Uniaxial ratchetting in rate-independent plasticity laws

Summary A discussion of uniaxial ratchetting in linear viscoelasticity is given in [1]. The purpose of this paper is to analyse uniaxial ratchetting in the context of rate-independent plasticity laws. The analysis is referred to constitutive models with kinematic hardening, as well as to sawtooth input functions for stress with constant absolute value for the stress rate. For such loading histories some analytical formulas and explicit expressions concerning the model response for a sufficiently large number of cycles are derived. It turns out that an analogy can be drawn between the constitutive models of linear viscoelasticity and those of rate-independent plasticity with kinematic hardening. The analogy is established by a correspondence between the line representing the elasticity law for the equilibrium stress in viscoelasticity and the axis of kinematic hardening for the plasticity models defined in this paper.     

A Damage-Mode Based Three Dimensional Constitutive Model for Fibre-Reinforced Composites

This article presents a three dimensional constitutive model for anisotropic damage to describe the elastic-brittle behavior of unidirectional fibrereinforced laminated composites. The primary objective of the article focuses on the three dimensional relationship between damage of the material and the effective elastic properties for the purpose of stress analysis of composite structures, in extension to the two dimensional model in Matzenmiller, Lubliner and Taylor (1995). A homogenized continuum is adopted for the constitutive theory of anisotropic damage and elasticity. Damage initiation criteria are based on Puck failure criterion for first ply failure and progressive micro crack propagation is based on the idea of continuum damage evolution. Internal variables are introduced to describe the evolution of the damage state under loading and as a subsequence the degradation of the material stiffness. Emphasis is placed on a suitable coupling among the equations for the rates of the damage variables with respect to the different damage modes.     

Nonlinear viscoelastic and viscoplastic constitutive equations with growing damage

Nonequilibrium thermodynamics, rate-process theory, viscoelastic fracture mechanics and various experimentally-motivated simplifications are used to develop constitutive equations that account for effects of viscoelasticity, viscoplasticity, growing damage and aging. Their form is more general than previously developed by the author, and allows for relatively general tensorial effects of damage. Some important special cases are then covered, with emphasis on viscoelasticity. Evolution equations for the damage expressed in terms of internal state variables (ISVs) are discussed, comparing formulations using scalar ISVs and tensor ISVs. Finally, some experimental support for the theory is described. An Appendix illustrates the theory for an aging, linear viscoelastic material with growing cracks. This revised version was published online in July 2006 with corrections to the Cover Date.     

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. Received 22 September 1999     

On the numerical handling of fractional viscoelastic material models in a FE analysis

The paper presents the finite element (FE) implementation of linear and nonlinear fractional viscoelasticity models. To this end, a short introduction on fractional calculus is given. In addition to the fractional operators, this includes analytical and numerical solution schemes for selected fractional integral and differential equations. The presented rheological model is based on state of the art approaches and has been adopted to model the strain rate dependent material behavior of polymers. To this end, two approaches with constant and overstress dependent viscous properties resulting in linear and nonlinear evolution equations are discussed. The uniaxial constitutive relations are generalized to the multiaxial case and processed to be implemented in a FE code. The model behavior of both approaches is demonstrated and compared for selected uniaxial and multiaxial load cases.     

Comparison of two viscoplastic flow laws as applied to a problem of crack growth

The finite element method is incorporated in the study of the mechanical response of an advanced nickel-based superalloy IN-100 with inelastic time-dependent material behavior at 1350°F (732°C). The material's time-dependent plastic deformation (viscoplasticity) is analytically modeled by both the Bodner-Partom equations and the Malvern overstress flow law. The Bodner-Partom constitutive equations involve the use of nine material parameters which are determined from a complex procedure. Various forms of the Malvern overstress flow law are studied since this flow law uses only a few material parameters which are easily determined from uniaxial experimental data. Both the Malvern and Bodner-Partom constitutive equations are formulated in multiaxial from in a two-dimensional finite element program incorporating the constant strain triangle. The program is applied to the analysis of a center crack plate of IN-100 under a monotonic tension load. The residual force method is utilized to handle variations in material stiffness due to time-dependent plastic deformation. A Hybrid Experimental Numerical (HEN) procedure is used in order to track crack opening displacement near the crack tip. This HEN procedure controls the crack growth rate in the finite element model by following the experimental displacement rates accurately. Thus, crack growth predictions become a byproduct of both the rate sensitive model and the near field experimental displacement rates. The predictions of the two constitutive models are compared with respect to total plastic work and effective stress profiles in addition to crack growth rate.     

Analytical forms of higher-order asymptotic elastic-plastic crack-tip fields in a linear hardening material under antiplane shear

An analytical study of the higher-order asymptotic solutions of the stress and strain fields near the traction-free crack tip under antiplane shear in a linear hardening material is investigated. The results show that every term of the asymptotic fields is controlled by both elasticity and plasticity and all the higher-order asymptotic fields are governed by linear nonhomogeneous equations. The first four term solutions are presented analytically and the first four terms are described by two independent parameters J and K ₂. The amplitude of the second order term solution is only dependent on the material properties, but independent of loading and geometry. This paper focuses on the case with traction-free crack surface boundary conditions. The effects of different crack surface boundary conditions, such as clamped and mixed surfaces, on the crack-tip fields are also presented. Comparison of multi-term solution with leading term solution, and finite element solution in an infinite strip with semi-infinite crack under constant displacements along the edges is provided.     

Damage-induced stress-softening and viscoelasticity of limited elastic materials

The rate-dependent behavior of filled natural rubber (NR) is investigated in tensile regime. In order to describe the viscosity-induced rate-dependent effects, a constitutive model of finite strain viscoelasticity is proposed on the basis of the multiplicative decomposition of the deformation gradient tensor into elastic and viscous parts. The total stress is decomposed into an equilibrium stress and a viscosity-induced overstress by following the rheological models of Poynting–Thomson and Zener types. To incorporate the Mullins stress-softening phenomenon into a viscoelastic material, an invariant-based stress-softening function is also proposed. To identify the constitutive equation for the viscosity from direct experimental observations, an analytical scheme is proposed that ascertains the fundamental relation between the viscous strain rate and the overstress tensor with limited elastic parent material model. Evaluation of the experimental results using the proposed analytical scheme confirms the necessity of considering both the current overstress and the current deformation as variables to describe the evolution of the rate-dependent phenomena. Based on this, an evolution equation is proposed to represent the effects of internal variables on viscosity phenomena. The proposed evolution equation has been incorporated into the finite-strain viscoelasticity model in a thermodynamically consistent way.     

A viscoplastic theory with thermodynamic considerations

Summary A thermodynamic foundation using the concept of internal state variables is given for a general theory of viscoplasticity for initially isotropic materials. Three, fundamental, internal, state variables are admitted; they are: a tensorial back stress for kinematic effects, and scalar drag and yield strengths for isotropic effects. All three are considered to evolve phenomenologically according to competitive processes between strain hardening, deformation induced dynamic recovery, and thermally induced static recovery. Within this phenomenological framework, a thermodynamically admissible set of evolution equations is proposed. The theory allows each of the three internal variables to be composed as a sum of independently evolving constituents. The evolution of internal state can also include terms that vary linearly with the external variable rates, whose presence affects the energy dissipation properties of a material.     

Thermodynamically consistent coupled viscoplastic damage model for perforation and penetration in metal matrix composite materials

Accurate modeling and efficient analysis of the metal matrix composite materials failure mechanism during high velocity impact conditions is still the ultimate goal for many researchers. The objective is to develop a micromechanical constitutive model that can effectively simulate the high impact damage problem of the metal matrix composite materials. Therefore in this paper, a multiscale micromechanical constitutive model that couples the anisotropic damage mechanism with the viscoplastic deformation is presented here as a solution to this situation. This coupled viscoplastic damage model is formulated based on thermodynamic laws. Nonlinear continuum mechanics is used for this heterogeneous media that assesses a strong coupling between viscoplasticity and anisotropic damage. It includes the strong directional effect of the fiber on the evolution of the back stress and the development of the viscoplastic strain in the material behavior for high velocity impact damage related problems.     

A stochastic plasticity approach to strength modeling of strand-based wood composites

A 3 dimensional stochastic finite element technique is presented herein for simulating the nonlinear behaviour of strand-based wood composites with strands of varying grain-angle. The approach is based on the constitutive properties of the individual strands to study the effects of varying strand characteristics (such as species or geometry) on the performance of the member. The constitutive properties of the strands are found empirically and are subsequently used in a 3 dimensional finite element program. The program is formulated in a probabilistic manner using random variable material properties as input. The constitutive model incorporates classic plasticity theory whereby anisotropic hardening and eventual failure of the material is established by the Tsai–Wu criterion with an associated flow rule. Failure is marked by an upper bound surface whereupon either perfect plasticity (i.e. ductile behavior) or an abrupt loss of strength and stiffness (i.e. brittle behavior) ensues. The ability of this technique to reproduce experimental findings for the stress–strain curves of angle-ply laminates in tension, compression as well as 3 point bending is validated.     

A density-dependent endochronic plasticity for powder compaction processes

This paper is concerned with the numerical modeling of powder cold compaction process using a density-dependent endochronic plasticity model. Endochronic plasticity theory is developed based on a large strain plasticity to describe the nonlinear behavior of powder material. The elastic response is stated in terms of hypoelastic model and endochronic plasticity constitutive equations are stated in unrotated frame of reference. A trivially incrementally objective integration scheme for rate constitutive equations is established. Algorithmic modulus consistent with numerical integration algorithm of constitutive equations is extracted. It is shown how the endochronic plasticity describes the behavior of powder material from the initial stage of compaction to final stage, in which material behaves as solid metals. It is also shown that some commonly used plasticity models for powder material can be derived as special cases of the proposed endochronic theory. Finally, the numerical schemes are examined for efficiency in the modeling of a plain bush, a rotational-flanged and a shaped tablet powder compaction component.     

The influence of loading rates on hardening effects in elasto/viscoplastic strain-hardening materials

In this paper the influence of increasing loading rates on hardening effects is analyzed for rate-dependent elastoplastic materials. The effects of different loading rates on hardening rules are discussed with regard to the constitutive behavior of strain-hardening materials in elasto/viscoplasticity. A suitable procedure for the numerical simulation of rate-sensitive material behavior is illustrated. A comparative analysis is presented on constitutive relations in strain-hardening plasticity without rate effects and with rate effects in order to show the different role played by hardening rules in the rate-sensitivity analysis of elasto/viscoplastic strain-hardening materials. By reporting suitable numerical simulations for the adopted constitutive relations it is shown that when the rate of application of the loading is increased the influence of hardening has a different effect in the mechanical behavior of structures. Computational results and applications are finally illustrated in order to show numerically the different role played by hardening on the plastic strains when the loading rates are incremented for elasto/viscoplastic strain-hardening materials and structures.     

Bifurcations in the mechanics of hypoelastic composite materials

This paper analyses the modification of a hypoelastic material behavior at the small variations of the material parameters, using elements of the bifurcation theory. The considered material is obtained by the combination of two granular hypoelastic materials, which have the memory of the initial stress state, and their stress work depends on stress history. Its constitutive equation is deduced by means of the constitutive equations of the component materials. In consequence, mechanical properties of those two materials are interpenetrated, generating, for the new material, domains of stability, as well as surfaces in stress space, surfaces on which the strain–stress system is not invertible. It results that it is necessary to choose correctly the component materials, their share and the process of forming the new material, so that the imposed solicitation by the operation conditions should be accessible to a composite material. When we are modelling, the choice of the component materials means the choice of their constitutive equations—the share of the component materials will fix the stability domain of the composite material—the forming process chosen correctly will determine that the initial stress state (from which the loading path will start) should be in the stability domain of the material.     

Variational eigenerosion for rate-dependent plasticity in concrete modeling at small strain

In the context of eigenfracture scheme, the work at hand introduces a variational eigenerosion approach for inelastic materials. The theory seizes situations where the material accumulates large amounts of plastic deformations. For these cases, the surface energy entering the energy balance equation is rescaled to favor fracture, thus energy minimization delivers automatically the crack-tracking solution also for inelastic cases. The minimization approach is sound and preserves the mathematical properties necessary for the Γ-limit proof, thus the existence of (local) minimizers is guaranteed by the Γ-convergence theory. Although it is not possible to demonstrate that the obtained minimizers are global, satisfactory results are obtained with the local minimizers provided by the method. Furthermore, with the goal of addressing the constitutive behavior of concrete, a Drucker-Prager viscoplastic consistency model is introduced in the microplane setting. The model delivers a rate-dependent three-surface smooth yield function that requires hardening and hardening-rate parameters. The independent evolution of viscoplasticity in different microplanes induces anisotropy in the mechanical response. The sound performance of the model is illustrated via numerical examples for both rate-independent and rate-dependent plasticity.     

The influence of microstructure-induced gradients on the localization of deformation in viscoplastic materials

Summary We suggest here a generalization of the conventional constitutive models of viscoplasticity. This is accomplished by the inclusion of spatial gradients of the equivalent stress and strain in the evolution equation for the equivalent plastic strain rate. We restrict attention to plane deformation and elastic effects are neglected for simplicity. The implications of the new terms in the constitutive model are discussed for the case of a general eigenvalue problem of an initially homogeneous and stationary viscous flow. It turns out that the nonclassical material parameters can be chosen in such a way that the governing differential equations are always strongly elliptic irrespective of whether the mateiral is strain softening. As it is well known, the latter typically leads to loss of ellipticity in the conventional theories. Explicit results are presented for the case of a shear band instability. Within the framework of the present theory, and in contrast to conventional models, the shear band kinematics have a well defined geometrical structure.     

Classical and Cosserat plasticity and viscoplasticity models for slow granular flow

We consider models for the rheology of dense, slowly deforming granular materials based of classical and Cosserat plasticity, and their viscoplastic extensions that account for small but finite particle inertia. We determine the scale for the viscosity by expanding the stress in a dimensionless parameter that is a measure of the particle inertia. We write the constitutive relations for classical and Cosserat plasticity in stress-explicit form. The viscoplastic extensions are made by adding a rate-dependent viscous stress to the plasticity stress. We apply the models to plane Couette flow, and show that the classical plasticity and viscoplasticity models have features that depart from experimental observations; the prediction of the Cosserat viscoplasticity model is qualitatively similar to that of Cosserat plasticity, but the viscosities modulate the thickness of the shear layer.     

Modeling of nonlinear viscoelasticity at large deformations

A constitutive model of finite strain viscoelasticity, based on the multiplicative decomposition of the deformation gradient tensor into elastic and inelastic parts, is presented. The nonlinear response of rubbers, manifested by the rate effect, cycling loading and stress relaxation tests was captured through the introduction of two internal variables, namely the constitutive spin and the back stress tensor. These parameters, widely used in plasticity, are applied in this work to model the nonlinear viscoelastic behaviour of rubbers. The experimental results, obtained elsewhere, related with shear deformation in monotonic and cyclic loading, as well as stress-relaxation, were simulated with a good accuracy.