### Surface waves in elastic half spaces coated with crystalline films [chapter]

David J. Steigmann
2013 CISM International Centre for Mechanical Sciences
Introduction Considered here is the general theory of surface wave propagation in elastic thin-film/substrate systems. Elasticity is of course an inherently nonlinear subject, although a great many applications are amenable to analysis using the linear theory, including those developed here. Thus for the sake of completeness and to establish the logical progression of our work we present a brief tutorial on the general nonlinear purely mechanical theory as a prelude to linearization. The main
more » ... ntribution of the present work is the derivation of and solution to an asymptotic twodimensional theory for the dynamics of a thin film bonded to a substrate, as distinct from the asymptotic treatment of the underlying three-dimensional equations (Fu, 2007) . Here the small parameter is the film thickness, and the considered model furnishes the rigorous leading-order system when this is small against the wavelength of a propagating surface wave. The purely elastic theory is developed first, followed by an extension to electroelasticity. We draw particular attention to some non-standard effects associated with the propagation of Love waves in conventional isotropic elastic half spaces coated with thin films having various kinds of crystalline symmetry. Standard notation is used throughout. Thus we use bold face for vectors and tensors and indices to denote their components. Latin indices take values in {1 2 3}; Greek in {1 2}. The latter are associated with surface coordinates and associated vector and tensor components. A dot between bold symbols is used to denote the standard inner product. Thus, if A 1 and A 2 are second-order tensors, is the trace and the superscript  is used to denote the transpose. The norm of a tensor A is |A| = √ A · A. The linear operator (·) delivers the symmetric part of its second-order tensor argument. The notation ⊗ identifies the standard tensor product of vectors. If C is a fourth-order tensor, then C[A] is the second-order tensor with orthogonal components C     The transpose C  is defined by B · C  [A] = A · C[B] and C is said to possess major symmetry if and A · C[B] = A · C[B  ] then C is said to possess minor symmetry. We use symbols such as  and  to denote the three-dimensional divergence and gradient operators, while  and ∇ are reserved, after Section 2, for their two-dimensional counterparts. Thus, for example, A =   e  and A =   e  , where {e  } is an orthonormal basis and subscripts preceded by commas are used to denote partial derivatives with respect to Cartesian coordinates. Finally, the notation  A stands for the tensor-valued derivative of a scalar-valued function  (A).