Deviator

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

Deviators or deviatoric tensors ( latin deviators ) are in the continuum mechanics tensors second stage, the trace disappears. 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 figure on the right.

The strain tensors , which describe the elongation , compression and shear of material lines and surfaces in a body when deformed , are of particular importance . The distortion tensors have an index number called "trace" ( main invariant ), which is a measure of the volume expansion at the place of its occurrence in such a way that it disappears if there is no volume expansion. The non-marking part of the distortion tensor, its deviator , describes (in linearized theory) the volume-preserving, shape-changing part of the deformation of a body.

Another field of application of deviators is in the theory of plasticity . With many metals it can be observed that they do not flow plastically under all-round hydrostatic pressure or, in other words, the plastic flow is only driven by the tensions freed from the hydrostatic component. The deviatoric part of a tensor is just the part that remains when its hydrostatic part is subtracted.

With deviators, the material behavior can be modeled under volume-preserving, shape-changing conditions.

definition

Deviators are second order tensors whose trace "Sp" disappears: ${\displaystyle \mathbf {T} }$

${\displaystyle \mathbf {T} :\quad \mathrm {Sp} (\mathbf {T} )=0}$.

The deviatoric part is denoted by a superscript "D" or "dev":

${\displaystyle \mathrm {dev} (\mathbf {T} )=\mathbf {T} ^{\mathrm {D} }=\mathbf {T} ^{\mathrm {dev} }:=\mathbf {T} -{\dfrac {\mathrm {Sp} (\mathbf {T} )}{\mathrm {Sp} (\mathbf {I} )}}\mathbf {I} =\mathbf {T} -{\dfrac {\mathrm {Sp} (\mathbf {T} )}{3}}\mathbf {I} }$.

The trace of the unit tensor is equal to the dimension of the underlying space, here and in the following three. ${\displaystyle \mathbf {I} }$

The subtrahend

${\displaystyle {\dfrac {\mathrm {Sp} (\mathbf {T} )}{3}}\mathbf {I} =:\mathbf {T} ^{\mathrm {K} }}$

is the spherical fraction of the tensor . ${\displaystyle \mathbf {T} }$

Deviators and volume expansion

A cuboid is stretched slightly in the x, y and z directions. The linear proportions of the increase in volume in the elongations are marked in blue, olive or red.

When stretching a body length to length , the elongation is called the length ratio ${\displaystyle L}$ ${\displaystyle l}$ ${\displaystyle \varepsilon }$

${\displaystyle \varepsilon ={\frac {l-L}{L}}\quad \leftrightarrow \quad l=L(1+\varepsilon )}$.

Are defined. During the deformation of a parallelepiped of length , width and height in x, y and z-direction (and thus volume ) to give analog strains and in x-, y- and z-direction, see figure to the right. The volume of the cuboid after the deformation is then calculated ${\displaystyle L}$ ${\displaystyle B}$ ${\displaystyle H}$ ${\displaystyle V=LBH}$ ${\displaystyle \varepsilon _{x},\varepsilon _{y}}$ ${\displaystyle \varepsilon _{z}}$

${\displaystyle {\begin{array}{rcl}v&=&L(1+\varepsilon _{x})B(1+\varepsilon _{y})H(1+\varepsilon _{z})\\&=&LBH+LBH\varepsilon _{x}+LBH\varepsilon _{y}+LBH\varepsilon _{z}+{\mathcal {O}}(\varepsilon ^{2})\\&=&V(1+\varepsilon _{x}+\varepsilon _{y}+\varepsilon _{z})+{\mathcal {O}}(\varepsilon ^{2})\end{array}}}$

The Landau symbol stands for terms that are at least square in the expansion and which can be neglected for small expansion. The sum of the strains in the x, y and z directions is the trace of the linearized strain tensor ${\displaystyle {\mathcal {O}}(\varepsilon ^{2})}$ ${\displaystyle \mathrm {Sp} }$

${\displaystyle {\boldsymbol {\varepsilon }}={\begin{pmatrix}\varepsilon _{x}&\varepsilon _{xy}&\varepsilon _{xz}\\\varepsilon _{xy}&\varepsilon _{y}&\varepsilon _{yz}\\\varepsilon _{xy}&\varepsilon _{yz}&\varepsilon _{z}\end{pmatrix}}\rightarrow \mathrm {Sp} ({\boldsymbol {\varepsilon }})=\varepsilon _{x}+\varepsilon _{y}+\varepsilon _{z}}$

and therefore the volume expansion results : ${\displaystyle v=V(1+\mathrm {Sp} ({\boldsymbol {\varepsilon }}))}$

${\displaystyle {\frac {v-V}{V}}=\mathrm {Sp} ({\boldsymbol {\varepsilon }})}$.

The connection is found for large deformations

${\displaystyle \ln \left({\frac {v}{V}}\right)=\mathrm {Sp} (\mathbf {E} _{H})}$

between the natural logarithm of the volume ratio and the trace of the Henky distortion tensor .

If the traces of the strain tensors or, in the case of small or large deformations, disappear and they are deviatorial , there is no volume expansion at the place of their occurrence. Conversely, the deviators of these strain tensors describe the volume-maintaining, shape-changing part of the deformation and can thus be used to model the material behavior under these conditions. ${\displaystyle {\boldsymbol {\varepsilon }}}$ ${\displaystyle \mathbf {E} _{H}}$

Areas in the eigenvalue space

Areas in the eigenvalue space

Symmetrical second- order tensors are considered . These have three real eigenvalues and represent a point in the eigenvalue space (the space in which the eigenvalues ​​are plotted on the three coordinate axes). ${\displaystyle \lambda _{1,2,3}}$

With the equation

${\displaystyle \parallel \mathbf {T} ^{\mathrm {D} }\parallel =C}$

an area is defined in this three-dimensional eigenvalue space with an area parameter , see figure on the right. This surface has the shape of an (infinitely long) cylinder, which is parallel to the hydrostatic axis ${\displaystyle C}$

${\displaystyle \lambda _{1}=\lambda _{2}=\lambda _{3}}$

is aligned. Because of

${\displaystyle \mathrm {Sp} (\mathbf {T} )=\lambda _{1}+\lambda _{2}+\lambda _{3}}$

all deviators lie in the deviatoric plane

${\displaystyle \lambda _{1}+\lambda _{2}+\lambda _{3}=0}$,

whose normal is the hydrostatic axis. The normals to the surface lie in planes that are parallel to the deviatoric plane, which is why the normals are also deviatoric. This is also calculated from the derivation ${\displaystyle \mathbf {N} }$ ${\displaystyle \parallel \mathbf {T} ^{\mathrm {D} }\parallel =C}$

${\displaystyle \mathbf {N} ={\dfrac {\mathrm {d} }{\mathrm {d} \mathbf {T} }}\parallel \mathbf {T} ^{\mathrm {D} }\parallel ={\dfrac {\mathbf {T} ^{\mathrm {D} }}{\parallel \mathbf {T} ^{\mathrm {D} }\parallel }}}$,

because the normals correspond exactly to this derivation.

A surface of this type is the Fließortfläche in the J ₂ - plasticity theory

${\displaystyle \sigma _{v}:={\sqrt {\dfrac {3}{2}}}\parallel {\boldsymbol {\sigma }}^{\mathrm {D} }\parallel =k}$.

The area parameter is the isotropic hardening , the (symmetrical) stress tensor and the von Mises equivalent stress . In the uniaxial case is ${\displaystyle k}$ ${\displaystyle {\boldsymbol {\sigma }}}$ ${\displaystyle \sigma _{v}}$ ${\displaystyle {\boldsymbol {\sigma }}=\sigma {\hat {e}}_{1}\otimes {\hat {e}}_{1}}$

${\displaystyle \sigma _{v}={\sqrt {\dfrac {3}{2}}}\parallel {\boldsymbol {\sigma }}^{\mathrm {D} }\parallel =|\sigma |}$

and models the yield stress . ${\displaystyle k}$

The hydrostatic axis is formed by the unit tensor and the spherical tensor .

Deviatorial rate

A small deformation, for which the rate of the linearized strain tensor is deviatorial, is volume-conserving because its trace is a measure of the compression at the place of its occurrence. This also applies to large deformations if the covariant Oldroyd derivative of the Euler-Almansi strain tensor ${\displaystyle {\boldsymbol {\varepsilon }}}$

${\displaystyle {\stackrel {\Delta }{\mathbf {e} }}={\dfrac {1}{2}}(\mathbf {L} +\mathbf {L} ^{\mathrm {T} })\quad {\text{mit}}\quad \mathbf {L} ={\dot {\mathbf {F} }}\cdot \mathbf {F} ^{-1}}$

is deviatorial. This deformation is volume-preserving, because because of

${\displaystyle {\begin{array}{rclcl}\mathrm {Sp} ({\stackrel {\Delta }{\mathbf {e} }})&=&\mathrm {Sp} (\mathbf {L} )=\mathrm {Sp} ({\dot {\mathbf {F} }}\cdot \mathbf {F} ^{-1})=\mathbf {F} ^{\mathrm {T} -1}:{\dot {\mathbf {F} }}&=&0\\\rightarrow \mathrm {det} (\mathbf {F} )\mathrm {Sp} ({\stackrel {\Delta }{\mathbf {e} }})&=&\mathrm {det} (\mathbf {F} )\mathbf {F} ^{\mathrm {T} -1}:{\dot {\mathbf {F} }}={\dfrac {\mathrm {d} }{\mathrm {d} t}}\mathrm {det} (\mathbf {F} )&=&0\end{array}}}$

the time derivative of the determinant "det" of the deformation gradient disappears . The determinant of the deformation gradient is equal to the volume expansion, which in this case is constant over time. ${\displaystyle \mathbf {F} }$

In J ₂ plasticity, in which the rate of plastic strains is deviatorial and the plastic strains are of the Euler-Almansi type, this has the effect that the plastic strains are volume-preserving, which is referred to as plastic incompressibility .

Invariants of deviators

The three main invariants of a deviator are ${\displaystyle \mathrm {I} _{1,2,3}}$

${\displaystyle {\begin{array}{rclcl}\mathrm {I} _{1}(\mathbf {T} ^{\mathrm {D} })&=&\mathrm {Sp} (\mathbf {T} ^{\mathrm {D} })&=&0\\\mathrm {I} _{2}(\mathbf {T} ^{\mathrm {D} })&=&{\dfrac {1}{2}}[\mathrm {Sp} {(\mathbf {T} ^{\mathrm {D} })}^{2}-\mathrm {Sp} (\mathbf {T} ^{\mathrm {D} }\cdot \mathbf {T} ^{\mathrm {D} })]&=&-{\dfrac {1}{2}}\mathrm {Sp} (\mathbf {T} ^{\mathrm {D} }\cdot \mathbf {T} ^{\mathrm {D} })\\\mathrm {I} _{3}(\mathbf {T} ^{\mathrm {D} })&=&\operatorname {det} (\mathbf {T} ^{\mathrm {D} })&=&{\dfrac {1}{3}}\operatorname {Sp} (\mathbf {T} ^{\mathrm {D} }\cdot \mathbf {T} ^{\mathrm {D} }\cdot \mathbf {T} ^{\mathrm {D} })\end{array}}}$.

The operator gives the determinant of its argument. The amount or Frobenius norm of a deviator is calculated using the Frobenius scalar product ":" ${\displaystyle \operatorname {det} }$

${\displaystyle \parallel \mathbf {T} ^{\mathrm {D} }\parallel ={\sqrt {\mathbf {T} ^{\mathrm {D} }:\mathbf {T} ^{\mathrm {D} }}}={\sqrt {\mathbf {T} :\mathbf {T} -{\dfrac {1}{3}}\mathrm {Sp} (\mathbf {T} )^{2}}}}$,

from what

${\displaystyle \parallel \mathbf {T} \parallel ^{2}=\parallel \mathbf {T} ^{\mathrm {D} }\parallel ^{2}+{\begin{Vmatrix}{\dfrac {{\text{Sp}}(\mathbf {T} )}{3}}\mathbf {I} \end{Vmatrix}}^{2}}$

follows. So three lines with the lengths of the magnitudes of a tensor, its deviator and its spherical part form a right triangle.

In the case of a symmetric tensor , its deviator is also symmetric and for its Frobenius norm we get: ${\displaystyle \mathbf {T} =\mathbf {T} ^{\mathrm {T} }}$

${\displaystyle \parallel \mathbf {T} ^{\mathrm {D} }\parallel :={\sqrt {\mathbf {T} ^{\mathrm {D} }:\mathbf {T} ^{\mathrm {D} }}}={\sqrt {-2\mathrm {I} _{2}(\mathbf {T} ^{\mathrm {D} })}}}$.

If the tensor has the representation

${\displaystyle \mathbf {T} =\sum _{i,j=1}^{3}T_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j}}$

with components with respect to the standard basis of the Euclidean vector space , then compute ${\displaystyle T_{ij}\in \mathbb {R} }$ ${\displaystyle {\hat {e}}_{1,2,3}}$ ${\displaystyle \mathbb {V} ^{3}}$

${\displaystyle {\begin{array}{lcl}\mathbf {T} ^{\mathrm {D} }&=&{\begin{pmatrix}{\frac {2}{3}}T_{11}-{\frac {1}{3}}T_{22}-{\frac {1}{3}}T_{33}&T_{12}&T_{13}\\T_{21}&{\frac {2}{3}}T_{22}-{\frac {1}{3}}T_{11}-{\frac {1}{3}}T_{33}&T_{23}\\T_{31}&T_{32}&{\frac {2}{3}}T_{33}-{\frac {1}{3}}T_{11}-{\frac {1}{3}}T_{22}\end{pmatrix}}\\\mathrm {I} _{2}(\mathbf {T} ^{\mathrm {D} })&=&{\dfrac {1}{3}}(T_{11}T_{22}+T_{11}T_{33}+T_{22}T_{33}-T_{11}^{2}-T_{22}^{2}-T_{33}^{2})-T_{12}T_{21}-T_{13}T_{31}-T_{23}T_{32}\\\operatorname {det} (\mathbf {T} ^{\mathrm {D} })&=&{\dfrac {1}{27}}[12T_{11}T_{22}T_{33}+2(T_{11}^{3}+T_{22}^{3}+T_{33}^{3})-3T_{11}^{2}(T_{22}+T_{33})-3T_{22}^{2}(T_{11}+T_{33})-3T_{33}^{2}(T_{11}+T_{22})]\\&&-{\dfrac {1}{3}}[(2T_{11}-T_{22}-T_{33})T_{23}T_{32}+(2T_{33}-T_{11}-T_{22})T_{12}T_{21}+(2T_{22}-T_{11}-T_{33})T_{13}T_{31}]\\&&+T_{13}T_{32}T_{21}+T_{12}T_{23}T_{31}\\\parallel \mathbf {T} ^{\mathrm {D} }\parallel &=&{\sqrt {{\dfrac {2}{3}}(T_{11}^{2}+T_{22}^{2}+T_{33}^{2}-T_{11}T_{22}-T_{11}T_{33}-T_{22}T_{33})+T_{12}^{2}+T_{21}^{2}+T_{13}^{2}+T_{31}^{2}+T_{23}^{2}+T_{32}^{2}}}\end{array}}}$

See also

• Voltage deviator
• Formula collection tensor algebra
• Tensoranalysis formula collection
• Gâteaux differential

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.
2. ↑ ^{a b} The second main invariant of the stress deviator is often referred to as J ₂ :

   ${\displaystyle {\mathrm {J} }_{2}:=\mathrm {I} _{2}({\boldsymbol {\sigma }}^{\mathrm {D} })}$

   and is because of

   ${\displaystyle \parallel \mathbf {T} ^{\mathrm {D} }\parallel ={\sqrt {-2\mathrm {I} _{2}(\mathbf {T} ^{\mathrm {D} })}}}$

   a measure of the amount of the voltage deviator.

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 .