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1.
C. Miehe F. Aldakheel S. Mauthe 《International journal for numerical methods in engineering》2013,94(11):1037-1074
This work outlines a theoretical and computational framework of gradient plasticity based on a rigorous exploitation of mixed variational principles. In contrast to classical local approaches to plasticity based on locally evolving internal variables, order parameter fields are taken into account governed by additional balance‐type PDEs including micro‐structural boundary conditions. This incorporates non‐local plastic effects based on length scales, which reflect properties of the material micro‐structure. We develop a unified variational framework based on mixed saddle point principles for the evolution problem of gradient plasticity, which is outlined for the simple model problem of von Mises plasticity with gradient‐extended hardening/softening response. The mixed variational structure includes the hardening/softening variable itself as well as its dual driving force. The numerical implementation exploits the underlying variational structure, yielding a canonical symmetric structure of the monolithic problem. It results in a novel finite element (FE) design of the coupled problem incorporating a long‐range hardening/softening parameter and its dual driving force. This allows a straightforward local definition of plastic loading‐unloading driven by the long‐range fields, providing very robust FE implementations of gradient plasticity. This includes a rational method for the definition of elastic‐plastic‐boundaries in gradient plasticity along with a post‐processor that defines the plastic variables in the elastic range. We discuss alternative mixed FE designs of the coupled problem, including a local‐global solution strategy of short‐range and long‐range fields. This includes several new aspects, such as extended Q1P0‐type and Mini‐type finite elements for gradient plasticity. All methods are derived in a rigorous format from variational principles. Numerical benchmarks address advantages and disadvantages of alternative FE designs, and provide a guide for the evaluation of simple and robust schemes for variational gradient plasticity. Copyright © 2013 John Wiley & Sons, Ltd. 相似文献
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F. Armero A. Prez‐Foguet 《International journal for numerical methods in engineering》2002,53(2):297-329
We present in this paper the characterization of the variational structure behind the discrete equations defining the closest‐point projection approximation in elastoplasticity. Rate‐independent and viscoplastic formulations are considered in the infinitesimal and the finite deformation range, the later in the context of isotropic finite‐strain multiplicative plasticity. Primal variational principles in terms of the stresses and stress‐like hardening variables are presented first, followed by the formulation of dual principles incorporating explicitly the plastic multiplier. Augmented Lagrangian extensions are also presented allowing a complete regularization of the problem in the constrained rate‐independent limit. The variational structure identified in this paper leads to the proper framework for the development of new improved numerical algorithms for the integration of the local constitutive equations of plasticity as it is undertaken in Part II of this work. Copyright © 2001 John Wiley & Sons, Ltd. 相似文献
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Daniel E. Hurtado Michael Ortiz 《International journal for numerical methods in engineering》2013,93(1):66-79
We present a finite element method for the analysis of ductile crystals whose energy depends on the density of geometrically necessary dislocations (GNDs). We specifically focus on models in which the energy of the GNDs is assumed to be proportional to the total variation of the slip strains. In particular, the GND energy is homogeneous of degree one in the slip strains. Such models indeed arise from rigorous multiscale analysis as the macroscopic limit of discrete dislocation models or from phenomenological considerations such as a line‐tension approximation for the dislocation self‐energy. The incorporation of internal variable gradients into the free energy of the system renders the constitutive model non‐local. We show that an equivalent free‐energy functional, which does not depend on internal variable gradients, can be obtained by exploiting the variational definition of the total variation. The reformulation of the free energy comes at the expense of auxiliary variational problems, which can be efficiently solved using finite element approximations. The addition of surface terms in the formulation of the free energy results in additional boundary conditions for the internal variables. The proposed framework is verified by way of numerical convergence tests, and simulations of three‐dimensional problems are presented to showcase its applicability. A performance analysis shows that the proposed framework solves strain‐gradient plasticity problems in computing times of the order of local plasticity simulations, making it a promising tool for non‐local crystal plasticity three‐dimensional large‐scale simulations. Copyright © 2012 John Wiley & Sons, Ltd. 相似文献
4.
Carsten Carstensen Antonio Orlando Jan Valdman 《International journal for numerical methods in engineering》2006,67(13):1851-1887
The boundary value problem representing one time step of the primal formulation of elastoplasticity with positive hardening leads to a variational inequality of the second kind with some nondifferentiable functional. This paper establishes an adaptive finite element algorithm for the solution of this variational inequality that yields the energy reduction and, up to higher order terms, the R‐linear convergence of the stresses with respect to the number of loops. Applications include several plasticity models: linear isotropic‐kinematic hardening, linear kinematic hardening, and multisurface plasticity as model for nonlinear hardening laws. For perfect plasticity, the adaptive algorithm yields strong convergence of the stresses. Numerical examples confirm an improved linear convergence rate and study the performance of the algorithm in comparison with the more frequently applied maximum refinement rule. Copyright © 2006 John Wiley & Sons, Ltd. 相似文献
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C. S. Ryoo R. P. Agarwal 《International journal for numerical methods in engineering》2002,54(11):1535-1556
We consider a numerical method that enables us to verify the existence of solutions for variational inequalities. This method is based on the infinite dimensional fixed point theorems and explicit error estimates for finite element approximations. Using the finite element approximations and explicit a priori error estimates, we present an effective verification procedure that through numerical computation generates a set which includes the exact solution. Copyright © 2002 John Wiley & Sons, Ltd. 相似文献
6.
Ramsharan Rangarajan Adrian J. Lew 《International journal for numerical methods in engineering》2011,88(6):556-585
We construct a method for the parameterization of a class of planar piecewise C2‐curves over a collection of edges in an ambient triangulation. The map from the collection of edges to the curve is the closest‐point projection. A distinguishing feature of the method is that edges in the ambient triangulation need not interpolate the curve. We formulate conditions on the ambient triangulations so that the resulting parameterization over its selected edges is (i) bijective, (ii) maps simple, connected collection of edges to simple, connected components of the curve, and (iii) is C1 within each edge of the collection. These properties of the parameterization make it particularly useful in the construction of high‐order finite element approximation spaces on planar curves immersed in triangulations. We discuss this application and illustrate it with numerical examples. The parameterization method applies to a large class of planar curves, including most ones of interest in engineering and computer graphics applications, and to a large family of triangulations, including acute‐angled triangulations. Copyright © 2011 John Wiley & Sons, Ltd. 相似文献
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Tina Liebe Paul Steinmann 《International journal for numerical methods in engineering》2001,51(12):1437-1467
The paper presents the theory and the numerics of a thermodynamically consistent formulation of gradient plasticity at small strains. Starting from the classical local continuum formulation, which fails to produce physically meaningful and numerically converging results within localization computations, a thermodynamically motivated gradient plasticity formulation is envisioned. The model is based on an assumption for the Helmholtz free energy incorporating the gradient of the internal history variable, a yield condition and the postulate of maximum dissipation resulting in an associated structure. As a result the driving force conjugated to the hardening evolution is identified as the quasi‐non‐local drag stress which incorporates besides the strictly local drag stress essentially the divergence of a vectorial hardening flux. At the numerical side, besides the balance of linear momentum, the algorithmic consistency condition has to be solved in weak form. Thereby, the crucial issue is the determination of the active constraints exhibiting plastic loading which is solved by an active set search algorithm borrowed from convex non‐linear programming. Moreover, different discretization techniques are proposed in order to compare the FE‐performance in local plasticity with the advocated gradient formulation both for hardening and softening. Copyright © 2001 John Wiley & Sons, Ltd. 相似文献
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John Y. Shu Wayne E. King Norman A. Fleck 《International journal for numerical methods in engineering》1999,44(3):373-391
A finite element implementation is reported of the Fleck–Hutchinson phenomenological strain gradient theory. This theory fits within the Toupin–Mindlin framework and deals with first‐order strain gradients and the associated work‐conjugate higher‐order stresses. In conventional displacement‐based approaches, the interpolation of displacement requires C1‐continuity in order to ensure convergence of the finite element procedure for higher‐order theories. Mixed‐type finite elements are developed herein for the Fleck–Hutchinson theory; these elements use standard C0‐continuous shape functions and can achieve the same convergence as C1 elements. These C0 elements use displacements and displacement gradients as nodal degrees of freedom. Kinematic constraints between displacement gradients are enforced via the Lagrange multiplier method. The elements developed all pass a patch test. The resulting finite element scheme is used to solve some representative linear elastic boundary value problems and the comparative accuracy of various types of element is evaluated. Copyright © 1999 John Wiley & Sons, Ltd. 相似文献
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Christian F. Niordson John W. Hutchinson 《International journal for numerical methods in engineering》2003,56(7):961-975
Significant increases in apparent flow strength are observed when non‐uniform plastic deformation of metals occurs at the scale ranging from roughly one to ten microns. Several basic plane strain problems are analysed numerically in this paper based on a new formulation of strain gradient plasticity. The problems are the tangential and normal loading of a finite rectangular block of material bonded to rigid platens and having traction‐free ends, and the normal loading of a half‐space by a flat, rigid punch. The solutions illustrate fundamental features of plasticity at the micron scale that are not captured by conventional plasticity theory. These include the role of material length parameters in establishing the size dependence of strength and the elevation of resistance to plastic flow resulting from constraint on plastic flow at boundaries. Details of the finite element method employed in the numerical analysis of the higher order gradient theory will be discussed and related to prior formulations having some of the same features. Copyright © 2003 John Wiley & Sons, Ltd. 相似文献
12.
Francisco Armero Edward Love 《International journal for numerical methods in engineering》2003,57(4):471-508
This paper presents a new arbitrary Lagrangian–Eulerian (ALE) finite element formulation for finite strain plasticity in non‐linear solid mechanics. We consider the models of finite strain plasticity defined by the multiplicative decomposition of the deformation gradient in an elastic and a plastic part ( F = F e F p), with the stresses given by a hyperelastic relation. In contrast with more classical ALE approaches based on plastic models of the hypoelastic type, the ALE formulation presented herein considers the direct interpolation of the motion of the material with respect to the reference mesh together with the motion of the spatial mesh with respect to this same reference mesh. This aspect is shown to be crucial for a simple treatment of the advection of the plastic internal variables and dynamic variables. In fact, this advection is carried out exactly through a particle tracking in the reference mesh, a calculation that can be accomplished very efficiently with the use of the connectivity graph of the fixed reference mesh. A staggered scheme defined by three steps (the smoothing, the advection and the Lagrangian steps) leads to an efficient method for the solution of the resulting equations. We present several representative numerical simulations that illustrate the performance of the newly proposed methods. Both quasi‐static and dynamic conditions are considered in these model examples. Copyright © 2003 John Wiley & Sons, Ltd. 相似文献
13.
F. T. Suttmeier 《International journal for numerical methods in engineering》2006,68(11):1180-1208
This work describes concepts for a posteriori error estimation and adaptive mesh design for finite element models where the solution is subjected to inequality constraints. These methods are developed here for several model problems. Based on these examples, unified frameworks are proposed, which provide a systematic way of adaptive error control for problems stated in form of variational inequalities. Copyright © 2006 John Wiley & Sons, Ltd. 相似文献
14.
T. Akita K. Nakashino M. C. Natori K. C. Park 《International journal for numerical methods in engineering》2007,71(10):1231-1259
A simple computer implementation of membrane wrinkle behaviour is presented within the classical elastic plane stress constitutive model. In the present method, a projection technique is utilized for modelling of the wrinkle mechanisms, in which the total strains in wrinkled membranes are decomposed into elastic and zero‐strain energy parts, and a projection matrix that extracts the elastic parts from the total strains is derived. The resulting modified elasticity matrix that represents the stress–strain relations in wrinkled membranes is thus obtained as product of the classical elasticity matrix and the projection matrix. The modified elasticity matrix is straightforward to implement within the context of the finite element method. Numerical examples are presented to demonstrate the accuracy and effectiveness of the proposed method. Copyright © 2007 John Wiley & Sons, Ltd. 相似文献
15.
An efficient numerical method is proposed for the valuation of American options
via the Black-Scholes variational inequality. A far field boundary condition is employed
to truncate the unbounded domain problem to produce the bounded domain
problem with the associated variational inequality, to which our finite element method
is applied. We prove that the matrix involved in the finite element method is symmetric
and positive definite, and solve the discretized variational inequality by the projection
and contraction method. Numerical experiments are conducted that demonstrate the
superior performance of our method, in comparison with earlier methods. 相似文献
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In this study, the crystal plasticity finite element method model was used to study the deformation in a single crystal aluminium processed by accumulative roll-bonding (ARB). The predicted textures match well with the experimental observations up to nine cycles. The texture and slip activities in representative layers were investigated, and cyclic transition of them was observed. It was found that the thickness position change caused by cutting and stacking was a basic reason for the cyclic transition. Finally, the effect of shear deformation to this transition was discussed. 相似文献
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Timothy J. Truster Omar Nassif 《International journal for numerical methods in engineering》2017,110(4):303-332
A computational method is developed for evaluating the plastic strain gradient hardening term within a crystal plasticity formulation. While such gradient terms reproduce the size effects exhibited in experiments, incorporating derivatives of the plastic strain yields a nonlocal constitutive model. Rather than applying mixed methods, we propose an alternative method whereby the plastic deformation gradient is variationally projected from the elemental integration points onto a smoothed nodal field. Crucially, the projection utilizes the mapping between Lie groups and algebras in order to preserve essential physical properties, such as orthogonality of the plastic rotation tensor. Following the projection, the plastic strain field is directly differentiated to yield the Nye tensor. Additionally, an augmentation scheme is introduced within the global Newton iteration loop such that the computed Nye tensor field is fed back into the stress update procedure. Effectively, this method results in a fully implicit evolution of the constitutive model within a traditional displacement‐based formulation. An elemental projection method with explicit time integration of the plastic rotation tensor is compared as a reference. A series of numerical tests are performed for several element types in order to assess the robustness of the method, with emphasis placed upon polycrystalline domains and multi‐axis loading. Copyright © 2016 John Wiley & Sons, Ltd. 相似文献