A new look at energy release rate in fracture mechanics

The energy balance for fracture in elastic/perfectly plastic solids is examined using the finite element method. An extension-release procedure that gives numerically converged solutions is employed in the numerical simulation of crack extensions in elastic/plastic solids. Increments of work and energy during crack extension are calculated for various loading conditions. Several conclusions are obtained. First, the elastic separation work of creating new crack surfaces is shown to be negligible, indicating that the Griffith-type energy release does not exist. Second, as the yield stress increases, the plastic dissipation work rate associated with crack extension converges to the energy release rate in the limiting elastic solid. The latter result can be adopted to interpret the classical energy release rate in elastic solids as plastic dissipation work rate taken in the limit as the yield stress approaches infinity during crack extension. Lastly, it is shown that the energy release rate obtained according to Irwin's plastic zone adjustment approach is equal to the plastic dissipation work rate for the original crack, provided the plastic zone size is less than 10% of the original crack size.     

A non‐linear programming approach to kinematic shakedown analysis of composite materials

Using a Representative volume element (RVE) to represent the microstructure of periodic composite materials, this paper develops a non‐linear numerical technique to calculate the macroscopic shakedown domains of composites subjected to cyclic loads. The shakedown analysis is performed using homogenization theory and the displacement‐based finite element method. With the aid of homogenization theory, the classical kinematic shakedown theorem is generalized to incorporate the microstructure of composites. Using an associated flow rule, the plastic dissipation power for an ellipsoid yield criterion is expressed in terms of the kinematically admissible velocity. By means of non‐linear mathematical programming techniques, a finite element formulation of kinematic shakedown analysis is then developed leading to a non‐linear mathematical programming problem subject to only a small number of equality constraints. The objective function corresponds to the plastic dissipation power which is to be minimized and an upper bound to the shakedown load of a composite is then obtained. An effective, direct iterative algorithm is proposed to solve the non‐linear programming problem. The effectiveness and efficiency of the proposed numerical method have been validated by several numerical examples. This can serve as a useful numerical tool for developing engineering design methods involving composite materials. Copyright © 2005 John Wiley & Sons, Ltd.     

非线性破坏准则对被动土压力的影响

在上限定理的基础上,根据非线性破坏准则,对墙后填土建立机动容许的速度场,运用流动法则以及速度边界条件求解被动土压力的上限解。首先通过“切线法”引进变量,然后对墙后填土建立三种含有变量的速度场,求出被动土压力的目标函数与约束条件,最后根据“序列二次规划算法”对该问题进行优化。数值结果表明:当非线性破坏准则变为线性破坏准则时,结果与前人的成果一致;非线性参数对被动土压力有重要影响。     

Solution for the general integral‐type nonlocal plasticity model with Tikhonov regularization

A new solution approach, based on Tikhonov regularization on the Fredholm integral equations of the first kind, is proposed to find the approximate solutions of the strain softening problems. In this approach, the consistency condition is regularized with the Tikhonov stabilizers along with a regularization parameter, and the internal variable increments are solved from the resulting Euler's equations. It is shown that, as the regularization parameter is increased, the solutions converge to a unique one. A nonlocal yield condition and a nonlocal return mapping algorithm are proposed to carry out the integration of constitutive equations in the time and spatial domains. A global plastic dissipation principle is proposed to relax the classical local plastic dissipation postulate. Numerical examples show that the proposed approach leads to objective, mesh‐independent solutions of the softening‐induced localization problems. A comparison of the results from the proposed approach with those from the gradient‐dependent plasticity model shows that the two models give close solutions.     

On a class of constitutive equations in viscoplasticity: Formulation and computational issues

The viscoplastic constitutive model is formulated based on the existence of the dissipation potential which embodies the notion of the gauge (Minkowski) function of the convex set. A perturbation method is used for a solution of stiff differential equations characterizing the associated problem of evolution. It relies on a discrete formulation of viscoplasticity which results from the regularized version of the principle of maximum plastic dissipation. The operator split methodology and the Newton-Raphson method are used to obtain the numerical solution of the discretized equations of evolution. The consistent tangent modulus is expressed in a closed form as a result of the exact linearization of the discretized evolution equations. For several variants of the flow potential function, including some representative stiff functional forms, numerical tests of the integration algorithm based on iso-error maps are provided. Finally, a numerical example is presented to illustrate the robustness and the effectiveness of the proposed approach.     

Numerical approximation of incremental infinitesimal gradient plasticity

We investigate a representative model of continuum infinitesimal gradient plasticity. The formulation is an extension of classical rate‐independent infinitesimal plasticity based on the additive decomposition of the symmetric strain tensor into elastic and plastic parts. It is assumed that dislocation processes contribute to the storage of energy in the material whereby the curl of the plastic distortion appears in the thermodynamic potential and leads to an additional nonlocal backstress tensor. The formulation is cast into a numerical framework by a saddle point approximation of the corresponding minimization problem in each incremental loading step. This allows one to reformulate the (nonlocal) dissipation inequality to a point‐wise flow rule and yields a solution scheme, which is a direct extension of the standard approach in classical plasticity. Our numerical results show the regularizing effects of the additional physically motivated terms. Copyright © 2008 John Wiley & Sons, Ltd.     

Limit analysis considering initial constant loadings and proportional loadings

This paper deals with the limit analysis of rigid perefectly plastic body under the combined action of initial constant loadings and proportional loadings. Using the von Mises yielding condition and the finite element technique, we present an efficient algorithm based on the mathematical programming formula of the classical kinematic theorem of plasticity theory. This algorithm includes an iterative procedure which produces a monotonically decrescent sequence converging to an upper bound of the real limit load. The results of some numerical examples are presented and show the stable convergency of the new algorithm.The work of the author was supported by Research Grants Council of the Hong Kong UPGC     

Dynamic frictional slip along an interface between plastically compressible solids

Dynamic frictional slip along an interface between plastically compressible solids is analyzed. The plane strain, small deformation initial/boundary value problem formulation and the numerical method are identical to those in Shi et al. (Int J Fract 162:51, 2010) except that here the material constitutive relation allows for plastic compressibility. The interface is characterized by a rate and state dependent friction law. The specimens have an initial compressive stress and are subject to shear loading by edge impact near the interface. Two loading conditions are analyzed, one giving rise to a crack-like mode of slip propagation and the other to a pulse-like mode of slip propagation. In both cases, the initial compressive stress is taken to vary with plastic compressibility such that the associated initial effective stress is the same for all values of plastic compressibility. The volume change for the crack-like slip mode is mainly plastic while the elastic volume change plays a larger role for the pulse-like mode. For the crack-like slip mode, the proportion of plastic dissipation in the material increases with the increasing plastic compressibility, but the effect of plastic compressibility on the energy partitioning for the pulse-like slip mode is much smaller. The predicted propagation speeds approach a speed about the dilational wave speed for both the crack-like and pulse-like slip modes and this speed is not sensitive to the value of the plastic compressibility parameter. Plastic dissipation is found to be mainly associated with the deformation induced by the loading wave rather than with the deformation arising from slip propagation. The amplitude of the slip rate in the slip pulses is found to be largely governed by the value of the initial compressive stress regardless of the value of plastic compressibility.

     

Heterogeneous upper-bound finite element limit analysis of masonry walls out-of-plane loaded

A heterogeneous approach for FE upper bound limit analyses of out-of-plane loaded masonry panels is presented. Under the assumption of associated plasticity for the constituent materials, mortar joints are reduced to interfaces with a Mohr–Coulomb failure criterion with tension cut-off and cap in compression, whereas for bricks both limited and unlimited strength are taken into account. At each interface, plastic dissipation can occur as a combination of out-of-plane shear, bending and torsion. In order to test the reliability of the model proposed, several examples of dry-joint panels out-of-plane loaded tested at the University of Calabria (Italy) are discussed. Numerical results are compared with experimental data for three different series of walls at different values of the in-plane compressive vertical loads applied. The comparisons show that reliable predictions of both collapse loads and failure mechanisms can be obtained by means of the numerical procedure employed.     

On the numerical implementation of 3D rate‐dependent single crystal plasticity formulations

In this paper, several important numerical issues are addressed for three‐dimensional (3D) rate‐dependent single crystal plasticity. After a thorough comparison of different constitutive algorithms, we classify the integration methods into three approaches, namely, the implicit elastic/plastic deformation gradient approach, the implicit slip‐rate approach, and the explicit slip‐rate approach. As part of this algorithmic study, we focus on five different schemes to enforce the plastic incompressibility, four ways to update the texture, and one convergence criteria. The numerical performance of these different methods is illustrated. The contribution of this study is three‐fold: a stable scheme for the incompressibility enforcement, an improved implicit algorithm, and a fully explicit algorithm. Copyright © 2005 John Wiley & Sons, Ltd.     

Transverse impact of free–free square aluminum beams: An experimental–numerical investigation

A combined experimental–numerical analysis was performed to model transverse impact of free–free square aluminum beams loaded at different locations along their length. The applied impact load was obtained from tests carried out on a single Hopkinson pressure bar. The 3D elastic–plastic numerical simulations show that the plastic deformation, adjacent to the impact location, is due to combined dominant bending and stretching modes. Most of the plastic deformation is confined to the impact zone but some partial additional plastic hinges are observed to develop. The plastic strain magnitude and distribution near the impact zone are similar for all tested impact locations, but higher for the more symmetrical impacts. The conversion of impact energy into kinetic, elastic strain energy and plastic dissipation work is characterized for various impact locations along the beam. It is observed that symmetrical impact results in higher plastic dissipation and lower kinetic energy as opposed to unsymmetrical impact. Between 52% and 76% of the applied energy is converted into plastic dissipation energy.     

Energy consistent algorithms for frictional contact problems

In this paper, the energy and momentum conserving algorithmic paradigm is extended to encompass a phenomenon featuring physical dissipation: dynamic frictional contact. Whereas in other works dealing with conservative systems the chief aim is often the maintenance of numerical stability in the non-linear regime, in this investigation we seek to achieve not only this benefit but also the accurate algorithmic production of physical dissipation associated with frictional processes. The approach here features a product formula algorithm for the evolution of local frictional conditions, with the associated operator split guided by an a priori energy estimate. The resulting algorithm is characterized by exact conservation of energy during stick friction, and positive dissipation consistent with the frictional model used during slip. Effectiveness of the algorithm is demonstrated by a series of finite element simulations involving large deformations and frictional slip, complete with appropriate comparisons to more traditional schemes. © 1998 John Wiley & Sons, Ltd.     

一族无条件稳定的显式结构动力学算法

利用离散控制理论分析HHT-α算法,提出了一族具有可控数值阻尼的无条件稳定显式结构动力学算法—显式HHT-α法,用于线性和非线性结构动力学分析。新算法采用显式的位移、速度递推式。研究了所提算法的精度,稳定性,数值色散和能量耗散特性。研究表明该算法对于线弹型和刚度软化型非线性系统是无条件稳定的,算法数值阻尼由单个参数控制,对于特定的参数值,所提算法不会产生数值能量耗散。此外所提出的显式算法的数值色散和能量耗散特性与隐式HHT-α算法相同。数值算例验证了理论分析的正确性。     

Limit analysis of cracked structures by mathematical programming and finite element technique

Limit analysis of cracked structures using mathematical programming and finite element method is presented. This direct algorithm is based on a proposition of the modified Markov variational principle. The obtained upper bound formula is applicable with any kinematic admissible finite elements. The regularization of plastic dissipation function is performed to overcome the indetermination of the objective gradient in the rigid region and to realize a smooth transition of the objective function between the plastified and non-plastified regions. In order to simulate the singularity of crack-tip strain field for an ideal plastic model, the separate-point degenerated finite elements are used around the crack tips. This improves the precision of limit solutions. Numerical results of plane and axisymmetric cracked structures are extensively compared with analytic ones. The application of limit analysis in fracture mechanics is illustrated by an example. Received 25 November 1998     

A microscopic nonlinear programming approach to shakedown analysis of cohesive–frictional composites

A microscopic approach together with nonlinear programming technique and finite element method is developed for shakedown analysis of a composite which has cohesive–frictional constituents. The macroscopic shakedown limit of a composite subject to cyclic loading is calculated in a direct way and the macro–micro relation is quantitatively evaluated. First, by means of the homogenization theory, the classical kinematic theorem of shakedown analysis is generalized to incorporate the microstructure – Representative Volume Element (RVE) chosen from a periodic heterogeneous material. Pressure-dependence and non-associated plastic flow of cohesive–frictional constituent materials are formulated into shakedown analysis. Based on the mathematical programming technique and the finite element method, the numerical micro-shakedown model is finally formulated as a nonlinear programming problem subject to only a few equality constraints, which is solved by a generalized Lagrangian-penalty iterative algorithm. The proposed approach provides a direct approach for determining the reduced macroscopic strength domain of heterogeneous or composite materials due to cyclic loading. Meanwhile, it can capture different plastic behaviors of materials and therefore the developed method has a wide applicability.     

Study of energy dissipation in previously deformed chromium-nickel steel

Preliminary testing for creep and stress relaxation is performed with static or asymmetrical cyclic loading with the maximum stress in the cycle below the fatigue limit. Energy dissipation is studied by a thermal method under symmetrical tension-compression conditions with a fixed stress amplitude. It is established that the main factor which governs energy dissipation is the amount of unidimensional plastic strain accumulated. The preliminary loading conditions studied do not have a significant effect on energy dissipation. A dependence is obtained for energy dissipation on preliminary plastic tensile strain accumulation for the case when the latter does not exceed 1%.Translated from Problemy Prochnosti, No. 11, pp. 82–87, November, 1991.     

Computation of limit and shakedown loads using a node‐based smoothed finite element method

This paper presents a novel numerical procedure for computing limit and shakedown loads of structures using a node‐based smoothed FEM in combination with a primal–dual algorithm. An associated primal–dual form based on the von Mises yield criterion is adopted. The primal‐dual algorithm together with a Newton‐like iteration are then used to solve this associated primal–dual form to determine simultaneously both approximate upper and quasi‐lower bounds of the plastic collapse limit and the shakedown limit. The present formulation uses only linear approximations and its implementation into finite element programs is quite simple. Several numerical examples are given to show the reliability, accuracy, and generality of the present formulation compared with other available methods. Copyright © 2011 John Wiley & Sons, Ltd.     

Classical plasticity and viscoplasticity models reformulated: theoretical basis and numerical implementation

The well-known phenomenological model of small strain rate-independent plasticity is reformulated in this paper. The main difference from the classical expositions concerns the absence of the plastic strain from the list of state variables. We show that with the proposed choice of state variables, including the total and the elastic strains and strain-like variables which control hardening, we recover all the ingredients of the classical model from a minimum number of hypotheses: instantaneous elastic response and the principle of maximum plastic dissipation. We also show that using a regularized, penalty-like form of the principle of maximum plastic dissipation, we can recover the classical viscoplasticity model. As opposed to the previous schemes used for the finite element implementation of this model (e.g. B-bar method), we propose an approach in which the basic set of equations need not be modified. The operator split method is used to simplify the details of the numerical implementation concerning both the computation of state variables and the incompatible mode based finite element approximations. The latter proves to be indispensable for accommodating the near-incompressible deformation patterns arising in the classical plasticity. An extensive set of numerical simulations is used to illustrate the proposed formulation. © 1998 John Wiley & Sons, Ltd.     

Upper bound limit analysis of plates utilizing the C1 natural element method

In this article, a novel numerical solution procedure is proposed to evaluate the upper bound limit load multipliers for thin plate problems, which incorporates the C¹ natural element method (C¹NEM) with a direct iteration algorithm. Due to its remarkable interpolation property to the nodal function and the nodal gradient values, the C¹NEM with the C¹- continuous trial function is used here to deal with the upper bound limit analysis problem of perfectly rigid-plastic plates. The relevant discrete mathematical programming formulation is established based on the kinematic theorem of plastic limit analysis, and a direct iteration algorithm with the advantages of simple solution formula and easy procedure implementation is presented to solve it. Several representative examples governed by the von Mises yield criterion are investigated. The numerical solutions obtained in this paper are reasonable and satisfactory, and are in good agreement with the previously reported results.     

A multi-grid enhanced GMRES algorithm for elasto-plastic problems

A combination of both GMRES and multi-grid (MG) methods is presented in this paper for solving large-scale two- and three-dimensional elasto-plastic problems, in which each MG iteration cycle serves as the preconditioning step for the GMRES procedure. A particular multi-grid approach, termed the Galerkin multi-grid scheme, is considered and the main effort is devoted to the implementation aspects of the proposed algorithm. Numerical examples, characterised by large-scale (up to 82145 DOF), strong non-linearity (nearly plastic limit state, necking and localization) and severe ill-conditioned states (presence of loading limit points), and also involving symmetric and unsymmetric as well as SPD and indefinite system matrices, are provided. The numerical results illustrate that the proposed method exhibits a remarkable performance in terms of efficiency and robustness in all circumstances. © 1998 John Wiley & Sons, Ltd.