Closed-form solutions for finite length crack moving in a strip under anti-plane shear stress

Summary In this paper exact expressions for the anti-plane dynamic stress distributions around finite length cracks propagating with constant velocity in infinitely long finite width strips are determined. Two cases of practical importance are investigated. Firstly, the lateral boundaries of the strip are clamped and displaced in equal and opposite directions, to produce anti-plane shear resulting in a tearing motion along the leading edge of the crack and, secondly, the lateral boundaries of the strip are subjected to shearing stresses. Employing Fourier transforms the solution of each problem is reduced to solving a pair of dual integral equations. Closed-form solutions of these integral equations are obtained leading to exact expressions for the stress intensity factors. Numerical results are presented in graphical form.

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Geschlossene Lösungen für einen Riß endlicher Länge, der sich in einem unter antiplaner Schubspannung stehenden Streifen bewegt

Zusammenfassung In dieser Arbeit werden exakte Ausdrücke für die antiplanen, dynamischen Spannungsverteilungen um Risse endlicher Länge, die sich mit konstanter Geschwindigkeit in einem unendlich langen Streifen begrenzter Breite ausbreiten, bestimmt. Zwei Fälle von praktischer Bedeutung werden untersucht. Erstens werden die Seitenränder des Streifens eingespannt und sowohl in dieser als auch in entgegengesetzter Richtung versetzt, um einen antiplanen Schub zu erzeugen, der eine Aufreißbewegung längs der Vorderkante des Risses bewirkt und zweitens werden die Seitenränder des Streifens einer Schubspannung unterworfen. Die Lösung jedes Problems wird durch die Verwendung der Fouriertransformationen auf die Lösung zweier dualer Integralgleichungen reduziert. Es werden Lösungen dieser Integralgleichungen in geschlossener Form erhalten, die auf exakte Ausdrücke für den Spannungsintensitätsfaktor führen. Numerische Ergebnisse werden in graphischer Form gezeigt.

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With 3 Figures     

Impact response of a finite crack in a finite strip under anti-plane shear

The response of a through-the thickness crack with finite dimensions to impact in a finite elastic strip is investigated in this study. The elastic strip is assumed to be subjected to anti-plane shear deformation. Laplace and Fourier transform were used to formulate the mixed boundary value problem. The dynamic stress intensity factor and crack opening displacement are obtained as a function of time and the strip width to crack length ratio, h/a. The results indicate that the intensity of the crack-tip stress field reaches a peak very quickly and then decreases in magnitude oscillating about the static value. In general, the dynamic stress intensity factor is higher for small $\text{h/a}$. Similar behavior has also been found for the crack surface displacement.     

Singular stress and electric fields of a piezoelectric ceramic strip with a finite crack under longitudinal shear

Summary Following the theory of linear piezoelectricity, we consider the problem of determining the singular stress and electric fields in an orthotropic piezoelectric ceramic strip containing a Griffith crack under longitudinal shear. The crack is situated symmetrically and oriented in a direction parallel to the edges of the strip. Fourier transforms are used to reduce the problem to the solution of a pair of dual integral equations. The solution of the dual integral equations is then expressed in terms of a Fredholm integral equation of the second kind. Numerical values on the stress intensity factor and the energy release rate for piezoelectric ceramics are obtained, and the results are graphed to display the influence of the electric field.     

Elastic solution of a two-dimensional functionally graded thick truncated cone with finite length under hydrostatic combined loads

In this paper, a thick truncated hollow cone with finite length made of two-dimensional functionally graded materials (2D-FGM) subjected to combined loads as internal, external and axial pressure is considered. The volume fraction distribution of materials and geometry are assumed to be axisymmetric but not uniform along the axial direction. The Finite Element Method based on the Rayliegh-Ritz energy formulation is applied to obtain the elastic behavior of the functionally graded thick truncated cone. By using this method, the effects of semi-vertex angle of the cone and the power law exponents on the distribution of different types of displacements and stresses are considered. The results show that using 2D-FGM leads to a more flexible design so that both the maximum stresses and stress distribution can be controlled by the material distribution.     

Asymptotic analysis and finite element calculation of a rubber wedge under tension

Summary In this paper, two kinds of elastic laws given by Knowles and Sternberg in 1973, and Gao in 1997 for rubber-like materials are used to investigate the problem of a rubber wedge tensioned by a line load at its tip. It was treated as a plane strain case, and large deformation was taken into account. Asymptotic solutions to the stress-strain field near the wedge tip are obtained and compared for different elastic laws. By means of the finite element method, the stress-strain fields for the two kinds of elastic laws are calculated, and the results are consistent well with those of the theoretical analysis.     

Elastodynamic response of a finite strip with two coplanar cracks under impact load

The elastodynamic response of two coplanar Griffith cracks in a finite elastic strip under in-plane compression and anti-plane shear impact is considered in this paper. Laplace and Fourier transforms are used to reduce two mixed boundary value problems to Cauchy-type singular integral equations in Laplace transform plane, which are solved numerically. The elastodynamic stress-intensity factors are obtained as functions of time and geometry parameters.     

The mechanics of splitting a strip through penetration with a sharp wedge

This paper deals with the mechanics involved in splitting a strip through penetration with a sharp wedge along the center line. First, a quasi-static problem is considered. The crack propagation can be associated with the loss in stability when the applied load reaches the critical value for bifurcation. Next, the static problem is extended to a dynamic case by including the inertia force of the wedge in the analysis. An initial value problem is formulated for the motion of the wedge. For any given weight of the wedge, the critical velocity of the wedge for indefinite crack propagation can be determined by means of the theory of dynamic stability. Finally, the case of splitting a strip under repeated applications of impulsive load is considered. The critical number of applications of impulsive load for indefinite crack propagation is determined numerically.     

Effects of tilt moment on the contact stresses for a wedge under sliding condition

The contact stresses between a tilted, shallow wedge and half-plane, under frictional sliding condition, are investigated analytically. The closed-form of contact law itself, together with the contact tractions and the size of the contact segments are explicitly found. The effects of tilt moment, material property, geometry variation and the applied loads on the contact pressure distributions are investigated. By making use of the Muskhelishvili??s potential function and the Plemelj formulae, the analytically derived contact pressure is the same as that obtained based on the singular integral equations. The result is verified by comparing with that in the literature. The in-plane stress field is evaluated from the standpoint of fatigue. It is shown that the Muskhelishvili??s potential function and the Plemelj formulae can be also used to solve this type of contact problem. This study is helpful for understanding the mechanism of frictional sliding contact problem with singular point and designing better high precision instruments and devices.     

Width of adiabatic shear bands formed under combined stresses

Abstract

An equation for the half width of an adiabatic shear band formed under combined stresses is derived. The importance of strain rate, stress, temperature rise, and thermal conductivity is described. Confirmation is found for the proposition that shear band width is independent of stress state and this is confirmed by comparing the theory with the experimental results of other workers.

MST/933     

End rotation of a strip when indented by a wedge

Experiments are described for the one-sided unconstrained indentation of half-hard and annealed aluminium strip with a wedge-shaped tool. The mechanics of the process are described and the end-rotation, which occurs after a certain critical penetration has been reached, is measured. At the present time there is no slip-line field solution which predicts this phenomenon.     

Impact response of a strip composite with a finite crack

The dynamic response of a central crack in a strip composite under normal impact is analyzed. The crack is oriented normally to the interfaces. Laplace and Fourier transform techniques are used to reduce the elastodynamic problem to a pair of dual integral equations. The integral equations are solved by using an integral transform technique and the result is expressed in terms of a Fredholm integral equation of the second kind. A numerical Laplace inversion routine is used to recover the time dependence of the solution. The dynamic stress intensity factor is determined and its dependence on time, the material properties and the geometrical parameters is discussed.     

On the plastic zone and residual stresses within a strip and a half plane under a moving load

A quasi-steady solution has been obtained for the determination of the extent of plastic zones and the amplitude of the residual stresses in (1) an elasto-plastic strip lying on a rigid frictionless base and (2) a half-plane, under the first passage of a moving load. The load is assumed to be applied with a semi-elliptic pressure distribution which is of sufficient magnitude to cause plastic yielding in the body. A work-hardening material obeying von-Mises' yield criterion is considered. A numerical solution procedure, based on discretizing the plastic strain field in the coordinate system moving with the load into uniformly strained rectangular or semi-infinite elements has been developed. Numerical results are obtained for a strip with linear work-hardening material properties. The effect on the distribution of residual stresses created by the ratio of load half-width to strip thickness is also discussed.     

Multilayered shell finite element with interlaminar continuous shear stresses: a refinement of the Reissner–Mindlin formulation

A finite element formulation for refined linear analysis of multilayered shell structures of moderate thickness is presented. An underlying shell model is a direct extension of the first‐order shear‐deformation theory of Reissner–Mindlin type. A refined theory with seven unknown kinematic fields is developed: (i) by introducing an assumption of a zig‐zag (i.e. layer‐wise linear) variation of displacement field through the thickness, and (ii) by assuming an independent transverse shear stress fields in each layer in the framework of Reissner's mixed variational principle. The introduced transverse shear stress unknowns are eliminated on the cross‐section level. At this process, the interlaminar equilibrium conditions (i.e. the interlaminar shear stress continuity conditions) are imposed. As a result, the weak form of constitutive equations (the so‐called weak form of Hooke's law) is obtained for the transverse strains–transverse stress resultants relation. A finite element approximation is based on the four‐noded isoparametric element. To eliminate the shear locking effect, the assumed strain variational concept is used. Performance of the derived finite element is illustrated with some numerical examples. The results are compared with the exact three‐dimensional solutions, as well as with the analytical and numerical solutions obtained by the classical, the first‐order and some representative refined models. Copyright © 2000 John Wiley & Sons, Ltd.     

Simulation of correlation of shear rates with pressure fluctuations in the equation for Reynolds stresses

A simulation of the correlation that contains pressure fluctuations and determines the process of redistribution of energy between the components of the tensor of Reynolds stresses is made. The results obtained can be used to construct new models of second-order turbulence.     

Twisted axially loaded rod with shear and compressibility

Summary Stability of a twisted and axially compressed elastic rod is analysed using the Euler method of adjacent equilibrium configuration. The constitutive equations of the rod are assumed in the form that takes into account both shear of the cross-sections and compressibility of the rod axis. It is shown that bifurcation points of the non-linear system of equations describing equilibrium are determined from the linearized system of equations. The influence of shear and compressibility on the critical values of twisting couple and compressive force is obtained. Moreover, the bifurcation pattern is examined by using the Liapunov-Schmidt method.     

Elastic–plastic behaviour in materials loaded with a spherical indenter

Certain ceramic materials display an indentation response similar to that observed for ductile metals when loaded with a spherical indenter. This unusual behaviour, for what are nominally brittle materials, influences the mode of contact damage in applications such as machining, wear, impact damage and hardness testing. The shape of the plastic zone beneath the indenter is typically fully contained within the circle of contact on the specimen surface and thus conventional hardness theories, such as the popular expanding cavity model, provide an inadequate account of indentation response of the material. The present work demonstrates, by experiment, finite element modelling and theoretical considerations, that the indentation response is determined by the interaction between the evolving plastic zone and the mechanical properties of the specimen material, in particular, the ratio of the elastic modulus to the yield stress. This revised version was published online in November 2006 with corrections to the Cover Date.