Analysis of cracked bars with rectangular cross-section and isotropic coating layer under torsion

The unrestrained torsional analysis of bars is an important theme in elasticity theory, first solved by Saint-Venant using semi-inverse methods. It has been considered and solved by several others using analytical methods and numerical procedures due to the importance in the design of machine parts under torsional moments. In this paper, the Saint Venant torsion problem is solved for rectangular prismatic bars using Galerkin-Kantorovich variational method (GKVM). The work presents a detailed theoretical framework of the problem, deriving using first principles considerations the stress compatibility equation in terms of the Prandtl stress function ϕ(x,y). The derived domain equation which is required to be satisfied over the rectangular cross-sectional domain is a partial differential equation of the Poisson type. GKVM is adopted as the solution method for finding the solution to the domain equation. The unknown Prandtl stress function ϕ(x,y) is assumed, following Kantorovich method to be a product of an unknown function for fx sought to minimize the Galerkin-Kantorovich variational functional (integral) (GKVF) and a known function (y2-b2) which satisfies the boundary conditions at all boundary points in the y-direction, that is, at y=±b. The resulting GKVF is a simplified functional whose integral is a second order inhomogeneous ordinary differential equation (ODE) in fx. The integrand is solved to find fx leading in a full determination of the Prandtl stress function. The expression for stresses, torsional moments and torsional parameters are then found and they satisfy the boundary conditions and the domain equation. The results for the torsional moments and torsional parameters are identical to previous results obtained using double finite sine transform method (DFSTM), and analytical methods. The merit of GKVM is that it has led to the exact solution of the unrestrained torsion problems.

The influence function method for fracture mechanics and residual fatigue life analysis of cracked components under complex stress fields

In order to show and illustrate the power of the theory proposed in Part I [J. Acoust. Soc. Am. 73, 1918–1922 (1983)] for the dynamic behavior of thermoelastic bars, the theory is applied to bars with circular and rectangular cross sections. In these applications, the order of the theory m is chosen to be 3. The general equations of the approximate theory governing all of the deformation modes (such as longitudinal, flexural, torsional, etc.) of circular and rectangular bars, and accomodating the thermal effects are presented. With the object of assessing the approximate theory, approximate and exact dispersion curves are compared for the flexural and longitudinal waves propagating in circular bars. A good match between these two is obtained without using any correction factors in the approximate theory. Further, experimental wave profiles are compared for longitudinal waves in rectangular bars. It is observed that the two agree quite well.

The Saint–Venant torsion problem of a circular cylinder containing a radial crack with surface elasticity is studied. The surface elasticity is incorporated into the crack faces by using the continuum-based surface/interface model of Gurtin and Murdoch. Both an internal crack and an edge crack are considered. By using the Green’s function method, the boundary value problem is reduced to a Cauchy singular integro-differential equation of the first order, which can be numerically solved by using the Gauss–Chebyshev integration formula, the Chebyshev polynomials and the collocation method. Due to the incorporation of surface elasticity, the stresses exhibit the logarithmic singularity at the crack tips. The torsion problem of a circular cylinder containing two symmetric collinear radial cracks of equal length with surface elasticity is also solved by using a similar method. The strengths of the logarithmic singularity and the size-dependent torsional rigidity are calculated.

A new microstructure-dependent Saint–Venant torsion model based on a modified couple stress theory

Analysis of an orthotropic circular bar weakened by multiple radial cracks under torsional transient loading

Application of fracture mechanics methods in safety analysis of piping components in subcreep and creep behavior

The ring ultrasonic transducers are widely used in the ocean engineering and medical fields. This paper employs an extended orthogonal polynomial approach to solve the guided wave propagation in two-dimensional structures, i.e. piezoelectric rings with rectangular cross-sections. The extended polynomial approach can overcome the drawbacks of the conventional orthogonal polynomial approach which can be used to solve wave propagation in one-dimensional structures. Through numerical comparison with the available results for a rectangular aluminum bar, the validity of the present approach is illustrated. The dispersion curves and displacement and electric potential distributions of various rectangular piezoelectric rings are calculated, and the effects of different radius to thickness ratios, width to height ratios and polarizing directions on the dispersion curves are illustrated.

This is a study of the formulation, some basic solutions, and applications of the Biot linearized quasistatic elasticity theory of fluid‐infiltrated porous materials. Whereas most previously solved problems are based on idealizing the fluid and solid constituents as separately incompressible, full account is taken here of constituent compressibility. Previous studies are reviewed and the Biot constitutive equations relating strain and fluid mass content to stress and pore pressure are recast in terms of new material parameters, more directly open to physical interpretation as the Poisson ratio and induced pore pressure coefficient in undrained deformation. Different formulations of the coupled deformation/diffusion field equations and their analogues in coupled thermoelasticity are discussed, and a new formulation with stress and pore pressure as basic variables is presented that leads, for plane problems, to a convenient complex variable representation of solutions. The problems solved include those of the suddenly introduced edge dislocation and concentrated line force and of the suddenly pressurized cylindrical and spherical cavity. The dislocation solution is employed to represent that for quasi‐static motions along a shear fault, and a discussion is given, based on fracture mechanics models for fault propagation, of phenomena involving coupled behavior between the rupturing solid and its pore fluid, which could serve to stabilize a fault against rapid spreading. Also, the solution for a pressurized cylindrical cavity leads to a time‐dependent stress field near the cavity wall, and its relevance to time effects in the inception of hydraulic fractures from boreholes, or from drilled holes in laboratory specimens, is discussed. Various limiting cases are identified, and numerical values of the controlling porous media elastic parameters are given for several rocks.

On the prediction of the stress field in adhesive joints using a combined analytical-numerical method

Guided wave in multilayered piezoelectric–piezomagnetic bars with rectangular cross-sections

Parametric features of an UIC60 rail at weld bottom crack in transverse direction under weld-wheel contact forces

An empirical stress intensity factor set of equations for a semi-elliptical crack in a semi-infinite body subjected to a polynomial stress distribution.

This paper investigates multiple cracking of a laminate composite subjected to anti-plane loads. The laminate composite is made up of a central layer sandwiched between two layers of different properties. There is a periodic array of cracks in the central layer along the central axis of the medium. A singular integral equation is formulated in terms of the crack-face displacement. The model developed is applied to the analysis of fibre multiple cracking in fibre-matrix laminate composites. It is also applied to the analysis of matrix cracking in fibre-matrix laminate composites. Numerical results have been given for the effects of crack spacing and constituent volume fraction on the crack tip field intensity factor and stress.

The stress field near a griffith crack at the interface of two bonded dissimilar elastic half-planes