Multiplicative Decomposition Of Deformation Gradient Research Articles

Constitutive Model Based on the Multiplicative Decomposition of Deformation Gradient in Arbitrary Orthogonal Coordinate System

According to general crystallographic slip constitutive laws, the stress-strain analysis of nickel based single crystal superalloy (NBSCS) in arbitrary coordinate system were based on material coordinate system (SRCS method). Ignoring the other symmetry of NBSCS, only the symmetry of plane families {001} were considered, which bring considerable errors into the stress-strain analysis. In this paper, the variation regulation of micro-physical systematical property about this alloy in arbitrary directions was investigated. From the arrangement of atoms in different crystal planes, NBSCS has four symmetry plane families, {001}, {110}, {112} and {111}. According to these symmetry planes, three reference orthogonal coordinate systems were established. Based on these coordinate systems and the coordinate rotation method (TRCS method), the stress-strain relationship of crystallographic slip constitutive model in arbitrary coordinate system was established. Meanwhile elastic constants in arbitrary directions were obtained. Comparing the results of the tensile stress-strain curves obtained from TRCS method with that from SRCS method, it is found that by the TRCS method the elastoplastic stress-strain simulation error of the NBSCS could be effectively reduced.

Modelling of dynamic behaviour of orthotropic metals including damage and failure

A physically based material model for metals, with elastic–plastic and damage/failure orthotropy is proposed in this paper. The model is defined within the frameworks of irreversible thermodynamics and configurational continuum mechanics and integrated in the isoclinic configuration. The use of the multiplicative decomposition of deformation gradient makes the model applicable to arbitrary plastic and damage deformations. To account for the physical mechanisms of failure, the concept of thermally activated damage initially proposed by Klepaczko (1990) was adopted as the basis for the new damage evolution model. This makes the proposed damage/failure model compatible with the Mechanical Threshold Strength (MTS) model (Follansbee and Kocks, 1988; Chen and Gray, 1996; Goto et al., 2000; Gray et al., 1999; Chen et al., 1998) which was used to control evolution of flow stress during plastic deformation. In addition the constitutive model is coupled with a shock equation of state which allows for modelling of shock wave propagation in the material. The new model was implemented in DYNA3D and our in-house non-linear transient SPH code, MCM (Meshless Continuum Mechanics).Parameters for the new constitutive model for AA7010 (a polycrystalline aluminium alloy, whose orthotropy is a consequence of grain morphology), were derived on the basis of the tensile tests and Taylor anvil tests. The tensile tests were performed for the range of temperatures between 223.15K and 413.15K, and strain rates between 6.4×10-4s-1 and 6.4×101s-1.The new model was validated in two stages. The first stage comprised a series of single element tests design to separately validate elasticity, plasticity and damage related parts of the model. The second stage comprised a series of numerical simulations of Taylor anvil and plate impact tests for AA7010 and comparison of the numerical results with the experimental data. The numerical results illustrate the ability of the new model to predict experimentally observed behaviour.

A Geometric Theory of Growth Mechanics

In this paper we formulate a geometric theory of the mechanics of growing solids. Bulk growth is modeled by a material manifold with an evolving metric. Time dependence of metric represents the evolution of the stress-free (natural) configuration of the body in response to changes in mass density and "shape". We show that time dependency of material metric will affect the energy balance and the entropy production inequality; both the energy balance and the entropy production inequality have to be modified. We then obtain the governing equations covariantly by postulating invariance of energy balance under time-dependent spatial diffeomorphisms. We use the principle of maximum entropy production in deriving an evolution equation for the material metric. In the case of isotropic growth, we find those growth distributions that do not result in residual stresses. We then look at Lagrangian field theory of growing elastic solids. We will use the Lagrange-d'Alembert's principle with Rayleigh's dissipation functions to derive all the governing equations. We make an explicit connection between our geometric theory and the conventional multiplicative decomposition of deformation gradient $\mathbf{F}=\mathbf{F}_e\mathbf{F}_g$ into growth and elastic parts. We linearize the nonlinear theory and derive a linearized theory of growth mechanics. Finally, we obtain the stress-free growth distributions in the linearized theory.

A stress driven growth model for soft tissue considering biological availability

Some of the key factors that regulate growth and remodeling of tissues are fundamentally mechanical. However, it is important to take into account the role of bioavailability together with the stresses and strains in the processes of normal or pathological growth. In this sense, the model presented in this work is oriented to describe the growth of soft biological tissue under "stress driven growth" and depending on the biological availability of the organism. The general theoretical framework is given by a kinematic formulation in large strain combined with the thermodynamic basis of open systems. The formulation uses a multiplicative decomposition of deformation gradient, splitting it in a growth part and visco-elastic part. The strains due to growth are incompatible and are controlled by an unbalanced stresses related to a homeostatic state. Growth implies a volume change with an increase of mass maintaining constant the density. One of the most interesting features of the proposed model is the generation of new tissue taking into account the contribution of mass to the system controlled through biological availability. Because soft biological tissues in general have a hierarchical structure with several components (usually a soft matrix reinforced with collagen fibers), the developed growth model is suitable for the characterization of the growth of each component. This allows considering a different behavior for each of them in the context of a generalized theory of mixtures. Finally, we illustrate the response of the model in case of growth and atrophy with an application example.

A geometric theory of thermal stresses

In this paper we formulate a geometric theory of thermal stresses. Given a temperature distribution, we associate a Riemannian material manifold to the body, with a metric that explicitly depends on the temperature distribution. A change in temperature corresponds to a change in the material metric. In this sense, a temperature change is a concrete example of the so-called referential evolutions. We also make a concrete connection between our geometric point of view and the multiplicative decomposition of deformation gradient into thermal and elastic parts. We study the stress-free temperature distributions of the finite-deformation theory using curvature tensor of the material manifold. We find the zero-stress temperature distributions in nonlinear elasticity. Given an equilibrium configuration, we show that a change in the material manifold, i.e., a change in the material metric will change the equilibrium configuration. In the case of a temperature change, this means that given an equilibrium configuration for a given temperature distribution, a change in temperature will change the equilibrium configuration. We obtain the explicit form of the governing partial differential equations for this equilibrium change. We also show that geometric linearization of the present nonlinear theory leads to governing equations that are identical to those of the classical linear theory of thermal stresses.

Verification of Compatibility of Isotropic Consolidation Characteristics of Soils to Multiplicative Decomposition of Deformation Gradient

It is verified that the linear relation of logarithmic volumetric strain vs. logarithm of pressure derived from the linear relation of both logarithms of volume and pressure (Hashiguchi, 1974) for isotropic consolidation of soils is exactly compatible to the multiplicative decomposition of deformation gradient (Lee, 1969) which would be one of the fundamental requirements for constitutive equations describing the finite elastoplastic deformation. On the other hand, it is emphasized that the adoption of the linear relation of void ratio vs. logarithm of pressure is impertinent for the constitutive equation describing the finite deformation although it is most widely used for constitutive equations of soils.

Constitutive relations for isotropic or kinematic hardening at finite elastic–plastic deformations

The rate-type constitutive relations of rate-independent metals with isotropic or kinematic hardening at finite elastic–plastic deformations were presented through a phenomenological approach. This approach includes the decomposition of finite deformation into elastic and plastic parts, which is different from both the elastic–plastic additive decomposition of deformation rate and Lee’s elastic–plastic multiplicative decomposition of deformation gradient. The objectivity of the constitutive relations was dealt with in integrating the constitutive equations. A new objective derivative of back stress was proposed for kinematic hardening. In addition, the loading criteria were discussed. Finally, the stress for simple shear elastic–plastic deformation was worked out.

Numerical implementation of a thermo-elastic–plastic constitutive equation in consideration of transformation plasticity in welding

Finite element analysis of welding processes, which entail phase evolution, heat transfer and deformations, is considered in this paper. Attention focuses on numerical implementation of the thermo-elastic–plastic constitutive equation proposed by Leblond et al. [J. Mech. Phys. Solids 34(4) (1986a) 395; J. Mech. Phys. Solids 34(4) (1986b) 411] in consideration of the transformation plasticity. Based upon the multiplicative decomposition of deformation gradient, hyperelastoplastic formulation is borrowed for efficient numerical implementation, and the algorithmic consistent moduli for elastic–plastic deformations including transformation plasticity are obtained in the closed form. The convergence behavior of the present implementation is demonstrated via a couple of numerical examples.

Symmetrization of the growth deformation and velocity gradients in residually stressed biomaterials

Some fundamental issues in the kinematic and kinetic analysis of the stress-modulated growth of residually stressed biological materials are addressed within the context of the multiplicative decomposition of deformation gradient into its elastic and growth parts. The symmetrizations of the growth part of the deformation gradient and the growth part of the velocity gradient are derived for isotropic pseudoelastic soft tissues. The significance of results in the formulation of the biomechanic constitutive theory is discussed.

Constitutive theories based on the multiplicative decomposition of deformation gradient: Thermoelasticity, elastoplasticity, and biomechanics

Some fundamental issues in the formulation of constitutive theories of material response based on the multiplicative decomposition of the deformation gradient are reviewed, with focus on finite deformation thermoelasticity, elastoplasticity, and biomechanics. The constitutive theory of isotropic thermoelasticity is first considered. The stress response and the entropy expression are derived in the case of quadratic dependence of the elastic strain energy on the finite elastic strain. Basic kinematic and kinetic aspects of the phenomenological and single crystal elastoplasticity within the framework of the multiplicative decomposition are presented. Attention is given to additive decompositions of the stress and strain rates into their elastic and plastic parts. The constitutive analysis of the stress-modulated growth of pseudo-elastic soft tissues is then presented. The elastic and growth parts of the deformation gradient and the rate of deformation tensor are defined and used to construct the corresponding rate-type biomechanic theory. The structure of the evolution equation for growth-induced stretch ratio is discussed. There are 112 references cited in this review article.

Associative coupled thermoplasticity at finite strain with temperature-dependent material parameters

The paper deals with associative coupled thermoplasticity at finite strains. J2 plasticity model is based on hyperelastic formulation with multiplicative decomposition of deformation gradient into elastic and plastic parts. As a novel aspect, temperature dependence of all material parameters is introduced. For such model, variational procedure is carried out and discretization by the mixed finite element method is performed. Consistent linearization is carried out and algorithmic elasto-plastic tangent moduli are given. Verification of algorithm is provided by two examples.

Simulation and interpretation of diffuse mode bifurcation of elastoplastic solids

The diffuse mode bifurcation of elastoplastic solids at finite strain is investigated. The multiplicative decomposition of deformation gradient and the hyperelasto-plastic constitutive relationship are adapted to the numerical bifurcation analysis of the elastoplastic solids. First, bifurcation analyses of rectangular plane strain specimens subjected to uniaxial compression are conducted. The onset of the diffuse mode bifurcations from a homogeneous state is detected; moreover, the post-bifurcation states for these modes are traced to arrive at localization to narrow band zones, which look like shear bands. The occurrence of diffuse mode bifurcation, followed by localization, is advanced as a possible mechanism to create complex deformation and localization patterns, such as shear bands. These computational diffuse modes and localization zones are shown to be in good agreement with the associated experimental ones observed for sand specimens to ensure the validity of this mechanism. Next, the degradation of horizontal sway stiffness of a rectangular specimen due to plane strain uniaxial compression is pointed out as a cause of the bifurcation of the first antisymmetric diffuse mode, which triggers the tilting of the specimen. Last, circular and punching failures of a footing on a foundation are simulated.

Finite-strain thermoelasticity based on multiplicative decomposition of deformation gradient

The constitutive formulation of the finite-strain thermoelasticity is revisited within the thermodynamic framework and the multiplicative decomposition of the deformation gradient into its elastic and thermal parts. An appealing structure of the Helmholtz free energy is proposed. The corresponding stress response and the entropy expressions are derived. The results are specified in the case of quadratic dependence of the elastic strain energy on the finite elastic strain. The specific and latent heats are discussed, and the comparison with the results of the classical thermoelasticity are given. .

On the partitioning of the rate of deformation gradient in phenomenological plasticity

The rate of deformation gradient is partitioned additively into its elastic and plastic parts within the framework of phenomenological plasticity based on the multiplicative decomposition of deformation gradient. The corresponding partition of the nonsymmetric nominal stress is also given. The results are compared with the well-known results of partitioning the rates of Lagrangian strain and its conjugate symmetric Piola–Kirchhoff stress. Extension to the framework of monocrystalline plasticity is then discussed.

Viscoplasticity model at finite deformations with combined isotropic and kinematic hardening

In this work, we construct a phenomenological constitutive model of viscoplasticity at finite strains, which generalizes the classical Perzyna or Duvaut–Lions models to finite strains. The latter is accomplished with a minimum number of hypothesis, including the multiplicative decomposition of deformation gradient, a definition of the elastic domain and finally a penalty-like, viscoplastic regularization of the principle of maximum plastic dissipation. The model is extended to include the isotropic and kinematic hardening of Prager–Ziegler type. Numerical computation of the viscoplastic flow at finite strain is also discussed, along with the corresponding simplification resulting from a convenient choice of the logarithmic strain measure. Several illustrative numerical examples are presented in order to demonstrate the ability of the proposed model to remove the deficiencies of some currently used models.

Performance of a finite element procedure for hyperelastic–viscoplastic large deformation problems

In this paper, the details of implementation of a formulation for hyperelastic–viscoplastic solids are discussed. The formulation employs the constitutive equation based on multiplicative decomposition of deformation gradient, incrementally objective integration, and closed-form tangent operator consistent with the constitutive evaluation. The standard updated Lagrangian framework for the virtual work equation is used. Different measures, taken to make computation efficient and stable, are discussed such as the solution of scalar nonlinear equations for rate-dependent plasticity using a hybrid method. The proposed method is numerically implemented and the computational aspects are examined in detail. A number of numerical examples are presented that illustrate the excellent performance of the proposed method, even with very large strain increments. The performance of the current implementation is compared with other closed-form elasto-viscoplastic tangent operators having hypoelastic or hyperelastic assumption reported in the literature.

EXPONENTIAL MAP ALGORITHM FOR STRESS UPDATES IN ANISOTROPIC MULTIPLICATIVE ELASTOPLASTICITY FOR SINGLE CRYSTALS

This paper presents a new stress update algorithm for large-strain rate-independent single-crystal plasticity. The theoretical frame is the well-established continuum slip theory based on the multiplicative decomposition of the deformation gradient into elastic and plastic parts. A distinct feature of the present formulation is the introduction and computational exploitation of a particularly simple hyperelastic stress response function based on a further multiplicative decomposition of the elastic deformation gradient into spherical and unimodular parts, resulting in a very convenient representation of the Schmid resolved shear stresses on the crystallographic slip systems in terms of a simple inner product of Eulerian vectors. The key contribution of this paper is an algorithmic formulation of the exponential map exp: sl(3) SL(3) for updating the special linear group SL(3) of unimodular plastic deformation maps. This update preserves exactly the plastic incompressibility condition of the anisotropic plasticity model under consideration. The resulting fully implicit stress update algorithm treats the possibly redundant constraints of single-crystal plasticity by means of an active set search. It exploits intrinsically the simple representation of the Schmid stresses by formulating the return algorithm and the associated consistent elastoplastic moduli in terms of Eulerian vectors updates. The performance of the proposed algorithm is demonstrated by means of a representative numerical example.

Multisurface thermoplasticity for single crystals at large strains in terms of eulerian vector updates

This paper presents a formulation of large-strain rate-independent multisurface thermoplasticity for single crystals and addresses aspects of its numerical implementation. The theoretical frame is the well-established continuum slip theory based on the multiplicative decomposition of the deformation gradient into elastic and plastic parts. A key feature of the present paper is the introduction and computational exploitation of a particularly simple hyperelastic stress response function based on a further multiplicative decomposition of the elastic deformation gradient into spherical and unimodular parts, resulting in a very convenient representation of the Schmid resolved shear stresses on the crystallographic slip systems in terms of a simple inner product of Eulerian vectors. This observation is intrinsically exploited on the numerical side by formulating a new fully implicit stress update algorithm and the associated consistent elastoplastic moduli in terms of these Eulerian vectors. The proposed return mapping algorithm treats the possibly redundant constraints of large-strain multisurface plasticity for single crystals by means of an active set search. Furthermore, it satisfies in an algorithmically exact way the plastic incompressibility condition in situations of multislip. The performance of the proposed formulation is demonstrated for two representative numerical simulations of thermoplastic deformation processes in single crystals with isotropic Taylor-type hardening.

Constitutive analysis of large elasto-plastic deformation based on the multiplicative decomposition of deformation gradient

By using Lee's (1969, J. Appl. Mech.36, 1–6) multiplicative decomposition of the deformation gradient into its elastic and plastic part of Hill and Rice's (1973, SIAM J. Appl. Math.25, 448–461) constitutive framework, explicit and consistent constitutive analysis of large elastoplastic deformation is given. Both isotropic and anisotropic material behaviour is considered, so that some earlier results come as particular cases of this more general formulation. The relationship with other related work is also given.

Some aspects of elasto-plastic constitutive analysis of elastically anisotropic materials

By utilizing the multiplicative decomposition of deformation gradient, we precisely identify the elastic and plastic contribution to the velocity strain of elasto-plastically deformed anisotropic materials. Corresponding elastic stiffness and compliance tensors are obtained to within the unknown rotation tensor, to be additionally determined. Orthotropic materials are considered in particular and constitutive structure for their plastic strain-rate established. Various kinematic aspects and relationships between different plastic strain-rate and spin tensors, as well as comparison to related work, are also given.