Continuum Mechanics · Motion, deformation gradients, strain, and transport
A nitrile gasket occupies the reference brick 0≤ X_1≤ 50 mm, 0≤ X_2≤ 20 mm, 0≤ X_3≤ 8.0…
Problem
A nitrile gasket occupies the reference brick \(0\le X_1\le 50\,\mathrm{mm}\), \(0\le X_2\le 20\,\mathrm{mm}\), \(0\le X_3\le 8.0\,\mathrm{mm}\) in a Cartesian material frame \(\{\mathbf{E}_1,\mathbf{E}_2,\mathbf{E}_3\}\). The motion (\(X\) in mm, \(t\) in s, spatial coordinates \(x\) in mm) is \(x_1=(1+0.080 t)X_1+0.25 X_2\), \(x_2=(0.92-0.030 t)X_2\), \(x_3=1.05 X_3\). At \(t=1.5\,\mathrm{s}\), evaluate the deformation gradient \(\mathbf{F}\) at the particle that occupied \(\mathbf{X}=(25,10,4.0)\,\mathrm{mm}\), compute \(J=\det\mathbf{F}\), and determine the current volume of the entire gasket in mm\(^3\).
Hint
The motion is linear in the material coordinates, so \(\mathbf{F}\) is the Jacobian matrix of partial derivatives and does not depend on \(\mathbf{X}\).
Check your work
Work the problem yourself first. Then open it in Training to check your answer and read the full worked solution.
Create a free account to check your answer and see the solution. Create a free account.
More Continuum Mechanics practice problems
- A brass coupon undergoes the homogeneous deformation x_1=1.25 X_1, x_2=0.15 X_1+0.80…Motion, deformation gradients, strain, and transport
- A material region currently occupies the cube 0≤ x_1≤ 0.050 m, 0≤ x_2≤ 0.050 m, 0≤ x_3≤…Motion, deformation gradients, strain, and transport
- A 110 mm aluminum calibration bar occupies |X|≤ 55.0 mm, |Y|≤ 4.00 mm, |Z|≤ 4.00 mmMotion, deformation gradients, strain, and transport
- A polyurethane cube occupies 0≤ X,Y,Z≤ 40.0 mm in the reference configuration and has…Motion, deformation gradients, strain, and transport
- An incompressible Mooney–Rivlin solid has C_(10)=118 kPa and C_(01)=29.5 kPa, with…Stress measures and local balance laws
- A solid circular phosphor-bronze shaft of radius R = 19.0 mm and length L = 920 mm is…Stress measures and local balance laws
- An incompressible neo-Hookean tensile bar has stored energy W=(μ/2)(I_1-3), μ=226 kPa…Stress measures and local balance laws
- A silicone damping fluid (ρ=970 kg/m^3, μ=0.290 Pa· s) is driven by the constant…Stress measures and local balance laws
- A one-dimensional nonlinear elastic bar has stored energy per unit reference volume…Stress measures and local balance laws