The Reissner-Sagoci problem for a non-homogeneous half-space with a penny-shaped crack

The present paper examines the elastostatic problem related to the axisymmetric rotation of a rigid circular disc bonded to a non-homogeneous half-space containing a penny-shaped crack. The shear modulus of the half-space is assumed to vary with depth according to the relation (z) = ₁(z + c), c > 0 and ₁, are constants. Using Hankel transforms, the solution of the problem is reduced to integral equations and finally to simultaneous Fredholm integral equations of the second kind. By solving numerically the simultaneous Fredholm integral equations, results are obtained which are used to estimate the stress intensity factor at the crack tip and the torque required to rotate the disc through an angle ₀.     

A problem of Reissner-Sagoci type for an elastic cylinder embedded in an elastic half-space

The problem considered is that of the torsion of an elastic cylinder which is embedded in an elastic half-space of different rigidity modulus. It is assumed that there is perfect bonding at the common cylindrical surface and also that the torque is applied to the cylinder through a rigid disk bonded to its flat surface. The problem is reduced, by means of the use of integral transforms and the theory of dual integral equations to that of solving a Fredholm integral equation of the second kind. The results obtained by solving this equation are exhibited graphically in Fig. 2.     

The impulsive Reissner-Sagoci problem in a composite medium

The small-time Reissner-Sagoci problem in a bi-material elastic half-space under an impulsive twist is investigated. The problem is reduced to an integral equation, which, by asymptotic analysis and application of the Wiener-Hopf technique, is further reduced to a Fredholm integral equation of the second kind. Approximate analytical results, whose accuracy can be improved by retaining higher order terms in the approximations, are obtained for the free surface displacement outside the disc and the reactive couple necessary to keep the disc in the twisted position, for a bi-material medium with small shear wave speed disparity between the two media.     

A problem of Reissner-Sagoci type for a semi-infinite composite elastic cylinder

Summary The problem considered is that of the torsion of a semi-infinite elastic cylinder which is embedded in a semi-infinite elastic cylindrical shell of different material. By the use of integral transforms and the theory of dual integral equations, the problem is reduced to the solution of a Fredholm integral equation of the second kind. Numerical solution of the integral equation is obtained and the numerical values of the torque are displayed graphically.     

A thermal problem of friction for a half-space with a crack

We investigate the effect of local frictional heating of the surface of a half-space on the stress intensity factors at the vertices of a surface cut.     

A contact problem for a nonhomogeneous half-space

A solution of a contact problem for a non-homogeneous half-space is given. The analysis is based on the assumption that the shearing displacement is prescribed within the band $\text{y = 0, ¦x¦<1, ¦z¦< ∞}$ and shearing stress is zero in the outside area $\text{|x|> 1, y = 0, ¦z¦<∞}$. The shear modulus of the non-homogeneous half-space is assumed in the following form μ = μ₀(1 + α₁x)² (1 + β₁y)², where α₁, β₁ and μ₀ are constants. Here x, y and z are referred to a rectangular coordinates system.     

The statical Reissner-Sagoci problem for an internally loaded transversely isotropic halfspace

The present paper examines the elastostatic problem related to the axisymmetric rotation of a rigid circular punch which is bonded to the surface of a transversely isotropic elastic halfspace region. The rotation of the punch is induced by a concentrated couple which acts at a finite distance from the punch along the axis of symmetry. The resulting rigid rotation of the punch is evaluated in exact closed form.     

A two-dimensional problem for a half-space in magneto-thermoelasticity with thermal relaxation

In this work we study a two-dimensional problem in electromagneto-thermoelasticity for a half-space whose surface is subjected to a non-uniform thermal shock and is stress free in the presence of a transverse magnetic field. The problem is in the context of the theory of generalized thermoelasticity with one relaxation time. Laplace and exponential Fourier transform techniques are used to obtain the solution by a direct approach. The solution of the problem in the physical domain is obtained by using a numerical method for the inversion of the Laplace transforms based on Fourier series expansions. The distributions of the temperature, the displacement, the stress and the induced magnetic and electric fields are obtained. The numerical values of these functions are represented graphically.     

A two-dimensional problem for a fibre-reinforced anisotropic thermoelastic half-space with energy dissipation

The theory of thermoelasticity with energy dissipation is employed to study plane waves in a fibre-reinforced anisotropic thermoelastic half-space. We apply a thermal shock on the surface of the half-space which is taken to be traction free. The problem is solved numerically using a finite element method. Moreover, the numerical solutions of the non-dimensional governing partial differential equations of the problem are shown graphically. Comparisons are made with the results predicted by Green–Naghdi theory of the two types (GNII without energy dissipation) and (GNIII with energy dissipation). We found that the reinforcement has great effect on the distribution of field quantities. Results carried out in this paper can be used to design various fibre-reinforced anisotropic thermoelastic elements under thermal load to meet special engineering requirements.     

Boundary element solution for half-space elasticity or stokes problem with a no-slip boundary

We report an implementation of the Boundary Element Method (BEM) for half-space elasticity or Stokes problems with a plane interface (the boundary of the half space). With a proper choice of the singularity solution this plane interface, on which the displacement or velocity vector is zero, does not need to be discretized. For a large class of problems involving translating or rotating bodies a simplification of the boundary element formulation is possible, with a resulting improvement in the accuracy of the numerical results. The three-dimensional boundary element program was tested with the moving sphere problem and was found to be satisfactory in all cases.     

A three-dimensional thermoelastic problem for a half-space without energy dissipation

A three-dimensional problem for a homogeneous, isotropic and thermoelastic half-space subjected to a time-dependent heat source on the boundary of the space, which is traction free, is considered in the context of Green and Naghdi model II (thermoelasticity without energy dissipation) of thermoelasticity. The normal mode analysis and eigenvalue approach techniques are used to solve the resulting non-dimensional coupled equations. Numerical results for the temperature, thermal stress, strain and displacement distributions are represented graphically and discussed.     

An approximate solution for a stationary heat-conduction problem in a half-space with mixed boundary conditions

A stationary axisymmetric problem with conditions of convective heat transfer is considered. A special method is employed to find an approximate solution in analytic form and its accuracy is evaluated.     

The elastic torsion problem for a nonhomogeneous and transversely isotropic half-space

Summary The statical Reissner-Sagoci problem for nonhomogeneous transversely isotropic elastic half-space is investigated. The modulus of rigidity is assumed asC _ii (r,z)= _ii r f(z),i=4, 6 and 0, where _ii are constants. The expressions for stresses, displacement and torque are given. The analysis is based on the assumption that the tangential displacement is prescribed within the areaz=0,r1 and shearing stress is zero in the outside arear>1,z=0. The integral equation which is obtained is solved in the case whenf(z)=cosh² kz and 0, wherek is a constant.With 7 Figures     

Axi-symmetric problem in micropolar elastic half-space with stretch

The present investigation is concerned with axisymmetric problem in micropolar elastic half-space with stretch. The displacement, force stress, couple stress and vector first moment are obtained for a half-space subjected to an arbitrary normal load and a particular case of impulsive force is studied in detail. The graphs for various quantities have been drawn to illustrate the effect of stretch in each particular case.     

Two-dimensional problem for a half-space with axi-symmetric distribution in the theory of generalized thermoelastic diffusion

In this work, we study a two-dimensional problem of axi-symmetric distribution of temperatures in a half-space with a permeating substance in contact with the bounding plane in the context of the theory of generalized thermoelastic diffusion with one relaxation time. The surface of the half-space is taken as traction free and subjected to axi-symmetric time-dependent thermal shock. The chemical potential is also assumed to be a known function of time on the bounding plane. The Laplace and Hankel transform techniques are used. The analytical solution in the transform domain is obtained by using a direct approach. The inverse of the double transform is obtained by using a numerical method based on Fourier expansion techniques. Numerical results for the temperature, displacement, stress, concentration, and chemical potential are carried out and represented graphically.     

Interaction of coplanar surface cracks in a half-space

Boundary integral equations are applied to the interaction of closely spaced surface-coplanar cracks with various geometrical dimensions in a half-space that is tensioned by forces perpendicular to the crack surfaces. The numerical results show that the crack depths and the distances between them greatly influence the stress intensity coefficients at the edges of the cracks, and the interaction between the cracks can be neglected only for shallow ones.Translated from Problemy Prochnosti, No. 10, pp. 12–16, October, 1994.     

Soft constraint handling for a real-world multiobjective energy distribution problem

Real-world optimisation problems usually involve some conflicting objectives and a number of constraints. In such cases, finding a feasible, Pareto-optimal solution poses a demanding challenge. In reality, constraints bear different importance levels to these conflicting objectives. If some constraints are relaxed within an acceptable degree, quality infeasible solutions could be found on the boundary from the infeasible side of the searching region. This paper formulates an energy distribution problem arising from a real-world iron and steel production as a multiobjective optimisation problem. During the course of the optimisation search, this paper attempts to handle certain constraints in a soft manner to find solutions with good balance among objective and constraints violation. Based on the analysis of constraints from the real-world perspective, different tolerance values are defined. The proposed constraint violation degree-based soft handling approach is incorporated into the advanced version of non-dominated sorting genetic algorithm framework, as a case study, to examine the efficiency of the proposed soft constraint handling approach for a real-world energy distribution problem. The proposed approach is also implemented in different ways of constraint handling and tested on some benchmark functions to further demonstrate the performance of soft constraint handling for multiobjective optimisation problems.