Quiz Strain Measures

Question 1

You can solve this question using the online tool.
A 2D displacement function of a unit square whose centroid is situated at the origin has the form:

x = (\begin{matrix} x_{1} \\ x_{2} \end{matrix}) = (\begin{matrix} 1.1 & 0.1 \\ 0.1 & 1.1 \end{matrix}) (\begin{matrix} X_{1} \\ X_{2} \end{matrix})

sketch the deformed shape by hand and calculate the small strain matrix.
Calculate the small strains along the directions

a = (1, 0)

and

b = (c o s 45^{\circ}, s i n 45^{\circ})

Solution:
Here is the solution with $d X = a$ and $d Y = b$

Question 2

You can use the oneline tool.
Find displacement functions that would satisfy each of the following:

\cdot

Vectors aligned with

e_{1}

are in compression, while the shear strain and

Î µ_{22}

are zero.

\cdot

Vectors aligned with

e_{2}

are in tension, while the shear strain and

Î µ_{11}

are zero.

\cdot

Î µ_{12}

is positive while all other strain components are zero.

Question 3

You can use the first tool under this link.

The following displacement function represents a clockwise rotation of around 11.6 degrees. Rotation should be accompanied by zero strains, however, the small strain matrix, erroneously, predicts some tensile and compressive strains. Calculate the strains associated with this rotation:

x = (\begin{matrix} x_{1} \\ x_{2} \end{matrix}) = (\begin{matrix} 0.98 & 0.2 \\ - 0.2 & 0.98 \end{matrix}) (\begin{matrix} X_{1} \\ X_{2} \end{matrix})

Question 4

The following displacement function is accompanied by shear strain. Find an orientation for the coordinate system where the shear strain is equal to zero:

x_{1} = 0.8 X_{1} + 0.1 X_{2}, x_{2} = 0.1 X_{1} + 0.9 X_{2}

The second tool under this link will perform the calculations.

Here are the steps:
1) Calculate the small strain matrix.
2) Calculate its eigenvalues and eigenvectors.
3) Use the eigenvectors as a new coordinate system.
4) Evaluate the strain matrix in the new coordinate system. In the new coordinate system, the strain matrix will be diagonal with the diagonal components being its eigenvalues!
Note: The eigenvectors are called the "principal directions of the strain matrix" while the eigenvalues are called: "principal strains".

Question 5

You can use the tool under this link to perform these calculations.
Compare the small strain matrix and the Green strain matrix in the following two rigid body rotations:

- A pure rotation of 11.6 Degrees.

x = (\begin{matrix} c o s {11.6}^{\circ} & s i n {11.6}^{\circ} \\ - s i n {11.6}^{\circ} & c o s {11.6}^{\circ} \end{matrix}) (\begin{matrix} X_{1} \\ X_{2} \end{matrix}) \approx (\begin{matrix} 0.98 & 0.2 \\ - 0.2 & 0.98 \end{matrix}) (\begin{matrix} X_{1} \\ X_{2} \end{matrix})

- A pure rotation of 30 Degrees.

x = (\begin{matrix} c o s 30^{\circ} & s i n 30^{\circ} \\ - s i n 30^{\circ} & c o s 30^{\circ} \end{matrix}) (\begin{matrix} X_{1} \\ X_{2} \end{matrix}) \approx (\begin{matrix} 0.87 & 0.5 \\ - 0.5 & 0.87 \end{matrix}) (\begin{matrix} X_{1} \\ X_{2} \end{matrix})