The Lamb Problem for an Inhomogeneous Elastic Half-Space

  • A. G. Alenitsyn
Part of the Topics in Mathematical Physics book series (TOMP, volume 1)


The following mixed boundary value problem for the system of equations of elasticity is considered: To find the displacement vector\(\vec u(x,y,z,t) = ({u_x},{u_y},{u_z})\) satisfying in −∞<x<+∞,-∞<y<+∞, z>0, t>0 the equations of dynamics of an isotropic elastic medium
$$ {\overrightarrow {p\mu } _{tt}} = (\lambda + 2\mu )\nabla \left( {\nabla ,\vec u} \right) - \mu \left[ {\nabla ,\left[ {\nabla ,\vec u} \right]} \right] + \nabla \lambda \left( {\nabla ,\vec u} \right) + 2\left( {\nabla \mu ,\nabla } \right)\vec u + \left[ {\nabla \mu ,\left[ {\nabla ,\vec u} \right]} \right],$$
where the boundary conditions at z = 0 are
$${\tau _{zx}} \equiv \mu \left( {\frac{{\partial {u_X}}}{{\partial z}} + \frac{{\partial {u_z}}}{{\partial x}}} \right) = 0,{\tau _{zy}} \equiv \mu \left( {\frac{{\partial {u_y}}}{{\partial z}} + \frac{{\partial {u_z}}}{{\partial y}}} \right) = 0,{\tau _{zy}} \equiv 2\mu \frac{{\partial {u_z}}}{{\partial z}} + \lambda \left( {\nabla ,\vec u} \right) = \delta \left( x \right)\delta \left( y \right)\varepsilon \left( t \right),$$
and the initial conditions at t = 0 are
$$\vec u = {\vec u_t} = 0.$$


Transverse Wave Longitudinal Wave Rayleigh Wave Asymptotic Formula Reversal Point 
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© Consultants Bureau 1967

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  • A. G. Alenitsyn

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