Finite Element Solutions for Plane Strain Mode I Crack with Strain Grading Effect

In this paper, a new phenomenological theory with strain gradient effects is proposed to account for the size dependence of plastic deformation at micro- and submicro-length scales. The theory fits within the framework of general couple stress theory and three rotational degrees of freedom omega(i) are introduced in addition to the conventional three translational degrees of freedom mu(i). omega(i) is called micro-rotation and is the sum of material rotation plus the particles' relative rotation. While the new theory is used to analyze the crack tip field or the indentation problems, the stretch gradient is considered through a new hardening law. The key features of the theory are that the rotation gradient influences the material character through the interaction between the Cauchy stresses and the couple stresses; the term of stretch gradient is represented as an internal variable to increase the tangent modulus. In fact the present new strain gradient theory is the combination of the strain gradient theory proposed by Chen and Wang (Int. J. Plast., in press) and the hardening law given by Chen and Wang (Acta Mater. 48 (2000a) 3997). In this paper we focus on the finite element method to investigate material fracture for an elastic-power law hardening solid. With remotely imposed classical K fields, the full field solutions are obtained numerically. It is found that the size of the strain gradient dominance zone is characterized by the intrinsic material length l(1). Outside the strain gradient dominance zone, the computed stress field tends to be a classical plasticity field and then K field. The singularity of stresses ahead of the crack tip is higher than that of the classical field and tends to the square root singularity, which has important consequences for crack growth in materials by decohesion at the atomic scale. (C) 2002 Elsevier Science Ltd. All rights reserved.

The scattering of general SH plane wave by interface crack between two dissimilar viscoelastic bodies

The scattering of general SH plane wave by an interface crack between two dissimilar viscoelastic bodies is studied and the dynamic stress,intensity factor at the crack-tip is computed. The scattering problem can be decomposed into two problems: one is the reflection and refraction problem of general SH plane waves at perfect interface (with no crack); another is the scattering problem due to the existence of crack. For the first problem, the viscoelastic wave equation, displacement and stress continuity conditions across the interface are used to obtain the shear stress distribution at the interface. For the second problem, the integral transformation method is used to reduce the scattering problem into dual integral equations. Then, the dual integral equations are transformed into the Cauchy singular integral equation of first kind by introduction of the crack dislocation density function. Finally, the singular integral equation is solved by Kurtz's piecewise continuous function method. As a consequence, the crack opening displacement and dynamic stress intensity factor are obtained. At the end of the paper, a numerical example is given. The effects of incident angle, incident frequency and viscoelastic material parameters are analyzed. It is found that there is a frequency region for viscoelastic material within which the viscoelastic effects cannot be ignored.

Flow-Induced Vibrations Of A Cylinder With Two Degrees Of Freedom Near Rigid Plane Boundary

Based on similarity analyses, the flow-induced vibrations of a near-wall cylinder with 2 degrees of freedom are investigated experimentally by employing a hydroelastic apparatus in conjunction with a flume. The cylinder's vibration amplitude, vibration frequency and vortex shedding frequency were measured and analyzed. The effects of gap-to-diameter ratio (e,ID) upon the vibration responses are further investigated. The experimental results indicate that, when the reduced velocity (Vr) is small (e.g. Vr = 1.2 similar to 2.6), only streamwise vibration occurs, and its frequency is quite close to its natural frequency in still water. When increasing Vr (e.g. Vr > 3.4), both streamwise and transverse vibrations of the near-wall cylinder may occur. In the examined range of gap-to-diameter ratio (0.42 < e(0)/D < 2.68), 2 vibration stages (in terms of Vr) of streamwise vibrations usually exist: First Streamwise Vibration (FSV) and Second Streamwise Vibration (SSV). In the SSV stage, the vortex shedding frequency may either undergo a jump to that of the streamwise vibration, or stay consistent with that of the transverse vibration. The amplitudes of transverse vibration are usually much larger than those of streamwise vibration for the same value of e(0)/D. The maximum amplitudes of both streamwise and transverse vibration get larger with the increase of e(0)/D (0.42 < e(0)/D < 2.68).

Instability Of Plane Poiseuille Flow In A Fluid-Porous System

The instability of Poiseuille flow in a fluid-porous system is investigated. The system consists of a fluid layer overlying porous media and is subjected to a horizontal plane Poiseuille flow. We use Brinkman's model instead of Darcy's law to describe the porous layer. The eigenvalue problem is solved by means of a Chebyshev collocation method. We study the influence of the depth ratio (d) over cap and the Darcy number delta on the instability of the system. We compare systematically the instability of Brinkman's model with the results of Darcy's model. Our results show that no satisfactory agreement between Brinkman's model and Darcy's model is obtained for the instability of a fluid-porous system. We also examine the instability of Darcy's model. A particular comparison with early work is made. We find that a multivalued region may present in the (k, Re) plane, which was neglected in previous work. Here k is the dimensionless wavenumber and Re is the Reynolds number. (C) 2008 American Institute of Physics. [DOI: 10.1063/1.3000643]

Plane strain asymptotic fields for cracks terminating at the interface between elastic and pressure-sensitive dilatant materials

The plane strain asymptotic fields for cracks terminating at the interface between elastic and pressure-sensitive dilatant material are investigated in this paper. Applying the stress-strain relation for the pressure-sensitive dilatant material, we have obtained an exact asymptotic solution for the plane strain tip fields for two types of cracks, one of which lies in the pressure-sensitive dilatant material and the other in the elastic material and their tips touch both the bimaterial interface. In cases, numerical results show that the singularity and the angular variations of the fields obtained depend on the material hardening exponent n, the pressure sensitivity parameter mu and geometrical parameter lambda.

Asymptotic analysis of plane-strain mode I steady-state crack growth in transformation toughening ceramics (II)

Based on a constitutive law which includes the shear components of transformation plasticity, the asymptotic solutions to near-tip fields of plane-strain mode I steadity propagating cracks in transformed ceramics are obtained for the case of linear isotropic hardening. The stress singularity, the distributions of stresses and velocities at the crack tip are determined for various material parameters. The factors influencing the near-tip fields are discussed in detail.

Stress intensity factors calculation in anti-plane fracture problem by orthogonal integral extraction method based on femol

For an anti-plane problem, the differential operator is self-adjoint and the corresponding eigenfunctions belong to the Hilbert space. The orthogonal property between eigenfunctions (or between the derivatives of eigenfunctions) of anti-plane problem is exploited. We developed for the first time two sets of radius-independent orthogonal integrals for extraction of stress intensity factors (SIFs), so any order SIF can be extracted based on a certain known solution of displacement (an analytic result or a numerical result). Many numerical examples based on the finite element method of lines (FEMOL) show that the present method is very powerful and efficient.

The mixed-type integral equations for transient elastodynamic plane crack problems

In the present paper, by use of the boundary integral equation method and the techniques of Green fundamental solution and singularity analysis, the dynamic infinite plane crack problem is investigated. For the first time, the problem is reduced to solving a system of mixed-typed integral equations in Laplace transform domain. The equations consist of ordinary boundary integral equations along the outer boundary and Cauchy singular integral equations along the crack line. The equations obtained are strictly proved to be equivalent with the dual integral equations obtained by Sih in the special case of dynamic Griffith crack problem. The mixed-type integral equations can be solved by combining the numerical method of singular integral equation with the ordinary boundary element method. Further use the numerical method for Laplace transform, several typical examples are calculated and their dynamic stress intensity factors are obtained. The results show that the method proposed is successful and can be used to solve more complicated problems.

Numerical analyses for the material bifurcation in plane sheet under tension

The various patterns (shear banding, surface wrinkling and necking) of material bifurcation in plane sheet under tension are investigated in this paper by means of a numerical method. It is found that numerical analysis can provide better ground for searching for the lowest critical loads. The inhomogeneity caused by void damage and the nonuniformity in the stress distribution across sheet thickness are proved to have detrimental effects on the material bifurcation. Nevertheless, material stability can be promoted by any means of depressing void damage or alleviating stress, even locally across the thickness. Besides, the peculiar behaviour of material bifurcation under slight biaxiality state is demonstrated. Copyright (C) 1996 Elsevier Science Ltd

Formation of shear bands in plane sheet

The formation of shear bands in plane sheet is studied, both analytically and experimentally, to enhance the fundamental understanding of this phenomenon and to develop a capability for predicting material failure. The evolution of voids is measured and its interaction with the process of shear banding is examined. The evolving dilatancy in plasticity is shown to have a vital role in analysing the shear-band type of bifurcation, and tremendously reduces the theoretical value of critical stresses. The analyses, referring to both localized and diffuse modes of bifurcation, fairly explain the corresponding observations obtained through testing a dual-phase steer sheet and provide a justification of the constitutive model used.

Unloading characteristics of anti-plane shear crack in softening materials

The local characteristics of the anti-plane shear stress and strain field are determined for a material where the stress increases linearly with strain up to a limit and then softens nonlinearly. Two unloading models are considered such that the unloading path always returns to the origin while the other assumes the unloading modulus to be that of the initial shear modulus. As the applied shear increases, an unloading zone is found to prevail between a zone in which the material softens and another zone in which the material is linear-elastic even though the crack does not propagate. The divisions of these zones are displayed graphically.

Higher-order analysis of crack-tip fields in power-law hardening materials

In this paper, we present an exact higher-order asymptotic analysis on the near-crack-tip fields in elastic-plastic materials under plane strain, Mode I. A four- or five-term asymptotic series of the solutions is derived. It is found that when 1.6 < n less-than-or-equal-to 2.8 (here, n is the hardening exponent), the elastic effect enters the third-order stress field; but when 2.8< n less-than-or-equal-to 3.7 this effect turns to enter the fourth-order field, with the fifth-order field independent. Moreover, if n>3.7, the elasticity only affects the fields whose order is higher than 4. In this case, the fourth-order field remains independent. Our investigation also shows that as long as n is larger than 1.6, the third-order field is always not independent, whose amplitude coefficient K3 depends either on K1 or on both K1 and K2 (K1 and K2 arc the amplitude coefficients of the first- and second-order fields, respectively). Firmly, good agreement is found between our results and O'Dowd and Shih's numerical ones[8] by comparison.

Asymptotic solution of mode-iii crack in damaged softening materials

In this paper, a constitutive model of elasticity coupled with damage suggested by Lemaitre et al, [1] is used. The macroscopic stress-strain response of the model includes two stages: strain hardening and strain softening. The basic equation is derived for the anti-plane shear problem. Several lowest order asymptotic solutions are obtained, and assembled for the crack-tip fields.

A simple approach to solve boundary-value problems in gradient elasticity

We outline a procedure for obtaining solutions of certain boundary value problems of a recently proposed theory of gradient elasticity in terms of solutions of classical elasticity. The method is applied to illustrate, among other things, how the gradient theory can remove the strain singularity from some typical examples of the classical theory.