Mathematics and Mechanics of Solids

The dynamic stability of a uniform cantilever column under a tip-concentrated subtangential follower force is examined. The governing two-point boundary value problem is converted to an initial value problem. A simple numerical iterative scheme is followed to generate a load versus frequency curve (eigencurve) in order to study the critical and post-critical behavior of the columns subjected to a tip-concentrated follower force. For the load rotation parameter greater than 1 2 , the coalesce frequency and the corresponding critical load are obtained. The deformed configurations of cantilever columns are generated for different load rotation parameters. The details of the critical load parameter, tip angle and tip coordinates of the columns for the specified load rotation parameter are presented. The critical load parameter increases with the load rotation parameter.

This paper presents a set of axisymmetric analytical solutions of the thermo-elasticity field induced by the external thermal load, in a heterogeneous circular plate. The plate can be either simply supported or clamped, and the properties of the transversely isotropic material are assumed to be functions of the thickness coordinate. Three typical types of thermal boundary conditions are considered and the corresponding temperature fields are determined. A direct displacement method is used to solve the governing equations for the elastic field in plate. The displacement functions pertinent to the current problem are solved by a step-by-step procedure of integration. The corresponding integral constants are determined by using the cylindrical boundary conditions in the Saint-Venant’s sense. In the present solutions, all of the material coefficients can vary independently and continuously along the thickness direction. Numerical calculations are performed to validate the present solutions and to show their applications to a heterogeneous model. The influence of the material heterogeneity is finally addressed. Since no ad hoc hypotheses on the elastic deformations of the plate are introduced, the present solution can be a nature benchmark to various plate theories and numerical codes.

In this paper, volume-average relations related to the multilevel modelling process in continuum mechanics are analysed and the concept of average consistency is investigated both analytically and numerically. These volume averages are used in the computational homogenization technique, where a transition of the mechanical properties from the local, microscopic, to the global, macroscopic, length scale is obtained. The representative volume element (RVE) is used as a reference placement and the solution, in terms of volume-averaged stress, will depend on which boundary conditions are chosen for the RVE. Three types of boundary conditions – periodic, affine and anti-periodic – are analysed with respect to the average consistence for the kinematical and stress relations used in continuum mechanics. The inconsistence is quantified by introducing the inconsistence ratio. It is shown analytically that some average stress relations are fulfilled, assuming the periodic boundary condition and anti-periodic traction vector, whereas the average relations connected to the deformation are in general not average consistent. The inconsistence is investigated in a plane model using the finite element technique. The numerical investigation has shown that the inconsistence ratios related to the deformation are also average consistent in the examples considered.

The present work is concerned with resolving a dynamical problem of an infinite type III thermoelastic space weakened by a finite linear mode I crack. The material of the medium is considered to be homogeneous and isotropic. The boundary of the crack is subjected to a prescribed stress distribution and temperature. The thermoelasticity theory of type III developed by Green and Naghdi has been employed, and integral transforms have been used to obtain the solution to the problem. A system of four dual integral equations have been obtained, the solution of which is shown to be equivalent to the solution of a Fredholm’s integral equation of the first kind. These integral equations are solved numerically by the regularization method. The inversion of the Laplace transform is also carried out numerically, and numerical values of the displacement components, temperature and stresses in the physical domain are computed for copper material by considering particular cases. The results are presented graphically.

A nonlinear programming approach, along with finite element implementation, has been developed to perform plastic limit analysis for materials under non-associated plastic flow, so the plastic stability condition of non-standard materials can be directly calculated. In the framework of Radenkovic’s theorems, a decoupled material model with non-associated plastic flow was introduced into kinematic limit analysis so that the classic limit theorems were extended for non-standard plastic flow materials, such as cohesive-frictional materials. A non-associated plastic dissipation power is derived, and a purely kinematic formulation is obtained for limit analysis. Based on the mathematical programming theory and the finite element method, the numerical implementation of kinematic limit analysis is formulated as a nonlinear programming problem subject only to one equality constraint. An extended direct iterative algorithm is proposed to solve the resulting programming problem. The developed method has a wide applicability for limit analysis. The effectiveness and efficiency of the proposed method are validated through numerical examples and the influence of non-associated plastic flow on stability conditions of structures is numerically investigated.

Anti-plane stress analysis of two different kinds of dissimilar sectors has been accomplished in our study. The first kind of dissimilar sectors consists of two isotropic sectors perfectly attached to each other along their circular edges. In the second kind, these isotropic sectors are joined perfectly in the radial interface. Solutions to the differential equations governing the dissimilar sector are found by utilizing finite Fourier cosine transform as well as the technique of the separation of variables. The closed-form solutions are obtained for the displacement and stress fields in each sector. Also the geometric and force singularities of the stress fields are studied and compared with those in the past research. In the special case, i.e. finite wedges, the results for the stress fields are in complete agreement with those in the literature.