Study on fast fracture and crack arrest

In this paper, the experimental results of brittle-crack propagation and arrest on structural steels are analyzed using the finite-difference method. The test specimens used are double-tension and double-cantilever type (DCB). It is found that dynamic consideration is indispensable for general interpretation of fast fracture and crack arrest and material toughness can be defined as a function of temperature and crack velocity. Using the defined fracture toughness, crack-propagation-arrest behavior can be predicted by crack-propagation simulation if the condition of boundary loading of a specimen is known. This implies a possibility to develop simple methods for crack-arrester design.     

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.     

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.     

Numerical analysis for a crack in piezoelectric material under impact

In this paper, a numerical analysis of impact interfacial fracture for a piezoelectric bimaterial is provided. Starting from the basic equilibrium equation, a dynamic electro-mechanical FEM formulation is briefly presented. Then, the path-independent separated dynamic J integral is extended to piezoelectric bimaterials. Based on the relationship of the path-independent dynamic J integral and the stress and electric displacement intensity factors, the component separation method is used to calculate the stress and electric displacement intensity factors for piezoelectric bimaterials in this finite-element analysis. The response curves of the dynamic J integral, the stress and electric displacement intensity factors are obtained for both homogeneous material (PZT-4 and CdSe) and CdSe/PZT-4 bimaterial. The influences of the piezoelectricity and the electro-mechanical coupling factor on these responses are discussed. The effects of an applied electric field are also discussed.     

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。     

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.     

Logarithm strain singularity at tip of mode I growing crack in elasticand perfectly-plastic material

Reanalyzed in detail is the stress and strain distribution near the tip of a Mode I steadily growing crack in an elastic and perfectly-plastic material. The crack tip region is divided into five angular sectors, one of which is singular in character and represents a rapid transition zone that becomes a line of strain discontinuity in the limit as crack tip is approached. It is shown for an incompressible material (ν=0.5) under plane strain that the local strain in all the angular sectors possesses the same logarithm singularity, i.e., In r where r is the radial distance measured from the crack tip. This result also prevails for the compressible material ( v < 0.5) and resolves a long standing controversy concerning the strain singularity in the sector just ahead of the crack tip.     

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.     

The stress field in the neighborhood of a branched crack in an infinite elastic sheet

The problem of a branched crack consisting of a main crack and a straight branch starting from one of its tip located in an infinite elastic sheet is considered under the assumptions of two-dimensional theory of Elasticity. Employing Kolosov-Muskhelishvili representation of the stress function and other well known techniques the problem is reduced to the solution of an integral equation. The nature of the stress singularity at the re-entrant corner, where the two branches of the crack meet, is discussed. Based upon a numerical solution of the integral equation the stress intensity factors at the two tips are computed for two types of prescribed traction at infinity and various geometric configurations of the branched crack.     

Numerical simulation of hydraulic fracture crack propagation

The plane problem of crack motion in an elastic medium under the pressure of a viscous fluid is considered. Under the condition of a constant fluid flow rate, the fluid is injected at the center of the crack. Contrary to other formulations of the problem, this paper attempts to take into account a possible fluid lag behind the crack tip. The resulting numerical solution is compared with a semianalytic one. It is found that the proposed numerical model can be used to predict the characteristics of a hydraulic fracture crack formed in a medium of a prescribed strength.     

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.     

Modeling the fracture process zone near a crack tip in a nonlinear elastic body

The model of a fracture process zone near the tip of a mode I crack in a nonlinear elastic body is proposed. Using the numerical solution of an appropriate boundary-value problem, the effect of the fracture process zone on a crack opening displacement is examined     

Diffraction of a plane stress wave by a semi-infinite crack in a general anisotropic elastic material

The problem of a semi-infinite crack subjected to an incident stress wave in a general anisotropic elastic solid is considered. The plane wave impinges the crack at a general oblique angle and is of any of the three types propagating in that direction. A related problem of a semi-infinite crack loaded by a pair of concentrated forces moving along the crack surfaces is also considered. In contrast to the conventional approach by Laplace transforms, a Stroh-like formalism is employed to construct the solution directly in the time domain. The solution is shown to depend on a Wiener–Hopf factorization of a symmetric matrix. Closed-form solution of the stress intensity factors is derived. A remarkably simple expression for the energy release rate is obtained for normal incidence.     

Perturbation approaches of a planar crack in linear elastic fracture mechanics: A review

One current challenge of linear elastic fracture mechanics (LEFM) is to take into account the non-linearities induced by the crack front deformations. For this, a suitable approach is the crack front perturbation method initiated by Rice (1985). It allows to update the stress intensity factors (SIFs) when the crack front of a planar crack is perturbed in its plane. This approach and its later extensions to more complex cases are recalled in this review. Applications concerning the deformation of the crack front when it propagates quasistatically in a homogeneous or heterogeneous media have been considered in brittle fracture, fatigue or subcritical propagation. The crack shapes corresponding to uniform SIF have been derived: cracks with straight or circular fronts, but also when bifurcations exist, with wavy front. For an initial straight crack, it has been shown that, in homogeneous media, in the quasistatic case, perturbations of all lengthscales progressively disappear unless disordered fracture properties yields Family and Vicsek (1985) roughness of the crack front. Extension of those perturbation approaches to more realistic geometries and to coalescence of cracks is also envisaged.     

The mixed-mode fracture mechanics analysis of an embedded arbitrary oriented crack in a two-dimensional functionally graded material plate

Mixed-mode fracture mechanics analysis of an embedded arbitrarily oriented crack in a two-dimensional functionally graded material using plane elasticity theory is considered. The material properties are assumed to vary exponentially in two planar directions. Then, employing Fourier integral transforms with singular integral equation technique, the problem is solved. The stress intensity factors (SIFs) at the crack tips are calculated under in-plane mechanical loads. Finally, the effects of crack orientation, material non-homogeneity, and other parameters are discussed on the value of SIF in mode I and mode II fracture.     

Dynamic fracture analysis of a penny-shaped crack in a magnetoelectroelastic layer

This paper analyzes the dynamic magnetoelectroelastic behavior induced by a penny-shaped crack in a magnetoelectroelastic layer subjected to prescribed stress or prescribed displacement at the layer surfaces. Two kinds of crack surface conditions, i.e., magnetoelectrically impermeable and permeable cracks, are adopted. The Laplace and Hankel transform techniques are employed to reduce the problem to Fredholm integral equations. Field intensity factors are obtained and discussed. Numerical results of the crack opening displacement (COD) intensity factors are presented and the effects of magnetoelectromechanical loadings, crack surface conditions and crack configuration on crack propagation and growth are examined. The results indicate that among others, the fracture behaviors of magnetoelectroelastic materials are affected by the sizes and directions of the prescribed magnetic and/or electric fields, and the effects are strongly dependent on the elastic boundary conditions.     

Fast numerical method for crack problem in the porous elastic material

The present paper discusses the crack problem in the linear porous elastic plane using the model developed by Nunziato and Cowin. With the help of Fourier transform the problem is reduced to an integral equation over the boundary of the crack. Some analytical transformations are applied to calculate the kernel of the integral equation in its explicit form. We perform a numerical collocation technique to solve the derived hyper-singular integral equation. Due to convolution type of the kernel, we apply, at each iteration step, the classical iterative conjugate gradient method in combination with the Fast Fourier technique to solve the problem in almost linear time. There are presented some numerical examples for materials of various values of porosity.     

Parametric excitation of flexural-gravity edge waves in the fluid beneath an elastic ice sheet with a crack

The properties of elastic-gravity oscillations of deep water beneath a thin elastic plate with a crack are investigated in the paper. The dependence of the reflection and transition coefficients of the waves through the crack on wave frequency and incident angle are found. The shape of the fluid surface deformed by edge waves, propagating along the crack and decreasing exponentially away from the crack, is investigated in the vicinity of the crack. The asymptotic equations describing the parametric excitation of counterpropagating edge waves by flexural-gravity waves which hit the crack at normal incidence are derived.     

Concentrated impact loading on one face of a semi-infinite crack in an anisotropic elastic material

The response of an unbounded anisotropic elastic body containing a semi-infinite crack subjected to a concentrated impact force on one of the crack faces is studied. An exact solution of the dynamic stress intensity factors is obtained from a linear superposition of the solution of Lamb’s problem and a solution of a dislocation emitting from the crack tip. The stress intensity factors exhibit square-root singularity upon the arrival of the Rayleigh wave at the crack tip. As the Rayleigh wave passes through the crack tip, the stress intensity factors either instantaneously assume the static values or gradually approach to zero. Several numerical examples are given for isotropic, cubic and orthotropic materials.