An implicit and incremental formulation for the solution of elastoplastic problems by the finite element method

In this paper, a new approach is developed to solve elastoplastic problems by the finite element method. This approach involves two steps: (1) A mechanical formulation using the principle of virtual work and an implicit incremental form of the constitutive equations. This form is obtained by an approximate integration of the flow rules over the increment and includes the yield criterion itself. (2) A resolution algorithm to solve the nonlinear equations obtained by the mechanical formulation: Two resolution algorithms based on the Newton-Raphson method are proposed and compared. As the mechanical formulation is no more bound to the resolution algorithm, the results obtained by these two algorithms are the same and are path independent. Two numerical examples are presented: A thick cylinder under an internal pressure and a tensile sample. The numerical results obtained by the presented approach are compared with those obtained by the classical I.S.M. The comparison shows that the accuracy of the results does not vary when the load increment size increases as in the I.S.M. For a given accuracy this method requires about 15 times less computer time than the I.S.M. for the same memory space. This approach is easy to implement in a program based on the I.S.M. and has been extended to compressible and viscoplastic materials.     

Efficient numerical integration of an elastic–plastic damage law within a mixed velocity–pressure formulation

This study focuses on numerical integration of constitutive laws in numerical modeling of cold materials processing that involves large plastic strain together with ductile damage. A mixed velocity–pressure formulation is used to handle the incompressibility of plastic deformation. A Lemaitre damage model where dissipative phenomena are coupled is considered. Numerical aspects of the constitutive equations are addressed in detail. Three integration algorithms with different levels of coupling of damage with elastic–plastic behavior are presented and discussed in terms of accuracy and computational cost. The implicit gradient formulation with a non-local damage variable is used to regularize the localization phenomenon and thus to ensure the objectivity of numerical results for damage prediction problems. A tensile test on a plane plate specimen, where damage and plastic strain tend to localize in well-known shear bands, successfully shows both the objectivity and effectiveness of the developed approach.     

The Absolute Coordinate Formulation with Elasto-Plastic Deformations

The present work contributes to the field of multibody systems with respect to the absolute coordinate formulation with a reduced expression of the strain energy and a non-linear constitutive model. Standard methods for multibody systems lead to highly non-linear terms either in the mass matrix or in the stiffness matrix and the most expensive part in the solution of the equations of motion is the assembling of these matrices, the computation of the Jacobian of the non-linear system and the solution of a linear system with the system matrices. In the present work, a consistent simplification of the equations of motion with respect to small deformations but large rigid-body motions is performed. The absolute coordinate formulation is used, therefore the total displacements are the unknowns. This formulation leads to a constant mass matrix while the non-linear stiffness matrix is composed of the constant small strain stiffness matrix rotated by the underlying rigid body rotation. Plastic strains are introduced by an additive split of the strain into an elastic and a plastic part, a yield condition and an associative flow rule. The decomposition of strain has to be performed carefully in order to obey the principle of objectivity for plasticity under large rigid body rotations. As an example, a two-dimensional plate which is hinged at one side and driven by a harmonic force at the opposite side is considered. Plastic deformation is assumed to occur due to extreme environmental influences or due to failure of some attached parts like a defect bearing.     

Finite element elastic-plastic-creep analysis of two-dimensional continuum with temperature dependent material properties

A technique is presented for performing finite element elastic-plastic-creep analysis of two-dimensional continuum composed of material with temperature dependent elastic, plastic, and creep properties. The plastic analysis utilizes the Prandtl-Reuss flow equations assuming isotropic material properties and linear strain-hardening. A power creep flow law formulated by Odquist is used to determine the steady state creep strain rate. The plastic and creep flow laws are employed to derive a ‘softened’ plastic-creep stress-strain matrix. These modified stress-strain relations are then used to formulate the element stiffness matrix in the usual manner. The differences in the elastic, plastic, and creep properties of the material due to the temperature change during the increment result in the formation of pseudo stresses, which in turn lead to load terms that appear on the right hand side of the equilibrium equations. The load terms resulting from these pseudo stresses not only keep the solution on the temperature dependent stress-strain curve of the material, but also correct for the elastic ‘overshoot’ that occurs when an element changes from an elastic to a plastic state. The effect of large displacements is included by the formulation of the geometric stiffness matrix for each element being used in the computer code. With this procedure it becomes economically feasible to perform elastic-plastic-creep stress analysis of two-dimensional continuum subjected to transient thermal and mechanical loadings. Several examples of both elastic-plastic and creep analyses are presented, and the finite element solutions are compared to either other theoretical solutions or experiment.     

A finite element formulation for transient analysis of viscoplastic solids with application to stress wave propagation problems

A brief note on a finite element formulation for the transient analysis of viscoplastic solids is presented. Attention is confined to small strains in the present discussion and the emphasis is on stress wave propagation problems. The algorithm is based on straightforward explicit integration of both the equations of motion and the plastic rate equations. The explicit central difference method, which appears suitable for wave propagation problems, is used for integration of the equation of motion. The stress update is accomplished by means of a forward gradient scheme based on an estimate of the plastic flow over a time increment (Peirce, Shih and Needleman, Comput. Struct.18, 875–887, 1984). A number of simple numerical examples are presented to illustrate the method.     

Finite element cyclic thermoplasticity analysis by the method of subvolumes

The subject of this paper is the development of an analytical tool capable of economically evaluating the cyclic plasticity which occurs in areas of strain concentration resulting from the combination of both mechanical and thermal stresses. The techniques developed are capable of handling large excursions in temperatures with the associated variations in material properties, including plasticity. The techniques are capable of reproducing real cyclic material behavior including Bauschinger effect, cross-hardening and memory.These analytical techniques have been implemented in a time-sharing finite element computer program. Cyclic plasticity has been introduced into this program using incremental loading and an iterative solution technique. The plasticity theory involved makes use of the von Mises yield criterion and the Prandtl-Reuss flow rule. The major portion of the developmental work in this effort was expended in the establishment of a temperature variable hardening rule and its finite element implementation. The plane stress, constant strain triangle is the finite element used in this work.The incremental plasticity solution is obtained by iteratively revising the right-hand side of the system of finite element equations by the addition of a vector of plastic pseudo forces. The method of subvolumes is used to generate the vector of plastic pseudo forces such that real material cyclic plasticity behavior is mathematically reproduced.The effects of the plastic deformations are introduced into the system of finite element equations by considering them as load terms in much the same way as thermal expansions are usually treated. The nonlinear solution is then attained through solution of a series of elastic problems and by variation of the plastic load terms until the requirements of compatibility, equilibrium and the specified nonlinear stress-strain relations are all met within a given tolerance.     

Finite element cyclic thermoplasticity analysis by the method of subvolumes

The subject of this paper is the development of an analytical tool capable of economically evaluating the cyclic plasticity which occurs in areas of strain concentration resulting from the combination of both mechanical and thermal stresses. The techniques developed are capable of handling large excursions in temperatures with the associated variations in material properties, including plasticity. The techniques are capable of reproducing real cyclic material behavior including Bauschinger effect, cross-hardening and memory.These analytical techniques have been implemented in a time-sharing finite element computer program. Cyclic plasticity has been introduced into this program using incremental loading and an interative technique. The plasticity theory involved makes use of the von Mises yield criterion and the Prandtl-Reuss flow rule. The major portion of the developmental work in this effort was expended in the establishment of a temperature variable hardening rule and its finite element implementation. The plane stress, constant strain triangle is the finite element used in this work.The incremental plasticity solution is obtained by interatively revising and right-hand side of the system of finite element equations by the addition of a vector of plastic pseudo forces. The method of subvolumes is used to generate the vector of plastic pseudo forces such that real material cyclic plasticity behavior is mathematically reproduced.The effects of the plastic deformations are introduced into the system of finite element equations by considering them as load terms in much the same way as thermal expansions are usually treated. The nonlinear solution is then attained through solution of a series of elastic problems and by variation of the plastic load terms until the requirements of compatibility, equilibrium and the specified non-linear stress-strain relations are all met within a given tolerance.     

Sheet metal forming and springback simulation by means of a new reduced integration solid-shell finite element technology

The paper deals with the validation of a recently proposed hexahedral solid-shell finite element in the field of sheet metal forming. Working with one integration point in the shell plane and an arbitrary number of integration points in thickness direction, highly non-linear stress states over the sheet thickness can be incorporated in an efficient way. In order to avoid volumetric locking and Poisson thickness locking at the level of integration points the enhanced assumed strain (EAS) concept with only one EAS degree-of-freedom is implemented. A key point of the formulation is the construction of the hourglass stabilization by means of different Taylor expansions. This leads to the advantage that the sensitivity with respect to mesh distortion is noticeably reduced. The hourglass stabilization includes the assumed natural strain (ANS) concept and a kind of B-Bar method. So transverse shear locking and volumetric locking are eliminated.The finite element formulation incorporates a finite strain material model for plastic anisotropy as well as non-linear (Armstrong–Frederick type) kinematic and isotropic hardening. In this context the plastic anisotropy can be modeled by representing the yield surface and the plastic flow rule as functions of so-called structural tensors. The integration of the evolution equations is performed by means of an exponential map exploiting the spectral decomposition. The element formulation and material model have been implemented into the commercial code ABAQUS/Standard by means of the UEL interface for user-defined elements. Using an implicit time integration scheme numerical results for classical deep drawing simulations as well as springback predictions are presented in comparison to experimental measurements.     

Hybrid strain based three node flat triangular shell elements—I. Nonlinear theory and incremental formulation

Aspects and theories of nonlinear analysis of structures, with special emphasis on structures that are discretized by the finite element method, are discussed. The updated Lagrangian formulation and the incremental Hellinger-Reissner variational principle are adopted. The independently assumed fields employed are the incremental displacements and incremental strains. Accordingly, the incremental second Piola-Kirchhoff stress and the incremental Washizu strain are selected as the incremental stress and strain measures. Various schemes for the transformation of the second Piola-Kirchhoff stress to Cauchy stress are included. Two versions of linear and nonlinear element stiffness and mass matrices are considered. These are the director and simplified versions. Variable thickness of the shell is considered so as to account for the ‘thinning effect’ due to large strain. Material nonlinearity studied in this paper is of elasto-plastic type with isotropic strain hardening. Cases in which small elastic but large plastic strain condition applies are considered and the J₂ flow theory of plasticity, in conjunction with Ilyushin's yield criterion, is employed. To simplify the derivation of (small displacement) stiffness matrix and to facilitate the derivation of explicit expressions for the element matrices, the non-layered approach has been applied.     

Numerical analysis of anisotropic ductile continuum damage

The paper deals with the numerical analysis of large elastic–plastic deformation behavior of anisotropically damaged ductile solids based on a generalized macroscopic theory within the framework of nonlinear continuum damage mechanics. Estimates of the stress and strain histories are obtained from a straightforward numerical integration algorithm based on operator split methodology which employs an inelastic (damage–plastic) predictor followed by an elastic corrector step. The finite element method is used to approximate the linearized variational problem. Furthermore, identification of material parameters is discussed. Numerical simulation of the elastic–plastic deformation behavior of damaged tension specimens demonstrate the efficiency of the formulation.     

作大范围运动弹塑性平面板的动力学分析

研究作大范围运动弹塑性平面板的动力学特性.考虑了几何非线性和材料非线性,基于平面应力假设、Mises屈服条件和流动法则,采用绝对节点坐标法,用虚功原理建立了作大范围运动弹塑性平面板的动力学方程.在数值计算时将各时刻的塑性应变储存在全局数组中,实现了塑性应变的迭代计算.通过对带集中质量、作大范围运动平面板的数值仿真研究塑性效应对系统的动力学特性的影响.     

A linear yield surface in plastic cyclic analysis

A three-parameter, uniaxial symmetric, linear yield surface suitable for tension-weak, as well as equal tension and compression yield stress material, is presented. This yield condition, along with a linear kinematic hardening rule and an associated flow rule, is used to formulate the constitutive laws for sides and comers of the yield surface. The formulation is based on incremental plasticity with the assumption of small displacements and suitable for plane stress problems, under monotonie or cyclic loading. Finite element formulation, numerical solution, and applications are discussed.     

Nonlinear bending of beams using the finite element method

In this paper a finite element formulation for determining the finite deflection of thin bars is presented. The nonlinear stiffness equations are generated after simple approximate expressions involving the nodal parameters are used to replace the nonlinear terms in the energy functional. The procedure used results in a simplified set of nonlinear algebraic equations which are more amenable to solution than the equations usually presented. The applicability and accuracy of the method together with an evaluation of three incremental solution techniques, a step by step method, a one step Newton-Raphson procedure, and a variable interpolation technique is demonstrated by solving a cantilever beam with a point load acting on the end. Curves showing the sensitivity to increment size and to the number of elements are also presented. The results indicate that the formulation is accurate and inexpensive in terms of computational effort.     

Nonlocal large deformation and localization behavior of metals

The present paper deals with a nonlocal continuum plasticity model which includes the dependence of the yield function on a nonlocal equivalent plastic strain measure. Particular attention is focused on the formulation of a generalized I₁–J₂ yield criterion to describe the effect of hydrostatic stress on the plastic flow properties of metals, and the nonlocal equivalent plastic strain is defined as a weighted average of the corresponding local measure taken over the neighboring material points of the body. The nonlocal yield condition leads to a partial differential equation which is solved using the finite difference method at each iteration of a loading step. Since this requires no additional boundary conditions, the displacement-based finite element procedure is governed by the standard principle of virtual work, and the associated linearized variational equations are obtained in the usual manner from a consistent linearization algorithm. Numerical simulations of the elastic–plastic deformation behavior of ductile metal specimens show the influence of the various model parameters on the deformation and localization prediction. The proposed nonlocal theory preserves well-posedness of the governing equations in the post-localization regime and prevents pathological mesh sensitivity of the numerical results. The internal length scale incorporated in the model determines the size of the localized shear bands.     

Total-FETI domain decomposition method for solution of elasto-plastic problems

In this paper we present an algorithm for the parallel solution of the rate-independent elasto-plastic problems with kinematic hardening. We assume the von Mises plastic criterion and the associated plastic flow rule. The time discretization is based on the implicit Euler method. The corresponding one-time-step problem is formulated in the incremental form with respect to the unknown displacement and discretized spatially by the finite element method. We use an ‘external’ algorithm based on a linearization of the elasto-plastic stress–strain relation by the corresponding tangential operator and we parallelize the arising linearized problem by the Total-FETI method. The numerical experiments were carried out using our novel C/C++ library FLLOP (FETI Light Layer On top of PETSc) at HECToR supercomputer located at EPCC, UK.     

Sensitivity analysis for the dynamic response of thermoviscoplastic shells of revolution

A computational procedure is presented for evaluating the sensitivity coefficients of the dynamic axisymmetric, fully-coupled, thermoviscoplastic response of shells of revolution. The analytical formulation is based on Reissner's large deformation shell theory with the effects of large-strain, transverse shear deformation, rotatory inertia and moments turning around the normal to the middle surface included. The material model is chosen to be viscoplasticity with strain hardening and thermal hardening, and an associated flow rule is used with a von Mises effective stress. A mixed formulation is used for the shell equations with the fundamental unknowns consisting of six stress resultants, three generalized displacements and three velocity components. The energy-balance equation is solved using a Galerkin procedure, with the temperature as the fundamental unknown.Spatial discretization is performed in one dimension (meridional direction) for the momentum and constitutive equations of the shell, and in two dimensions (meridional and thickness directions) for the energy-balance equation. The temporal integration is performed by using an explicit central difference scheme (leap-frog method) for the momentum equation; a predictor-corrector version of the trapezoidal rule is used for the energy-balance equation; and an explicit scheme consistent with the central difference method is used to integrate the constitutive equations. The sensitivity coefficients are evaluated by using a direct differentiation approach. Numerical results are presented for a spherical cap subjected to step loading. The sensitivity coefficients are generated by evaluating the derivatives of the response quantities with respect to the thickness, mass density, Young's modulus, two of the material parameters characterizing the viscoplastic response and the three parameters characterizing the thermal response. Time histories of the response and sensitivity coefficients are presented, along with spatial distributions of some of these quantities at selected times.     

Coupled viscoelastic–viscoplastic modeling of homogeneous and isotropic polymers: Numerical algorithm and analytical solutions

A coupled viscoelastic–viscoplastic (VE–VP) model is implemented and studied. The total strain is the sum of VE and VP parts, and the Cauchy stress is given by a linear VE model as a Boltzmann integral of the history of VE strains. The proposed computational algorithm features fully implicit integration, return mapping based on a two-step VE predictor/VP corrector strategy, and a consistent tangent operator. The algorithm is applied to J₂ VP coupled with VE. Very compact expressions are obtained which are form-identical to classical elasto-viscoplasticity (EVP) provided that the constant linear elastic shear and bulk moduli are replaced with incremental relaxation moduli which are appropriate functions of the time increment. Two different integration methods to obtain the incremental moduli are proposed and assessed. Closed-form solutions for uniaxial tension and simple shear are developed, based on an original solution method for integro-differential equations. The analytical results enable to illustrate the constitutive model and provide unambiguous benchmarks for numerical algorithms. Model predictions are compared with experimental data and reasonable correlation is obtained.     

大规模图上具有约束的多起点路径规划算法

路径规划查询是图数据上的一个基本问题,在众多的领域都有重要的应用价值。通常在实际问题中查询的路径是具有约束的,例如在外卖配送和共享出行问题中路径具有节点约束,其路径需要满足节点之间的先后关系约束。目前对于具有节点约束的路径查询问题,大多数的工作都在研究单起点的节点约束路径查询,但很难拓展到多起点节点约束问题中。因为具有节点约束的多起点路径查询问题是NP-hard的,所以该问题的大多数已有方法是使用贪心增量处理,但对于处理静态规则集拓展性不足。因此,提出了基于子路径的启发式算法和基于约束集拓展的精确算法,并在真实数据集上验证了算法的有效性。实验结果表明,启发式算法能够给出问题的精确解,而启发式算法能快速给出较好的近似解。     

Numerical concepts for rate-independent single crystal plasticity

We provide a new time integration algorithm for rate-independent single crystal plasticity. The algorithm is based on an augmented Lagrangian formulation of the principle of maximum plastic dissipation. It is a synthesis of Lagrangian multiplier and penalty formulations avoiding their drawbacks (active set search, pseudo-inverse and approximate fulfillment of the yield condition respectively). The algorithm is physically motivated and especially provides a clear concept how to determine the set of active slip systems. The method is stable and efficient, several representative numerical examples are shown.     

Comparative study of pressure-correction and Godunov-type schemes on unsteady compressible cases

Two pressure-correction algorithms are studied and compared to an approximate Godunov scheme on unsteady compressible cases. The first pressure-correction algorithm sequentially solves the equations for momentum, mass and enthalpy, with sub-iterations which ensure conservativity. The algorithm also conserves the total enthalpy along a streamline, in a steady flow. The second pressure-correction algorithm sequentially solves the equations for mass, momentum and energy without sub-iteration. This scheme is conservative and ensures the discrete positivity of the density. Total enthalpy is conserved along a streamline, in a steady flow. It is numerically verified that both pressure-correction algorithms converge towards the exact solution of Riemann problems, including shock waves, rarefaction waves and contact discontinuities. To achieve this, conservativity is compulsory. The two pressure-correction algorithms and the approximate Godunov scheme are finally compared on cases with heat source terms: all schemes converge towards the same solution as the mesh is refined.