# Analysis of simple shear

1. May 9, 2013

### Bertbos

1. The problem statement, all variables and given/known data
Let {ii}, i = 1, 2, 3 be an orthonormal basis as shown in Figure 2 and consider a
simple shear deformation from R to R′ defined as
$\hat{y}$(x) = x + γ(x · i2)i1 , where the scalar γ corresponds to amount of shear.

2. Relevant equations
For this homogeneous deformation, compute the deformation gradient F
and the translation vector c. Compute the change in length of fibers of
unit length in R aligned with the basis vectors ii (i = 1, 2, 3). What can
you say about fibers aligned with i1 and i3? Compute the change in angle
of pairs of fibers aligned with ii, ij (for i, j = 1, 2, 3, i ≠ j).

3. The attempt at a solution
If I could get $\hat{y}$(x) in a form of $\hat{y}$(x) = Fx + c I could compute the different variables, but I don't know how to get the equation in the right format. Computing F can be done by F = ∇y, with ∇=∇x ; however, I don't know how to compute the translation vector c. Could anyone help me?

2. May 9, 2013

### Staff: Mentor

Let x = x1i1+x2i2+x3i3

y=y1i1+y2i2+y3i3

So, $$x\centerdot i_2=x_2$$

and γ(x · i2)i1=γx2i1

so
y1i1+y2i2+y3i3
=x1i1+x2i2+x3i3
+γx2i1
So now, resolve this equation into its 3 components (i.e., into the coefficients of the three unit vectors). Also, write out y = Fx + c in component form, and compare the expressions.

3. May 10, 2013

### Bertbos

Thank you very much. Sometimes I've trouble to write the equations is the right form. Your explanation was very clear and now it's easy to me to solve the problem!

4. May 10, 2013

### Staff: Mentor

There's an even simpler way of doing it that might appeal to you even more.

Write $$x=(i_1i_1+i_2i_2+i_3i_3)\centerdot x$$
and $$(x \centerdot i_2)i_1=i_1i_2\centerdot x$$

So $$y=x+γ(x \centerdot i_2)i_1=(i_1i_1+i_2i_2+i_3i_3)\centerdot x+γi_1i_2\centerdot x=(i_1i_1+γi_1i_2+i_2i_2+i_3i_3)\centerdot x$$

So, $F=i_1i_1+γi_1i_2+i_2i_2+i_3i_3$