A polyurethane cube occupies 0≤ X,Y,Z≤ 40.0 mm in the reference configuration and has…

A polyurethane cube occupies \(0\le X,Y,Z\le 40.0\,\mathrm{mm}\) in the reference configuration and has uniform reference density \(\rho_0=1240\,\mathrm{kg/m}^3\). With \(t\) in seconds and reference coordinates in metres, the homogeneous motion is \[ x=(1+0.080 t)X+(0.15 t)Y,\qquad y=(1-0.040 t)Y,\qquad z=Z. \] The body force per unit mass is \(\mathbf{b}=-9.81\,\mathbf{e}_z\,\mathrm{m/s}^2\) in both the spatial and referential descriptions. At \(t=2.00\,\mathrm{s}\) the Cauchy stress throughout the cube is the uniform field \[ \sigma=\begin{bmatrix}8.40&1.20&0\\1.20&-3.60&0\\0&0&2.00\end{bmatrix}\,\mathrm{MPa}. \] Compute \(F\), \(J\), the current density \(\rho=\rho_0/J\), the material acceleration \(\mathbf{a}\) of every particle, and \(P=J\sigma F^{-T}\). Evaluate the spatial linear-momentum residual \(\operatorname{div}\sigma+\rho\mathbf{b}-\rho\mathbf{a}\) and the referential residual \(\operatorname{Div} P+\rho_0\mathbf{b}-\rho_0\mathbf{a}\). Report the two residual vectors in \(\mathrm{N/m}^3\) and in \(\mathrm{N/m}^3\) of reference volume, respectively, and state the exact algebraic relation between them. Conclude whether local linear-momentum balance holds at this instant.

Read \(F\) off the motion at \(t=2\), then obtain \(J\), \(\rho=\rho_0/J\), and the material acceleration by differentiating \(\boldsymbol{\chi}(\mathbf{X},t)\) with \(\mathbf{X}\) held fixed.