Bounds on stress concentration factors in finite anti-plane shear

This paper provides an analytical approach for obtaining bounds on elastic stress concentration factors in the theory of finite anti-plane shear of homogeneous, isotropic, incompressible materials. The problem of an infinite slab with traction-free circular cavity subject to a state of finite simple shear deformation is considered. Explicit estimates are obtained for the maximum shearing stress in terms of the applied stress at infinity and constitutive parameters. The analysis is based on application of maximum principles for second-order quasilinear uniformly elliptic equations.     

The anti-plane shear field in an infinite slab of elasto-damaged material with a semi-infinite crack

This paper deals with an infinite slab with a semi-infinite crack, which is subjected to the anti-plane sheark _III field at infinity. The slab is made of an elasto-damaged material. Analytical solution is obtained by use of conformal mapping. The shape of damaged-zone, the dissipative energy, the shear opening displacement on the crack surface and several stress distribution curves are given. The far field condition is checked, The asymptotic behavior near the crack-tip is given. The project supported by National Natural Science Foundation of China     

Strain energy density of a circular cavity buried in a semi-infinite slab of functionally graded materials subjected to anti-plane shear waves

The paper presents a theoretical method to investigate the multiple scattering of shear waves and strain energy density in a semi-infinite slab of functionally graded materials with a circular cavity. The analytical solutions of wave fields are expressed by employing wave function expansion method and the expanded mode coefficients are determined by satisfying the boundary conditions of the cavity. Image method is used to satisfy the free boundary condition of the semi-infinite structure. The analytical solution of the problem is derived, and the numerical solutions of the strain energy density factors around the cavity are also graphically presented. The effects of the distances between the cavity and the boundaries of the semi-infinite slab, the wave number and the non-homogeneous parameter of materials on strain energy density factors are analyzed. Analyses show that the strain energy density around the cavity increases with increasing non-homogeneous parameter of materials and incident wave number. The boundaries of the semi-infinite slab have great effect on both the maximum strain energy density and the distribution around the circular cavity, and the effect increases with increasing incident wave number. When the distance between the semi-infinite boundary and the cavity varies, the effect of the upper and lower boundaries on the distribution of the strain energy density factors around the cavity is also examined.     

Magneto-elastic stress in a thin infinite plate with an elliptical hole under uniform magnetic field

Two-dimensional magnetic field and magneto-elastic stress solutions are presented for a magnetic material of a thin infinite plate with an elliptical hole under uniform magnetic field. The linear constitutive equation is used for the magnetic field and the stress analyses. The magneto-elastic stress is analyzed using Maxwell stress since only Maxwell stress is caused as a body force according to the electro magneto theory. Except the approximation of the plane stress state in which the plate is thin, no further assumption is made for the stress analysis, though Maxwell stress components are expressed by nonlinear terms. The rigorous boundary condition expressed by Maxwell stress is completely satisfied without any linear assumptions on the boundary. First, magnetic field and stress for soft ferromagnetic material is analyzed and then those for paramagnetic and diamagnetic materials are analyzed. It is stated that the stress components are the same expressions for those materials and the difference is only the magnitude of the permeability, though the magnetic fields are different each other in the plates. If the analysis of magnetic field of paramagnetic materials is easier than that of soft ferromagnetic material, the stress analysis may be carried out using the magnetic field for paramagnetic material. Shear deflection as well as stress in the direction of the plate thickness arises and the solutions are also obtained. Figures of the magnetic field and stress distribution are shown. Stress intensity factors are also derived.     

Magneto-elastic stress induced by an electric current in an infinite thin plate with an elliptical hole

Two-dimensional magnetic field and stress analyses have been presented for soft ferromagnetic, paramagnetic, and diamagnetic materials of an infinite thin plate with an elliptical hole under steady electric current. The magnetic stress has been analyzed in the Maxwell Stress Model. Except for the approximation of the plane stress state since the plate is the thin plate, any assumption is not made for the stress analysis, though the Maxwell stress components are expressed by nonlinear terms. The boundary condition expressed by Maxwell’s stress is completely satisfied without any linear assumptions on the boundary. Two ways for the boundary condition are stated. The analysis of σ _z in the direction of the plate thickness is also carried out. Figures of the magnetic field and stress distribution are shown. Stress intensity factors are also derived, and the magnitude of the stress intensity factor for the magnetic stress and thermal stress due to the Joule heat caused by the electric current is discussed.     

Magneto elastic stress subjected to uniform magnetic field over a thin infinite plate containing an elliptical hole with an edge crack

Two dimensional solutions of the magnetic field and magneto elastic stress are presented for a magnetic material of a thin infinite plate containing an elliptical hole with an edge crack subjected to uniform magnetic field. Using a rational mapping function, each solution is obtained as a closed form. The linear constitutive equation is used for these analyses. According to the electro-magneto theory, only Maxwell stress is caused as a body force in a plate. In the present paper, it raises a plane stress state for a thin plate, the deformation of the plate thickness and the shear deflection. Therefore the magneto elastic stress is analyzed using Maxwell stress. No further assumption of the plane stress state that the plate is thin is made for the stress analysis, though Maxwell stress components are expressed by nonlinear terms. The rigorous boundary condition expressed by Maxwell stress components is completely satisfied without any linear assumptions on the boundary. First, magnetic field and stress analyses for soft ferromagnetic material are carried out and then those analyses for paramagnetic and diamagnetic materials are carried out. It is stated that those plane stress components are expressed by the same expressions for those materials and the difference is only the magnitude of the permeability, though the magnetic fields H_x, H_y are different each other in the plates. If the analysis of magnetic field of paramagnetic material is easier than that of soft ferromagnetic material, the stress analysis may be carried out using the magnetic field for paramagnetic material to analyze the stress field, and the results may be applied for a soft ferromagnetic material. It is stated that the stress state for the magnetic field H_x, H_y is the same as the pure shear stress state. Solutions of the magneto elastic stress are nonlinear for the direction of uniform magnetic field. Stresses in the direction of the plate thickness and shear deflection are caused and the solutions are also obtained. Figures of the magnetic field and stress distribution are shown. Stress intensity factors are also derived and investigated for the crack length.     

Antiplane shear crack in an infinite plate with a circular inclusion

Summary The effect of an elastic circular inclusion of a different material on the stress state of a single-cracked infinite sheet subjected to anti-plane shear is investigated. The proposed method makes use of complex variables in conjunction with Muskhelishvili's technique and has given interesting results for the stress field in the cracked plate. Numerical results, derived from this technique, investigate the influence of the regional inhomogeneity, produced by the inclusion in the structure, on the variation of stress-intensity factors at the crack tips.

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Nichtebener Schubspannungsriß in einer unendlichen Platte mit einem kreisförmigen Einschluß

Übersicht Es wird der Einfluß untersucht, den ein elastischer, kreisförmiger Einschluß aus verschiedenem Material in einer unendlichen Platte mit einem Riß, welche einer longitudinalen Schubspannung unterworfen ist, auf den Spannungszustand der Platte ausübt. Die vorgeschlagene Methode benutzt komplexe Variablen in Zusammenhang mit der Muskhelishvilischen Technik und gibt für den Spannungszustand in der Platte interessante Resultate. Numerische Ergebnisse wurden gefunden, um den Einfluß der Inhomogenität auf die Spannungsintensitätsfaktoren des Risses zu bestimmen.

     

Discontinuous deformation gradients near the tip of a crack in finite anti-plane shear: an example

Summary This investigation aims at the elastostatic field near the edges (tips) of a plane crack of finite width in an all-round infinite body, which — at infinity — is subjected to a state of simple shear parallel to the crack edges. The analysis is carried out within the fully nonlinear equilibrium theory of homogeneous and isotropic, incompressible elastic solids. Further, the particular constitutive law employed here gives rise to a loss of ellipticity of the governing displacement equation of equilibrium in the presence of sufficiently severe anti-plane shear deformations.The study reported in this paper is asymptotic in the sense that the actual crack is replaced by a semi-infinite one, while the far field is required to match the elastostatic field predicted near the crack tips by the linearized theory for a crack of finite width. The ensuing global boundary-value problem thus characterizes the local state of affairs in the vicinity of a crack-tip, provided the amount of shear applied at infinity is suitably small.An explicit exact solution to this problem, which is deduced with the aid of the hodograph method, exhibits finite shear stresses at the tips of the crack, but involves two symmetrically located lines of displacement-gradient and stress discontinuity issuing from each crack-tip.The results communicated in this paper were obtained in the course of an investigation supported by Contract N00014-75-C-0196 with the Office of Naval Research in Washington, D.C.     

Magnetic field and magneto elastic stress in an infinite plate containing an elliptical hole with an edge crack under uniform electric current

Two-dimensional solutions of the electric current, magnetic field and magneto elastic stress are presented for a magnetic material of a thin infinite plate containing an elliptical hole with an edge crack under uniform electric current. Using a rational mapping function, the each solution is obtained as a closed form. The linear constitutive equation is used for the magnetic field and the stress analyses. According to the electro-magneto theory, only Maxwell stress is caused as a body force in a plate which raises a plane stress state for a thin plate and the deformation of the plate thickness. Therefore the magneto elastic stress is analyzed using Maxwell stress. No further assumption of the plane stress state that the plate is thin is made for the stress analysis, though Maxwell stress components are expressed by nonlinear terms. The rigorous boundary condition expressed by Maxwell stress components is completely satisfied without any linear assumptions on the boundary. First, electric current, magnetic field and stress analyses for soft ferromagnetic material are carried out and then those analyses for paramagnetic and diamagnetic materials are carried out. It is stated that the stress components are expressed by the same expressions for those materials and the difference is only the magnitude of the permeability, though the magnetic fields H_x, H_y are different each other in the plates. If the analysis of magnetic field of paramagnetic material is easier than that of soft ferromagnetic material, the stress analysis may be carried out using the magnetic field for paramagnetic material to analyze the stress field, and the results may be applied for a soft ferromagnetic material. It is stated that the stress state for the magnetic field H_x, H_y is the same as the pure shear stress state. Solving the present magneto elastic stress problem, dislocation and rotation terms appear, which makes the present problem complicate. Solutions of the magneto elastic stress are nonlinear for the direction of electric current. Stresses in the direction of the plate thickness are caused and the solution is also obtained. Figures of the magnetic field and stress distribution are shown. Stress intensity factors are also derived and investigated for the crack length and the electric current direction.     

Heat conduction and thermal stress induced by an electric current in an infinite thin plate containing an elliptical hole with an edge crack

A uniform electric current at infinity was applied to a thin infinite conductor containing an elliptical hole with an edge crack. The electric current gives rise to two states, i.e., uniform and uneven Joule heat. These two states must be considered to analyze the heat conduction problem. The uneven Joule heat gives rise to uneven temperature and thus to heat flux, and to thermal stress.Using a rational mapping function, problems of the electric current, the Joule heat, the temperature, the heat flux, the thermal stress are analyzed, and each of their solutions is obtained as a closed form. The distributions of the electric current, the Joule heat, the temperature, the heat flux and the stress are shown in figures.The heat conduction problem is solved as a temperature boundary value problem. Solving the thermal stress problem, dislocation and rotation terms appear, which complicates this problem. The solutions of the Joule heat, the temperature, the heat flux and the thermal stress are nonlinear in the direction of the electric current. The crack problems are also analyzed, and the singular intensities at the crack tip of each problem are obtained. Mode II (sliding mode) stress intensity factor (SIF) is produced as well as Mode I (opening mode) SIF, for any direction of the electric current. The relations between the electric current density and the melting temperature and between the electric current density and SIF are investigated for some crack lengths in an aluminum plate.     

Dynamic stress concentration in an elastic half space with a semi-circular cavity excited by SH waves

Subject of the investigation is the stress distribution and the dynamic stress concentration factor at the surface of a semi-circular cavity in a half space excited by plane harmonic SH waves. Using wave function expansion for the incident wave and the reflected waves, a closed form solution is obtained. Numerical results are represented graphically.     

Analysis of anisotropic sector with a radial crack under anti-plane shear loading

In this paper, the anti-plane shear deformation of an anisotropic sector with a radial crack is investigated. The traction–traction boundary conditions are imposed on the radial edges and the traction-free condition is considered on the circular segment of the sector. A novel mathematical technique is employed for the solution of the problem. This technique consists of the use of some recently proposed finite complex transforms (Shahani, 1999), which have complex analogies to the standard finite Mellin transforms of the first and second kinds. However, it is essential to state the traction-free condition of the crack faces in the form of a singular integral equation which is done in this paper by describing an exact analytical method. The resultant dual integral equations are solved numerically to determine the stress intensity factors at the crack tips. In the special cases, the obtained results coincide with those cited in the literature.     

Generalized thermoelastic problem of an infinite body with a spherical cavity under dual-phase-lags

The aim of the present contribution is the determination of the thermoelastic temperatures, stress, displacement, and strain in an infinite isotropic elastic body with a spherical cavity in the context of the mechanism of the two-temperature generalized thermoelasticity theory (2TT). The two-temperature Lord–Shulman (2TLS) model and two-temperature dual-phase-lag (2TDP) model of thermoelasticity are combined into a unified formulation with unified parameters. The medium is assumed to be initially quiescent. The basic equations are written in the form of a vector matrix differential equation in the Laplace transform domain, which is then solved by the state-space approach. The expressions for the conductive temperature and elongation are obtained at small times. The numerical inversion of the transformed solutions is carried out by using the Fourier-series expansion technique. A comparative study is performed for the thermoelastic stresses, conductive temperature, thermodynamic temperature, displacement, and elongation computed by using the Lord–Shulman and dual-phase-lag models.     

Finite viscoplasticity of non-affine networks: stress overshoot under shear

A constitutive model is derived for the viscoplastic behavior of polymers at finite strains. A polymer is treated as an equivalent network of chains bridged by permanent junctions. The elastic response of the network is attributed to elongation of strands, whereas its plastic behavior is associated with sliding of junctions with respect to their reference positions. A new kinetic equation is proposed that expresses the rate of sliding of junctions in terms of the Cauchy stress tensor. Constitutive equations for an equivalent non-affine network are developed by using the laws of thermodynamics, where internal dissipation of energy reflects two processes at the micro-level: sliding of chains along entanglements and friction of strands between junctions. A similarity is revealed between these relations and the pom-pom model. The governing equations are applied to study stress overshoot at simple shear of an incompressible medium. Adjustable parameters in the stress-strain relations are found by fitting experimental data on polycarbonate melt reinforced with short glass fibers and polystyrene solution. Fair agreement is demonstrated between the observations and the results of numerical simulation.Received: 6 January 2003, Accepted: 28 April 2003