Hysteresis in multi-stable lattices with non-local interactions

We study a model for martensitic phase transitions represented by a lattice of mass points connected by bi-stable nearest neighbor (NN) springs and harmonic next to nearest neighbor (NNN) springs. Our main assumption of weak NNN interactions allows us to obtain a fully analytical representation of the quasistatic evolution of the overdamped system, including both the ‘non-local’ interaction with the external load and the presence of imperfections.This simple model reproduces the experimental observation of different evolution strategies, with internal or boundary nucleation and with the possibility of one or more coherently propagating phase fronts. The model describes also the observation of a Peierls stress higher or lower than the nucleation stress. We show that all these properties are also preserved in the ‘continuum’ limit.     

A less hypothetical perspective on rate-independent continuum theory of metal plasticity

In the framework of rate-independent theory of metal plasticity the hypothesis of maximum plastic dissipation is commonly applied to derive the associated flow-rule and to prove convexity of the yield surface in stress space. Alternatively, Drucker’s postulate of material stability may be used to prove convexity. Both hypotheses appear reasonable, but it is appealing to derive the basic framework for rate-independent metal plasticity without involving additional hypotheses.In the present article it will be shown that indeed both hypotheses can be dropped and an alternative way is established to derive the associated flow-rule and to prove convexity of the yield surface. It turns out that the associated flow-rule as well as the convexity of the yield surface are intrinsic features of the proposed theoretical framework.     

On the stored and dissipated energies in heterogeneous rate-independent systems: theory and simple examples

The aim of the present work is to determine the amount of dissipated and stored energies in structures containing frictional cracks and elasto-plastic zones. The proposed theory combines micromechanical and thermodynamic tools to calculate both energies. Using simple examples, it is shown that the Taylor–Quinney coefficient is not a constant, and can be much less than the values usually considered (i.e. close to unity).      

Modeling rate-independent hysteresis in large deformations of preconditioned soft tissues

A phenomenological model is proposed for characterizing rate-independent hysteresis exhibited by preconditioned soft tissues. The preconditioned tissue is modeled as an isotropic composite of a hyperelastic component and a dissipative (inelastic) component. Specifically, the constitutive equations are hyperelastic in the sense that the stress is determined by derivatives of a strain energy function. Inelasticity of the dissipative component is controlled by a yield function with different functional forms for the hardening variable during deformation loading and unloading. The constitutive equations proposed in this paper are simple. In particular, they depend on only seven material constants: three controlling the response of the elastic component and the remainder controlling the response of the dissipative component. More importantly, the material constants can be determined to match rather general loading and unloading behavior. It is observed that the hysteretic response of the model compares well with experimental data for passive uniaxial loading/unloading of Manduca muscle. Moreover, the present model treats partial loading and reloading of preconditioned tissue as elastic–plastic response, which is different from the treatment of pseudo-elastic models used in the literature.     

A simple theory of strain gradient plasticity based on stress-induced anisotropy of defect diffusion

A new strain gradient plasticity theory is formulated to accommodate more than one material length parameter. The theory is an extension of the classical J₂ flow theory of metal plasticity to the micron scale. Distinctive features of the proposed theory as compared to other existing theories are the simplicities of mathematical formulation, numerical implementation and physical interpretation.     

Discrete dislocation plasticity analysis of the wedge indentation of films

The plane strain indentation of single crystal films on a rigid substrate by a rigid wedge indenter is analyzed using discrete dislocation plasticity. The crystals have three slip systems at ±35.3^° and 90^° with respect to the indentation direction. The analyses are carried out for three values of the film thickness, 2, 10 and , and with the dislocations all of edge character modeled as line singularities in a linear elastic material. The lattice resistance to dislocation motion, dislocation nucleation, dislocation interaction with obstacles and dislocation annihilation are incorporated through a set of constitutive rules. Over the range of indentation depths considered, the indentation pressure for the 10 and thick films decreases with increasing contact size and attains a contact size-independent value for contact lengths . On the other hand, for the films, the indentation pressure first decreases with increasing contact size and subsequently increases as the plastic zone reaches the rigid substrate. For the 10 and thick films sink-in occurs around the indenter, while pile-up occurs in the film when the plastic zone reaches the substrate. Comparisons are made with predictions obtained from other formulations: (i) the contact size-independent indentation pressure is compared with that given by continuum crystal plasticity; (ii) the scaling of the indentation pressure with indentation depth is compared with the relation proposed by Nix and Gao [1998. Indentation size effects in crystalline materials: a law for strain gradient plasticity. J. Mech. Phys. Solids 43, 411-423]; and (iii) the computed contact area is compared with that obtained from the estimation procedure of Oliver and Pharr [1992. An improved technique for determining hardness and elastic-modulus using load and displacement sensing indentation experiments, J. Mater. Res. 7, 1564-1583].     

Hyperbolicity in Extended Thermodynamics of Fermi and Bose gases

We consider the balance system of Extended Thermodynamics with 13 Moments in the case of Fermi and Bose gases, for processes not far from equilibrium. In this case, the hyperbolicity of the differential system holds only in a neighborhood of the equilibrium state. The main aim of the paper is to evaluate the hyperbolicity region of the differential system. The knowledge of this region in the state variables is mandatory to check the admissibility of the solutions and the corresponding boundary and Cauchy data in the limit of the approximation considered. The results are obtained through numerical evaluations of the Fermi and Bose integral functions that appear in the characteristic polynomial. Particular attention is devoted to the completely degenerate case when Fermi gas reaches the 0 K and when the Bose gas is in proximity of the transition temperature T _c . In these limiting cases, the hyperbolicity requirement is lost according to previous results. In the last section we make use of the Maxwellian iteration in order to evaluate the heat conductivity and the viscosity for the degenerate Fermi and Bose gas.Received: 2 March 2004, Accepted: 26 March 2004, Published online: 25 June 2004 Correspondence to: T. Ruggeri     

Discrete dislocation plasticity analysis of the grain size dependence of the flow strength of polycrystals

The grain size dependence of the flow strength of polycrystals is analyzed using plane strain, discrete dislocation plasticity. Dislocations are modeled as line singularities in a linear elastic solid and plasticity occurs through the collective motion of large numbers of dislocations. Constitutive rules are used to model lattice resistance to dislocation motion, as well as dislocation nucleation, dislocation annihilation and the interaction with obstacles. The materials analyzed consist of micron scale grains having either one or three slip systems and two types of grain arrangements: either a checker-board pattern or randomly dispersed with a specified volume fraction. Calculations are carried out for materials with either a high density of dislocation sources or a low density of dislocation sources. In all cases, the grain boundaries are taken to be impenetrable to dislocations. A Hall–Petch type relation is predicted with Hall–Petch exponents ranging from ≈0.3 to ≈1.6 depending on the number of slip systems, the grain arrangement, the dislocation source density and the range of grain sizes to which a Hall–Petch expression is fit. The grain size dependence of the flow strength is obtained even when no slip incompatibility exists between grains suggesting that slip blocking/transmission governs the Hall–Petch effect in the simulations.     

A consistent plasticity theory of incompressible and hydrostatic pressure sensitive metals

In this article a rate-independent plasticity theory is presented that aims at describing the plastic behavior of incompressible, but hydrostatic pressure sensitive metals as experimentally observed by Spitzig and Richmond [Spitzig, W.A., Richmond, O., 1984. The effect of pressure on the flow stress of metals. Acta Metallurgica 32, 457–463]. The presented theory leads consistently to a non-associated flow-rule, without the need of introducing an arbitrary potential function.     

An Increasing Number of Moments in Relativistic Extended Thermodynamics

Thermodynamic models with an increasing number of moments are here compared, in the frame-work of relativistic extended thermodynamics. They constitute a family of models depending on a parameter n. In these models, some constitutive functions occur; a part of them are determined by imposing that they satisfy exactly the entropy and the relativity principles. It is pleasant to see how many elegant symmetries, which are present in the pioneering work by Liu, Müller and Ruggeri about the 14 fields model, nicely extend also to the general case.The other constitutive functions are production terms and they must satisfy a residual inequality which is consequence of the entropy principle; other restrictions on their generality are imposed by considering a well-known iterative procedure, and one of its possible generalizations, and by imposing that the results dont depend on n. It is proved that all these conditions are consistent.     

On the uniqueness of a rate-independent plasticity model for single crystals

This paper presents the uniqueness and existence conditions for a rate-independent plasticity model for single crystals under a general stress state. The model is based on multiple slips on three-dimensional slip systems. The uniqueness condition for the plastic slips in a single crystal with nonlinear hardening is derived using the implicit function theorem. The uniqueness condition is the non-singularity of a matrix defined by the Schmid tensors, the elasticity, and the hardening rates of the slip systems. When this matrix becomes singular, the limitations on the loading paths that can be accommodated by the active slip systems (i.e., the existence conditions) are also given explicitly. For the compatible loading paths, a particular solution is selected by requiring the solution vector to be orthogonal to the null space of the singular coefficient matrix. The paper also presents a fully implicit algorithm for the plasticity model. Numerical examples of an fcc copper single crystal under cyclic loadings (pure shear and uniaxial strain) are presented to demonstrate the main features of the algorithm.     

Thermodynamic and relaxation-based modeling of the interaction between martensitic phase transformations and plasticity

This paper focuses on the issue plasticity within the framework of a micromechanical model for single-crystal shape-memory alloys. As a first step towards a complete micromechanical formulation of such models, we work with classical J₂-von Mises-type plasticity for simplicity. The modeling of martensitic phase transitions is based on the concept of energy relaxation (quasiconvexification) in connection with evolution equations derived from inelastic potentials. Crystallographic considerations lead to the derivation of Bain strains characterizing the transformation kinematics. The model is derived for arbitrary numbers of martensite variants and thus can be applied to any shape-memory material such as CuAlNi or NiTi. The phase transition model captures effects like tension/compression asymmetry and transformation induced anisotropy. Additionally, attention is focused on the interaction between phase transformations and plasticity in terms of the inheritance of plastic strain. The effect of such interaction is demonstrated by elementary numerical studies.     

A simple plasticity model for prediction of non-coaxial flow of sand

A bounding surface plasticity model for non-coaxiality, another aspect of anisotropic behavior of sands under rotation of principal stress axes; is developed in the critical state framework. Numerous experimental evidences exist that corroborate dependence of plastic shear strain rate direction on inherent fabric anisotropy. At first, general expressions for plastic strain rate with respect to possible emerge of non-coaxial flow are obtained. Consequently, using an anisotropy state parameter that is specially developed for this model and accounts for the interaction between imposed loading and soil fabric; effect of anisotropy on plastic flow direction is taken into account. Besides, novel circumstances are proposed for plastic modulus and dilatancy under rotation of principal stress axes. Finally, it is shown that the model is able to simulate successfully the non-coaxial behavior of sands subjected to principal stress axes rotation.     

A review of some plasticity and viscoplasticity constitutive theories

The purpose of the present review article is twofold:

•

recall elementary notions as well as the main ingredients and assumptions of developing macroscopic inelastic constitutive equations, mainly for metals and low strain cyclic conditions. The explicit models considered have been essentially developed by the author and co-workers, along the past 30 years;     

Phase-field slip-line theory of plasticity

A variational approach to determine the deformation of an ideally plastic substance is proposed by solving a sequence of energy minimization problems under proper conditions to account for the irreversible character of plasticity. The flow is driven by the local transformation of elastic strain energy into plastic work on slip surfaces, once that a certain energetic barrier for slip activation has been overcome. The distinction of the elastic strain energy into spherical and deviatoric parts is used to incorporate in the model the idea of von Mises plasticity and isochoric plastic strain. This is a “phase field model” because the matching condition at the slip interfaces is substituted by the evolution of an auxiliary phase field that, similar to a damage field, is unitary on the elastic phase and null on the yielded phase. The slip lines diffuse in bands, whose width depends upon a material length-scale parameter.Numerical experiments on representative problems in plane strain give solutions with noteworthy similarities with the results from classical slip-line field theory, but the proposed model is much richer because, accounting for elastic deformations, it can describe the formation of slip bands at the local level, which can nucleate, propagate, widen and diffuse by varying the boundary conditions. In particular, the solution for a long pipe under internal pressure is very different from the one obtainable from the classical macroscopic theory of plasticity. For this case, the location of the plastic bands may be an insight to explain the premature failures that are sometimes encountered during the manufacturing process. This practical example enhances the importance of this new theory based on the mathematical sciences.     

Reproduction of solutions of bidimensional ideal plasticity

The symmetries of a system of differential equations allowed the transformation of its solutions to a solution of this system. New analytical exact solutions of a system of two-dimensional ideal plasticity equations were constructed from two well-known solutions, that for a circular cavity stressed by normal pressure, and Prandtl's solution for a block compressed between perfectly rough plates, for the case where the thickness of the block was rather small. A mechanical sense of new solutions was discussed.     

The bimodal theory of plasticity: A form-invariant generalisation

The bimodal plasticity model of fibre-reinforced materials is currently available and applicable only in association with thin-walled fibrous composites containing a family of straight fibres which are conveniently assumed parallel with the x₁-axis of an appropriately chosen Cartesian co-ordinate system. Based on reliable experimental evidence, the model suggests that plastic slip in the composite operates in two distinct modes; the so-called matrix dominated mode (MDM) which depends on a matrix yield stress, and the fibre dominated mode (FDM) which depends also on the fibre yield stress. Each mode is activated by different states of applied stress, has its own yield surface (or surfaces) in the stress space and has its own segment on the overall yield surface of the composite. This paper employs theory of tensor representations and produces a form-invariant generalisation of both modes of the model. This generalisation furnishes the model with direct applicability to relevant plasticity problems, regardless of the shape of the fibres or the orientation of the co-ordinate system. It thus provides a proper mathematical foundation that underpins important physical concepts associated with the model while it also elucidates several technical relevant issues. A most interesting of those issues is the revelation that activation of the MDM plastic regime is possible only if the applied stress state allows the fibres to act like they are practically inextensible. Moreover, activation of the more dominant, between the two MDM plastic slip branches is possible only if conditions of material incompressibility hold, in addition to the implied condition of fibre inextensibility. A direct mathematical connection is thus achieved between basic, experimentally verified concepts of the bimodal plasticity model and a relevant mathematical model originated earlier from the theory of ideal fibre-reinforced materials. An additional issue of discussion involves the number of independent yield stress parameters that the bimodal theory needs to take into consideration. Moreover, an analytical expression is provided of a relatively simple mathematical surface that possesses all known features of the FDM yield surface; currently captured with the aid of both experimental and computational means. The present study is guided by the existing relevant experimental evidence which, however, is principally associated with the plastic behaviour of solids reinforced by strong fibres. Nevertheless, several of the outlined developments are expected to be applicable to composite materials containing a single family of more compliant or even weak fibres.     

The role of the weakest-link mechanism in controlling the plasticity of micropillars

We present a computational study on the effects of sample size on the strength and plastic flow characteristics of micropillars under compression loading. We conduct three-dimensional simulations using the parametric dislocation dynamics coupled with the boundary element method. Two different loading techniques are performed. The plastic flow characteristics as well as the stress-strain behavior of simulated micropillars are shown to be in general agreement with experimental observations. The flow strength versus the diameter of the micropillar follows a power law with an exponent equal to -0.69. A stronger correlation is observed between the flow strength and the average length of activated dislocation sources. This relationship is again a power law, with an exponent -0.85. Simulation results with and without the activation of cross-slip are compared. Discontinuous hardening is observed when cross-slip is included. Experimentally observed size effects on plastic flow and work-hardening are consistent with a “weakest-link activation mechanism”.     

Dry Friction Model of Percussive Drilling

It is postulated that the main mechanism of the enhancement of material removal rate (MRR) in percussive drilling is associated with generating impact forces, which act on the workpiece and help to develop micro-cracking in the cutting zone. The inherent non-linearity of the discontinuous impact process is modelled as a frictional pair, to generate the pattern of the impact forces. A novel formula for calculating the MRR is proposed, which explains the experimentally observed fall in MRR at higher static forces.