Fundamentals of Generalized Rigidity Matrices for Multi-layered Media

Rigidity matrices for multi-layered media are derived for isotropic and orthotropic layers by a simple direct procedure which brings to light their fundamental mathematical structure. The method was introduced many years ago by the author in the more general context of dynamics and stability of multi-layers under initial stress. Other earlier results are also briefly recalled such as the derivation of three-dimensional solutions from plane strain modes, the effect of initial stresses, gravity, and couple stresses for thinly laminated layers. The extension of the same mathematical structure and symmetry to viscoelastic media is valid as a consequence of fundamental principles in linear irreversible thermodynamics. INTRODUCTION In a recent paper (Kausel and Roesset, 1981), stiffness matrices were derived for isotropic multi-layered media starting from the Thomson-Haskell transfer matrices. However, such stiffness matrices were already derived many years ago (Biot, 1963, 1965) and discussed in great detail in the more general context of orthotropic multilayered solids under initial stress. The derivation was not based on the ThomsonHaskell matrices but used instead a direct approach which is extremely simple and takes advantage of the physical symmetry of the layers. The fundamental mathematical structure of the 4 X 4 stiffness matrix is thus brought to light, forming a symmetric matrix of six independent terms. The structure is valid for a wide range of physical systems and embodies the basic reciprocity properties of cross impedances of linear conservative systems. In the present paper, the method has been used to derive the stiffness matrices directly in the particular c&e of isotropic and orthotropic layers without initial stress, and it is shown how these results are also obtained from the earlier ones. The basic equations for multi-layers are obtained in the form of recurrence equations for displacements at three successive interfaces, and it is recalled how they may be written in compact form as a variational principle including the effect of initial stresses, gravity forces and couple stresses. Using the earlier developments (Biot, 1966, 1972b, 1974), it is shown how simple plane strain solutions lead to a large class of three-dimensional solutions for transverse isotropy without any additional evaluation of matrix elements. Finally, for viscoelastic materials whose heredity properties are based on linear thermodynamics with Onsager’s reciprocity relations and internal coordinates, it is recalled that all results remain formally valid with complex functions of the frequency replacing the elastic coefficients. The complex matrices exhibit the same mathematical structure as in the elastic case. STIFFNESS MATRIX OF SYMMETRIC AND ANTISYMMETRIC MODES A single isotropic elastic layer is analyzed for plane strain in the x y plane normal to the layer. The thickness of the layer is h, and the x axis is parallel to the layer and equidistant from the faces. These faces are thus represented by the planes 749 750 MAURICE A. BIOT y = 2 h/2. We denote by u and u the displacements along x and y defining the plane strain components as au au exx = ax e,, = ay (1) The corresponding stresses are u xx = He,, + (H 2L)e, a,, = (H 2L)e, + Heyy a,, = 2Le, (2) where H = X + 2p, L = p, and A, p denote the Lam6 constants. We use L instead of p in order to avoid a change of notation in the more general cases. For harmonic time dependence, solutions are proportional to the factor exp(iot). This factor may be omitted in all formulas. The dynamic equilibrium equations are then _+++ppw2v=o achy ax ay (3) where p is the density. For the particular case of an isotropic medium, equations (1) to (3) are solved by the classical procedure of decoupling of dilatational and rotational waves. The values of u and u are then obtained by putting aq a+ acp ali, UC---_ u=-+ax ay ay ax (4) where the scalars ‘p and 4 satisfy the two wave equations HQ2, + pw2rp = 0 LV2, + p&P+ = 0