A loaded block has the non-uniform displacement field u_1=kx_1x_2

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

Bending, Ringing, Breaking and Flowing, Ch Strain and stress in a deformed solid

Problem

A loaded block has the non-uniform displacement field u1=kx1x2,u2=kx12,u3=0,u_1=kx_1x_2,\qquad u_2=kx_1^2,\qquad u_3=0, with k=2×10−3 m−1k=2\times10^{-3}\,\mathrm{m^{-1}} and the coordinates in metres. Consider the point P=(0.05, 0.20, 0) mP=(0.05,\,0.20,\,0)\,\mathrm m. ((a)) Find the nine components of the displacement gradient ∂ui/∂xj\partial u_i/\partial x_j at PP, and check that they are small. ((b)) Split the gradient at PP into the strain tensor εij\varepsilon_{ij} and the local rotation ωij\omega_{ij}. By what angle, and in which sense about x3x_3, has the neighbourhood of PP turned as a rigid body? ((c)) Find the dilatation at PP and the fractional extension (change of length per unit original length) of a short line element through PP along (1,1,0)/2(1,1,0)/\sqrt2. ((d)) Find the three principal strains at PP and their axes, and check your answer against the dilatation.

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