Complex potentials in two-dimensional elasticity

The complex potentials are found for two elastic problems, involving body forces, for a circular disk.

In the present paper a stress general solution is obtained for the generalized plane stress problem with planar body forces, and it is demonstrated that only body force of biharmonic type ensures the compatibility with the generalized plane stress assumptions (σ 33=0). Inspired by the Filon perspective of average values, two more generalized plane stress problems with weak assumptions on the out-of-plane stress averages ( $\bar{\sigma}_{33}=0$ or $\nabla^{2} \bar{\sigma}_{33}=0$ ) are studied, and the averages of the corresponding stress fields are expressed by the Airy stress functions. The authors also provide an alternative proof of the Gregory decomposition theory.

This paper deals with a method of approach by means of three stress functions to axially symmetric problems of the dynamical theory of elasticity in cylindrical co-ordinates. The principal results are as follows : (1) The solution presented in the paper is a complete one for problems associated with both torsional deformations and body forces as well. (2) In the case of no body forces, it is possible to drop a stress function (not arbitrary) form three stress functions without loss of generality. (3) Stress functions for dynamical body force problems and for initial value problems are presented and examples of their application are investigated. (4) As a special case of the solution shown in the paper Love's solution for elastostatics can be obtained.

An analytical method for solving gravity-induced stresses in slope

Plane strain and generalized plane stress problems for fibre-reinforced materials

On the development of compatible elements for the solution of three-dimensional elasticity problems

On anisotropic plasticity models using linear transformations on the deviatoric stress: Physical constraints on plastic flow in generalized plane strain

This paper extends the numerical study of the structure and development of a concentrated vortex by Leslie (1971), in which a vortex is simulated by suddenly imposing an upwards body force along a section of the vertical axis of a contained rotating fluid, initially in a state of uniform rotation. Whereas the former paper was concerned primarily with demonstrating the prediction of Morton (1969) that a concentrated vortex may be generated only for a restricted range of the flow parameters, the present paper investigates the important role of boundaries on vortex behaviour. Particular interest is focused on the boundary which is normal to the vortex core and ‘behind’ the body force. On this boundary the surface stress is related to the surface velocity by a drag coefficient CD and experiments are performed in which CD is infinite, unity and zero corresponding with a no-slip, a partially yielding and a free-slip boundary respectively. These calculations are motivated by the desire to assess what differences, if any, between tornadoes (which develop over land) and waterspouts (which develop over the sea) can be attributed to the different surface constraint. We also study the effect on a vortex due to an abrupt change in surface condition as this is relevant to the behaviour of a tornado which happens to cross a water surface, or even one which traverses ground with varying roughness characteristics, and conversely to the behaviour of a waterspout which moves over land. It is shown that the strength of the meridional circulation associated with the vortex, and hence the strength of the upflow in the vortex itself, are increasing functions of the surface stress. On the other hand, the azimuthal kinetic energy, and in particular the strength of the vortex as measured by the maximum swirling velocity attained, decreases as the surface stress increases. Moreover, if the drag coefficient is suddenly increased, the meridional circulation increases, the azimuthal kinetic energy decreases and the vortex width (as measured by the radius of the maximum swirling velocity at a given height) increases. These effects are reversed if the drag coefficient is decreased and do not depend on the frictional effect of the container side wall. The role of the side wall itself is briefly explored. The results accord with the behaviour of laboratory vortices formed in air over surfaces of different roughness as studied by Dessens (1972). They also appear consistent with the observation reported by Golden (1971, p. 146) concerning the behaviour of a waterspout whose circulation decreased rapidly and visible funnel expanded during a traverse of about a kilometre over land and which subsequently reformed on moving back over water.

The present paper gives a theoretical solution for a semi-infinite plate with a circular hole, which is filled with an elastic inclusion of another material, and subjected to uniform heat flow of the temperature gradient μ in the direction of the straight edge. The analysis is developed on the basis of the Airy's stress function in the generalized plane stress and by applying the bipolar coordinates. Here the plate is supposed to be insulated and the boundaries are free from the traction. The method of perturbation is adopted for the determination of unknown coefficients involved in the solution. Numerical calculations about the stress distributions along main parts of the semi-infinite plate and the inclusion are worked out in some detail, in order to clarify the effects of an inclusion. The maximum stresses on the common boundary are also calculated and compared with the results available.

The elastic field induced by an edge dislocation located inside a lamellar inhomogeneity is evaluated, the resulting formulas applying as well to any internal stress source within a lamella in generalized plane stress or plane strain, and the Peach–Koehler force on the dislocation derived. The features of the interaction are determined by the mismatch in both the shear moduli and the Poisson’s ratios of lamella and matrix. The differences from the case of an edge dislocation in either a free or a clamped strip turn out to be mainly of a quantitative nature and to become relevant when the lamella is up to one order of magnitude harder or softer than the matrix.

Drilling new infill wells in depleted reservoirs is extremely problematic and costly due to low formation fracture pressure and narrow mud window resulting from in-situ stress changes due to fluid extraction. This is of paramount importance especially for drilling operations in deep-water reservoirs, which requires precise prediction of formation fracture pressure. In turn, this entails accurate prediction of reservoir stress changes with pore pressure depletion, i.e., the stress path. Currently-used models assume a transient flow regime with reservoir depletion. However, flow regime in depleted reservoirs is dominantly pseudo-steady state (PSS). Shahri and Miska (2013) proposed a model under plane-strain assumption. However, subsea subsidence measurements confirm that depletion-induced reservoir deformation mainly occurs in axial direction. We provide analytical solutions for stress path prediction under different deformational conditions namely, plane strain-traction and displacement boundary conditions, generalized-plane-stress, generalized uniaxial strain, and uniaxial-strain. For this purpose, constitutive relations of poroelasticity are combined with equilibrium equations, and pore pressure profile is described by PSS flow regime. In a numerical example, we examine the effects of different deformational conditions on depletion-induced in-situ stress changes. Interestingly, results indicates that stress path in reservoir is significantly affected by reservoir’s boundary conditions. The stress path under plane strain-displacement assumption overestimates the stress path predicted under uniaxial strain state by almost a factor of two. However, the generalized plane stress and traction plane strain conditions underestimates the results of uniaxial strain assumption. The order of stress path values for different boundary conditions can be summarized as: SPps-disp > SPuniaxial > SPps-trac > SPgps.

Explicit analytical expressions, written in terms of complex variables and suitable for rapid computer evaluation, are presented for the Percus-Yevick hard-sphere radial distribution function, g(R), for R ˇ- 5σ. Some effects of truncating g(R) to unity past R = 5 σ are discussed.

Although it is well known that Bernoulli's equation is obtained as the first integral of Euler's equations in the absence of vorticity and that in the case of non-vanishing vorticity a first integral of them can be found using the Clebsch transformation for inviscid flow, generalization of the procedure for viscous flow has remained elusive. Accordingly, in this paper, a first integral of the Navier–Stokes equations for steady flow is constructed. In the case of a two-dimensional flow, this is made possible by formulating the governing equations in terms of complex variables and introducing a new scalar potential. Associated boundary conditions are considered, and an extension of the theory to three dimensions is proposed. The capabilities of the new approach are demonstrated by calculating a Reynolds number correction to the laminar shear flow generated in the narrow gap between a flat moving and a stationary wavy wall, as is often encountered in lubrication problems. It highlights the first integral as a suitable tool for the development of new analytical and numerical methods in fluid dynamics.

We propose a new method of attacking problems in rigid body rotation, focusing on the heavy symmetric top. The technique is a direct extension of the method traditionally applied to the free symmetric top. We write Euler's equations in a frame which is attached to the top and thus shares its entire angular velocity. The structure of the resulting equations is such that it is advantageous to cast and solve them in terms of complex variables (space phasors). Through this formalism, we obtain a direct link between the initial conditions at the time of launch and the subsequent behavior of the top. The insertion of a damping term allows us to further explain the behavior of a top where the pivot is non-ideal and has friction. Finally, we make some suggestions regarding experimental verification of our results.

The article presents a solution to the bending problem for an infinite isotropic plate with a circular rigid inclusion and arbitrarily oriented straight through cracks. It is assumed that under the action of an external load at infinity, the faces of all cracks are in smooth contact along a region of constant width near the upper surface of the plate. The solution is obtained using methods of the theory of functions of complex variable and complex potentials and is reduced to a system of singular integral equations, which is numerically solved by the mechanical quadrature method. On the basis of numerical analysis, the graphical dependences of the contact force, forces and moments intensity factors are constructed at various parameters of the problem.