### A Galerkin spectral approximation in linearized beam theory

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The concept of reduced plastic dissipation is introduced for a perfectly plastic rate-independent material not obeyng the associated normality rule and characterized by a strictly convex plastic potential function. A maximum principle is provided and shown to play the role of variational statement for the nonassociative constitutive equations. The Kuhn-Tucker conditions of this principle describe the actual material behaviour as that of a (fictitious) composite material with two plastic constituents,...

In the context of the linear, dynamic problem for elastic bodies with voids, a minimum principle in terms of mechanical energy is stated. Involving a suitable (Reiss type) function in the minimizing functional, the minimum character achieved in the Laplace-transform domain is preserved when going back to the original time domain. Initial-boundary conditions of quite general type are considered.

The elastoplastic rate problem is formulated as an unconstrained saddle point problem which, in turn, is obtained by the Lagrange multiplier method from a kinematic minimum principle. The finite element discretization and the enforcement of the min-max conditions for the Lagrangean function lead to a set of algebraic governing relations (equilibrium, compatibility and constitutive law). It is shown how important properties of the continuum problem (like, e.g., symmetry, convexity, normality) carry...

The thermoelastic stresses created in a solid phase domain in the course of solidification of a molten ingot are investigated. A nonlinear behaviour of the solid phase is admitted, too. This problem, obtained from a real situation by many simplifications, contains a moving boundary between the solid and the liquid phase domains. To make the usage of standard numerical packages possible, we propose here a fixed-domain approximation by means of including the liquid phase domain into the problem (in...

It is proved that, as in three-dimensional elasticity, Betti's theorem represents a criterion for the existence of a stored-energy function for a Cosserat elastic shell.

The paper deals with a class of optimal shape design problems for elastic bodies unilaterally supported by a rigid foundation. Cost and constraint functionals defining the problem depend on contact stresses, i.e. their control is of primal interest. To this end, the so-called reciprocal variational formulation of contact problems making it possible to approximate directly the contact stresses is used. The existence and approximation results are established. The sensitivity analysis is carried out....

We explicitly introduce and exploit div-curl Young measures to examine optimal design problems governed by a linear state law in divergence form. The cost is allowed to depend explicitly on the gradient of the state. By means of this family of measures, we can formulate a suitable relaxed version of the problem, and, in a subsequent step, put it in a similar form as the original optimal design problem with an appropriate set of designs and generalized state law. Many of the issues involved has been...

One may produce the qth harmonic of a string of length π by applying the 'correct touch' at the node $\pi /q$ during a simultaneous pluck or bow. This notion was made precise by a model of Bamberger, Rauch and Taylor. Their 'touch' is a damper of magnitude b concentrated at $\pi /q$. The 'correct touch' is that b for which the modes, that do not vanish at $\pi /q$, are maximally damped. We here examine the associated spectral problem. We find the spectrum to be periodic and determined by a polynomial of degree $q-1$....