crystal orientations. The computed responses are in qualitative agreement with experimental results reported in the literature. In terms of the numerical model, a variation study of the mesh density shows that the numerical results tend to converge upon mesh refinement.

Size-dependent effective properties of a heterogeneous material with interface energy effect: from finite deformation theory to infinitesimal strain analysis

In this paper, the change of the elastic fields induced by the interface energies and the interface stresses from the reference configuration to the current configuration is considered. It is emphasized that the governing equations taking into account the interface energy effect should be established within the framework of finite deformation in the first place, and then the approximations of governing equations for a finitely deformed multi-phase elastic medium by an infinitesimal strain analysis can be formulated. Hence it can be seen that the asymmetric interface stress has to be used in the Young-Laplace equation. According to the above mentioned formalism, analytical expressions of the size-dependent effective moduli of a particle-filled composite material with interface energy effect are derived. It is shown that, different from the results obtained by previous researchers, the liquid-like surface/interface tension, as a residual stress-type term, also influences the effective property of the composite.

3D computational modeling of powder compaction processes using a three-invariant hardening cap plasticity model

In this paper, a three-invariant cap plasticity is developed for description of powder behavior under cold compaction process. The constitutive elasto-plastic matrix and its components are derived as the nonlinear functions of powder relative density. Different aspects of 2D and 3D cap plasticity models are illustrated and the procedure for determination of powder parameters is described. It is shown how the proposed model could generate the elliptical yield surface, double-surface cap plasticity and the irregular hexagonal pyramid of the Mohr–Coulomb and cone-cap yield surface, as special cases. The single-cap plasticity is performed within the framework of large finite element deformation, in order to predict the nonuniform relative density distribution during powder die pressing. Finally, the applicability of the proposed model for description of powder behavior is demonstrated in numerical simulation of triaxial and confining pressure tests. The numerical schemes are examined for efficiency in the modeling of an automotive component, a conical shaped-charge liner and a connecting-rod.

A theory of hyperelasticity of multi-phase media with surface/interface energy effect

In addition to the classical governing equations in continuum mechanics, two kinds of governing equations are necessary in the solution of boundary-value problems for the stress fields in multi-phase hyperelastic media with the surface/interface energy effect. The first is the interface constitutive relation, and the second are the discontinuity conditions of the traction across the interface, namely, the Young-Laplace equations. In this paper, the interface consitutive relations are presented in terms of the interface energy in both Lagrangian and Eulerian descriptions within the framework of finite deformation, and the expressions of the interface stress for an isotropic interface are given as a special case. Then, by introducing a fictitious stress-free configuration, a new energy functional for multi-phase hyperelastic media with interface energy effect is proposed. The functional takes into account the interface energy and the interface stress-induced ``residual'' elastic field, which reflects the intrinsic physical properties of the material. All field equations, including the generalized Young-Laplace equation, can be derived from the stationary condition of this functional. The present theory is illustrated by simple examples. The results in this paper provide a theoretical framework for studying the elastostatic problems of multi-phase hyperelastic bodies that involve surface/interface energy effects at finite deformation.

A model for the consolidation of hexagonal array matrix coated fibre composites

Physically based constitutive equations for the consolidation of matrix coated fibres (MCFs) arranged in hexagonal arrays are presented in this paper. The constitutive equations have firstly been developed using a variational method for a unit cell representing hexagonal array packing of the coated fibres under symmetric in-plane compressive load. The resulting constitutive equations have subsequently been generalized to multiaxial stress states and implemented into finite element software within a finite deformation framework to enable simulations of practical manufacturing processes. The model has been validated by comparing predicted results with those obtained from independently developed explicit micromechanical finite element models. The ability of the model to predict the observed behaviour has been tested by comparing the results of simulations with those of experiments. The model predictions compare well with the measured results. In addition the results of simulations suggest that the densification curves for square and hexagonal array packing effectively provide the lower and upper bounds, respectively, for practical consolidation behaviour of MCFs under uniaxial constrained compression.

Computational modelling of plasticity induced by martensitic phase transformations

A time integration scheme is presented for the martensitic phase transformation model developed in recent theoretical work of Turteltaub and Suiker (A multi-scale thermomechanical model for cubic to tetragonal martensitic phase transformations 2005; Transformation-induced plasticity in ferrous alloys 2005). The phase transformation model can be used for analysing transformation-induced plasticity (TRIP) phenomena in ferrous alloys. The microstructural information for the phase transformation model is provided by the crystallographic theory of martensitic transformations. The transformation characteristics depend on the specific transformation systems activated during a loading process. The time integration scheme is formulated within a framework of finite deformations, where the stress-update algorithm is based on a fully implicit Euler backward discretization. A robust search algorithm is used for detecting the transformation systems activated during loading. The completion of the transformation process is prescribed by a constraint on the total martensitic volume fraction, which is accurately satisfied in the converged state using a sub-stepping algorithm. The computation of the consistent tangent operator is performed through a numerical differentiation method, which avoids the determination of extensive analytical derivatives and allows the model to be easily adapted if necessary. The ability of the algorithm to solve complex transformation-induced plasticity problems is illustrated with the aid of three-dimensional analyses, in which an aggregate of single-crystal grains of retained austenite embedded in a ferrite-based matrix is subjected to uniaxial tension. The response characteristics are in agreement with experimental findings of Jacques et al. (Philos. Mag. 2001; 81(7):1789; Metall. Mater. Trans. A 2001; 32A:2759) and Oliver et al. (Appl. Phys. A 2002; 74:S1143). Copyright © 2005 John Wiley & Sons, Ltd.

Interface effect on the effective bulk modulus of a particle-reinforced composite

Classical micromechanical methods for calculating the effective moduli of a heterogeneous material are generalized to include the interface (surface) effect. By using Hashin's Composite Sphere Assemblage (CSA) model, a new expression of the bulk modulus for a particle-reinforced composite is derived. It is emphasized that the present study is within the finite-deformation framework such that the effective properties are not influenced by the interface stress itself solely, but influenced by the change of the interface stress due to changes of the shape and size of the interface. Hence some inadequacies in previous papers are pointed out.

Fracture Simulation Using an Elasto-Viscoplastic Virtual Internal Bond Model With Finite Elements

A virtual internal bond (VIB) model for isotropic materials has been recently proposed by Gao (Gao, H., 1997, “Elastic Waves in a Hyperelastic Solid Near its Plane Strain Equibiaxial Cohesive Limit,” Philos. Mag. Lett. 76, pp. 307–314) and Gao and Klein (Gao, H., and Klein, P., 1998, “Numerical Simulation of Crack Growth in an Isotropic Solid With Randomized Internal Cohesive Bonds,” J. Mech. Phys. Solids 46(2), pp. 187–218), in order to describe material deformation and fracture under both static and dynamic loading situations. This is made possible by incorporating a cohesive type law of interaction among particles at the atomistic level into a hyperelastic framework at the continuum level. The finite element implementation of the hyperelastic VIB model in an explicit integration framework has also been successfully described in an earlier work by the authors. This paper extends the isotropic hyperelastic VIB model to ductile materials by incorporating rate effects and hardening behavior of the material into a finite deformation framework. The hyperelastic VIB model is formulated in the intermediate configuration of the multiplicative decomposition of the deformation gradient framework. The results pertaining to the deformation, stress-strain behavior, loading rate effects, and the material hardening behavior are studied for a plate with a hole problem. Comparisons are also made with the corresponding hyperelastic VIB model behavior.

Modeling of deformation and rotation bands and of deformation induced grain boundaries in IF steel aggregate during large plane strain compression

A computation using crystal plasticity modeling of an actual IF steel aggregate plane strain compression deformation, underlines the formation of different deformation bands morphologies and grain splitting occurrence, already experimentally observed by different authors. The model based on dislocation densities as internal variables, developed in the framework of finite deformation and implemented in the Finite Element Method, is able to capture the main characteristics of different inhomogeneities and to analyze their formation and further development with strain, from the determination of the active and latent slip systems, and also from the quantification of their dislocation densities and corresponding glide rates evolutions. The respective boundary conditions and material properties effects are discussed.

Non-local coupling of viscoplasticity and anisotropic viscodamage for impact problems using the gradient theory

A general framework for the analysis of heterogeneous media that assesses a strong coupling between viscoplasticity and anisotropic viscodamage evolution is formulated for-impact related problems within the framework of thermodynamic laws and nonlinear continuum mechanics. The proposed formulations include thermo-elasto-viscoplastici- ty with anisotropic thermo-elasto-viscodamage, a dynamic yield criterion of a von Mises type and a dynamic viscodamage criterion, the associated flow rules, non-linear strain hardening, strain-rate hardening, and temperature softening. The constitutive equations for the damaged material are written according to the principle of strain energy equivalence between the virgin material and the damaged material. That is, the damaged material is modeled using the constitutive laws of the effective undamaged material in which the nominal stresses are replaced by the effective stresses. The evolution laws are impeded in a finite deformation framework based on the multiplicative decomposition of the deformation gradient into elastic, viscoplastic, and viscodamage parts. Since the material macroscopic thermomechanical response under high-impact loading is governed by different physical mechanisms on the macroscale level, the proposed three-dimensional kinematical model is introduced with manifold structure accounting for discontinuous fields of dislocation interactions (plastic hardening), and crack and void interactions (damage hardening). The non-local theory of viscoplasticity and viscodamage that incorporates macroscale interstate variables and their higher-order gradients is used here to describe the change in the internal structure and in order to investigate the size effect of statistical inhomogeneity of the evolution-related viscoplasticity and viscodamage hardening variables. The gradients are introduced here in the hardening internal state variables and are considered to be independent of their local counterparts. It also incorporates the thermomechanical coupling effects as well as the internal dissipative effects through the rate-type covariance constitutive structure with a finite set of internal state variables. The model presented in this paper can be considered as a framework, which enables one to derive various non-local and gradient viscoplasticity and viscodamage theories by introducing simplifying assumptions.

On a geometrically nonlinear damage model based on a multiplicative decomposition of the deformation gradient and the propagation of microcracks

We aim to derive a damage model for materials damaged by microcracks. The evolution of the cracks shall be governed by the maximum energy release rate, which was recently shown to be a direct consequence of the variational principle of a body with a crack (Arch. Appl. Mech. 69 (5) (1999) 337). From this, we get the path of the growing crack by introducing a series of thermodynamically equivalent straight cracks. The equivalence of the energy dissipated by microcrack growth and the damage dissipation leads to our damage evolution law. This evolution law will be embedded in a finite deformation framework based on a multiplicative decomposition into elastic and damage parts. As a consequence of this, we can present the anisotropic damaged elasticity tensor with the help of push and pull operations. The connection of this approach to other well known damage theories will be shown and the advantages of a finite element framework will be worked out. Numerical examples show the possibilities of the proposed model.

Methods in entropic thermomechanics

Elements of constitutive model formulation for simple materials within a thermodynamic, finite deformation framework are reviewed. Recent developments in addressing the second law and entropy existence are discussed. Caratheodory-based approaches are emphasized for their simplicity and ease of interpretation. An original treatment of finite thermoviscoelasticity is developed in this context. A broadened Caratheodory-based entropy is proposed, and compared to recent work of Casey (Casey J., On elastic–thermo-plastic materials at finite deformations. Int J Plast, 14, 173–91.). Frame indifference, objective rates, and spins are briefly addressed, introducing application of the broadened framework to (cited work in) finite thermoviscoplasticity. Fundamental references are provided.

Formulation du contact avec adhérence en élasticité non linéaire entre deux solides déformables

Within the framework of finite deformations and using an approach to the kinematics of contact due to A. Curnier, Q.C. He and J.J. Téléga, we propose a spatial thermodynamic formulation for the problem coupling unilateral condition, friction and adhesion. The adhesion is characterized by its intensity introduced by M. Frémond. In the case of frictionless contact between an hyperelastic body and a plane rigid support, with a particular `static' law for the evolution of the intensity of adhesion, the problem can be reduced to a minimization one for which we can show the existence of a solution.

Finite deformation plasticity and viscoplasticity laws exhibiting nonlinear hardening rules

This paper deals with plasticity and viscoplasticity laws exhibiting nonlinear kinematic hardening as well as nonlinear isotropic hardening rules. In Tsakmakis (1996a, b) a constitutive theory has been formulated within the framework of finite deformations, which is based on the concept of so-called dual variables and associated time derivatives. Within two families of dual variables, two different formulations have been proposed for kinematic hardening, referred to as Models 1 and 2. In particular, rigid plastic deformations without isotropic hardening have been considered. In the present paper, the constitutive theory of Tsakmakis (1996a, b) is appropriately extended to take into account isotropic hardening as well as elastic deformations. Care is taken that the evolution equations governing the hardening response fulfill the intrinsic dissipation inequality in every admissible process. For the case of small elastic strains combined with a simplification concerning kinematic hardening, to be explained in the paper, an efficient, implicit time-integration algorithm is presented. The algorithm is developed with a view to implementation in the ABAQUS Finite Element code. Also, explicit formulas for the consistent tangent modulus are derived.

Solution of the slump test using a finite deformation elasto‐plastic Drucker‐Prager model

AbstractDevelopment of a finite deformation elasto‐plasticity model based on the multiplicative decomposition of the deformation gradient is presented and discussed in detail. The formulation presented in this paper includes the derivation of the full set of equations for the Drucker‐Prager yield criterion. The equations, which are not available elsewhere, are developed within a framework using a spectral decomposition approach. Further, expressions for the consistent (algorithmic) tangent moduli in the finite strain regime are developed. Since the finite deformation framework employed to obtain the expressions presented here collapses to the classical infinitesimal plasticity framework when the finite strain assumption is no longer necessary, the finite deformation consistent tangent moduli are compared to the consistent tangent moduli valid for use with infinitesimal plasticity. Validation of the implemented finite deformation elasto‐plastic Drucker‐Prager model is performed through the solution of the concrete slump test. Comparisons between an existing approximate analytical solution and experimental data are presented, and results are discussed in detail.

Quasi-impact damage initiation and growth of thick-section and toughened composite materials

Results are presented for a study of damage initiation and growth in thick-section and toughened continuous fiber-reinforced composites resulting from a spherical indenter/impactor. The proposed analytical/computational model consists of several sub-models. A three-dimensional mesh generator is used for generating initial meshes for laminated composites containing various matrix cracks and delaminations. A stress analysis is developed in the framework of finite deformation for calculating stresses and strains of the composites with and without internal damage. A robust contact analysis proposed for handling the contact condition of cracked interfaces for matrix cracks/delaminations induced during indentation loading. A failure analysis is developed for predicting the damage initiation and propagation. A fracture mechanics based, simplified criterion is proposed for predicting the critical load for delamination growth.The proposed model predicts that the damage of thick-section composites may be initiated from the layer right below the indenter due to the compressive load, or the interior shear matrix cracks due to transverse shear. This contradicts the conclusions for thin, brittle laminated composites, in which surface cracking is in general the initial damage mode. It is shown that the delamination induced by the shear cracks is dominated by fracture modes II and III for growth of thick-section composites due to indentation loading. It is also interesting to observe that complicated contact zone exists for different delaminations.

Objective corotational rates and shear oscillation

Influence of objective corotational rates to shear oscillation is discussed here under the framework of finite deformation, J 2 -flow theory. Under an elastic-plastic, mixed hardening constitutive law, the active stress s , as defined in eqn (8) of the text, and back stress b are shown to be governed by two sets of equations formally uncoupled to each otther. Formal solutions of complex stresses are obtained for plane strain, isochoric deformation, with the conclusion that the shear oscillation behavior for an elastic-plastic solid can be asymptotically reduced to a rigid-plastic solution and a higher order correction, provided the initial shear strain is small. Consequently, the active stress s appears to be nonoscillatory under a realistic but arbitrary corotational rate, whereas the oscillatory behavior of back stress b is solely determined by the maximum corotation angle, defined in eqns (18) and (41), as well as the variation of material hardening with respect to the corotation angle. A spectrum of back stress responses could be predicted by the incorporation of different objective corotational rates.

On the influence of material properties on penetration of a thick elastic-plastic body

Within the frame-work of finite deformations of continuous media, the quasi-static deformations are considered that ensues upon applying internal pressure and axial shear stress, uniformly distributed along the surface of a cylindrical hole of infinitesimal initial radius in an elastic-plastic body of unbounded extent. With internal pressure only, also dynamic effects are taken into account. For each considered case of material and loading analytic expressions are given for the state of stress in the body and the amount of energy being dissipated in plastic working during the deformation process. The obtained results furnish the basis for a discussion of the influence of material parameters on the mechanical behavior of armor materials.