理想弹塑性压力敏感性材料中球形孔洞的动态扩展研究

通过椭圆形的屈服方程和自相似假设,结合Hopkins三区模型,研究了理想弹塑性压力敏感性材料中球形孔洞的动态扩展问题,得到了塑性区场量的非线性控制微分方程组。通过弹性区的应力场以及弹塑性边界的塑性屈服条件,给出了塑性区的初值,并应用打靶法给出了问题的数值解。结果表明:与幂硬化材料中的应力场变化不同,理想弹塑性压力敏感性材料中,应力场的变化较小,且基本不受参数孔洞膨胀速度m和压力敏感性参数α1、α2的影响;随到孔边距离的减小,应变明显增大,同时相对密度稍有增大。     

Fields near a rapidly propagating crack-tip in an elastic perfectly-plastic material

Dynamic effects are investigated for the steady-state fields of stress and deformation in the immediate vicinity of a rapidly propagating crack-tip in an elastic perfectly-plastic material. Both the cases of antiplane strain and in-plane strain have been considered. The governing equations in the plastic regions are hyperbolic in nature. Simple wave solutions together with uniform fields provide explicit asymptotic expressions for the stresses and the strains in the near-tip regions. The dynamic solutions describe a region of plastic loading which completely surrounds the propagating crack-tip.     

On non-local and non-homogeneous elastic continua

A thermodynamic framework endowed with the concept of non-locality residual is adopted to derive non-local models of integral-type for non-homogeneous linear elastic materials. Two expressions of the free energy are considered: the former yields a one-component non-local stress, the latter leads to a two-component local–non-local stress since the stress is expressed as the sum of the classical local stress and of a non-local component identically vanishing in the case of constant strains. The attenuation effects are accounted for by a symmetric space weight function which guarantees the constant strain requirement as well as the dual constant stress condition everywhere in the body. The non-local and non-homogeneous elastic structural boundary-value problem under quasi-static loads is addressed in a geometrically linear range. The complete set of variational formulations for the structural problem is then provided in a unitary framework. The solution uniqueness of the non-local structural model is proved and the non-local FEM is addressed starting from the non-local counterpart of the total potential energy. Numerical applications are provided with reference to a non-homogeneous bar in tension using the Fredholm integral equation and the non-local FEM. The solutions show no pathological features such as numerical instability and mesh sensitivity for degraded bar conditions.     

Crack growth under plane stress conditions in an elastic perfectly-plastic material

Mode-I crack growth under conditions of generalized plane stress has been investigated. It has been assumed that near the plane of the crack in the loading zone, the simple stress components corresponding to a central fan field maintain validity up to the elastic-plastic boundary. By the use of expansions of the particle velocities in the coordinate y, and by matching of the relevant stress components and particle velocities to the dominant terms of appropriate elastic fields at the elastic-plastic boundary, a complete solution has been obtained for ε_y in the plane of the crack. The solution applies from the propagating crack tip up to the moving elastic-plastic boundary. The strain fields for a self-similar crack nucleating at a point and for steady-state propagation of a crack have been considered as special cases.     

Some possible alternative solutions of near-tip fields for steady crack growth in elastic perfectly-plastic media

Some possible alternative solutions of near-tip fields are studied for plane-strain Mode—I quasi-static steady crack growth in incompressible (ν=1/2) elastic perfectly-plastic media. A group of four-sector solutions and a three-sector solution in which the elastic-unloading region vanishes are given. Stress functions, plastic flow factors and plastic strains in each region are also given. Project supported by the State Education Commission under a funding program for Excellent University Young Faculties and National Natural Science Foundation of China.     

Numerical analysis of fast fracture and crack arrest in a sheet of elastic perfectly-plastic material

The dynamic fields for acceleration, deceleration and arrest of a crack tip have been investigated numerically. We consider cracks which start to extend rapidly under brittle conditions. The crack-tips then enter regions of elasto-plastic constitutive behavior and they are subsequently arrested. Results have been obtained for a symmetrically expanding central crack and for an edge crack, both in thin sheets. The elasto-plastic behavior has been described by J₂ flow theory, with the von Mises yield criterion and a bilinear relation between effective stress and effective strain. Numerical results are presented for stress and strain components at a short distance ahead of the propagating and arresting crack tips.     

A kinematic elastic nonshakedown theorem for implicit standard materials

Summary  Criteria for a priori recognition of the type of steady-state response induced by cyclic loads and prediction whether a structure will shakedown elastically or not, without the necessity of performing a step-by-step full analysis, have considerable importance. Melan and Koiter theorems provide criteria that guarantee whether elastic shakedown occurs or not under cyclic loads in case of perfect plasticity. However, there remain some aspects of the shakedown theory which deserve further study. One of these, concerned with more realistic nonassociative elastic–plastic constitutive material models, allowing for nonlinear kinematic and isotropic hardening suitable to describe the cyclic plastic behaviour of metallic materials, has strong motivation. Koiter's elastic nonshakedown theorem is reconsidered here, with the objective of extending it to the de Saxcé's implicit standard material class, which contains a wide class of nonassociative elastic–plastic material behaviours. Shakedown analysis is formulated by a kinematic approach based on the plastic accumulation mechanism concept due to Polizzotto. A sufficient condition for elastic nonshakedown and a distinct necessary condition are established. Then, an upper bound to the shakedown multiplier is evaluated. Received 15 February 2001; accepted for publication 18 October 2001     

Periodic oscillations of a class of non-autonomous non-linear elastic continua

A modified perturbation method for obtaining periodic solutions to a class of non-autonomous non-linear partial differential equations is developed. The classic small divisor is discussed in detail and a general method for its elimination is presented. New terminology is introduced for the purpose of discussing forcing functions that produce in a system a response that is of the same form as a non-linear periodic mode for the same system. Specific examples are examined to verify the results of this work.     

Variational formulations and a consistent finite-element procedure for a class of nonlocal elastic continua

The structural boundary-value problem in the context of nonlocal (integral) elasticity and quasi-static loads is considered in a geometrically linear range. The nonlocal elastic behaviour is described by the so-called Eringen model in which the nonlocality lies in the constitutive relation. The diffusion processes of the nonlocality are governed by an integral relation containing a recently proposed symmetric spatial weight function expressed in terms of an attenuation function. A firm variational basis to the nonlocal model is given by providing the complete set of variational formulations, composed by ten functionals with different combinations of the state variables. In particular the nonlocal counterpart of the classical principles of the total potential energy, complementary energy and mixed Hu–Washizu principle and Hellinger–Reissner functional are recovered. Some examples concerning a piecewise bar in tension are provided by using the Fredholm integral equation and the proposed nonlocal FEM.     

A Betti-Rayleigh theorem for elastic bodies exhibiting temperature dependent properties

Summary The reciprocal theorem of Betti and Rayleigh is extended to thermoelastic problems concerning temperature dependent properties of the bodies. Illustrative examples are solved concerning extension and flexural rotation of a bar, change of volume of a body without and with a cavity, and the thermoelastic displacement. This work has been sponsored by the United States Army under contract No. DA-11-022-ORD-2059.     

On the work and energy theorem for unbounded elastic bodies

The main scope of the paper is the statement and the proof of the work and energy theorem for elastic bodies that occupy an unbounded region of space and whose acoustic tensor A may suitably grow at large spatial distance from a fixed origin.     

Matched asymptotic expansions in nonlinear fracture mechanics—I. longitudinal shear of an elastic perfectly-plastic specimen

A method of analysis based upon matched asymptotic expansions is proposed for a cracked specimen which is subjected to longitudinal shear (mode III) loading. This gives the small-scale yielding estimate of linear fracture mechanics as a first approximation, and provides systematic refinements which take account of the nonlinear interaction between the elastic and the plastic regions. Explicit solutions can be generated for any specimen which is amenable to a linear elastic analysis. Fracture parameters, such as crack opening displacement and the Jintegral, are expressed as power series in the ratio of applied stress to yield stress, and three terms are given explicitly. These are defined from linear elastic solutions alone. The edge-cracked strip and cracking from a semi-circular notch are studied as examples. Comparison with an exact solution for the former geometry suggests that the three-term expansions give useful results up to 75 % of limit load. The latter example is new and shows the effect of a notch on a crack at loads beyond the normal range of validity of linear elastic fracture mechanics.     

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.     

A time-domain energy theorem for the scattering of plane elastic waves

A time-domain energy theorem for the scattering of plane elastic waves by an obstacle of bounded extent is derived. The obstacle is embedded in a homogeneous, isotropic, perfectly elastic medium. As to the elastodynamic behavior of the obstacle no assumptions have to be made; so, lossy, non-linear and time-variant behavior is included. As to the wave motion, three different kinds of time behavior are distinguished: (a) transient, (b) periodic, and (c) pertuating, but with finite mean power flow density. For these cases, the total energy (case (a)) or the time-averaged power (cases (b) and (c)) that is both absorbed and scattered by the obstacle is related to a certain time interaction integral of the incident plane wave (P or S) and the spherical-wave amplitude of the scattered wave of the same type (P or S) in the far-field region, when observed in the direction of propagation of the incident wave.     

Matched asymptotic expansions in nonlinear fracture mechanics—III. In-plane loading of an elastic perfectly-plastic symmetric specimen

The matched expansion method, introduced by the authors in two earlier papers (1976) devoted to mode III loading, is applied to the practically important case of mode I loading of a symmetric specimen. The method allows the linear elastic far-field to be considered separately from the elasto-plastic near-tip field, except for coupling through a set of parameters that are determined explicitly in the matching. The effects of the plasticity are thus found, once and for all, from the solution of a set of standard elasto-plastic problems for a semi-infinite crack in an infinite body, whose properties may be tabulated. The solution for any particular specimen geometry and loading then follows from a small set of linear elastic solutions for the specimen, which define, through coefficients γ_ij appearing in their near-tip expansions, all the parameters in the “inner” and “outer” solutions. The effects of plasticity appear in these parameters only through a set of constants C_ti that define the far-field expansions of the “inner” (near-tip) solutions: they are material constants, depending upon the constitutive relation for the material, but not upon specimen geometry and loading. The J-integral, being obtainable from the far-field, is expressed as an explicit asymptotic series in the loading parameter ε, whose coefficients are given as functions of the “elastic” parameters γ_ij and the material constants C_i. It is demonstrated that a plastic-zone correction term, r_y, can be chosen to yield a two-term asymptotic expansion for J; the value of r_y depends upon the yielding model only through the constant C₁.The Dugdale (1960) model of yielding is treated, as a simple example for which all calculations can be performed analytically, and for which exact solutions are available for comparison.Finally, the near-tip solutions are constructed for a material obeying the Mises yield criterion and associated flow-rule, using a specially developed finite element program. The first eight of the constants C_i are tabulated, which suffice to define the J-integral up to terms of order ε⁶ (where ε is a loading parameter) and some representative near-tip features are displayed graphically. The computed value of C₁ shows that the conventionally adopted value for the plastic-zone correction r_y is too large by a factor of roughly 2.8, if it is to yield a genuine asymptotic estimate for J. As an example, the “elastic” parameters γ_ij are found, from a boundary collocation program, for a centre-cracked square plate subjected to tensile loading; and a plot of J versus load, and the plastic-zone shape at a particular load level, are displayed.     

On a theory of micropolar protoelastic material bodies and constitutive equations for nonlocal micropolar elastic continua

In this paper the definition of micropolar protoclastic material bodies is given and with the help of the principle of virtual power, the variational principle of those bodies is derived. In terms of that same idea and the definition of micropolar protopotential presented here, the constitutive equations for nonlocal micropolar elastic continua are naturally derived.     

The reciprocal theorem and linear elastic variational principles

In this paper classical linear elastic variational principles are systematically derived from the reciprocal theorem and mixed variational principles of variations of boundary conditions are given.     

On the adjacent equilibrium method in the stability theory of conservative elastic continua

The relationship of the adjacent equilibrium method, the regular perturbation method and the energy method for neutral equilibrium is studied. It is shown that unlike the adjacent equilibrium method, the regular perturbation method yields, for the problems under consideration, non-homogeneous perturbation equations and that adjacent states of equilibrium do not exist at the bifurcation point. These results are then compared with the result of the energy criterion for neutral equilibrium V₂[u] = 0. It is found that although the physical arguments are different in the three methods, the resulting stability equations are identical; thus showing why the adjacent equilibrium argument, even for cases when it is incorrect, yields correct critical loads. This is followed by a discussion of an incorrect derivation of a stability condition and a notion about a load type introduced in the stability literature, which are consequences of the assumption of the general existence of adjacent equilibrium states at bifurcation points.