A critical review of the state of finite plasticity

The object of this paper is to provide a critical review of the current state of plasticity in the presence of finite deformation. In view of the controversy regarding a number of fundamental issues between several existing schools of plasticity, the areas of agreement are described separately from those of disagreement. Attention is mainly focussed on the purely mechanical, rate-independent, theory of elastic-plastic materials, although closely related topics such as rate-dependent behavior, thermal effects, experimental and computational aspects, microstructural effects and crystal plasticity are also discussed and potentially fruitful directions are identified.A substantial portion of this review is devoted to the area of disagreement that covers a detailed presentation of argument(s), bothpro andcon, for all of the basic constitutive ingredients of the rate-independent theory such as the primitive notion or definition of plastic strain, the structure of the constitutive equation for the stress response, the yield function, the loading criteria and the flow and the hardening rules. The majority of current research in finite plasticity theory, as with its infinitesimal counterpart, still utilizes a (classical) stress-based approach which inherently possesses some shortcomings for the characterization of elastic-plastic materials. These and other anomalous behavior of a stress-based formulation are contrasted with the more recent strain-based formulation of finite plasticity. A number of important features and theoretical advantages of the latter formulation, along with its computational potential and experimental interpretation, are discussed separately.     

The generalized Masing models for deteriorating hysteresis and cyclic plasticity

The Masing model originally proposed for one-dimensional cyclically stabilized hysteretic behavior is generalized for deteriorating hysteresis and for cyclic plasticity, with system degradation and cyclic hardening effects taken into account. The generalization into the multi-dimensional case is based on the concept of a universal stress–strain curve and the associated effective stress and effective strain. Numerical simulations confirm the validity of the generalized models that are conceptually simple and parametrically parsimonious. The generalization method employed also provides a unifying way of extending 1-D hysteretic models to multi-dimensional constitutive models in cyclic plasticity.     

Simulation of a chassis bushing with regard to strain induced softening of filled rubber

A material model is presented, which describes the material behavior of rubbery materials in the case of uniaxial tension/compression. It includes the characteristic nonlinearity of rubber at large strains as well as strain induced softening called Mullins effect. The generalization to arbitrary deformations is done with the concept of representative directions, published by Freund & Ihlemann [2] utilizing a numerical integration on a unit sphere. An FE simulation of a chassis bushing shows that the Mullins effect is reproduced appropriately, verifying the preceding material characterization. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Microplane modeling of cyclic behavior of concrete: a gradient plasticity-damage formulation

A microplane model is developed to simulate the behavior of concrete under cyclic loading conditions. Pure damage mechanics or pure plasticity models yield satisfactory results for concrete under monotonic loading but cannot capture correctly the unloading and reloading response. Therefore, coupling damage and plasticity is necessary for accurate constitutive modeling of concrete. The microplane model offers a straightforward approach to simulate induced anisotropy by formulating the material laws on many randomly oriented planes. Distinguishing between compression and tension response using the proper plastic yield function and damage laws is considered. Furthermore, gradient enhancement is employed to handle the pathological mesh sensitivity related to strain softening. The new formulation is implemented within a 3D finite element code and a numerical example is simulated and compared to experiments in order to evaluate the capabilities of the model. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A finite element model of localized deformation in frictional materials taking a strong discontinuity approach

A finite element model of localized deformation in frictional materials taking a strong discontinuity approach is presented. A rate-independent, non-associated, strain-softening Drucker–Prager plasticity model is formulated in the context of strong discontinuities and implemented along with an enhanced quadrilateral element within the framework of an assumed enhanced strain finite element method. For simple model problems such as uniform compression, the strong discontinuity approach has been shown to lead to mesh-independent finite element solutions when localized deformation is present. In this paper, a finite element analysis of localized deformation occurring in a more complex model problem of slope stability is conducted in a nearly mesh-independent manner. The effect of dilatancy on the orientation of slip lines is demonstrated for the slope stability problem.     

A constitutive model for continuous fiber reinforced polymer plies

In the present paper a constitutive model is reviewed which can be used to predict the non-linear behavior of continuous fiber reinforced laminates with polymeric matrix materials. The constitutive model considers stiffness degradation and plastic strain accumulation at the length scale of the individual plies (laminae). These effects are modeled via two different phenomenological approaches, however, their interaction is considered when the constitutive equations are solved by an implicit integration scheme. To demonstrate the predictive capabilities of the individual model parts, examples are given where the above mentioned effects are decoupled. This way, their impact on the laminate's response can be studied independently. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The effect of precipitates on the evolution of a martensitic phase boundary

In this article, we study the role of defects in the quasistatic evolution of a martensitic phase boundary. The formulation of the model gives rise to a nonlocal free boundary problem, for which we present an implicit finite-time discretization. For an approximate model, assuming a nearly flat interface, we show that the phase boundary exhibits a sick-slip behavior in the presence of a heterogeneous environment, thus leading to a transition from viscous kinetics to rate-independent behavior. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Multiscale Modeling of Nanocrystalline Materials: A Variational Approach

This paper presents a variational multi-scale constitutive model in the finite deformation regime capable of capturing the mechanical behavior of nanocrystalline (nc) fcc metals. The nc-material is modeled as a two-phase material consisting of a grain interior (GI) phase and a grain boundary (GB) phase. A rate-independent isotropic porous plasticity model is employed to describe the GB phase, whereas a crystal-plasticity model which accounts for the transition from partial dislocation to full dislocation mediated plasticity is employed for the GI phase. Assuming the rule of mixtures, the overall behavior of a given grain is obtained via volume averaging. The scale transition from a single grain to a polycrystal is achieved by Taylor-type homogenization. It is shown that the proposed model is able to capture the inverse Hall-Petch effect. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On Implicit Constitutive Theories

In classical constitutive models such as the Navier-Stokes fluid model, and the Hookean or neo-Hookean solid models, the stress is given explicitly in terms of kinematical quantities. Models for viscoelastic and inelastic responses on the other hand are usually implicit relationships between the stress and the kinematical quantities. Another class of problems wherein it would be natural to develop implicit constitutive theories, though seldom resorted to, are models for bodies that are constrained. In general, for such materials the material moduli that characterize the extra stress could depend on the constraint reaction. (E.g., in an incompressible fluid, the viscosity could depend on the constraint reaction associated with the constraint of incompressibility. In the linear case, this would be the pressure.) Here we discuss such implicit constitutive theories. We also discuss a class of bodies described by an implicit constitutive relation for the specific Helmholtz potential that depends on both the stress and strain, and which does not dissipate in any admissible process. The stress in such a material is not derivable from a potential, i.e., the body is not hyperelastic (Green elastic).     

Specimen design for high precision tension-compression tests

In this contribution a new specimen, which enables high precision tension-compression testing, is presented. Due to a special mounting geometry, tests from a compression strain of −45 % up to a tension strain of 400 % can be performed with a nearly homogeneous deformation field within the measuring zone. Consequently, the mechanical behavior of rubber, which exhibits phenomena like Mullins effect, Payne effect, recovery and relaxation behavior, can be characterized within experimental investigations. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Micro-Mechanical Modelling of Anisotropic Time-Dependent Healing Effects in Rubber-like Materials

Filler-reinforced rubber-like materials demonstrate deformation induced softening, known as the Mullins effect. This softening can be reversible in a certain range (healing). The previously introduced concept of network evolution (NE) explains the Mullins effect as the result of a filler-polymer network rearrangement [1]. It is assumed that debonding of polymer chain segments from rigid aggregates does not effect the number of active segments. Furthermore, debonded chains are still active as a part of a longer chain. Now, considering the entropic vibration of polymer chains, entanglement of chains to the rough aggregate surfaces is assumed. This process inverse to the NE leads to a micro-mechanical model of time-dependent recovery of the deformation induced softening. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A finite elasto-plasticity model for thick adhesive layers based on generalized stress-strain measure

The collective term adhesives includes a wide field of materials with a diversity of different material properties. Regarding high-strength adhesives, the assumption of small strains often holds according to their brittle behavior. The experience with plasticity models based on the additive decomposition into elastic and inelastic strains indicates an appropriate approach to characterize such materials. In some cases, due to a more ductile material response, the assumption of infinitesimal strains is not valid anymore. In particular this is the case for high-strength adhesives with additives like rubber. But ductile behavior is also observed for specific stress states in one adhesive, e.g. when the behavior for tensile is quite brittle while large shear-strains could appear. The objective of this work is to overcome the theoretical restriction of small strains and to archive the practical experiences. For the failure criterion two stress invariants are used, which involves the hydrostatic pressure as well as the deviator stress state. The flowrule is introduced for the evolution of the inelastic variables. Herein the flow rule has to be of non-associated type to ensure the thermodynamical consistency of the model. The plasticity model also includes hardening as well as softening. The presented finite strain model makes use of the fact that the eigenvalues for Green-Lagrange strains and generalized strains are the same. Thus the limit of applicability is extended to finite strains. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Simulation of Asymmetric Effects in Plasticity

In the framework of plasticity the simulation of materials is addressed, which experimentally exhibit different behavior in different loading scenarios, such as tension, compression and shear. To this end an additive decomposition of the yield function and the plastic potential, is assumed into a sum of weighted stress mode related quantities. The characterization of the stress modes is obtained in the octahedral plane of the deviatoric stress space in terms of the Lode angle, such that stress mode dependent scalar weighting functions can be constructed. Verification of the proposed methodology is succeeded for simulation of the pseudoelastic behavior of shape memory alloys with different hardening characteristics in tension and shear. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A micromechanical model for the transformation induced plasticity in polycrystalline steels

Due to the effect of transformation induced plasticity (TRIP) , TRIP-steels are very promising materials, e.g. for the automobile industry. The material behavior is characterized by very complex inner processes, namely phase transformation coupled with plastic deformation and kinematic hardening. We establish a micromechanical model which uses the volume fractions of the single phases, the plastic strain and the hardening parameter in every grain of the polycrystalline material as internal variables. Furthermore, we apply the Principle of the Minimum of the Dissipation Potential to derive the associated evolution equations. The use of a coupled dissipation functional and a combined Voigt/Reuss bound directly results in coupled evolution equations for the internal variables and in one combined yield function. Additionally, we show numerical results which prove our model's ability to give a first prediction of the TRIP-steels' complex material behavior. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Incompatibility Induced Nonlocal Crystal Plasticity

Motivated by the fact that locally inhomogeneous elastic or plastic deformations may result in incompatibilities of the fictitious intermediate configuration a strain gradient crystal plasticity model is developed. Thereby incompatibilities can be accounted for and scale dependent material behavior, as also observed experimentally, is predictable. A nonlocal extension of existing local formulations is proposed which does not require additional boundary conditions and thus maintains the classical BVP structure. On the numerical side key developments are an extended FE-formulation for rate-(in)dependent strain gradient plasticity and a local FE-formulation which bases the gradient computation on an operator split combined with a smoothing algorithm. Comparative numerical studies for classical examples proove the superior efficiency of the second approach. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Existence and stability for a visco‐plastic material with a fractional constitutive equation

The well posedness of the evolutive problem for visco‐plastic materials represented by two different fractional constitutive equations is proved. We show that, for these materials, we can observe permanent deformations. So that, as it is usual in plasticity, when the stress goes to zero, then the strain assumes a constant nonzero behavior. Moreover, we prove the compatibility of our models with the classical laws of thermodynamics. For the second model, described through a fractional derivative with an exponential kernel, we obtain the exponential decay of the solutions by means of the semigroup theory.     

Mesoscopic model for ferromagnets with isotropic hardening

The proposed model combines tendency for minimization of Gibbs magnetic energy with the rate-independent maximum-dissipation mechanism that reflects the macroscopic quantity of energy required to change one pole of a magnet to another. This energy can increase within the evolution which is the effect like hardening in plasticity. The microstructure is described on a mesoscopic level in terms of Young measures. Such a mesoscopic, distributed-parameter model is formulated (and, after a suitable regularization), analyzed, discretized, implemented, and eventually tested computationally on a uni-axial magnet. The desired hysteretic macroscopic response, including effects as virgin curves and minor loops, is demonstrated.Received: December 3, 2002; revised: March 25, 2003     

Mesoscopic model for ferromagnets with isotropic hardening

The proposed model combines tendency for minimization of Gibbs magnetic energy with the rate-independent maximum-dissipation mechanism that reflects the macroscopic quantity of energy required to change one pole of a magnet to another. This energy can increase within the evolution which is the effect like hardening in plasticity. The microstructure is described on a mesoscopic level in terms of Young measures. Such a mesoscopic, distributed-parameter model is formulated (and, after a suitable regularization), analyzed, discretized, implemented, and eventually tested computationally on a uni-axial magnet. The desired hysteretic macroscopic response, including effects as virgin curves and minor loops, is demonstrated.     

Investigation of elastoplastic effects of cables under large spatial deformation

Cables are complex components consisting of a multi-layer structure and various materials. The structural setup includes for example conducting wires, isolating shields and protecting sheaths. This leads to several inelastic effects under large deformations like pull-out of wires, delamination of layers or friction between the constituents. The materials used in cables belong to different material classes and consequently show different behavior under load. Elastoplastic behavior has to be expected for metallic wires, whereas polymer layers behave viscoelastically. The combination of these inelastic effects caused by the structure and constituents of cables motivates the inclusion of inelasticity in the material model on a phenomenological level. Since cables are flexible, slender structures, they can be described physically correctly by the theory of Cosserat rods. In this context, the constitutive equations are formulated in terms of the sectional quantities. The related model parameters have to be determined in suitable experiments. As cables undergo large multiaxial deformations in applications, uniaxial experiments are not sufficient for their characterization. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A fluid mechanical model for granular flow in silos

Granular materials may display both solid and fluid like behaviour. For low densities and high strain rates as in avalanches or during the discharge of silos the behaviour is mainly governed by interparticle collisions. On the other hand, frictional contacts characterise the solid state which is represented within the framework of plasticity theory. A fluid like constitutive model describes granular materials when subjected to large deformations and high strain rates. It bases upon a modified viscoplastic model that is valid for both yielded and unyielded regions. The central idea is the distinction between fluid and solid regions by means of comparing actual shear stress and Coulomb yield stress. The application to the simultion of the discharge of silos shows the feasibility of the chosen method. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)