Minimum principles for linear elastodynamics

Abstract

Two minimum principles which take into account inhomogeneous initial conditions are presented within the context of the linear dynamic theory of elasticity. One principle, formulated in terms of displacements alone, is the dynamic counterpart to the static principle of minimum potential energy; the other principle is formulated in terms of stresses alone, but has no counterpart in elastostatics. Both principles are motivated by taking Laplace transforms of the field equations and boundary values and then using established minimum functionals in the "transform domain". The introduction of an appropriate "weight function" enables one to get back to the original time domain while preserving the minimum character of the transformed functionals. RESUME Deux principes minimum qui fiennent compte de conditions initiales non lin6aires sont pr6sent6s sans le contexte de la th6orie dynamique lin6alre de l'61asticit6. Un des principes, exprim6s en fonction seulement des d6placements, est la contrepartie dynamique du principe statique de l'energie potentieUe minimum; le second principe, exprim6 en fonction des efforts seulement, n'a pas de contrepartie en 61asticit6 statique. Les deux principes sont obtenus en prenant les transform6s de Laplace des 6quations differentielles et des conditions aux limites, puis en utilisant dans le "domaine transform6" des fonctionnels minimum connus. En introduisant une "weight function" appropri6e, il est possible de retourner au dornaine temporel d'origine tout en pr6servant le caractere minimum des fonctionnels transform6s.

Some results concerning the state of bending for transversely isotropic plates

“…The right-hand terms of Equations (17), (18) In what follows, we only consider functions defined on¯ ×[0, ∞) having the following properties: (11), (13), (14) and initial-boundary conditions (17), (18) for some given data.…”

“…The boundary-initial-value problem (11), (13), (14), (17), (18) related to the state of bending for elastic plates has at most one admissible solution.…”

“…Now, we establish a minimum principle of Reiss type using the procedure from [18,19]. We introduce an admissible weight function such that…”

“…Starting from the reciprocal relation, we formulate a variational theorem of Gurtin type [17]. Moreover, we derive a minimum principle using the arguments of Reiss and Haug presented in [18,19]. In the last section, we give a Galerkin representation of solution of the field equations.…”

New variational principles in quasi-static viscoelasticity

“…Indeed, as noticed in [6], the convexity of the convolution-type functio introduced by CHRISTENSEN follows from thermodynamic restrictions, but yet it does turn out to be stationary at the solution unless we assume the same time dependence all displacement fields. As a consequence, the converse statement of the principle cannot proved, that is to say, the whole solution u 0 to QSP cannot be characterized as a minim A new technique to obtain stationary and minimum principles for linear evolut equations was early introduced by REISS [9] and further developed by many authors ( [10], [6], [2]). It rests upon the introduction of suitable bilinear forms of convolution t involving Laplace transformation with respect to time and then applies to initial bounda value problems only.…”