Continuum Mechanics · Motion, deformation gradients, strain, and transport
A brass coupon undergoes the homogeneous deformation x_1=1.25 X_1, x_2=0.15 X_1+0.80…
Problem
A brass coupon undergoes the homogeneous deformation \(x_1=1.25 X_1\), \(x_2=0.15 X_1+0.80 X_2\), \(x_3=X_3\), with coordinates in millimeters. Form \(\mathbf{F}\), invert it, and compute the Almansi strain \(\mathbf{e}=\frac12(\mathbf{I}-\mathbf{b}^{-1})\), where \(\mathbf{b}=\mathbf{F}\mathbf{F}^{T}\). For the current unit vector \(\mathbf{n}=\mathbf{e}_2\), evaluate the Eulerian strain \(\mathbf{n}\cdot\mathbf{e}\cdot\mathbf{n}\), and also compute the stretch of the spatial line element that is currently along \(\mathbf{e}_2\) by using \(\lambda=1/\sqrt{\mathbf{n}\cdot\mathbf{c}\cdot\mathbf{n}}\) with \(\mathbf{c}=\mathbf{b}^{-1}\).
Hint
\(\mathbf{F}\) is the Jacobian of the given homogeneous map; because it is triangular, inversion is immediate from the diagonal entries and a single off-diagonal correction.
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