Some qualitative results on the dynamic viscoelasticity of the reissner–mindlin plate model

SOME QUALITATIVE RESULTS ON THE DYNAMIC VISCOELASTICITY OF THE REISSNER–MINDLIN PLATE MODEL by MAURO FABRIZIO† (Department of Mathematics, University of Bologna, Piazza di Porta S. Donato 5, 40127 Bologna, Italy) (Faculty of Mathematics, University of Iaşi, 6600 Iaşi, Romania) [Received 12 December 2002. Revise 25 August 2003] Summary This paper is concerned with a linear viscoelastic plate model based on the Reissner–Mindlin assumption on the displacements. The initial-boundary-value problems are formulated and some qualitative results are established concerning the solutions of such problems. In fact, appropriate uniqueness and continuous data dependence results are established under various constraint restrictions upon the relaxation functions. The spatial behaviour of solutions is also studied. By assuming that the external given data have a compact support D̂T on the time interval [0, T ], the spatial behaviour of the solution is completely described throughout the plate without the support D̂T . 1. Introduction The classical linear elastic plate model based on the Kirchhoff–Love assumption concerning strain–displacement relations has been well studied (see, for example, (1 to 3)). Such theory neglects completely the effects of transverse shear forces and so gives rise to a few mathematical discrepancies. Moreover, the unknown deflection can satisfy only two boundary conditions instead of the physically expected three. A more refined, Mindlin-type theory (4, 5), which takes account of transverse shear deformation, makes no such simplifying assumption. The study of deformation of a Mindlin-type thin elastic plate is based on the general kinematic assumption that the displacement field is of the form u α (x1 , x2 , x3 , t) = u 0α (x1 , x2 , t) + x3 vα (x1 , x2 , t), u 3 (x1 , x2 , x3 , t) = v3 (x1 , x2 , t), (1.1) with respect to Cartesian coordinates (x1 , x2 , x3 ) such that the (x1 , x2 )-plane coincides with the middle plane of the plate. Because of linearity, the general problem splits into two simpler ones: that of extensional motions, characterized by u 01 and u 02 , and that of bending, characterized by † ‡
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 c Oxford University Press 2004; all rights reserved. Q. Jl Mech. Appl. Math. (2004) 57 (1), 59–78  and STAN CHIRIŢÇ 60 M . FABRIZIO AND S . CHIRIŢ Ă v1 , v2 and v3 . If transverse shear deformation is neglected, that is, if it is further assumed that vα = −∂v3 /∂ xα , then the problem of bending reduces to Kirchhoff–Love classical theory. Another model was developed by Reissner (6) by starting with an assumption like (1.1), but regarding the form of the stresses. These two refined theories are known collectively as Reissner–Mindlin theory: it finds wide practical application, from aircraft engineering to microchip production. Such a theory also better represents boundary conditions (7) because it offers at least some approximation of the boundary layer and it is advantageous for numerical approximation. The bending of elastic plates with transverse shear deformation has been subjected to an intensive study in the recent years (see, for example, (4, 8 to 10)). It was shown recently in (11) that the Reissner–Mindlin plate bending model has a wider range of applicability than the Kirchhoff–Love model for the approximation of clamped linearly elastic plates. In fact, under the assumption that the body force is constant in the transverse direction, it was shown that the Reissner–Mindlin approximation is convergent for the full range of surface loads while the Kirchhoff–Love approximation is divergent if the surface loads induce a significant transverse shear. Thermal effects in the bending of thin plates with transverse shear deformation are considered in (12). For such a model the bending of a Mindlin-type thermoelastic plate is examined in (13) when the source terms are harmonic in time and sufficient time has elapsed for the system to have reached a steady state. On the other hand, the viscoelastic model of the Kirchhoff plate was studied by Lagnese (14) in order to prove that the solution of the viscoelastic plate equation decays to zero as time t → ∞. The asymptotic temporal behaviour of the solution of the viscoelastic plate equation was studied in (15) for establishing various rates of temporal decay. A thermoelastic plate model has been studied in (16) in connection with the asymptotic behaviour of the solution as t → ∞ and the domain of dependence for the solution. In this paper we consider a linear viscoelastic plate model based on the Reissner–Mindlin assumption on the displacements. Some initial-boundary-value problems are formulated. Then we study the uniqueness and continuous data dependence problems under various constraint restrictions upon the relaxation functions. The continuous dependence estimates established in the paper are in terms of comparatively weak measures and furthermore restrict the solution to lie in various constrained sets equivalent to supposing that the solutions belong to certain Sobolev function spaces. It is possible to include in our treatment suitably defined weak solutions, but since this generalization is somewhat standard, we prefer instead to emphasize the technique itself by admitting only classical solutions. These, of course, are always assumed to exist. The existence problem can be treated by various methods (as, for example, by means of the semigroup of linear operators theory), but is not pursued here; it will constitute the subject of a future work. We also study the spatial behaviour of solutions. Under the assumption that the external given data have a compact support D̂T on the time interval [0, T ], we establish the spatial estimates giving a complete description for the spatial behaviour of the solution throughout the plate domain without the support D̂T . With this aim we prove that, for all fixed t ∈ [0, T ], the solution vanishes in the part of the plate situated at distances to the support D̂T larger than ct, where c is a constant depending on the relaxation functions and the mass density. Moreover, we establish an exponential decay of Saint-Venant’s type for the part of the plate situated at distances smaller than ct. The plan of the paper is as follows. In section 2 we present the viscoelastic plate model with transverse shear deformation and then we formulate the initial-boundary-value problems associated with the model. Section 3 is devoted to the study of the uniqueness problem under various constraints upon the relaxation functions. In section 4 we use Gronwall’s inequality and logarithmic VISCOELASTICITY OF THE REISSNER – MINDLIN PLATE 61 convexity methods to establish estimates describing the continuous data dependence of solutions with respect to the given data. In section 5 we present the spatial behaviour of the solution throughout the plate without the support of the external given data. 2. Viscoelastic plate model with transverse s (...truncated)