Introduction to Elasticity/Kinematics example 4

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[edit] Example 4

Given:

Displacement field \mathbf{u} = \kappa X_2 \widehat{\mathbf{e}}_{1} + \kappa X_1 \widehat{\mathbf{e}}_{2}.

Find:

  1. The Lagrangian Green strain tensor \boldsymbol{E}\,.
  2. The infinitesimal strain tensor \boldsymbol{\varepsilon}\,.
  3. The infintesimal rotation tensor \boldsymbol{\omega}\,.
  4. The infinitesimal rotation vector \boldsymbol{\theta}\,.
  5. The exact longitudinal strain in the reference material direction \mathbf{e}_1\,.
  6. The approximate longitudinal strain in the direction \mathbf{e}_1\, based on the infinitesimal strain tensor \boldsymbol{\varepsilon}\,.

[edit] Solution

The Maple output of the computations are shown below:


  with(linalg): with(LinearAlgebra): 
  X := array(1..3): x := array(1..3):
  e1 := array(1..3,[1,0,0]): 
  e2 := array(1..3,[0,1,0]): 
  e3 := array(1..3,[0,0,1]):
  u := evalm(k*X[2]*e1 + k*X[1]*e2);
u := \left[ \! k\,{X_{2}}, \,k\,{X_{1}}, \,0 \! \right] 


  x := evalm(u + X);
x := \left[ \! k\,{X_{2}} + {X_{1}}, \,k\,{X_{1}} + {X_{2}}, \, {X_{3}} \! \right] 


  F := linalg[matrix](3,3):
  for i from 1 to 3 do
    for j from 1 to 3 do
      F[i,j] := diff(x[i],X[j]);
    end do;
  end do;
  evalm(F);
F := \begin{bmatrix} 1 & k & 0 \\ k & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} 


  Id := IdentityMatrix(3): C := evalm(transpose(F)&*F); 
  E := evalm((1/2)*(C - Id));
C := \begin{bmatrix} 1 + k^{2} & 2\,k & 0 \\ 2\,k & 1 + k^{2} & 0 \\ 0 & 0 & 1 \end{bmatrix}
E := \begin{bmatrix} { \frac {k^{2}}{2}} & k & 0 \\ [2ex] k & { \frac {k^{2}}{2}} & 0 \\ [2ex] 0 & 0 & 0 \end{bmatrix} 


  gradu := linalg[matrix](3,3):
  for i from 1 to 3 do
    for j from 1 to 3 do
      gradu[i,j] := diff(u[i],X[j]);
    end do;
  end do;
  evalm(gradu);
gradu := \begin{bmatrix} 0 & k & 0 \\ k & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} 


  epsilon := evalm((1/2)*(gradu + transpose(gradu)));
\varepsilon  := \begin{bmatrix} 0 & k & 0 \\ k & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} 


  omega := evalm((1/2)*(gradu - transpose(gradu)));
\omega  := \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} 


  stretch1 :=  sqrt(evalm(evalm(e1&*C)&*transpose(e1))[1,1]):
  longStrain1 := stretch1 - 1;
\mathit{stretch1} := \sqrt{1 + k^{2}}
\mathit{longStrain1} := \sqrt{1 + k^{2}} - 1 


  approxLongStrain1 := evalm(evalm(e1&*epsilon)&*transpose(e1))[1,1];
approxLongStrain1: = 0

The geometrical difference between the large strain and small strain cases can be observed by looking at the figures from the previous examples.

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