A uniform isotropic elastic solid of density obeys Cauchy's equation of motion (held…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Fluids & Continuum Physics · Solid continuum mechanics and complex fluids

Invariance, Ch Index notation and Cartesian tensors

Problem

A uniform isotropic elastic solid of density ρ\rho obeys Cauchy's equation of motion ρ ∂2ui/∂t2=∂jσij\rho\,\partial^2u_i/\partial t^2=\partial_j\sigma_{ij} (held from the continuum-mechanics course), where ui(x,t)u_i(\mathbf x,t) is the displacement field, no body force acting. For small strains eij=12(∂iuj+∂jui)e_{ij}=\tfrac12(\partial_iu_j+\partial_ju_i) an isotropic solid obeys Hooke's law in Lamé form, σij=λekkδij+2μeij\sigma_{ij}=\lambda e_{kk}\delta_{ij}+2\mu e_{ij}, with λ,μ\lambda,\mu the Lamé constants. ((a)) State, in one sentence and citing the isotropic tensor result for rank four, why the elastic tensor of an isotropic solid can involve only the two scalars λ,μ\lambda,\mu. ((b)) Substitute the Lamé form into Cauchy's equation and use the epsilon-delta identity for ∇×(∇×u)\nabla\times(\nabla\times\mathbf u) to show ρ u¨=(λ+2μ)∇(∇⋅u)−μ ∇×(∇×u).\rho\,\ddot{\mathbf u}=(\lambda+2\mu)\nabla(\nabla\cdot\mathbf u)-\mu\,\nabla\times(\nabla\times\mathbf u). ((c)) Hence show that a purely irrotational disturbance (∇×u=0\nabla\times\mathbf u=\mathbf 0 everywhere) propagates as a wave with speed vL=(λ+2μ)/ρv_L=\sqrt{(\lambda+2\mu)/\rho}.

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