Isotropic function

Linear mapping of a vector by a tensor . ${\displaystyle {\vec {v}}}$ ${\displaystyle \mathbf {T} }$

In continuum mechanics, an isotropic function is a function that is dependent on one or more scalars , geometric vectors or tensors , the value of which is transformed in the same way as its arguments when its arguments are rotated . Second-level tensors are used here as linear mapping of geometric vectors onto geometric vectors, which are generally rotated and stretched in the process, see the figure on the right. The tensors consist of dyads of two geometric vectors and are rotated by rotating both vectors in the dyad in the same way. An isotropic function follows this rotation of its arguments.

Isotropic functions play an important role in defining properties of isotropic materials, e.g. B. in hyperelasticity .

definition

The three-dimensional Euclidean vector space , the vector space of the linear tensorial mappings of this space and the special orthogonal group are given ${\displaystyle \mathbb {V} }$ ${\displaystyle {\mathcal {L}}(\mathbb {V} ,\mathbb {V} )}$

${\displaystyle {\mathcal {SO}}=\{\mathbf {Q} \in {\mathcal {L}}|\mathbf {Q} ^{-1}=\mathbf {Q} ^{\mathrm {T} }\;\wedge \;\det(\mathbf {Q} )=+1\}}$

the actually orthogonal tensors , which embody pure rotations without reflections. Then the transformation equations apply for a rotation

size	Transformed size
Scalar ${\displaystyle y\in \mathbb {R} }$	${\displaystyle y'=y}$
vector ${\displaystyle {\vec {v}}\in \mathbb {V} }$	${\displaystyle {\vec {v}}'=\mathbf {Q} \cdot {\vec {v}}}$
Tensor ${\displaystyle \mathbf {T} \in {\mathcal {L}}}$	${\displaystyle \mathbf {T} '=\mathbf {Q\cdot T\cdot Q} ^{\mathrm {T} }}$

Scalar function

A scalar function of real, vector or tensor valued arguments is isotropic if for every orthogonal tensor from the special orthogonal group:

${\displaystyle {\begin{array}{l}f(y_{1},y_{2},\ldots ,{\vec {v}}_{1},{\vec {v}}_{2},\ldots ,\mathbf {T} _{1},\mathbf {T} _{2},\dots )=\ldots \\\ldots =f(y_{1},y_{2},\ldots ,\mathbf {Q} \cdot {\vec {v}}_{1},\mathbf {Q} \cdot {\vec {v}}_{2},\ldots ,\mathbf {Q\cdot T} _{1}\cdot \mathbf {Q} ^{\mathrm {T} },\mathbf {Q\cdot T} _{2}\cdot \mathbf {Q} ^{\mathrm {T} },\ldots )\quad \forall \;\mathbf {Q} \in {\mathcal {SO}}\\{\textsf {mit}}\quad y_{1},y_{2},\ldots \in \mathbb {R} \,,\quad {\vec {v}}_{1},{\vec {v}}_{2},\ldots \in \mathbb {V} \,,\quad \mathbf {T} _{1},\mathbf {T} _{2},\ldots \in {\mathcal {L}}\end{array}}}$

Tensor-valued function or tensor function

A tensor function of tensors is isotropic if for every orthogonal tensor from the special orthogonal group:

${\displaystyle {\begin{array}{l}\mathbf {f} (\mathbf {Q\cdot T} _{1}\cdot \mathbf {Q} ^{\mathrm {T} },\mathbf {Q\cdot T} _{2}\cdot \mathbf {Q} ^{\mathrm {T} },\ldots )=\mathbf {Q} \cdot \mathbf {f} (\mathbf {T} _{1},\mathbf {T} _{2},\dots )\cdot \mathbf {Q} ^{\mathrm {T} }\quad \forall \;\mathbf {Q} \in {\mathcal {SO}}\\{\textsf {mit}}\quad \mathbf {T} _{1},\mathbf {T} _{2},\ldots \in {\mathcal {L}}\end{array}}}$

Examples

Scalar functions

All major invariants and other invariants of the tensors are by definition isotropic functions of their tensor, for example:

${\displaystyle \operatorname {Sp} (\mathbf {Q\cdot T\cdot Q} ^{\mathrm {T} })=\operatorname {Sp} (\mathbf {Q} ^{\mathrm {T} }\cdot \mathbf {Q\cdot T} )=\operatorname {Sp} (\mathbf {T} )}$.

Tensor functions

The derivatives of the invariants according to their tensor are isotropic tensor functions, for example:

${\displaystyle {\begin{array}{rcl}{\dfrac {\mathrm {d} \operatorname {I} _{2}(\mathbf {T} )}{\mathrm {d} \mathbf {T} }}&=&\operatorname {Sp} (\mathbf {T} )\mathbf {I} -\mathbf {T} ^{\mathrm {T} }\\\rightarrow {\dfrac {\mathrm {d} \operatorname {I} _{2}}{\mathrm {d} \mathbf {T} }}(\mathbf {Q\cdot T\cdot Q} ^{\mathrm {T} })&=&\operatorname {Sp} (\mathbf {Q\cdot T\cdot Q} ^{\mathrm {T} })\mathbf {I} -(\mathbf {Q\cdot T\cdot Q} ^{\mathrm {T} })^{\mathrm {T} }\\&=&\operatorname {Sp} (\mathbf {T} )\mathbf {Q\cdot I\cdot Q} ^{\mathrm {T} }-\mathbf {Q\cdot T} ^{\mathrm {T} }\cdot \mathbf {Q} ^{\mathrm {T} }\\&=&\mathbf {Q} \cdot (\operatorname {Sp} (\mathbf {T} )\mathbf {I} -\mathbf {T} ^{\mathrm {T} })\cdot \mathbf {Q} ^{\mathrm {T} }\\&=&\mathbf {Q} \cdot {\dfrac {\mathrm {d} \operatorname {I} _{2}(\mathbf {T} )}{\mathrm {d} \mathbf {T} }}\cdot \mathbf {Q} ^{\mathrm {T} }\end{array}}}$

A polynomial of a tensor-valued variable with constant real coefficients

${\displaystyle \mathbf {f} (\mathbf {T} )=a_{0}\mathbf {I} +\sum _{n=1}^{N}a_{n}\underbrace {\mathbf {T\cdot T\ldots \cdot T} } _{\textsf {n-mal}}}$

is an isotropic tensor function, because

${\displaystyle {\begin{array}{rcl}\mathbf {f} (\mathbf {Q\cdot T\cdot Q} ^{\mathrm {T} })&=&\displaystyle a_{0}\mathbf {I} +\sum _{n=1}^{N}a_{n}\underbrace {(\mathbf {Q\cdot T\cdot Q} ^{\mathrm {T} })\cdot (\mathbf {Q\cdot T\cdot Q} ^{\mathrm {T} })\cdot \ldots \cdot (\mathbf {Q\cdot T\cdot Q} ^{\mathrm {T} })} _{\textsf {n-mal}}\\&=&\displaystyle a_{0}\mathbf {Q\cdot I\cdot Q} ^{\mathrm {T} }+\sum _{n=1}^{N}a_{n}\mathbf {Q} \cdot \underbrace {\mathbf {T\cdot T\cdot \ldots \cdot T} } _{\textsf {n-mal}}\cdot \mathbf {Q} ^{\mathrm {T} }\\&=&\displaystyle \mathbf {Q} \cdot \left(a_{0}\mathbf {I} +\sum _{n=1}^{N}a_{n}\underbrace {\mathbf {T\cdot T\ldots \cdot T} } _{\textsf {n-mal}}\right)\cdot \mathbf {Q} ^{\mathrm {T} }\\&=&\mathbf {Q} \cdot \mathbf {f} (\mathbf {T} )\cdot \mathbf {Q} ^{\mathrm {T} }\end{array}}}$

Isotropic tensor functions of a symmetric argument

The stress , strain and strain sensors play an outstanding role in the formulation of material models in continuum mechanics and are symmetrical . If the arguments of an isotropic tensor function are symmetrical, then this function has special and important properties.

Eigensystem

The eigenvectors of an isotropic tensor function of a symmetrical tensor agree with those of the tensor. If so

${\displaystyle \mathbf {T} \cdot {\vec {v}}=\lambda {\vec {v}}}$

holds, then is

${\displaystyle \mathbf {f} (\mathbf {T} )\cdot {\vec {v}}=\eta {\vec {v}}}$,

ie the eigenvectors agree, but not so - in general - the eigenvalues. This is one of the starting points for the following display set.

Representation set

Every isotropic tensor function of a symmetric argument can be expressed in the form

${\displaystyle \mathbf {f} (\mathbf {T} )=\phi _{0}(\operatorname {I} _{1},\operatorname {I} _{2},\operatorname {I} _{3})\mathbf {I} +\phi _{1}(\operatorname {I} _{1},\operatorname {I} _{2},\operatorname {I} _{3})\mathbf {T} +\phi _{2}(\operatorname {I} _{1},\operatorname {I} _{2},\operatorname {I} _{3})\mathbf {T\cdot T} }$

reproduce. Therein are scalar functions of the main invariants of the tensor. According to Cayley-Hamilton's theorem, can be synonymous ${\displaystyle \phi _{0,1,2}}$ ${\displaystyle \operatorname {I} _{1,2,3}(\mathbf {T} )}$

${\displaystyle \mathbf {f} (\mathbf {T} )=\psi _{0}(\operatorname {I} _{1},\operatorname {I} _{2},\operatorname {I} _{3})\mathbf {I} +\psi _{1}(\operatorname {I} _{1},\operatorname {I} _{2},\operatorname {I} _{3})\mathbf {T} +\psi _{-1}(\operatorname {I} _{1},\operatorname {I} _{2},\operatorname {I} _{3})\mathbf {T} ^{-1}}$

are written with other scalar functions of the main invariants. ${\displaystyle \psi _{-1,0,1}}$

Commutativity

In the tensor product of an isotropic tensor function of a symmetric tensor with its argument, the order of the factors can be reversed:

${\displaystyle \mathbf {f} (\mathbf {T} )\cdot \mathbf {T} =\mathbf {T} \cdot \mathbf {f} (\mathbf {T} )}$,

which is a direct consequence of the above notation.

Footnotes

1. ↑ The Fréchet derivative of a scalar function with respect to a tensor is the tensor for which - if it exists - applies: ${\displaystyle f(\mathbf {T} )}$ ${\displaystyle \mathbf {T} }$ ${\displaystyle \mathbf {A} }$

   ${\displaystyle \mathbf {A} :\mathbf {H} =\left.{\frac {\mathrm {d} }{\mathrm {d} s}}f(\mathbf {T} +s\mathbf {H} )\right|_{s=0}=\lim _{s\rightarrow 0}{\frac {f(\mathbf {T} +s\mathbf {H} )-f(\mathbf {T} )}{s}}\quad \forall \;\mathbf {H} }$

   There is and ":" the Frobenius scalar product . Then will too ${\displaystyle s\in \mathbb {R} }$

   ${\displaystyle {\frac {\partial f}{\partial \mathbf {T} }}=\mathbf {A} }$

   written.

literature

• H. Altenbach: Continuum Mechanics . Springer, 2012, ISBN 978-3-642-24118-5 .
• P. Haupt: Continuum Mechanics and Theory of Materials . Springer, 2000, ISBN 3-540-66114-X .