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What is the equilibrium deformation of an infinite half-space (that is, an isotropic and homogeneous linearly-elastic three-dimensional medium, with a single planar surface) produced by a force which is localised on an interior point (a.k.a. Mindlin's problem; a.k.a. Green's function for infinite half-space)?

In other words, I'm looking for the solution ($\mathbf u$) to:

  • $\partial_j \sigma_{ij}=0$ (Newton's law for static continuous materials)
  • where $\sigma_{ij}=\lambda \delta_{ij} \varepsilon_{kk} + 2\mu \varepsilon_{ij}$ (linear isotropic relationship between stress and strain)
  • and $\varepsilon_{ij}=\frac 1 2 (\partial_i u_j + \partial_j u_i)$ (deformation strain)
  • $\lambda,\mu$ constant (homogeneous material properties)

under the boundary conditions:

  • $\sigma_{i3}=0$ for $\forall i$ where $x_3=0$ (no pressure or traction upon surface)
  • $\partial_j \sigma_{ij} = -f_i$ in an infinitesimal neighbourhood around $\mathbf x = [0,0,h]$ (applied force)
  • $\mathbf u \rightarrow 0$ far from the origin.

Is there any straight-forward approach (perhaps a Fourier method) for solving this analytically? Alternatively, is the solution written correctly in full someplace accessible? I've tried googling for the solution to Mindlin's problem, but it is usually only expressed (a bit vaguely) as a correction to other (Boussinesque/Cerrutti) solutions; I haven't been able to piece together yet (and plot) an expression which correctly gives vanishing stresses at the boundary.

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This problem is solved using image methods in the book "Elasticity" by Barber (3rd edition, 2010), ch. 23.4.1, pg 373-374.

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