Viscoelastic composite with a cracked layer under anti-plane shear

A viscoelastic cracked sandwich composite under anti-plane load is presented via the application of the Laplace transform and the complex variable formulation. Using a special technique of analytical continuation associated with the method of image, the triple-layered problem is reduced to a singular function by simple algebraic manipulations. The superposition method and the singular integral equation is applied to deal with the concentrated load and crack problem. Finally, some typical viscoelastic models and its corresponding stresses and stress intensity factors are also discussed. The results show that the time dependent stress intensity factor is affected by the relative strength of creep compliances of each layer.     

Numerical method for elastic-plastic waves in cracked solids,Part 1: anti-plane shear problem

Summary The paper deals with a numerical method combining a characteristic-based scheme and Zwas' method in order to solve the hyperbolic PDE's of elastic and elastic-plastic anti-plane shear waves in two space dimensions. First, the need of new physically reasonable numerical methods for stress waves in solids is demonstrated by numerical applications to problems with impulsive loading, where defects of some standard methods are shown. Then, the new secon-dorder accurate method is derived. A suitable procedure to model the elastic-plastic behaviour of materials by simple waves is included. The capability of the methods is demonstrated by application to several examples. Additionally, for comparison with numerical results, a similarity solution for the semi-infinite crack undergoing an elastic-plastic shock loading is derived in the appendix.

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Numerisches Verfahren für elastisch-plastische Wellen in gerissenen Festkörpern, Teil 1: Das dynamische Querschubproblem

Übersicht In der Arbeit wird eine numerische Methode vorgestellt, in der das Charakteristiken erfahren mit der Methode von Zwas kombiniert wurde, um die hyperbolischen, partiellen Differentialgleichungen der elastisch-plastischen Wellenausbreitung für das dynamische Querschubproblem zu lösen. Um die Notwendigkeit neuer, physikalisch brauchbarer Methoden für Spannungswellenprobleme zu begründen, werden zunächst an Beispielen typische Schwächen einiger Standardmethoden gezeigt. Danach wird die neue Methode dargestellt. Dabei wird auch ein geeigneter Weg im Spannungsraum angegeben, um elastisch-plastische Wellen durch sog. einfache Wellen zu beschreiben. Die Fähigkeiten der Methode werden an einigen Beispielen überprüft. Ferner wird im Anhang, um mit numerischen Ergebnissen zu vergleichen, eine Ähnlichkeitslösung hergeleitet, die für den halbunendlichen Riß bei stoßartig aufgebrachter Belastung und Beanspruchung bis in den elastisch-plastischen Bereich gilt.

     

Dynamic anti-plane shear of a cracked piezoelectric ceramic

The dynamic theory of antiplane piezoelectricity is applied to solve the problem of a line crack subjected to horizontally polarized shear waves in an arbitrary direction. The problem is formulated by means of integral transforms and reduced to the solution of a Fredholm integral equation of the second kind. The path-independent integral G is extended here to include piezoelectric effects, and is evaluated at the crack tip to obtain the dynamic energy release rate. Numerical calculations are carried out for the dynamic stress intensity factor and energy release rate. The material is piezoelectric ceramic.     

An elastic-plastic analysis of an asymmetric cracked specimen subjected to longitudinal shear

Some recent elastic-plastic analyses of cracked specimens subjected to symmetric mode III loading are extended to include asymmetric loading and geometry. Solutions are given for arbitrary work hardening behaviour in any specimen that is amenable to a linear elastic analysis. It is shown that asymmetry has a major influence on the shape of the plastic zone, but does not affect the J-integral unil the loading is well into the large scale yielding range. In particular the “plastic zone corrected” estimate of J, obtained by elastically solving a problem for a crack longer than the actual one, is shown to remain a valid two-term asymptotic expansion in the presence of asymmetry. The general results are applied to a crack at an angle to a uniform stress field in a power law hardening material. The growth of the plastic zone is displayed graphically for various hardening exponents and crack orientations. No other asymmetric solution is available, but values of J are compared with those obtained from a fully plastic analysis in the symmetric case.     

Analytical solutions to general anti-plane shear problems in finite elasticity

This paper presents a pure complementary energy variational method for solving a general anti-plane shear problem in finite elasticity. Based on the canonical duality–triality theory developed by the author, the nonlinear/nonconvex partial differential equations for the large deformation problem are converted into an algebraic equation in dual space, which can, in principle, be solved to obtain a complete set of stress solutions. Therefore, a general analytical solution form of the deformation is obtained subjected to a compatibility condition. Applications are illustrated by examples with both convex and nonconvex stored strain energies governed by quadratic-exponential and power-law material models, respectively. Results show that the nonconvex variational problem could have multiple solutions at each material point, the complementary gap function and the triality theory can be used to identify both global and local extremal solutions, while the popular convexity conditions (including rank-one condition) provide mainly local minimal criteria and the Legendre–Hadamard condition (i.e., the so-called strong ellipticity condition) does not guarantee uniqueness of solutions. This paper demonstrates again that the pure complementary energy principle and the triality theory play important roles in finite deformation theory and nonconvex analysis.     

Fully plastic center-cracked strip under anti-plane shear

The problem of a center-cracked strip subjected to uniform remote anti-plane shear stress is transformed to a problem in a hodograph plane which is solved exactly by Mellin transform and Wiener-Hopf technique. The material of the strip satisfies a pure power hardening stress strain relation and the results are valid for both deformation and flow theories of placticity. Numerical values are given for the crack opening displacement δ and Rice's path independent J integral for several values of the power hardening exponent n and crack width to strip width ratios. Approximate asymptotic formulas are presented for J and δ for large n.     

Transient response of a cracked magnetoelectroelastic strip under anti-plane impact

The transient anti-plane problem of a magnetoelectroelastic strip containing a crack vertical to the boundary is considered. Singular integral equations for the impermeable crack are obtained by using Fourier and Laplace transforms. Numerical results show the effects of the relative loading parameters κ_D and κ_B, and the crack configuration on the dynamic fracture behavior. The results obtained indicate that for the impermeable crack, the electric and magnetic impacts have significant influences on the dynamic stress intensity factor and the dynamic energy density factor.     

Fatigue crack growth under anti-plane shear mode deformation

In this paper, the theory of the steady growth of fatigue crack in an infinite medium under the periodic anti-plane remote shear loading has been examined. The criterion of accumulative plastic work for material failure associated with the slip displacement in the fracture process zone of Dugdale type ahead of the crack tip is employed in the analysis. The effect of the locked dislocation in the fracture process zone is considered. Under the assumption that the speed of fatigue crack propagation remains uniform through the fracture process zone, the steady speed of fatigue crack can be expressed as a function of the range of the applied shearing stress and the maximum shearing stress. The effect of the crack size on the fatigue crack speed is discussed. The effect of the finite width of specimen on the speed of fatigue crack speed is investigated. The differences between the present work and the previous studies on fatigue crack speed are discussed.     

Layered piezoelectric medium with interface crack under anti-plane shear

The theory of linear piezoelectricity is applied to solve the anti-plane shear problem of a piezoelectric layer sandwiched by two dissimilar homogeneous materials with a crack at the interface. Both mechanical and electrical loads are applied to the piezoelectric laminate. By the use of Fourier transforms, the mixed boundary value problem is reduced to a singular integral equation which is solved numerically to determine the stress intensity factors for several layered piezoelectric media, and the results are presented in graphical form.     

Elastic-plastic mechanics of steady crack growth under anti-plane shear

For a crack with steady growth under anti-plane shear, analysis shows a primary plastic zone included in an angle of ±19.7° ahead of the crack tip, and two very thin secondary (reverse) plastic zones along the crack flanks, each included in an angle of 0.37°. Numerical solutions give the shape of the plastic zones which determine the active and residual plastic strains, and give the crack tip displacement, which is approximately 0.07 of that for monotonic loading without growth. The length of the primary plastic zone is almost the same as that without growth, but the thickness is about 3/5 as great. Coupled with ductile fracture criteria, the present results predict initially stable crack growth, whereas analyses based on the simplification of yielding on just one plane predict unstable fracture immediately following initiation.     

A solution for an isotropic sector under anti-plane shear loadings

In this study, the problem of an isotropic sector subjected to anti-plane shear loadings is investigated. The loadings were applied to the arc of the sector, and the radial edges of the sector were under traction-free or fixed conditions. Depending on these conditions, three problems, namely, free–free, fixed–free, and fixed–fixed edges were studied. A procedure using the finite Mellin transform combined with the Laplace transform was proposed for solving these problems. Explicit closed form solutions for the displacement and stress fields throughout the sector were obtained. The stress intensity factor (SIF) for each problem was analyzed using the obtained stress fields. It was determined that the SIF disappeared under the special condition of a fixed–fixed edge. Other special cases having anti-symmetric conditions were deduced from the derived solutions, and the results of these verified those cited in the literature as well as those obtained using finite element analysis (FEA).     

Finite-deformation crack in an infinite body under anti-plane simple shear

The anti-plane shear crack problem in an infinite body is studied within the context of finite-deformation elastostatics. The governing equations are found to be identical to those of certain previously studied small strain plastic problems under anti-plane shear. For a particular strain energy function, an exact solution of the problem is obtained through use of results in the literature.     

Fully plastic crack in an infinite body under anti-plane shear

The problem of a semi-infinite body with an edge crack subjected to far out-of-plane shear is solved by a transformation to a hodograph plane and the Wiener-Hopf technique. The material stress-strain behavior is governed by a pure power hardening relation and the results are valid for both deformation theory and flow theory of plasticity. Results are presented for crack opening displacement, path independent J integral and crack tip singularities for all finite values of the power hardening parameter.     

Fracture dynamics problems of orthotropic solids under anti-plane shear loading

By application of the approaches of the theory of complex functions, fracture dynamics problems of orthotropic solids under anti-plane shear loading were researched. Universal representation of analytical solutions was obtained by means of self-similar functions. The problems dealt with can be facilely transformed into Riemann–Hilbert problems by this technique, and analytical solutions of the stress, the displacement and dynamic stress intensity factor under the actions of moving increasing loads Px ²/t ² and Pt ³/x ² for the edges of asymmetrical mode III crack, respectively, were acquired. In the light of corresponding material properties, the variable rule of dynamic stress intensity factor was illustrated very well.     

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.     

Elastic-viscoplastic fields near the tip of a propagating crack under anti-plane shear

The elastic-viscoplastic model proposed by Bingham was used to analyse the stress and strain surrounding the tip of a propagating crack under antiplane shear. The proper displacement pattern was given ; the asymptotic equations were derived and solved numerically. The analysis and calculation show that for smaller viscosity the crack-tip possesses logarthmic singularity, and for larger viscosity it possesses power-law singularity.In critical case, the two kinds of singularity are consistent with each other. The result revealed the important role of viscosity for crack-tip field.     

Elastic solutions of the end problem for a nonhomogeneous semi-infinite strip in anti-plane shear

The end problem referring to anti-plane shear deformation of a nonhomogeneous semi-infinite strip is investigated here, by using the analogous methodology proposed by Papkovich and Fadle in plane problem. Two types of nonhomogeneity are considered: (i) the shear modulus varies with the thickness coordinate x exponentially; (ii) it varies with the length coordinate y exponentially. The closed elastic solutions in trigonometric series form are derived by the eigenfunctions expansion, and the completeness of the solutions is also proved. Therefore, the elastic field caused by a self-equilibrating traction on the end could be solved in an arbitrary accuracy by taking a necessary number of terms in the series, approximatively, which is usually neglected by invoking Saint-Venant principle. By considering the biggest negative eigenvalue, the Saint-Venant Decay rates of the problem is also estimated in the last section.     

A nonlinearly thermoelastic half-space under time-dependent normal and shear loading

A nonlinearly thermoelastic half-space is subjected to combined time-dependent normal and shear loading. The solution is obtained by a numerical method which is shown to yield accurate results by comparison with some known analytical solutions which can be obtained in some special cases. When shocks are involved, it is shown that the numerical results satisfy all the Rankine-Hugoniot jump conditions as well as the entropy condition across the shock.     

Basic singular solutions for a cracked circular hole in an infinite medium

Summary This article provides basic elastic solutions for an infinite medium containing a cracked circular hole, subject to singular external loads in form of a dilatation center, a concentrated force and a point couple. The complex function method in conjunction with a rational mapping function is used to solve the problem. Analytic expressions for stress functions, stresses and stress intensity factors are obtained. Numerical values of stress intensity factors are presented in graphic and tabular forms.

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Grundlegende singuläre Lösungen eines angerissenen kreisförmigen Loches in einem unbegrenzten Medium

Übersicht In diesem Artikel werden die grundlegenden elastischen Lösungen für einen unendlichen Körper hergeleitet, der ein angerissenes kreisförmiges Loch enthält und mit singulären äußeren Lasten in Form einer punktförmigen Aufweitung, einer Einzelkraft und eines Einzelmoments beaufschlagt wird. Zur Problemlösung werden komplexe Funktionen und rationale Abbildungsfunktionen verwendet. Man erhält analytische Ausdrücke für die Spannungsfunktionen, die Spannungen und die Spannungsintensitätsfaktoren. Numerische Werte für die Spannungsintensistätsfaktoren werden in graphischer und tabellarischer Form angegeben.