Singularity characteristics for a lip-shaped crack subjected to remote biaxial loading

In this paper, the conformal mapping method was adopted to solve the problem of an infinite plate containing a central lip-shaped crack subjected to remote biaxial loading. A kind of leaf-shaped configuration was also constructed in order to solve the problem. The analytical result showed that the singularity order of the stress field at the tip of a lip-shaped crack remains -1/2, despite the difference in notch-crack width.

Two critical crack propagating velocities for PMMA fracture surface

Ultrasonic fractography and scanning electronic microscopy (SEM) are used to determine the direct relationship between the fracture surface morphology and the main crack velocity during the rapid rupture of polymethylmethacrylate (PMMA). Two critical crack velocities are found for the fracture. Quasi-parabolic markings will appear when the crack speed exceeds the first critical speed. Crack propagating at speed above the second critical speed leaves a thicket of small branches penetrating the surface behind them. Both critical speeds are functions of the thickness of the specimens.

Mechanism on Progressive Failure of a Faulted Rock Slope Due to Slip-Weakening

Slip-weakening is one of the characteristics of geological materials under certain loadings. Non-uniform rock structure may exist in the vicinity of the slip surface for a rock slope. Some portion of the slip surface may be penetrated but the other not. For the latter case, the crack or the fault surface will undergo shear deformation before it becomes a successive surface under a certain loading. As the slipped portion advances,slip-weakening occurs over a distance behind the crack tip. In the weakening zone, the shear strength will decrease from its peak value to residual friction level. The stress will redistribute along the surface of crack and in the weakening zone. Thus the changed local stress concentration leads the crack to extend and the ratio of penetration of the slip surface to increase. From the view of large-scale for the whole slip surface, the shear strength will decrease due to the damage of interior rock structure, and the faulted rock behaves as a softening material. Such a kind of mechanism performs in a large number of practical landslides in the zones experienced strong earthquakes. It should be noted that the mechanism mentioned above is different from that of the breakage of structural clay,in which the geological material is regarded as a medium containing structural lumps and structural bands. In this paper, the softening behavior of a faulted rock should be regarded as a comprehensive result of the whole complicated process including slip-weakening, redistribution of stress, extension of crack tip, and the penetration of the slip surface. This process is accompanied by progressive failure and abrupt structural damage. The size of slip-weakening zone is related to the undergoing strain. Once the relative slide is initiated (local or integrated), the effect of slip-weakening will behave in a certain length behind the crack tip until the formation of the whole slip surface.

Cell shape changes indicate a role for extrinsic tensile forces in Drosophila germ-band extension.

Drosophila germ-band extension (GBE) is an example of the convergence and extension movements that elongate and narrow embryonic tissues. To understand the collective cell behaviours underlying tissue morphogenesis, we have continuously quantified cell intercalation and cell shape change during GBE. We show that the fast, early phase of GBE depends on cell shape change in addition to cell intercalation. In antero-posterior patterning mutants such as those for the gap gene Krüppel, defective polarized cell intercalation is compensated for by an increase in antero-posterior cell elongation, such that the initial rate of extension remains the same. Spatio-temporal patterns of cell behaviours indicate that an antero-posterior tensile force deforms the germ band, causing the cells to change shape passively. The rate of antero-posterior cell elongation is reduced in twist mutant embryos, which lack mesoderm. We propose that cell shape change contributing to germ-band extension is a passive response to mechanical forces caused by the invaginating mesoderm.

Dynamic SIF of interface crack between two dissimilar viscoelastic bodies under impact loading

In this paper, the transient dynamic stress intensity factor (SIF) is determined for an interface crack between two dissimilar half-infinite isotropic viscoelastic bodies under impact loading. An anti-plane step loading is assumed to act suddenly on the surface of interface crack of finite length. The stress field incurred near the crack tip is analyzed. The integral transformation method and singular integral equation approach are used to get the solution. By virtue of the integral transformation method, the viscoelastic mixed boundary problem is reduced to a set of dual integral equations of crack open displacement function in the transformation domain. The dual integral equations can be further transformed into the first kind of Cauchy-type singular integral equation (SIE) by introduction of crack dislocation density function. A piecewise continuous function approach is adopted to get the numerical solution of SIE. Finally, numerical inverse integral transformation is performed and the dynamic SIF in transformation domain is recovered to that in time domain. The dynamic SIF during a small time-interval is evaluated, and the effects of the viscoelastic material parameters on dynamic SIF are analyzed.

A crack perpendicular to and terminating at a bimaterial interface

Using dislocation simulation approach, the basic equation for a finite crack perpendicular to and terminating at a bimaterial interface is formulated. A novel expansion method is proposed for solving the problem. The complete solution to the problem, including the explicit formulae for the T stresses ahead of the crack tip and the stress intensity factors are presented. The stress held characteristics are analysed in detail. It is found that normal stresses sigma(x) and sigma(y) ahead of the crack tip, are characterised by Q fields if the crack is within a stiff material and the parameters \p(T)\ and \q(T)\ are very small, where Q is a generalised stress intensity factor for a crack normal to and terminating at the interface. If the crack is within a weak material, the normal stresses sigma(x) and sigma(y) are dominated by the Q field plus T stress.

Ice induced low temperature fatigue crack propagation in offshore structural steel A131

To investigate the low temperature fatigue crack propagation behavior of offshore structural steel A131 under random ice loading, three ice failure modes that are commonly present in the Bohai Gulf are simulated according to the vibration stress responses induced by real ice loading. The test data are processed by a universal software FCPUSL developed on the basis of the theory of fatigue crack propagation and statistics. The fundamental parameter controlling the fatigue crack propagation induced by random ice loading is determined to be the amplitude root mean square stress intensity factor K-arm. The test results are presented on the crack propagation diagram where the crack growth rate da/dN is described as the function of K-arm. It is evident that the ice failure modes have great influence on the fatigue crack propagation behavior of the steel in ice-induced vibration. However, some of the experimental phenomena and test results are hard to be physically explained at present. The work in this paper is an initial attempt to investigate the cause of collapse of offshore structures due to ice loading.

Large strain field near a crack tip in a rubber sheet

The distribution of stress-strain near a crack tip in a rubber sheet is investigated by employing the constitutive relation given by Gao (1997). It is shown that the crack tip field is composed of two shrinking sectors and one expanding sector. The stress state near the crack tip is in uniaxial tension. The analytical solutions are obtained for both expanding and shrinking sectors.

An Extension of 2D Janbu’s Generalized Procedure of Slices for 3D Slope Stability Analysis II—Numerical Method and Applications

This paper provides a numerical approach on achieving the limit equilibrium method for 3D slope stability analysis proposed in the theoretical part of the previous paper. Some programming techniques are presented to ensure the maneuverability of the method. Three examples are introduced to illustrate the use of this method. The results are given in detail such as the local factor of safety and local potential sliding direction for a slope. As the method is an extension of 2D Janbu's generalized procedure of slices (GPS), the results obtained by GPS for the longitudinal sections of a slope are also given for comparison with the 3D results. A practical landslide in Yunyang, the Three Gorges, of China, is also analyzed by the present method. Moreover, the proposed method has the advantages and disadvantages of GPS. The problem frequently encountered in calculation process is still about the convergency, especially in analyzing the stability of a cutting corner. Some advice on discretization is given to ensure convergence when the present method is used. However, the problem about convergency still needs to be further explored based on the rigorous theoretical background.

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.

Torsional impact of transversely isotropic solid with functionally graded shear moduli and a penny-shaped crack

The torsional impact response of a penny-shaped crack in an unbounded transversely isotropic solid is considered. The shear moduli are assumed to be functionally graded such that the mathematics is tractable. Laplace transform and Hankel transform are used to reduce the problem to solving a Fredholm integral equation. The crack tip stress fields are obtained. Investigated are the influence of material nonhomogeneity and orthotropy on the dynamic stress intensity factor. The peak value of the dynamic stress intensity factor can be suppressed by increasing the shear moduli's gradient and/or increasing the shear modulus in a direction perpendicular to the crack surface.

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.

Interface crack problems with strain gradient effects

In this paper, the strain gradient theory proposed by Chen and Wang (2001 a, 2002b) is used to analyze an interface crack tip field at micron scales. Numerical results show that at a distance much larger than the dislocation spacing the classical continuum plasticity is applicable; but the stress level with the strain gradient effect is significantly higher than that in classical plasticity immediately ahead of the crack tip. The singularity of stresses in the strain gradient theory is higher than that in HRR field and it slightly exceeds or equals to the square root singularity and has no relation with the material hardening exponents. Several kinds of interface crack fields are calculated and compared. The interface crack tip field between an elastic-plastic material and a rigid substrate is different from that between two elastic-plastic solids. This study provides explanations for the crack growth in materials by decohesion at the atomic scale.

Asymptotic analysis for a crack on interface of damaged materials

An asymptotic analysis for a crack lying on the interface of a damaged plastic material and a linear elastic material is presented in this paper. The present results show that the stress distributions along the crack tip are quite similar to those with HRR singularity field and the crack faces open obviously. Material constants n, mu and mo are varied to examine their effects on the resulting stress distributions and displacement distributions in the damaged plastic region. It is found that the stress components sigma(rr), sigma(theta theta), sigma(r theta) and sigma(e) are slightly affected by the changes of material constants n, mu and m(0), but the damaged plastic region are greatly disturbed by these material parameters.

Mode I crack growth resistance of metallic foams

A Dugdale-type cohesive zone model is used to predict the mode I crack growth resistance (R-curve) of metallic foams, with the fracture process characterized by an idealized traction-separation law that relates the crack surface traction to crack opening displacement. A quadratic yield function, involving the von Mises effective stress and mean stress, is used to account for the plastic compressibility of metallic foams. Finite element calculations are performed for the crack growth resistance under small scale yielding and small scale bridging in plane strain, with K-field boundary conditions. The following effects upon the fracture process are quantified: material hardening, bridging strength, T-stress (the non-singular stress acting parallel to the crack plane), and the shape of yield surface. To study the failure behaviour and notch sensitivity of metallic foams in the presence of large scale yielding, a study is made for panels embedded with either a centre-crack or an open hole and subjected to tensile stressing. For the centre-cracked panel, a transition crack size is predicted for which the fracture response switches from net section yielding to elastic-brittle fracture. Likewise, for a panel containing a centre-hole, a transition hole diameter exists for which the fracture response switches from net section yielding to a local maximum stress criterion at the edge of the hole.