A framework for multiplicative elastoplasticity with kinematic hardening coupled to anisotropic damage

The objective of this contribution is the formulation and algorithmic treatment of a phenomenological framework to capture anisotropic geometrically nonlinear inelasticity. We consider in particular the coupling of viscoplasticity with anisotropic continuum damage whereby both, proportional and kinematic hardening are taken into account. As a main advantage of the proposed formulation standard continuum damage models with respect to a fictitious isotropic configuration can be adopted and conveniently extended to anisotropic continuum damage. The key assumption is based on the introduction of a damage tangent map that acts as an affine pre-deformation. Conceptually speaking, we deal with an Euclidian space with respect to a non-constant metric. The evolution of this field is directly related to the degradation of the material and allows the modeling of specific classes of elastic anisotropy. In analogy to the damage mapping we introduce an internal variable that determines a back-stress tensor via a hyperelastic format and therefore enables the incorporation of plastic anisotropy. Several numerical examples underline the applicability of the proposed finite strain framework.     

On the Comparison of Two Strategies to Formulate Orthotropic Hyperelasticity

The main goal of this work is to clarify the relation between two strategies to formulate constitutive equations for orthotropic materials at large strains. On the one hand, the classical approach is based on the incorporation of structural tensors into the free energy function via an enriched set of invariants. On the other hand, a fictitious isotropic configuration is introduced which renders an anisotropic, undeformed reference configuration via an appropriate linear tangent map. This formulation results in a reduced (with respect to the more general setting based on structural tensors) but nevertheless physically motivated set of invariants which are related to the invariants defined by structural tensors. As a main conceptual advantage standard isotropic constitutive equations can be applied and moreover, due to the reduced set of physically motivated invariants, the numerical treatment within a finite element setting becomes manageable.     

A model framework for anisotropic damage coupled to crystal (visco)plasticity

We develop a model framework for anisotropic damage coupled to crystal (visco)plasticity, which is based on the concept of a fictitious (undamaged) configuration. The theoretical setting is that of finite strains, which is natural when studying crystal inelasticity even in the case of actual small strains. It turns out that the evolution law for damage, which reflects degradation in the slip planes and which is the key new relation, bears strong resemblance with the inelastic flow rule. Some numerical results showing qualitatively the anisotropic development of damage concludes the paper.     

Strain localization analysis using a large deformation anisotropic elastic–plastic model coupled with damage

Sheet metal forming processes generally involve large deformations together with complex loading sequences. In order to improve numerical simulation predictions of sheet part forming, physically-based constitutive models are often required. The main objective of this paper is to analyze the strain localization phenomenon during the plastic deformation of sheet metals in the context of such advanced constitutive models. Most often, an accurate prediction of localization requires damage to be considered in the finite element simulation. For this purpose, an advanced, anisotropic elastic–plastic model, formulated within the large strain framework and taking strain-path changes into account, has been coupled with an isotropic damage model. This coupling is carried out within the framework of continuum damage mechanics. In order to detect the strain localization during sheet metal forming, Rice’s localization criterion has been considered, thus predicting the limit strains at the occurrence of shear bands as well as their orientation. The coupled elastic–plastic-damage model has been implemented in Abaqus/implicit. The application of the model to the prediction of Forming Limit Diagrams (FLDs) provided results that are consistent with the literature and emphasized the impact of the hardening model on the strain-path dependency of the FLD. The fully three-dimensional formulation adopted in the numerical development allowed for some new results – e.g. the out-of-plane orientation of the normal to the localization band, as well as more realistic values for its in-plane orientation.     

Nonlocal continuum theory of anisotropically damaged metals

The paper deals with a consistent and systematic general framework for the development of anisotropic continuum damage in ductile metals based on thermodynamic laws and nonlocal theories. The proposed model relies on finite strain kinematics based on the consideration of damaged as well as fictitious undamaged configurations related via metric transformation tensors which allow for the interpretation of damage tensors. The formulation is accomplished by rate-independent plasticity using a nonlocal yield condition of Drucker–Prager type, anisotropic damage based on a nonlocal damage growth criterion as well as non-associated flow and damage rules. The nonlocal theory of inelastic continua is established to be able to take into account long-range microstructural interaction. The approach incorporates macroscopic interstate variables and their higher-order gradients which properly describe the change in the internal structure and investigate the size effect of statistical inhomogeneity of the heterogeneous material. The idea of bridging length-scales is made by using higher-order gradients in the evolution equations of the equivalent inelastic strain measures which leads to a system of elliptic partial differential equations which is solved using the finite difference method at each iteration of the loading step and the displacement-based finite element procedure is governed by the standard principle of virtual work. Numerical simulations of the elastic–plastic deformation behavior of damaged solids demonstrate the efficiency of the formulation. Tension tests undergoing large strains are used to investigate the damage growth in high strength steel. The influence of various model parameters on the prediction of the deformation and localization of ductile metals is discussed.     

Thermal convection around obstacles: the case of Sierpinski carpets

Measurements of free convection velocity profiles by laser Doppler velocimetry in a cavity containing Plexiglas reconstructed Sierpinski carpets are compared with computed profiles using the SIMPLER numerical code applied to the Navier–Stokes equations. This first step validates the numerical code into which two thermal conductivities are used (that of the liquid and that of the solid), together with two viscosities (that of the liquid and a fictitious high viscosity of the order of 10³⁰ for the solid). Next, the code is used for a network of Sierpinski carpets, allowing the evaluation of a seepage velocity from the Navier–Stokes equations.     

Equations of the mechanics of porous media saturated with liquid and gas

The approach proposed by Podil'chuk [1] is used to derive a system of equations of motion for saturated porous media, allowance being made for the mutual influence of the solid, liquid, and gas phases. The permeabilities of the anisotropic porous medium are assumed to depend on the direction. It is shown that when there are no gas phases and the liquid is incompressible the system of equations reduces to the general equations of the theory of elasticity of an anisotropic body with fictitious stress components. For a porous medium saturated with liquid, the relationships between the permeabilities and the anisotropy constants are obtained. The motion of liquid in an elastic porous medium in the form of an orthotropic cylindrical region with a cavity in the form of a circular cylinder is considered as an example.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 82–87, July–August, 1981.     

On the modelling of anisotropic elastic and inelastic material behaviour at large deformation

The purpose of this work is the formulation and discussion of an approach to the modelling of anisotropic elastic and inelastic material behaviour at large deformation. This is done in the framework of a thermodynamic, internal-variable-based formulation for such a behaviour. In particular, the formulation pursued here is based on a model for plastic or inelastic deformation as a transformation of local reference configuration for each material element. This represents a slight generalization of its modelling as an elastic material isomorphism pursued in earlier work, allowing one in particular to incorporate the effects of isotropic continuum damage directly into the formulation. As for the remaining deformation- and stress-like internal variables of the formulation, these are modelled in a fashion formally analogous to so-called structure tensors. On this basis, it is shown in particular that, while neither the Mandel nor back stress is generally so, the stress measure thermodynamically conjugate to the plastic “velocity gradient”, containing the difference of these two stress measures, is always symmetric with respect to the Euclidean metric, i.e., even in the case of classical or induced anisotropic elastic or inelastic material behaviour. Further, in the context of the assumption that the intermediate configuration is materially uniform, it is shown that the stress measure thermodynamically conjugate to the plastic velocity gradient is directly related to the Eshelby stress. Finally, the approach is applied to the formulation of metal plasticity with isotropic kinematic hardening.     

An Eulerian formulation for large deformations of anisotropic elastic and viscoelastic solids and viscous fluids

An Eulerian formulation has been developed for the constitutive response of a group of materials that includes anisotropic elastic and viscoelastic solids and viscous fluids. The material is considered to be a composite of an elastic solid and a viscous fluid. Evolution equations are proposed for a triad of vectors m _i that represent the stretches and orientations of material line elements in the solid component. Evolution equations for an orthonormal triad of vectors s _i are also proposed to characterize anisotropy of the fluid component. In particular, for an elastic solid it is shown that the material response is totally characterized by the functional form of the strain energy and by the current values of m _i , which are measurable in the current state of the material. Moreover, it is shown that the proposed Eulerian formulation removes unphysical arbitrariness of the choice of the reference configuration in the standard formulation of constitutive equations for anisotropic elastic solids.     

Two-phase flow over a surface with the formation of a liquid film by particle deposition

A consistent asymptotic theory of wall flow with film formation is constructed with reference to subsonic two-phase flow over a blunt body. The external flow problem and the film equations are solved simultaneously. This formulation of the problem supplements the investigation carried out in [4] in which particles deposited on the surface were assumed to disappear from the flow. It is shown that depending on the values of the governing parameters the flow in the film should be described either by the boundary layer equations or by the equations of creeping flow in a layer of unknown thickness. At the outer edge of the film the mass, momentum and energy fluxes found from the numerical solution of the flow problem are given. The case of isothermal film flow on the front of a sphere is investigated. The thickness of the film and the friction and heat transfer coefficients near the axis of symmetry are found for nonisothermal flows. The conditions under which the presence of a film significantly reduces the heat flow to the wall are determined. A similar formulation of the problem (but with another type of mass, momentum and energy sources at the outer edge) is encountered in problems of film condensation on a cold surface [5, 6].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 85–92, July–August, 1989.     

准脆性材料的弹塑性各向异性损伤耦合本构模型研究

针对准脆性材料的非线性特征：强度软化和刚度退化、单边效应、侧限强化和拉压软化、不可恢复变形、剪胀及非弹性体胀,在热动力学框架内,建立了准脆性材料的弹塑性与各向异性损伤耦合的本构关系。对准脆性材料的变形机理和损伤诱发的各向异性进行了诠释,并给出了损伤构形和有效构形中各物理量之间的关系。在有效应力空间内,建立了塑性屈服准则、拉压不同的塑性随动强化法则和各向同性强化法则。在损伤构形中,采用应变能释放率,建立了拉压损伤准则、拉压不同的损伤随动强化法则和各向同性强化法则。基于塑性屈服准则和损伤准则,构建了塑性势泛函和损伤势泛函,并由正交性法则,给出了塑性和损伤强化效应内变量的演化规律,同时,联立塑性屈服面和损伤加载面,给出了塑性流动和损伤演化内变量的演化法则。将损伤力学和塑性力学结合起来,建立了应变驱动的应力-应变增量本构关系,给出了本构数值积分的要点。以单轴加载-卸载往复试验识别和校准了本构材料常数,并对单轴单调试验、单轴加载-卸载往复试验、二轴受压、二轴拉压试验和三轴受压试验进行了预测,并与试验结果作了比较,结果表明,所建本构模型对准脆性材料的非线性材料性能有良好的预测能力。     

Micromorphic approach for finite gradient-elastoplasticity fully coupled with ductile damage: Formulation and computational aspects

It is well established that the use of inelastic constitutive equations accounting for induced softening, leads to pathological space (mesh) and time discretization dependency of the numerical solution of the associated Initial and Boundary Value Problem (IBVP). To avoid this drawback, many less or more approximate solutions have been proposed in the literature in order to regularize the IBVP and to obtain numerical solutions which are, at convergence, much less sensitive to the space and the time discretization. The basic idea behind these regularization techniques is the formulation of nonlocal constitutive equations by introducing some effects of characteristic lengths representing the materials microstructure. In this work, using the framework of generalized nonlocal continua, a thermodynamically-consistent micromorphic formulation using appropriate micromorphic state variables and their first gradients, is proposed in order to extend the classical local constitutive equations by incorporating appropriate characteristic internal lengths. The isotropic damage, the isotropic and the kinematic hardenings are supposed to carry the targeted micromorphic effects. First the theoretical aspects of this fully coupled micromorphic formulation is presented in details and the proposed generalized balance equations as well as the fully coupled micromorphic constitutive equations deduced. The associated numerical aspects in the framework of the classical Galerkin-based FE formulation are briefly discussed in the special case of micromorphic damage. Specifically, the formulation of 2D finite elements with additional degrees of freedom (d.o.f.), the dynamic explicit global resolution scheme as well as the local integration scheme, to compute the stress tensor and the state variables at each integration point of each element, are presented. Application is made to the typical uniaxial tension specimen under plane strain conditions in order to chow the predictive capabilities of the proposed micromorphic model, particularly against its ability to give (at convergence) a mesh independent solution even for high values of the ductile damage (i.e., the macroscopic cracks).     

A co‐ordinate system independent streamwise upwind method for fluid flow computation

A new co‐ordinate invariant streamwise upwind formulation for convection dominated flows is developed. The eddy diffusivity/viscosity is added directly to the equations in order to remove the oscillations in the solution. The equations then can be solved by any high‐order scheme and the solution retains the accuracy of the high‐order scheme. The accuracy and reduced lateral thickness growth rate are demonstrated with several numerical examples, including pure convective flows and lid‐driven cavity flow. The lateral spreading due to the numerical diffusion is controlled by the anisotropic tensor. Copyright © 2004 John Wiley & Sons, Ltd.     

Non-isothermal polymer melt flow in sudden expansions

This paper presents numerical results for laminar, incompressible and non-isothermal polymer melt flow in sudden expansions. The mathematical model includes the mass, momentum and energy conservation laws within the framework of a generalized Newtonian formulation. Two constitutive relations are adopted to describe the non-Newtonian behavior of the flow, namely Cross and Modified Arrhenius Power-Law models. The governing equations are discretized using the finite difference method based on central, second-order accurate formulas for both convective and diffusive terms. The pressure–velocity coupling is treated by solving a Poisson equation for pressure. The results are presented for two commercial polymers and demonstrate that important flow parameters, such as pressure drop and viscosity distribution, are strongly affected by heat transfer features.     

Invariant formulation of hyperelastic transverse isotropy based on polyconvex free energy functions

In this paper we propose a formulation of polyconvex anisotropic hyperelasticity at finite strains. The main goal is the representation of the governing constitutive equations within the framework of the invariant theory which automatically fulfill the polyconvexity condition in the sense of Ball in order to guarantee the existence of minimizers. Based on the introduction of additional argument tensors, the so-called structural tensors, the free energies and the anisotropic stress response functions are represented by scalar-valued and tensor-valued isotropic tensor functions, respectively. In order to obtain various free energies to model specific problems which permit the matching of data stemming from experiments, we assume an additive structure. A variety of isotropic and anisotropic functions for transversely isotropic material behaviour are derived, where each individual term fulfills a priori the polyconvexity condition. The tensor generators for the stresses and moduli are evaluated in detail and some representative numerical examples are presented. Furthermore, we propose an extension to orthotropic symmetry.     

A FE formulation for elasto-plastic materials with planar anisotropic yield functions and plastic spin

In this article a stress integration algorithm for shell problems with planar anisotropic yield functions is derived. The evolution of the anisotropy directions is determined on the basis of the plastic and material spin. It is assumed that the strains inducing the anisotropy of the pre-existing preferred orientation are much larger than subsequent strains due to further deformations. The change of the locally preferred orientations to each other during further deformations is considered to be neglectable. Sheet forming processes are typical applications for such material assumptions. Thus the shape of the yield function remains unchanged. The size of the yield locus and its orientation is described with isotropic hardening and plastic and material spin.The numerical treatment is derived from the multiplicative decomposition of the deformation gradient and thermodynamic considerations in the intermediate configuration. A common formulation of the plastic spin completes the governing equations in the intermediate configuration. These equations are then pushed forward into the current configuration and the elastic deformation is restricted to small strains to obtain a simple set of constitutive equations. Based on these equations the algorithmic treatment is derived for planar anisotropic shell formulations incorporating large rotations and finite strains. The numerical approach is completed by generalizing the Return Mapping algorithm to problems with plastic spin applying Hill’s anisotropic yield function. Results of numerical simulations are presented to assess the proposed approach and the significance of the plastic spin in the deformation process.     

A multi-field incremental variational framework for gradient-extended standard dissipative solids

The paper presents a constitutive framework for solids with dissipative micro-structures based on compact variational statements. It develops incremental minimization and saddle point principles for a class of gradient-type dissipative materials which incorporate micro-structural fields (micro-displacements, order parameters, or generalized internal variables), whose gradients enter the energy storage and dissipation functions. In contrast to classical local continuum approaches to inelastic solids based on locally evolving internal variables, these global micro-structural fields are governed by additional balance equations including micro-structural boundary conditions. They describe changes of the substructure of the material which evolve relatively to the material as a whole. Typical examples are theories of phase field evolution, gradient damage, or strain gradient plasticity. Such models incorporate non-local effects based on length scales, which reflect properties of the material micro-structure. We outline a unified framework for the broad class of first-order gradient-type standard dissipative solids. Particular emphasis is put on alternative multi-field representations, where both the microstructural variable itself as well as its dual driving force are present. These three-field settings are suitable for models with threshold- or yield-functions formulated in the space of the driving forces. It is shown that the coupled macro- and micro-balances follow in a natural way as the Euler equations of minimization and saddle point principles, which are based on properly defined incremental potentials. These multi-field potential functionals are outlined in both a continuous rate formulation and a time-space-discrete incremental setting. The inherent symmetry of the proposed multi-field formulations is an attractive feature with regard to their numerical implementation. The unified character of the framework is demonstrated by a spectrum of model problems, which covers phase field models and formulations of gradient damage and plasticity.     

Modeling stress state dependent damage evolution in a cast Al–Si–Mg aluminum alloy

Internal state variable rate equations are cast in a continuum framework to model void nucleation, growth, and coalescence in a cast Al–Si–Mg aluminum alloy. The kinematics and constitutive relations for damage resulting from void nucleation, growth, and coalescence are discussed. Because damage evolution is intimately coupled with the stress state, internal state variable hardening rate equations are developed to distinguish between compression, tension, and torsion straining conditions. The scalar isotropic hardening equation and second rank tensorial kinematic hardening equation from the Bammann–Chiesa–Johnson (BCJ) Plasticity model are modified to account for hardening rate differences under tension, compression, and torsion. A method for determining the material constants for the plasticity and damage equations is presented. Parameter determination for the proposed phenomenological nucleation rate equation, motivated from fracture mechanics and microscale physical observations, involves counting nucleation sites as a function of strain from optical micrographs. Although different void growth models can be included, the McClintock void growth model is used in this study. A coalescence model is also introduced. The damage framework is then evaluated with respect to experimental tensile data of notched Al–Si–Mg cast aluminum alloy specimens. Finite element results employing the damage framework are shown to illustrate its usefulness.