Deformation gradient

The deformation gradient is the derivative of the deformed coordinates with respect to the undeformed coordinates.

\[F = \frac{\partial (X + U)}{\partial X} = I + \frac{\partial U}{\partial X}\]

Note

In the 2D case, a plane strain state is assumed. This implies \(F_{zz}=1\).

>>> import numpy as np
>>> import sympy as sy
>>> import conforce_gen.expressions as expr

Undeformed

The deformation gradient of the undeformed state in 2D is the identity matrix.

>>> F_undeformed = expr.eval_F(2, sy.ZeroMatrix(2, 2))
>>> F_undeformed
Matrix([
[1, 0],
[0, 1]])

In order to integrate over a volume, the volume change is required. The change in volume is computed using the determinant of the deformation gradient, which is

>>> sy.det(F_undeformed)
1

for the undeformed state. So the volume does not change.

Deformed

However, for a deformation of \(\varepsilon_{xx}=0.2; \gamma_{xy}=0.1\), the deformation gradient is

>>> F_deformed = expr.eval_F(2, sy.Matrix([[0.2, 0.1], [0, 0]]))
>>> F_deformed
Matrix([
[1.2, 0.1],
[  0,   1]])

and its determinat

>>> float(sy.det(F_deformed))
1.2

states that the volume increases by a factor of 1.2. Note, that the tensile strain \(\varepsilon_{xx}=0.2\) results in a volume change. The simple shear \(\gamma_{xy}=0.1\) does not contribute to the volume change.

Pure shear

However, a pure shear deformation of \(\gamma_{xy}=0.1\) without a tensile strain would decrease the volume by a factor of

>>> float(sy.det(expr.eval_F(2, sy.Matrix([[0.0, 0.1], [0.1, 0]]))))
0.99

Transformation

The deformation gradient is a linear transformation from the undeformed to the deformation state. Considering a simple shear deformation,

>>> F = expr.eval_F(2, sy.Matrix([[0.0, 0.1], [0.0, 0]]))
>>> F
Matrix([
[1, 0.1],
[0,   1]])

the points of the undeformed unit square can be transformed to the deformed state by a vector matrix multiplication. The “@”-symbol stands for a matrix multiplication.

>>> X_unit_square = np.array([
...     [0, 0],
...     [0, 1],
...     [1, 1],
...     [1, 0],
... ])
>>> X_deformed_unit_square = X_unit_square @ F.T
>>> X_deformed_unit_square
Matrix([
[  0, 0],
[0.1, 1],
[1.1, 1],
[  1, 0]])

For the back transformation from the deformed to the undeformed state, the inverse of the deformation gradient is used.

>>> np.array(X_deformed_unit_square @ F.inv().T, dtype=float)
array([[0., 0.],
       [0., 1.],
       [1., 1.],
       [1., 0.]])