Tensor derivative (continuum mechanics)

The derivatives of scalars, vectors, and second-order tensors with respect to second-order tensors are of considerable use in continuum mechanics. These derivatives are used in the theories of nonlinear elasticity and plasticity, particularly in the design of algorithms for numerical simulations. J. C. SImo and T. J. R. Hughes, 1998, Computational Inelasticity, Springer]

The directional derivative provides a systematic way of finding these derivatives. J. E. Marsden and T. J. R. Hughes, 2000, Mathematical Foundations of Elasticity, Dover.]

Derivatives with respect to vectors and second-order tensors

The definitions of directional derivatives for various situations are given below. It is assumed that the functions are sufficiently smooth that derivatives can be taken.

Derivatives of vector valued functions of vectors

Let $mathbf\{f\}(mathbf\{v\})$ be a vector valued function of the vector $mathbf\{v\}$. Then the derivative of $mathbf\{f\}(mathbf\{v\})$ with respect to $mathbf\{v\}$ (or at $mathbf\{v\}$) in the direction $mathbf\{u\}$ is the second order tensor defined as:$frac\{partial\; mathbf\{f\{partial\; mathbf\{vcdotmathbf\{u\}\; =\; Dmathbf\{f\}(mathbf\{v\})\; [mathbf\{u\}]\; =\; left\; [frac\{d\; \}\{d\; alpha\}~mathbf\{f\}(mathbf\{v\}\; +\; alpha~mathbf\{u\})\; ight]\; \_\{alpha\; =\; 0\}$for all vectors $mathbf\{u\}$.

"Properties:"

1) If $mathbf\{f\}(mathbf\{v\})\; =\; mathbf\{f\}\_1(mathbf\{v\})\; +\; mathbf\{f\}\_2(mathbf\{v\})$ then $frac\{partial\; mathbf\{f\{partial\; mathbf\{vcdotmathbf\{u\}\; =\; left(frac\{partial\; mathbf\{f\}\_1\}\{partial\; mathbf\{v\; +\; frac\{partial\; mathbf\{f\}\_2\}\{partial\; mathbf\{v\; ight)cdotmathbf\{u\}$

2) If $mathbf\{f\}(mathbf\{v\})\; =\; mathbf\{f\}\_1(mathbf\{v\})\; imesmathbf\{f\}\_2(mathbf\{v\})$ then $frac\{partial\; mathbf\{f\{partial\; mathbf\{vcdotmathbf\{u\}\; =\; left(frac\{partial\; mathbf\{f\}\_1\}\{partial\; mathbf\{vcdotmathbf\{u\}\; ight)\; imesmathbf\{f\}\_2(mathbf\{v\})\; +\; mathbf\{f\}\_1(mathbf\{v\})\; imesleft(frac\{partial\; mathbf\{f\}\_2\}\{partial\; mathbf\{vcdotmathbf\{u\}\; ight)$

3) If $mathbf\{f\}(mathbf\{v\})\; =\; mathbf\{f\}\_1(mathbf\{f\}\_2(mathbf\{v\}))$ then $frac\{partial\; mathbf\{f\{partial\; mathbf\{vcdotmathbf\{u\}\; =\; frac\{partial\; mathbf\{f\}\_1\}\{partial\; mathbf\{f\}\_2\}cdotleft(frac\{partial\; mathbf\{f\}\_2\}\{partial\; mathbf\{vcdotmathbf\{u\}\; ight)$

Derivatives of scalar valued functions of second-order tensors

Let be a real valued function of the second order tensor . Then the derivative of with respect to (or at ) in the direction is the second order tensor defined as:for all second order tensors .

"Properties:"

1) If then

2) If then

3) If then

Derivatives of tensor valued functions of second-order tensors

Let be a second order tensor valued function of the second order tensor . Then the derivative of with respect to (or at ) in the direction is the fourth order tensor defined as:for all second order tensors .

"Properties:"

1) If then

2) If then

3) If then

4) If then

Derivative of the determinant of a tensor

The derivative of the determinant of a second order tensor is given by:In an orthonormal basis, the components of can be written asa matrix $mathbf\{A\}$. In that case, the right hand side corresponds the cofactors of the matrix.

:

References

See also

* Tensor derivative
* Directional derivative