Continuing crack-tip deformation and fracture for plane-strain crack growth in elastic-plastic solids

Analysis of the deformation field consistent with a Prandtl stress distribution travelling with an advancing plane-strain crack reveals the functional form of the near tip crack profile in an elastic-plastic solid. The crack opening δ is shown to have the form δ ～ r In (const./r) at a distance r from the tip. This observation coupled with data generated from finite element investigations of growing cracks in small-scale yielding permits the construction of a relation characterizing the deformation at an extending crack tip. A ductile crack-growth criterion consisting of the attainment of a critical opening at a small characteristic material distance from the tip is adopted. Predictions of the stability of a growing crack for both small-scale yielding specimens and those subject to general yielding are discussed.     

Non-elliptic elastic materials and the modeling of elastic-plastic behavior for finite deformation

For illustrative purposes this paper treats a special problem in the theory of finite deformations of elastic materials whose associated displacement equations of equilibrium do not remain elliptic at all strains. The typical deformation arising in this problem possesses a discontinuous gradient, so that quasi-static motions involving such equilibrium states may be dissipative. For a special class of such “non-elliptic” elastic materials, it is shown that the macroscopic response in the problem treated may be precisely of the form associated with elastic—perfectly plastic behavior. The counterparts of yield, plastic strain and plastic strain rate are determined by the underlying elastic strain energy function.     

Theory of elastic-plastic deformation of randomly reinforced composite materials

Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 120–124, September–October, 1991.     

On the deformation and fracture of porous materials

The constitutive behavior of porous materials (including the yield loci, the void growth rate, the macro stress-strain relation and the strain to localization instability) is examined based on the lower bound approach proposed by the present authors. These results are then compared with the experimental and the finite element results as well as those predicted by Gurson's equations. Emphasis is placed on approaching the real behavior from the upper and the lower bound analysis. Calculation is also made on the influence of void nucleation on the critical strain to instability and a modified strain-controlled nucleation criterion is proposed. Finally the instability and fracture of AISI4340 steel in plane strain tension is examined and comparison is made between theoretical and experimental results.     

Finite deformation solution of a dynamic problem of combined compressive and shear loading for an elastic-plastic half-space

A self-similar problem involving combined compressive and shear loading of a non work-hardening elastic-plastic half-space is solved, for large deformations, within the framework proposed earlier by J.R. Willis (1969). The thermodynamic system, which was left open in the general framework, is taken to be that proposed recently by G.W. Swan and C.K. Thornhill (1974). Results are presented for oblique loading of a block of low carbon steel. For an imposed normal velocity of the order of 1,100 m s^?1, a smooth shear wave behind the shock that is formed is possible only for small transverse velocities, perhaps no greater than 14 m s^?1, depending upon the value assumed for the yield stress. Larger transverse velocities would give rise to a thin layer of intense shear near the surface of the block, whose study would require allowance for both work-hardening and temperature-dependent yield stress.     

A finite strain theory for elastic-plastic deformation

An elastic-plastic theory that is applicable when the elastic part of the strain is finite is proposed. A flow rule for an incompressible solid is obtained from Drucker's postulate [1]. Isothermal simple shear of a material which is neo-Hookean both before yielding and during elastic unloading after yielding is considered as an application of the theory. The problem is solved for two yield conditions and associated flow rules.     

Electromechanical deformation and fracture of piezoelectric/ferroelectric materials

This review presents the progress and current status of the investigation on electromechamical deformation and fracture of piezoelectric/ferroelectric materials. An attempt is made to summarize a few fundamental aspects, which include electromechanical constitutive relations, piezoelectric micromechanics and electric fracture and fatigue, instead of describing all technological backgrounds, basic physics, experimental findings, and theoretical developments. A number of open questions and future prospective are presented. It is hoped that this review will encourage people to joint the exploration of this important and interesting field. The project supported by the National Natural Science Foundation of China (100025209)     

Plane-stress crack-tip stress fields for orthotropic perfectly-plastic materials

Under the hypothesis that the stress components of crack-tip fields are only thefunctions ofθ,the differential equations of plane-stress crack-tip stress fields fororthotropic perfectly-plastic materials are obtained by using Hill’s yield condition andequilibrium equations.By combining the general analytical expression with the numericalmethod the crack-tip stress fields for orthotropic perfectly-plastic materials for plane stressare presented.     

Plane-strain crack-tip stress field for anisotropic perfectly-plastic materials

Plane-strain crack-tip stress solutions for anisotropic perfectly-plastic materials are presented. These solutions are obtained using the plane-strain slip-line theory developed by Rice (1973). The plastic anisosotropy is described by the Hill quadratic yield condition. The crack-tip stress solutions under symmetric (Mode I) and anti-symmetric (Mode II) conditions agree well with the low-hardening solutions for the corresponding power-law hardening materials. The crack-tip stress solutions under mixed Mode I and II conditions are also presented. All the solutions indicate that the general features of the slip-line field near a crack tip in orthotropic plastic materials with the elliptical yield contours in the Mohr plane are the same as those associated with isotropic plastic materials. However, the angular variations of the crack-tip stress fields for the materials with large plastic orthotropy differ substantially from those for isotropic plastic materials. Modifications due to polygonal yield contours are outlined and implications of solutions to the fracture analysis of ductile composite materials containing macroscopic flaws are discussed.     

Finite-element formulations for problems of large elastic-plastic deformation

An Eulerian finite element formulation is presented for problems of large elastic-plastic flow. The method is based on Hill's variational principle for incremental deformations, and is ideally suited to isotropically hardening Prandtl-Reuss materials. Further, the formulation is given in a manner which allows any conventional finite element program, for “small strain” elastic-plastic analysis, to be simply and rigorously adapted to problems involving arbitrary amounts of deformation and arbitrary levels of stress in comparison to plastic deformation moduli. The method is applied to a necking bifurcation analysis of a bar in plane-strain tension.The paper closes with a unified general formulation of finite element equations, both Lagrangian and Eulerian, for large deformations, with arbitrary choice of the conjugate stress and strain measures. Further, a discussion is given of other proposed formulations for elastic-plastic finite element analysis at large strain, and the inadequacies of some of these are commented upon.     

直杆碰撞刚性壁弹塑性动力后屈曲有限元分析

利用显式动力学有限元方法对直杆弹塑性动力后屈曲进行了分析,模拟了直杆轴向碰撞动力屈曲的变形及发展过程。分析中在直杆碰撞端局部临界屈曲长度范围内引入半正弦波形式的初始缺陷,计算结果与文献中的实验数据获得了很好的一致。分析结果表明,随着碰撞过程中所产生的应力波逐渐向前传播,后屈曲变形过程中所呈现的多个半波形式的高阶屈曲模态由初始具有单个半波形式的简单屈曲模态迅速演变而成。分析结果同时也揭示了直杆动力屈曲变形发展的机理,以及轴向应力波和屈曲变形的相互作用规律。     

On shakedown theory for elastic-plastic materials and extensions

The idea that an elastic-plastic structure under given loading history may shake down to some purely elastic state (and hence to a safe state) after a finite amount of initial plastic deformation, can apply to many sophisticated material models with possible allowable changes of additional material characteristics, as has been done in the literature. Despite some claims to the contrary, it is shown; however, that the shakedown theorems in a Melan-Koiter path-independent sense have been extended successfully only for certain elastic-plastic hardening materials of practical significance. Shakedown of kinematic hardening material is determined by the ultimate and initial yield stresses, not the generally plastic deformation history-dependent hardening curve between. The initial yield stress is no longer the convenient one (corresponding to the plastic deformation at the level of 0.2%) as in usual elastic-plastic analysis but to be related to the shakedown safety requirement of the structure and should be as small as the fatigue limit for arbitrary high-cycle loading. Though the ultimate yield strength is well defined in the standard monotonic loading experiment, it also should be reduced to the so-called “high-cycle ratchetting” stress for the path-independent shakedown analysis. A reduced simple form of the shakedown kinematic theorem without time integrals is conjectured for general practical uses. Application of the theorem is illustrated by examples for a hollow cylinder, sphere, and a clamped disk, under variable (including quasiperiodic dynamic) pressure.     

Analysis of elastic-plastic deformation in TiAl polycrystals

A computational model is described for analyzing stress variations within polycrystals of γ-TiAl, including the effect of anisotropic yielding and small-scale plastic flow. Interlamellar (soft mode) slip behavior is controlled by a separate collection of slip systems whose properties are derived from measurements on polysynthetically twinned (PST) specimens. When used to represent several hundred randomly oriented material grains, the model provides distributions and statistical data about the local stress, strain, and plastic deformation resulting from a prescribed macroscopic loading.     

Nonlinear mode II crack-tip fields for some Hookean materials

This paper presents a modified nonlinear Mode II crack model which is shown to satisfy the nonpenetrating crack surface boundary condition for homogeneous isotropic Hookean materials taking into account finite deformations. A recent investigation of the problem by Knowles [1] reveals apparent interpenetration of the crack surfaces which is considered nonphysical and therefore invalid. This observation is confirmed when a general solution based on Knowles's perturbation boundary layer method to characterize the finite deformation effects on Mode II crack-tip fields for the materials is derived. By deducing complete 2nd order solutions of the problem, the Poynting nonlinear effect becomes self-evident at the crack tip and for Hookean materials with there always exists the penetrating phenomenon between upper and lower crack surfaces.     

The singularity analysis of the stress and strain fields for Mode I fracture in elastic-plastic state

The stress and strain singularities of power hardening material for Mode I fracrure are analysed according to the fundamental equations of elastic-plastic mechanics. It is found that the singularities of all stress and strain components do not change in the thick direction, and neither the six stress components nor the six strain components have the same singularity.     

正交异性材料平面裂纹尖端应力场

根据正交各向异性材料力学性能确定出了用应力函数表示的弹性力学基本方程,利用坐标变换和复变函数方法求解了正交异性材料平面裂纹体的应力边值问题。借鉴一般断裂力学解法构造了I型和II型裂纹问题的应力函数,推导出了正交各向异性板裂纹尖端区的奇异应力场。通过数值计算说明了裂纹尖端应力表达式的正确性,验证了裂尖前沿应力变化规律,即σx与材料特征参数h2成正比,而σy和τxy不随材料特性变化。     

Numerical analysis of large elastic-plastic deformation of beams due to dynamic loading

Numerical calculations were performed for two examples of the response of elastic-plastic beams subjected to dynamic loads. These were a simply supported, axially restrained beam under suddenly applied uniform pressure, and an axially restrained, clamped beam with a central mass that is impacted by a projectile. Large elastic-plastic deflections were considered, and the method of finite differences was used. Two different constitutive equations were assumed: the elástic-perfectly plastic relation, and a special elastic-viscoplastic, strain hardening model. Analysis of the results included examining the interaction between the bending moment and the axial force, the variation of the axial force, bending moment and deflection with time, and the propagation velocities of the various phenomena during motion. Experiments were carried out in which a rifle projectile hit a central mass which had been fastened to a clamped beam. Comparison between the theoretical and experimental dynamic deflections shows good agreement for relatively short response times.     

Electromechanical fracture analysis for corners and cracks in piezoelectric materials

Corners and cracks are usually studied separately in the literature. To build a bridge connecting these two different but similar topics, in this paper the solutions for piezoelectric multi-wedges, which cover corners and interface corners, are used to study the cracks and interface cracks in piezoelectric materials. Moreover, the stress/electric intensity factors defined for cracks, interface cracks and interface corners are also extended to the general corners. By taking the special feature of Stroh formalism for anisotropic elasticity, all the solutions presented in this paper for piezoelectric materials preserve the same matrix form as those of the corresponding anisotropic problems. To see more clearly about the piezoeffects on the corners and cracks, most of the complex matrix form solutions are expanded in real component form for two typical piezoelectric ceramics with different poling directions.     

粘弹性大变形动力响应的有限元分析

本文基于ＴｏｔａｌＬａｇｒａｎｇｉａｎ增量叠加方法，采用Ｋｉｒｃｈｈｏｆｆ应力增量和Ｇｒｅｅｎ应变增量表示的动力虚功方程和Ｋｉｒｃｈｈｏｆｆ应力－Ｇｒｅｅｎ应变的单积分型本构关系，导出粘弹性大变形的动力变分方程。依此采用Ｎｅｗｍａｒｋ法和八节点轴对称等参数元与二十节点三维等参数元编制了轴对称及三维问题的动力响应计算程序，典型例题的计算结果表明分析符合结构的物理性质。