Internal State Variable Theory Research Articles

A multiscale analysis of fixed-end simple shear using molecular dynamics, crystal plasticity, and a macroscopic internal state variable theory

We examine FCC nickel undergoing simple shear by using three different numerical frameworks formulated at three different size scales. The three frameworks included embedded atom method potentials used in molecular dynamics simulations, crystal plasticity used in finite element simulations, and a macroscale internal state variable formulation used in finite element simulations. Simple shear simulations were performed in which the specimen aspect ratio was varied to give insight into the homogeneous and inhomogeneous aspects of large deformation. This study revealed that the `apparent' yield stress was sensitive to the specimen aspect ratio except when the length-to-height ratio reached about 8 : 1, the yield stress remained constant. The three numerical frameworks gave similar qualitative responses related to inhomogeneous stress and strain distributions in the corner regions of the specimens and also similar responses in the centralized homogeneous deformation region. However, when comparing the shear stress distribution for the finite element analyses to the atomistic simulations, a much narrower distribution arose for the finite element analyses due to the lack of thermal vibrations experienced in the atomistic simulations at 300 K. A 10 K an atomistic simulation which dampened out the high frequency thermal vibrations verified this reasoning. Three different sizes of blocks of atoms were also used in the atomistic simulations and the results showed very similar stress and strain distributions with respect to each other indicating that no size scale effect is evidenced when normalized by the global shear stress. However, a size scale effect exists related to the global (volume average) shear stress in the specimen. As the specimen size increased, the yield stress decreased. Finally, when comparing the three different numerical frameworks, the location of maximum dislocation nucleation occurred at the location of the maximum plastic spin, stress gradients, and strain gradients.

Anisotropic damage model under continuum slip crystal plasticity theory for single crystals

Modern concepts in the safe and economical design of components and structures need modern material models, which describe the material behavior more correctly. In high temperature design, the study of creep damage accumulation and its influence on the deformation behavior is very important. For ductile materials large deformation takes place at the level of damage appearance. Damage is anisotropic in nature. In this paper an anisotropic damage mechanics model based on a continuum damage mechanics (CDM) has been developed to model creep behavior of single crystal superalloys. Using the theory of CDM, the slip system model coupling with anisotropic damage model is developed. The model is formulated in the context of irreversible thermodynamics and the internal state variable theory. The distinguishing characteristic of the proposed constitutive model is that, by construction, the corresponding incremental stress–strain relations including damage evolution equation derive from a thermodynamical framework. The finite element program ABAQUS has been used and the slip system model is written using a user material subroutine. The numerical simulations show that the developed damage crystal model can reflect the microstructure such as the lattice orientation, self- and latent-hardening and rate sensitivity has great influence on the creep and damage development.

A fatigue damage model for seismic response of RC structures

Numerous damage models have been developed in order to analyze seismic behavior. Among the different possibilities existing in the literature, it is very clear that models developed along the lines of continuum damage mechanics are more consistent with the definition of damage as a phenomenon with mechanical consequences because they include explicitly the coupling between damage and mechanical behavior. On the other hand, for seismic processes, phenomena such as low cycle fatigue may have a pronounced effect on the overall behavior of the frames and, therefore, its consideration turns out to be very important. However, most of existing models evaluate the damage only as a function of the maximum amplitude of cyclic deformation without considering the number of cycles. In this paper, a generalization of the simplified model proposed by Cipollina et al. [Cipollina A, López-Hinojosa A, Flórez-López J. Comput Struct 1995;54:1113–26] is made in order to include the low cycle fatigue. Such a model employs in its formulation irreversible thermodynamics and internal state variable theory.

Discussion of coupled elastoplasticity and damage constitutive equations for small and finite deformations

Three small deformation plasticity models taking into account isotropic damage effects are presented and discussed. The models are formulated in the context of irreversible thermody-namics and the internal state variable theory. They exhibit nonlinear isotropic and nonlinear kinematic hardening. The aim of the paper is first to give a comparative study of the three models with reference to homogeneous and inhomogeneous deformations by using a general damage law. Secondly, and this is the main objective of the paper, we generalize the constitutive models to finite deformations by applying a thermodynamical framework based on the Mandel stress tensor. The responses of the obtained finite deformation models are then discussed for loading processes with homogeneous deformations.

A self-consistent analysis for coupled elastoplastic damage problems

Based on irreversible thermodynamics and internal state variable theory, the volume-averaged Clausius-Duhem inequality is presented. In contrast to former investigations on damage-elastoplasticity, our evalustions are founded on the volume-averaged field equations of the analyzed elements and the self-consistent method. Hence, our results not only include the influence of void shapes but also consider the interaction among voids. Further, previous work about coupled elastoplastic damage problems only takes into account small initial void volume fractions. Our work, however, will be able to deal with elastoplastic damage problems with larger initial void volume fractions.

Constitutive Model for Coupled Inelasticity and Damage

A constitutive model to describe a coupling between deformation and damage due to creep of polycrystalline metallic materials is developed from phenomenological and continuum mechanics points of view. The constitutive modeling is based on the irreversible thermodynamics for internal state variable theories, where the thermodynamic potentials, i.e., free energy and dissipation energy functions, are defined using hardening and damage variables. The material damage is assumed to be isotropic. We first derive a damage coupled kinematic-hardening model in the invariant form on the basis of the Malinin-Khadjinsky model. Then, an isotropic-hardening model which includes a coupling with damage is formulated by assuming a particular representation of the kinematic hardening variable. The evolution equation of the hardening variable is prescribed by the Bailey-Orowan format which includes the effect of damage. The damage rate is governed by the magnitude of the assumed strain hardening variable. These models can describe a transition from primary to tertiary creep stages, and it is applicable to variable loading conditions. In a particular case the expression for the creep rupture time has a similar form to the Kachanov-Rabotnov type, although it depends on the time and damage at the instant of a hardening saturation under the applied stress condition.

Damage-Coupled Constitutive Model for Metal Matrix Composites.

Constitutive models to describe creep deformation and creep damage of unidirectional fiber-reinforced metal matrix composites are developed from a phenomenological point of view. The constitutive modeling is based on the well-established thermodynamic formalism for internal state variable theories. A damage-coupled kinematic hardening model for transversely isotropic materials is first formulated in the invariant form on the basis of the Malinin-Khadjinsky model. An isotropic hardening model is then developed by assuming a particular representation of the kinematic hardening variable. The extent of the composite damage is described in terms of a scalar variable, and a directional effect of the damage on the composite behavior is characterized by a fourth-rank anisotropic tensor. The evolution equation of the creep damage is analogous to the model developed by Kachanov and Rabotnov, but it includes a coupling between creep deformation and creep damage. The analytical expression of the creep rupture time is derived in a particular case of the isotropic hardening model, and for a constant stress creep it depends on both the time and degree of damage at the moment of hardening saturation.

Tensile cracks in polycrystalline ice under transient creep: Part II — Numerical simulations

Abstract This paper discusses predictions of a numerical model presented in the companion paper (Nanthikesan and Shyam Sunder, 1995) to analyze tensile cracks in polycrystalline ice undergoing transient creep. The numerical model is based on the internal state variable constitutive theory of transient creep in ice developed by Shyam Sunder and Wu (1989a,b, 1990). The finite element model uses the boundary layer approach of Rice (1968), in conjunction with a mid-point crack-tip element and reduced integration, to simulate the asymptotic stress and deformation fields in the vicinity of the crack tip, including incompressible creep deformations. The problem of a stationary, traction-free, tensile (mode I) crack is analyzed to predict the size, shape and time evolution of the creep-dominated fracture process zone surrounding the crack-tip. The numerical simulations quantify the effects of transient creep, material strain hardening, fabric anisotropy, loading rate, temperature, and finite fracture test-specimen boundary on the development of the creep zone. A range of stress-intensity rates from 1 to 100 kPa m 1 2 s−1 and temperatures from −5° to −25°C is considered in the simulations. The results from a comprehensive numerical simulation study show that: (i) transient creep increases the creep zone size by more than an order of magnitude over that for a power-law creeping material, i.e., about 40 times for the isotropic, equiaxed granular ice tested by Jacka (1984); (ii) material strain hardening significantly affects the creep zone size, i.e., the creep zone for the transversely-isotropic columnar-grained ice tested by Sinha (1978), with the crack loaded in the plane of isotropy, is about 4 times smaller than that for the granular isotropic ice; (iii) fabric anisotropy increases the size of the creep zone by a factor of at least two for cracks in the transversely-isotropic, columnar-grained ice loaded in the plane of isotropy; (iv) the Riedel and Rice (1980) equation, which was derived for an isotropic power-law creeping material subjected to a suddenly applied constant stress-intensity, overestimates the creep zone size by a factor of 4.2 for a constant stress-intensity rate loading; and (v) as the crack size increases, linear elastic fracture mechanics becomes increasingly applicable at lower loading rates and higher temperatures.

Large strain constitutive modelling and computation for isotropic, creep elastoplastic damage solids

AbstractA three‐dimensional fully coupled creep elastoplastic damage model at finite strain for isotropic non‐linear material is developed. The model is based on the thermodynamics of an irreversible process and the internal state variable theory. A hyperelastic form of stress–strain constitutive relation in conjunction with the multiplicative decomposition of the deformation gradient into elastic and inelastic parts is employed. The pressure‐dependent plasticity with strain hardening and the damage model with two damage internal variables are particularly considered. The rounding of stress–strain curves appearing in cycling loading is reproduced by introduction of the creep mechanism into the model. A numerical integration procedure for the coupled constitutive equations with three hierarchical phases is proposed. A consistent tangent matrix with consideration of the fully coupled effects at finite strain is derived. Numerical examples are tested to demonstrate the capability and performance of the present model at large strain.

Constitutive Model for Coupled Inelasticity and Damage.

Constitutive models to describe a coupling between deformation and damage due to creep of polycrystalline metallic materials are discussed from phenomenological and continuum mechanics points of view. The constitutive modeling is based on the irreversible thermodynamics for internal state variable theories, where the thermodynamic potentials, i. e., free energy and dissipation energy functions, are defined using hardening and damage variables. The material damage is assumed to be isotropic. First, a new damage-coupled kinematic-hardening model is developed in the invariant form on the basis of the Malinin-Khadjinsky model. The evolution equation of the hardening variable is prescribed by the Bailey-Orowan format which includes the effects of isotropic damage. Then, an isotropic-hardening model taking damage into consideration is formulated by assuming a particular representation of the kinematic hardening variable. The evolution equation of the isotropic creep damage is analogous to that developed by Kachanov and Rabotnov. However, it takes into account a coupling with creep hardening and softening. The present models can describe primary, secondary and tertiary creep behavior, and they are applicable to variable loading conditions. The creep rupture time predicted, even in the simplest case, depends on the time and degree of damage at which the hardening variable reaches its saturation state under the applied stress conditions.

Formulation of Inelastic Constitutive Model for Metal Matrix Composites. 1st Report, Transversely Isotropic Model .

A constitutive model to describe time-dependent inelastic behavior of unidirectionally fiber-reinforced metal matrix composites is developed from phenomenological and continuum mechanics points of view. The fibrous composite is treated as a homogeneous medium which hardens with inelastic deformation. The inherent anisotropy is assumed to be transversely isotropic. The constitutive modeling is based on the well-established thermodynamic formalism for internal state variable theories, where the thermodynamic potentials are defined using a transversely isotropic tensor of the fourth order. First, a kinematic hardening model is derived in the invariant form. In this model, the evolution of the internal state variable is prescribed by the Bailey-Orowan type so that both the transient and steady-state creep behavior can be described. Then, an isotropic hardening model is formulated by assuming a particular representation of the kinematic hardening variable. It is found that the isotropic hardening model can yield the power law relation for steady state responses.

Formulation of Inelastic Constitutive Model for Fiber-Reinforced Composite Materials.

Constitutive model to describe rate-dependent nonlinear inelastic behavior of unidirctionally fiber-reinforced composites is developed from phenomenological and continuum mechanics points of view. The fibrous composite is treated as a homogeneous medium which hardens with inelastic deformation. The inherent anisotropy due to the fibers is assumed to be transversely isotropic. We first derive a kinematic hardening model in invariant form. The constitutive modeling is based on the well-established thermodynamic formalism for internal state variable theories, where the thermodynamic potentials are defined by using a transversely isotropic tensor of fourth rank. In this model the evolution of the internal state variable is prescribed by the nonlinear kinematic hardening format of Armstrong-Frederick type. This enables us to describe the rate-dependence of the nonlinear anisotropic inelastic behavior of fibrous composites. Then, an isotropic hardening model is derived from this model by using an empirical method which assumes a particular representation of the kinematic hardening variable. Finally, some generalized expressions which are implied by these formulations are presented.

On energy-based coupled elastoplastic damage theories: Constitutive modeling and computational aspects

Novel energy-based coupled elastoplastic damage theories are presented in this paper. The proposed formulation employs irreversible thermodynamics and internal state variable theory for ductile and brittle materials. At variance with Lemaitre's work on damage-elastoplasticity, the present formulation renders rational thermodynamic potential and damage energy release rate. In contrast to previous work by Simo and Ju (featuring an additive split of the stress tensor), current formulation assumes an additive split of the strain tensor. It is shown that the “strain split” damage-elastoplasticity formulation leads to more robust tangent moduli than the “stress split” formulation. The plastic flow rule and hardening law are characterized in terms of the effective quantities; viz. the effective stress space plasticity. This mechanism is both physically well-motivated and computationally efficient. Further, a fourth-order anisotropic damage mechanism is proposed for brittle materials. Rational mechanisms are also presented to account for the microcrack opening and closing operations as well as the strain-rate dependency of microcrack growth.Efficient computational algorithms for proposed elasloplaslic damage models are subsequently explored by making use of the “operator splitting” methodology. In particular, new three-step operator split algorithms are presented Application is made to a class of inviseid and rate-dependent cap-damage models for concrete and mortar. Experimental validations are also given to illustrate the applicability of the proposed damage models.

Visco‐damage tension model for rocks in ground‐shock applications

AbstractA simple rate‐dependent tension model, tailored primarily for rocks in ground‐shock calculations, has been constructed. It is formulated as an internal state variable theory with spherical void development taken to approximate the underlying dissipative micromechanism. This simplifying assumption of ductile, rather than brittle, damage growth has been made to enhance the practicality of the model for large‐scale problems. The model has been restated within the fromework of (strain‐softening) viscoplastcity to associate it with the desirable mathematical properties of the latter theory. The tension option, extended to include healing in the post‐damage compressive phase, has been implemented in a standard cap constitutive routine and is referred to as the visco‐damage tension model. Model parameters are selected for a shale‐like material. Results of uniaxial shock‐loading strain‐rate calculations exhibit desired characteristics in tension. Damage‐related hysteretic effects are illustrated through cyclic loading calculations. It is planned to develop more effecient computational procedures to facilitate adoption of the model into large‐scale ground shock calculations. Anisotropic extensions of the model will then be considered.