Variational principles in the linear dynamic theory of viscoelasticity

Abstract

Introduction.In a recent paper Gurtin [1] deduces variational principles which characterize the standard initial-value problems of linear elastodynamics.Here we extend** these principles to dynamic viscoelasticity theory. Notational agreements and mathematical preliminaries are given in Section 2. In Section 3 we formulate the initial-history problem for the dynamic linear theory of viscoelasticity and deduce two equivalent characterizations of its solutions. These alternate formulations, aside from being essential to the results presented here, are of interest in themselves. In Sections 4 and 5 we prove three variational principles applicable to the mixed boundary-value problem. The first of these is quite general and the admissible states are required to satisfy only the initial history condition. In the second, the admissible states, in addition, must satisfy the strain-displacement relations. The third variational principle is concerned only with the stresses.We allow the state history to be of infinite duration. Those cases for which the history may be considered finite fall as a special case of the results presented. We also remark that for histories of finite duration variational principles may be deduced which have a simpler form. They will, however, require separate proof.We do not, in this paper, give all the counterparts of the results in [1], since it is clear from those presented how the others may be obtained.2. Notation and mathematical preliminaries. We will try to use, whenever possible, the notation developed in [1], Indicial notation is used throughout. Thus, subscripts have the range of the integers 1, 2, 3 and denote the Cartesian components of vector-and tensor-valued functions; summation over repeated subscripts is implied and subscripts preceded by a comma indicate differentiation with respect to the corresponding Cartesian coordinate. Parentheses about a pair of free subscripts denote symmetry with respect to these subscripts, e.g. «(,-,,■) = i(ui,i + R is an open bounded region in three-dimensional Euclidean space with the closure R° and boundary dR. Bu and B, denote disjoint sets whose union is dR and n{ is the unit outward normal to dR. We write xt for a point of Rc and use the abbreviation

One-dimensional theories of motion for beams

“…If Gurtin's principle is replaced by that of Leitman [19], analogous theories describing the motion of viscoelastic rods can be derived through the same procedure. If the structure of hyperbolic system already exhibited in the case of elastic materials is to be kept, attention should be restricted to viscoelastic materials with finite discrete spectrum and instantaneous elasticity, as shown in [14].…”

Two-Temperature Theory in Linear Micropolar Thermoviscoelastic Anisotropic Solid

“…Using equations of motion (20) and (21) and the boundary conditions (4), Eqs. (43) and (51) lead to (45) Using Equations (31) the Reciprocity relation (34) results.…”

Incremental viscoelastic formulation using generalized variables for thin structures: relaxation differential approach

“…A simplified principle was obtained for elastodynamics by taking the second time derivative of a functional developed by Gurtin[11]. This simple method can also be applied to the linear dynamic theory of viscoelasticity and, therefore, simplified variational principles can be obtained by taking second-order time derivatives of the functionals presented by Leitman [12].…”