Introduction to Elasticity/Kinematics example 4

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Example 4[edit | edit source]

Given:

Displacement field .

Find:

  1. The Lagrangian Green strain tensor .
  2. The infinitesimal strain tensor .
  3. The infintesimal rotation tensor .
  4. The infinitesimal rotation vector .
  5. The exact longitudinal strain in the reference material direction .
  6. The approximate longitudinal strain in the direction based on the infinitesimal strain tensor .

Solution[edit | edit source]

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);
  x := evalm(u + X);
  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);
  Id := IdentityMatrix(3): C := evalm(transpose(F)&*F); 
  E := evalm((1/2)*(C - Id));
  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);
  epsilon := evalm((1/2)*(gradu + transpose(gradu)));
  omega := evalm((1/2)*(gradu - transpose(gradu)));
  stretch1 :=  sqrt(evalm(evalm(e1&*C)&*transpose(e1))[1,1]):
  longStrain1 := stretch1 - 1;
  approxLongStrain1 := evalm(evalm(e1&*epsilon)&*transpose(e1))[1,1];

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