Suppose we have a simple tension deformation which maps points X[I] to new points x[i] following...

 

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Suppose we have a simple tension deformation which maps points X[I] to new points x[i] following x₁ = X₁ + yX₁, x2 = X2 - VyX2, X3 = X3- VEX3. For each part, write the result in symbolic notation. Problem 5.1 (2 points) Compute the displacement field ui] = x[i] - X[i] as a function of Xi] and x[i]. Problem 5.2 (3 points) Compute, in symbolic form, the deformation gradient, ax[i] Fax Problem 5.3 (7 points) Compute, in symbolic form, the infinitesimal strain tensor, (du[i] Juj] Əx[i] + E[ij] = Note that the derivatives are with respect to the lower-case coordinate variables, so you may need to perform a further step on Equation (5.1) in order to obtain the appropriate expression. Problem 5.4 (3 points) In truth, the deformation may not necessarily be infinitesimal, and so one should use an appropriate measure of strain. The Green-Lagrange strain tensor is defined as Em₁ = (Cm₁-801). where C = FTF is the right Cauchy-Green tensor. Compute the Green-Lagrange strain tensor, in symbolic form. (5.4) Problem 5.5 (5 points) Make two plots - the 11-component, and the 22- or 33-component of the infinitesimal and Green-Lagrange strain measures. Choose some v = 0, v| ≤ 1/2 (report which value you chose), and plot over a range of 0 ≤ y ≤ 1/2. Suppose we have a simple tension deformation which maps points X[I] to new points x[i] following x₁ = X₁ + yX₁, x2 = X2 - VyX2, X3 = X3- VEX3. For each part, write the result in symbolic notation. Problem 5.1 (2 points) Compute the displacement field ui] = x[i] - X[i] as a function of Xi] and x[i]. Problem 5.2 (3 points) Compute, in symbolic form, the deformation gradient, ax[i] Fax Problem 5.3 (7 points) Compute, in symbolic form, the infinitesimal strain tensor, (du[i] Juj] Əx[i] + E[ij] = Note that the derivatives are with respect to the lower-case coordinate variables, so you may need to perform a further step on Equation (5.1) in order to obtain the appropriate expression. Problem 5.4 (3 points) In truth, the deformation may not necessarily be infinitesimal, and so one should use an appropriate measure of strain. The Green-Lagrange strain tensor is defined as Em₁ = (Cm₁-801). where C = FTF is the right Cauchy-Green tensor. Compute the Green-Lagrange strain tensor, in symbolic form. (5.4) Problem 5.5 (5 points) Make two plots - the 11-component, and the 22- or 33-component of the infinitesimal and Green-Lagrange strain measures. Choose some v = 0, v| ≤ 1/2 (report which value you chose), and plot over a range of 0 ≤ y ≤ 1/2.

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Solutions to the problems you requested Problem 51 The displacement field ui is the difference betwe... View the full answer