Tensor derivative (continuum mechanics)

These derivatives are used in the theories of nonlinear elasticity and plasticity, particularly in the design of algorithms for numerical simulations.

It is assumed that the functions are sufficiently smooth that derivatives can be taken.

be a real valued function of the second order tensor

in the direction of an arbitrary constant vector c is defined as:

Since the basis vectors do not vary in a Cartesian coordinate system we have the following relations for the gradients of a scalar field

From this definition we have the following relations for the gradients of a scalar field

{\displaystyle {\begin{aligned}{\boldsymbol {\nabla }}\phi &={\frac {\partial \phi }{\partial \xi ^{i}}}~\mathbf {g} ^{i}\\{\boldsymbol {\nabla }}\mathbf {v} &={\frac {\partial \left(v^{j}\mathbf {g} _{j}\right)}{\partial \xi ^{i}}}\otimes \mathbf {g} ^{i}=\left({\frac {\partial v^{j}}{\partial \xi ^{i}}}+v^{k}~\Gamma _{ik}^{j}\right)~\mathbf {g} _{j}\otimes \mathbf {g} ^{i}=\left({\frac {\partial v_{j}}{\partial \xi ^{i}}}-v_{k}~\Gamma _{ij}^{k}\right)~\mathbf {g} ^{j}\otimes \mathbf {g} ^{i}\\{\boldsymbol {\nabla }}{\boldsymbol {S}}&={\frac {\partial \left(S_{jk}~\mathbf {g} ^{j}\otimes \mathbf {g} ^{k}\right)}{\partial \xi ^{i}}}\otimes \mathbf {g} ^{i}=\left({\frac {\partial S_{jk}}{\partial \xi _{i}}}-S_{lk}~\Gamma _{ij}^{l}-S_{jl}~\Gamma _{ik}^{l}\right)~\mathbf {g} ^{j}\otimes \mathbf {g} ^{k}\otimes \mathbf {g} ^{i}\end{aligned}}}

where tensor index notation for partial derivatives is used in the rightmost expressions.

For a symmetric second-order tensor, the divergence is also often written as[4]

Note that such a definition is not consistent with the rest of this article (see the section on curvilinear co-ordinates).

The difference stems from whether the differentiation is performed with respect to the rows or columns of

In a Cartesian coordinate system the second order tensor (matrix)

The last equation is equivalent to the alternative definition / interpretation[4]

Hence, using the definition of the curl of a first-order tensor field,

The most commonly used identity involving the curl of a tensor field,

This identity holds for tensor fields of all orders.

In that case, the right hand side corresponds the cofactors of the matrix.

Then, from the definition of the derivative of a scalar valued function of a tensor, we have

The determinant of a tensor can be expressed in the form of a characteristic equation in terms of the invariants

For the derivatives of the other two invariants, let us go back to the characteristic equation

In index notation with respect to an orthonormal basis

where the symmetric fourth order identity tensor is

In index notation with respect to an orthonormal basis

{\displaystyle {\frac {\partial A_{ij}^{-1}}{\partial A_{kl}}}=-{\cfrac {1}{2}}\left(A_{ik}^{-1}~A_{jl}^{-1}+A_{il}^{-1}~A_{jk}^{-1}\right)}

Another important operation related to tensor derivatives in continuum mechanics is integration by parts.

are differentiable tensor fields of arbitrary order,

is the unit outward normal to the domain over which the tensor fields are defined,

represents a generalized tensor product operator, and

is equal to the identity tensor, we get the divergence theorem

We can express the formula for integration by parts in Cartesian index notation as