A 110 mm aluminum calibration bar occupies |X|≤ 55.0 mm, |Y|≤ 4.00 mm, |Z|≤ 4.00 mm

A 110 mm aluminum calibration bar occupies \(|X|\le 55.0\,\mathrm{mm}\), \(|Y|\le 4.00\,\mathrm{mm}\), \(|Z|\le 4.00\,\mathrm{mm}\). The recorded displacement is the infinitesimal rigid motion \(\mathbf{u}=\mathbf{w}\times\mathbf{X}+\mathbf{b}\), with axial vector \(\mathbf{w}=(0.012\,\mathbf{e}_X-0.008\,\mathbf{e}_Y+0.015\,\mathbf{e}_Z)\) (radians) and translation \(\mathbf{b}=(0.40\,\mathbf{e}_X-0.25\,\mathbf{e}_Y+0.10\,\mathbf{e}_Z)\,\mathrm{mm}\), taking \(\mathbf{X}\) in millimetres so that \(\mathbf{u}\) is in millimetres. Write the Cartesian matrix of the infinitesimal rotation tensor \(\omega\) (the unique skew tensor with axial vector \(\mathbf{w}\)) and of \(\mathrm{Grad}\,\mathbf{u}\). Compute the infinitesimal strain \(\varepsilon=\mathrm{sym}(\mathrm{Grad}\,\mathbf{u})\) and confirm that it vanishes identically. Evaluate the displacement of the end point originally at \(\mathbf{X}=(55.0\,\mathbf{e}_X+4.00\,\mathbf{e}_Y-4.00\,\mathbf{e}_Z)\,\mathrm{mm}\). Then form \(F_{\mathrm{inf}}=I+\mathrm{Grad}\,\mathbf{u}\), compute \(C_{\mathrm{inf}}=F_{\mathrm{inf}}^TF_{\mathrm{inf}}\), and report the residual \(C_{\mathrm{inf}}-I\).

The skew tensor of an axial vector \(\mathbf{w}\) has the standard cross-product matrix; for \(\mathbf{u}=\mathbf{w}\times\mathbf{X}+\mathbf{b}\) with constant \(\mathbf{w}\) and \(\mathbf{b}\), this matrix is exactly \(\mathrm{Grad}\,\mathbf{u}\).