Ordinary state-based peridynamics for plastic deformation according to von Mises yield criteria with isotropic hardening

This study presents the ordinary state-based peridynamic constitutive relations for plastic deformation based on von Mises yield criteria with isotropic hardening. The peridynamic force density–stretch relations concerning elastic deformation are augmented with increments of force density and stretch for plastic deformation. The expressions for the yield function and the rule of incremental plastic stretch are derived in terms of the horizon, force density, shear modulus, and hardening parameter of the material. The yield surface is constructed based on the relationship between the effective stress and equivalent plastic stretch. The validity of peridynamic predictions is established by considering benchmark solutions concerning a plate under tension, a plate with a hole and a crack also under tension.     

Novel variational formulations for nonlocal plasticity

A nonlocal structural model of softening plasticity is considered in the framework of the internal variable theories of inelastic behaviours of associative type. The finite-step nonlocal structural problem in a geometrically linear range is formulated according to a backward difference scheme for time integration of the flow rule. The related finite-step variational formulation in the complete set of local and nonlocal state variables is recovered. A family of mixed nonlocal variational formulations, with different combinations of state variables, is provided starting from the general variational formulation. The specialization of a mixed variational formulation to existing nonlocal models of softening plasticity, assuming both linear and nonlinear constitutive behaviour, is provided to show the effectiveness of the theory.     

Three-step relaxed hybrid steepest-descent methods for variational inequalities

The classical variational inequality problem with a Lipschitzian and strongly monotone operator on a nonempty closed convex subset in a real Hilbert space is studied. A new three-step relaxed hybrid steepest-descent method for this class of variational inequalities is introduced.Strong convergence of this method is established under suitable assumptions imposed on the algorithm parameters.     

An improved proximal-based decomposition method for structured monotone variational inequalities

The proximal-based decomposition method was originally proposed by Chen and TebouUe(Math.Programming,1994,64:81-101 for solving convex minimization problems.This paper extends it to solving monotone variational inequalities associated with separable structures with the improvements that the restrictive assumptions on the involved parameters are much relaxed,and thus makes it practical to solve the subprob- lems easily.Without additional assumptions,global convergence of the new method is proved under the same mild assumptions on the problem's data as the original method.     

A variational multiscale constitutive model for nanocrystalline materials

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 phase and a grain boundary effected zone (GBAZ). A rate-independent isotropic porous plasticity model is employed to describe the GBAZ, whereas a crystal-plasticity model which accounts for the transition from partial dislocation to full dislocation mediated plasticity is employed for the grain interior. The constitutive models of both phases are formulated in a small strain framework and extended to finite deformation by use of logarithmic and exponential mappings. 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 where a log-normal grain size distribution is assumed. It is shown that the proposed model is able to capture the inverse Hall-Petch effect, i.e., loss of strength with grain size refinement. Finally, the predictive capability of the model is validated against experimental results on nanocrystalline copper and nickel.     

AN ITERATIVE METHOD FOR THE DISCRETE PROBLEMS OF A CLASS OF ELLIPTICAL VARIATIONAL INEQUALITIES

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Dual variational formulation for Trefftz finite element method of elastic materials

A dual variational principle is presented for Trefftz finite element analysis. The proof of the stationary conditions of the variational functional and the theorem on the existence of extremum are provided in this paper. They are boundary displacement condition, surface traction condition and interelement continuity condition. Based on the assumed intraelement and frame fields, element stiffness matrix equation is obtained which can easily be implemented into computer programs for numerical analysis with Trefftz finite element method. Two numerical examples are considered to illustrate the effectiveness and applicability of the proposed element model.     

The existence and uniqueness of solutions of generalized variational inequalities arising from elasticity with friction

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NEW SIMPLE SMOOTH MERIT FUNCTION FOR BOX CONSTRAINED VARIATIONAL INEQUALITIES AND DAMPED NEWTON TYPE METHOD

IntroductionLetX Rn,F:X→Rnbe given, the variational inequality, denoted byVI(X,F),is tofind a vectorx∈ Xsuch thatF(x)T(y-x)≥0, y∈ X. (1)DenoteN∶={1,2,…,n},whenX =[a,b]∶={x∈Rn|ai≤xi≤bi,i∈N},theVI(X,F)is called the box constrained variational ine     

On variational formulations with rigid-body constraints for finite elasticity: Applications to 2D and 3D finite element simulations

We investigate a recently proposed variational principle with rigid-body constraints and present an extension of its implementation in three dimensional finite elasticity problems. Through numerical examples, we illustrate that the proposed variational principle with rigid-body constraints is applicable to both single field and mixed finite elements of arbitrary order and geometry, e.g. triangular/tetrahedral and quadrilateral/hexagonal elements, in two and three dimensions. Moreover, we demonstrate that, as compared to the commonly adopted approach of discretizing the rigid domains with standard finite elements, the proposed formulation requires neither discretization nor numerical integration in the interior of each rigid domain. As a comparative result, the variational formulation may reduce the total number of degrees of freedom of the resulting finite element system and provide improved accuracy.     

A model-reduction approach in micromechanics of materials preserving the variational structure of constitutive relations

In 2003 the authors proposed a model-reduction technique, called the Nonuniform Transformation Field Analysis (NTFA), based on a decomposition of the local fields of internal variables on a reduced basis of modes, to analyze the effective response of composite materials. The present study extends and improves on this approach in different directions. It is first shown that when the constitutive relations of the constituents derive from two potentials, this structure is passed to the NTFA model. Another structure-preserving model, the hybrid NTFA model of Fritzen and Leuschner, is analyzed and found to differ (slightly) from the primal NTFA model (it does not exhibit the same variational upper bound character). To avoid the “on-line” computation of local fields required by the hybrid model, new reduced evolution equations for the reduced variables are proposed, based on an expansion to second order (TSO) of the potential of the hybrid model. The coarse dynamics can then be entirely expressed in terms of quantities which can be pre-computed once for all. Roughly speaking, these pre-computed quantities depend only on the average and fluctuations per phase of the modes and of the associated stress fields. The accuracy of the new NTFA-TSO model is assessed by comparison with full-field simulations. The acceleration provided by the new coarse dynamics over the full-field computations (and over the hybrid model) is then spectacular, larger by three orders of magnitude than the acceleration due to the sole reduction of unknowns.     

A large deformation theory for rate-dependent elastic–plastic materials with combined isotropic and kinematic hardening

We have developed a large deformation viscoplasticity theory with combined isotropic and kinematic hardening based on the dual decompositions F=F^eF^p [Kröner, E., 1960. Allgemeine kontinuumstheorie der versetzungen und eigenspannungen. Archive for Rational Mechanics and Analysis 4, 273–334] and [Lion, A., 2000. Constitutive modelling in finite thermoviscoplasticity: a physical approach based on nonlinear rheological models. International Journal of Plasticity 16, 469–494]. The elastic distortion F^e contributes to a standard elastic free-energy ψ^(e), while , the energetic part of F^p, contributes to a defect energy ψ^(p) – these two additive contributions to the total free energy in turn lead to the standard Cauchy stress and a back-stress. Since F^e=FF^p-1 and , the evolution of the Cauchy stress and the back-stress in a deformation-driven problem is governed by evolution equations for F^p and – the two flow rules of the theory.We have also developed a simple, stable, semi-implicit time-integration procedure for the constitutive theory for implementation in displacement-based finite element programs. The procedure that we develop is “simple” in the sense that it only involves the solution of one non-linear equation, rather than a system of non-linear equations. We show that our time-integration procedure is stable for relatively large time steps, is first-order accurate, and is objective.     

A phase-field theory of dislocation dynamics, strain hardening and hysteresis in ductile single crystals

A phase-field theory of dislocation dynamics, strain hardening and hysteresis in ductile single crystals is developed. The theory accounts for: an arbitrary number and arrangement of dislocation lines over a slip plane; the long-range elastic interactions between dislocation lines; the core structure of the dislocations resulting from a piecewise quadratic Peierls potential; the interaction between the dislocations and an applied resolved shear stress field; and the irreversible interactions with short-range obstacles and lattice friction, resulting in hardening, path dependency and hysteresis. A chief advantage of the present theory is that it is analytically tractable, in the sense that the complexity of the calculations may be reduced, with the aid of closed form analytical solutions, to the determination of the value of the phase field at point-obstacle sites. In particular, no numerical grid is required in calculations. The phase-field representation enables complex geometrical and topological transitions in the dislocation ensemble, including dislocation loop nucleation, bow-out, pinching, and the formation of Orowan loops. The theory also permits the consideration of obstacles of varying strengths and dislocation line-energy anisotropy. The theory predicts a range of behaviors which are in qualitative agreement with observation, including: hardening and dislocation multiplication in single slip under monotonic loading; the Bauschinger effect under reverse loading; the fading memory effect, whereby reverse yielding gradually eliminates the influence of previous loading; the evolution of the dislocation density under cycling loading, leading to characteristic ‘butterfly’ curves; and others.     

关于变分不等式问题近似解的数值证明

基于文献[1]给出了一种数值证明变分不等式解的存在性方法。通过Hilbert空间中的Riesz表示定理,首先将变分不等式问题的迭代过程转化为一种不动点形式,再利用Schauder不动点定理构造了一个高效率的数值证明过程,即通过数值计算产生一个包含近似解的有界闭凸子集。非线性Helmholtz方程的算例说明这一方法的可行性和高效性。     

Non-local and numerical formulations for dry sliding friction and wear at high velocities

Severe contact stress problems generate high temperature and create thermomechanical gouging and wear due to high velocity sliding between two materials staying in contact. In order to improve the facilitation of the design of particular components and improve performance of these engineering applications, it is necessary to better understand the physical behavior of high speed environment. As presented here this environment is made up of two components in contact. Therefore, basing on the experimental approach ( [Lodygowski, 2010] and [Lodygowski et al., submitted for publication]) the major consideration of this paper is aimed to develop an experimental/theoretical model for the material constitutive behavior in order to better characterize and predict the internal failure surrounding the gouging and wear events.This research is to be carried out in two stages. First, by investigating the phenomenon of wear and later it will be extended to incorporate gauging problems. The principle of virtual power is used by introducing the contributions from damage and its corresponding gradients as a measure of micro motion of damage within the bulk. In addition two internal state variables are introduced on the frictional contact interface, one measuring the tangential slip and another measuring the wear. By using these internal state variables together with displacement and temperature, the constitutive model is formulated with state laws based on the free energies and the complimentary laws based on the dissipation potentials. The proposed theoretical model is implemented as user defined subroutine VUMAT in the explicit finite element code ABAQUS to analyze the structural response of the ultra high speed sliding experiment between Steel and VascoMax steel at Ecole de’Nationale Institut der Mechanic, at Metz France.This model provides a potential feature for enabling one to relate the non-local continuum plasticity and damage of the bulk material to friction and wear at the contact interfaces. The findings of this research effort is invaluable in providing a multiscale material model and numerical procedure that will be used within a hydrocode to better facilitate the design components of the severe contact stress applications.     

A variational formulation for the incremental homogenization of elasto-plastic composites

This work addresses the micro–macro modeling of composites having elasto-plastic constituents. A new model is proposed to compute the effective stress–strain relation along arbitrary loading paths. The proposed model is based on an incremental variational principle (Ortiz, M., Stainier, L., 1999. The variational formulation of viscoplastic constitutive updates. Comput. Methods Appl. Mech. Eng. 171, 419–444) according to which the local stress–strain relation derives from a single incremental potential at each time step. The effective incremental potential of the composite is then estimated based on a linear comparison composite (LCC) with an effective behavior computed using available schemes in linear elasticity. Algorithmic elegance of the time-integration of J₂ elasto-plasticity is exploited in order to define the LCC. In particular, the elastic predictor strain is used explicitly. The method yields a homogenized yield criterion and radial return equation for each phase, as well as a homogenized plastic flow rule. The predictive capabilities of the proposed method are assessed against reference full-field finite element results for several particle-reinforced composites.     

Generalized gap functions and error bounds for generalized variational inequalities

We consider some classes of generalized gap functions for two kinds of generalized variational inequality problems. We obtain error bounds for the underlying variational inequalities using the generalized gap functions under the condition that the involved mapping F is g-strongly monotone with respect to the solution, but not necessarily continuous differentiable, even not locally Lipschitz.     

Unified formulae of variational bounds for multiphase anisotropic elastic composites

Using the spherical and deviator decomposition of the polarization and strain tensors, we present a general algorithm for the calculation of variational bounds of dimension d for any type of anisotropic linear elastic composite as a function of the properties of the comparison body. This procedure is applied in order to obtain analytical expressions of bounds for multiphase, linear elastic composites with cubic symmetry where the geometric shapes of the inclusions are arbitrary. For the validation, it can be proved that for the isotropic particular case, the bounds coincide with those recently reported by Gibiansky and Sigmund. On the other hand, based on this general procedure some, classical bounds reported by Hashin for transversely isotropic composites, are reproduced. Numerical calculations and some comparisons with other models and experimental data are shown.     

Stationary and extremum variational formulations for the elastostatics of cable networks

Summary The static analysis of elastic cable networks submitted to generic, conservative loads and prescribed dislocations is considered in this paper. The cables are assumed as stress-unilateral (only tensile stresses are admitted), and represented according to a Lagrangian standpoint, under the customary large displacements hypothesis. A variational formulation of the problem is given, as the stationary of a saddle functional with respect to displacements and (signconstrained) tractions. Uniqueness properties for the solution are derived from this statement, together with two complementary (constrained) minimum formulations, which correspond to the well-known extremum principles of the total potential energy and the complementary energy. The case of a network loaded only at the nodes is exposed as a specialization.

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Sommario Si considera il problema dell’equilibrio delle reti di funi elastiche sottoposte a carichi conservativi, distribuiti o concentrati, e a distorsioni impresse. Alla singola fune è attribuito comportamento unilaterale (reagenza a sola trazione). La trattazione è condotta in forma lagrangiana, sotto la naturale ipotesi di grandi spostamenti. Si perviene ad una formulazione variazionale di stazionarietà di un funzionale sella nelle variabili di spostamento e di sforzo, queste ultime vincolate in segno, e sulla base di questa formulazione vengono discusse le proprietà di unicità della soluzione. Si deducono inoltre due formulazioni complementari di estremo vincolato, corrispondenti agli usuali principi di minimo dell’energia complementare e dell’energia potenziale totale. Il problema delle reti caricate solo nei nodi è esposto come caso particolare.

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The content of this paper was partly presented at the sixth congress AIMETA, held in Genova (Italy), October 7–9, 1982.