Strain tensor

Distortion tensors are dimensionless second-level tensors that describe the relationship between the instantaneous configuration and the initial configuration during the deformation of continuous bodies and thus the change in the mutual positional relationships of the material elements. This change ( deformation ) of the internal arrangement corresponds to a change in the external shape of the solid and is thus visible , for example, as expansion , compression , shear , etc. The strain tensors are an essential parameter in the description of the kinematics of deformation. In continuum mechanics a number of different strain tensors are defined, the naming of which is not uniform.

The strain tensors are mainly used for the formulation of material models, e.g. B. the hyperelasticity used, which establish a relationship between the stresses in the material and its deformations. Such material models are used to calculate deformations of bodies.

introduction

A large number of strain tensors which are formed from the deformation gradient are known in the literature . The displacements

${\displaystyle {\vec {u}}({\vec {X}},t)={\vec {\chi }}({\vec {X}},t)-{\vec {X}}=\sum _{i=1}^{3}u_{i}{\vec {e}}_{i}={\begin{pmatrix}u({\vec {X}},t)\\v({\vec {X}},t)\\w({\vec {X}},t)\end{pmatrix}}}$

introduced as a difference vector between the current position of a particle and its starting position , with as the material coordinates of the particle with respect to the standard basis . The displacement gradient ${\displaystyle {\vec {x}}={\vec {\chi }}({\vec {X}},t)}$ ${\displaystyle {\vec {X}}=\sum _{i=1}^{3}X_{i}{\vec {e}}_{i}}$ ${\displaystyle X_{i}}$

${\displaystyle \mathbf {H} :=\operatorname {GRAD} ({\vec {u}}):=\sum _{i,j=1}^{3}{\frac {\mathrm {d} u_{i}}{\mathrm {d} X_{j}}}{\vec {e}}_{i}\otimes {\vec {e}}_{j}={\frac {\mathrm {d} {\vec {u}}}{\mathrm {d} {\vec {X}}}}}$

is then the derivative of the displacement vector according to the material coordinates and contains the derivatives of the displacements u _i according to the coordinates X _j . This gives the deformation gradient its shape ${\displaystyle {\vec {u}}}$ ${\displaystyle {\vec {X}}}$

${\displaystyle \mathbf {F} :={\frac {\mathrm {d} {\vec {\chi }}}{\mathrm {d} {\vec {X}}}}={\frac {\mathrm {d} }{\mathrm {d} {\vec {X}}}}({\vec {u}}+{\vec {X}})=\mathbf {H} +\mathbf {1} }$

where 1 is the unit tensor . First of all we can use it to calculate the right Cauchy-Green tensor

${\displaystyle \mathbf {C} :=\mathbf {F^{\top }\cdot F} }$

regarding the initial configuration and the left Cauchy-Green tensor

${\displaystyle \mathbf {b} :=\mathbf {F\cdot F^{\top }} }$

with respect to the current configuration. These two distance sensors are symmetrical and, in the case of a non-deformation, they are equal to the unit tensor.

For engineering applications, however, quantities are usually desired which represent a zero in the event of non-deformation . This leads to definitions of the Green-Lagrange strain tensor

${\displaystyle \mathbf {E} :={\frac {1}{2}}(\mathbf {F^{\top }\cdot F-1} )}$

or the Euler-Almansi strain tensor

${\displaystyle \mathbf {e} :={\frac {1}{2}}(\mathbf {1-(F\cdot F^{\top })} ^{-1})\,.}$

In addition, there is a large number of other similar definitions, each of which has its justification and advantages in different theories, see below. This explains the factor ½ that occurs above.

Linearized strain tensor

To describe small strains is in the engineering mechanics usually the linearized strain tensor used. This strain tensor is also called engineering strain , because in many technical applications there are small strains or they have to be kept small for safety reasons. The linearized strain tensor is created by linearizing the quantities or for this purpose, the definition of the deformation gradient is inserted into the strain tensor: ${\displaystyle {\boldsymbol {\varepsilon }}}$ ${\displaystyle \mathbf {E} }$ ${\displaystyle \mathbf {e} \,.}$

${\displaystyle \mathbf {E} ={\frac {1}{2}}{\Bigl [}\mathbf {F^{\top }\cdot F} -\mathbf {1} {\Bigr ]}={\frac {1}{2}}{\Bigl [}{\Bigl (}\mathbf {H+1} {\Bigr )}^{\top }\cdot {\Bigl (}\mathbf {H} +\mathbf {1} {\Bigr )}-\mathbf {1} {\Bigr ]}={\frac {1}{2}}{\Bigl [}\mathbf {H+H^{\top }} +\mathbf {H^{\top }\cdot H} {\Bigr ]}\,.}$

In the case of small distortions, the last term can be neglected and this creates the linearized distortion tensor

${\displaystyle {\boldsymbol {\varepsilon }}:={\frac {1}{2}}{\Bigl [}\mathbf {H+H^{\top }} {\Bigr ]}={\begin{pmatrix}\varepsilon _{xx}&\varepsilon _{xy}&\varepsilon _{xz}\\\varepsilon _{yx}&\varepsilon _{yy}&\varepsilon _{yz}\\\varepsilon _{zx}&\varepsilon _{zy}&\varepsilon _{zz}\end{pmatrix}},}$

with the components

${\displaystyle {\begin{matrix}&\varepsilon _{xx}={\frac {\partial u_{x}}{\partial X_{x}}},\;\varepsilon _{yy}={\frac {\partial u_{y}}{\partial X_{y}}},\;\varepsilon _{zz}={\frac {\partial u_{z}}{\partial X_{z}}},\;\varepsilon _{xy}=\varepsilon _{yx}={\frac {1}{2}}\left({\frac {\partial u_{x}}{\partial X_{y}}}+{\frac {\partial u_{y}}{\partial X_{x}}}\right),\\&\varepsilon _{yz}=\varepsilon _{zy}={\frac {1}{2}}\left({\frac {\partial u_{y}}{\partial X_{z}}}+{\frac {\partial u_{z}}{\partial X_{y}}}\right),\;\varepsilon _{zx}=\varepsilon _{xz}={\frac {1}{2}}\left({\frac {\partial u_{z}}{\partial X_{x}}}+{\frac {\partial u_{x}}{\partial X_{z}}}\right).\end{matrix}}}$

general definition

A tensor E is a suitable distortion measure if it fulfills three requirements:

1. E disappears with rigid body movements (displacement and / or rotation without changing shape)
2. E is a monotonic, continuous and continuously differentiable function of the displacement gradient H and
3. In the case of small distortions, E goes over to the linearized strain tensor ε .

The polar decomposition of the deformation gradient F = R · U = v · R cleaves the deformation locally in a pure rotation, mediated by the orthogonal rotation tensor R (with R · R ^T and the determinant det ( R ) = 1), and a pure stretch, mediated by the symmetrical positive definite right and left distance sensors U and v, respectively . The latter are used to define a large number of strain tensors.

In its natural representation in convective coordinates , the right stretch tensor U is covariant and the left stretch tensor v is contravariant. This property is carried over to the strain tensors formed with them. By inversion, covariant tensors become contravariant and vice versa.

Seth-Hill family of strain tensors

The strain tensors

${\displaystyle \mathbf {E} _{(m)}={\frac {1}{2m}}(\mathbf {U} ^{2m}-\mathbf {1} )={\frac {1}{2m}}(\mathbf {C} ^{m}-\mathbf {1} )}$

and

${\displaystyle \mathbf {e} _{(m)}={\frac {1}{2m}}(\mathbf {v} ^{2m}-\mathbf {1} )={\frac {1}{2m}}(\mathbf {b} ^{m}-\mathbf {1} )\,,}$

which result for different values ​​of the parameter satisfy the conditions of the general definition. The following table lists some common values ​​of corresponding tensors: ${\displaystyle m}$ ${\displaystyle m}$

m	Strain tensor	${\displaystyle \nabla \Delta }$	Names
1	${\displaystyle \mathbf {E} _{(1)}={\frac {1}{2}}(\mathbf {U} ^{2}-\mathbf {1} )={\frac {1}{2}}(\mathbf {C-1} )}$	${\displaystyle \Delta }$	Green-Lagrange strain tensor , Green or St. Venant strains
½	${\displaystyle \mathbf {E} _{(1/2)}=\mathbf {U} -\mathbf {1} ={\sqrt {\mathbf {C} }}-\mathbf {1} }$	${\displaystyle \Delta }$	Biot strain tensor , material Biot, Cauchy or Swainger strain tensor
0	${\displaystyle \mathbf {E} _{(0)}=\ln(\mathbf {U} )={\frac {1}{2}}\ln(\mathbf {C} )}$	${\displaystyle \Delta }$	Hencky expansion , material logarithmic expansion
−1	${\displaystyle \mathbf {E} _{(-1)}={\frac {1}{2}}(\mathbf {1} -\mathbf {U} ^{-2})={\frac {1}{2}}(\mathbf {1} -\mathbf {C} ^{-1})}$	${\displaystyle \nabla }$	negative Piola strain tensor , Lagrange-Karni-Reiner strain tensor

The names used here come first in italics . In the spatial description the correspondences result:

m	Strain tensor	${\displaystyle \nabla \Delta }$	Names
1	${\displaystyle \mathbf {e} _{(1)}={\frac {1}{2}}(\mathbf {v} ^{2}-\mathbf {1} )={\frac {1}{2}}(\mathbf {b} -\mathbf {1} )}$	${\displaystyle \nabla }$	negative finger tensor , Euler-Karni-Reiner strain tensor
0	${\displaystyle \mathbf {e} _{(0)}=\ln(\mathbf {v} )={\frac {1}{2}}\ln(\mathbf {b} )}$	${\displaystyle \nabla }$	Hencky spatial expansion , logarithmic spatial expansion
-½	${\displaystyle \mathbf {e} _{(-1/2)}=\mathbf {1-v} ^{-1}=\mathbf {1} -{\sqrt {\mathbf {b} }}^{\,-1}}$	${\displaystyle \Delta }$	Swainger strain tensor , Biot spatial strain tensor
−1	${\displaystyle \mathbf {e} _{(-1)}={\frac {1}{2}}(\mathbf {1-v} ^{-2})={\frac {1}{2}}(\mathbf {1-b} ^{-1})}$	${\displaystyle \Delta }$	Euler-Almansi strain tensor , Almansis or Hamels strain tensor

In the tables, " " means covariance and " " means contravariance. The function value of a tensor (e.g. ) is calculated using the principal axis transformation , the formation of the function values ​​of the diagonal elements and the inverse transformation. ${\displaystyle \Delta }$ ${\displaystyle \nabla }$ ${\displaystyle {\sqrt {\mathbf {C} }}\,,\;\ln(\mathbf {C} )}$

Description of some strain tensors

Because the strain tensors of the Seth-Hill family pass over to the linearized strain tensor for small strains, what has been said here also applies to the linearized strain tensor for small strains.

The Green-Lagrange strain tensor

Stretching and shearing of the tangents (red and blue) on material lines (black) in the course of a deformation

The Green-Lagrange strain tensor is motivated by comparing two material line elements and in the point , see figure on the right: ${\displaystyle \mathrm {d} {\vec {X}}}$ ${\displaystyle \mathrm {d} {\vec {Y}}}$ ${\displaystyle {\vec {X}}}$

${\displaystyle \mathrm {d} {\vec {x}}\cdot \mathrm {d} {\vec {y}}-\mathrm {d} {\vec {X}}\cdot \mathrm {d} {\vec {Y}}=(\mathbf {F} \cdot \mathrm {d} {\vec {X}})\cdot (\mathbf {F} \cdot \mathrm {d} {\vec {Y}})-\mathrm {d} {\vec {X}}\cdot \mathrm {d} {\vec {Y}}=2\mathrm {d} {\vec {X}}\cdot \mathbf {E} \cdot \mathrm {d} {\vec {Y}}\,.}$

In one direction there is about ${\displaystyle {\vec {e}}_{1}={\tfrac {\mathrm {d} {\vec {X}}}{|\mathrm {d} {\vec {X}}|}}}$

${\displaystyle {\begin{aligned}\mathrm {d} {\vec {x}}\cdot \mathrm {d} {\vec {x}}=&2\mathrm {d} {\vec {X}}\cdot \mathbf {E} \cdot \mathrm {d} {\vec {X}}+\mathrm {d} {\vec {X}}\cdot \mathrm {d} {\vec {X}}=(2{\vec {e}}_{1}\cdot \mathbf {E} \cdot {\vec {e}}_{1}+1)(\mathrm {d} {\vec {X}}\cdot \mathrm {d} {\vec {X}})\\\rightarrow \quad |\mathrm {d} {\vec {x}}|=&{\sqrt {2{\vec {e}}_{1}\cdot \mathbf {E} \cdot {\vec {e}}_{1}+1}}\;|\mathrm {d} {\vec {X}}|\end{aligned}}}$

the elongation :

${\displaystyle \varepsilon :={\frac {|\mathrm {d} {\vec {x}}|-|\mathrm {d} {\vec {X}}|}{|\mathrm {d} {\vec {X}}|}}={\sqrt {1+2{\vec {e}}_{1}\cdot \mathbf {E} \cdot {\vec {e}}_{1}}}-1}$

If is in the initial configuration, the following is calculated: ${\displaystyle \mathrm {d} {\vec {X}}\cdot \mathrm {d} {\vec {Y}}=0}$

${\displaystyle {\begin{aligned}\mathrm {d} {\vec {x}}\cdot \mathrm {d} {\vec {y}}=&2\mathrm {d} {\vec {X}}\cdot \mathbf {E} \cdot \mathrm {d} {\vec {Y}}\\{\frac {\mathrm {d} {\vec {x}}\cdot \mathrm {d} {\vec {y}}}{|\mathrm {d} {\vec {x}}||\mathrm {d} {\vec {y}}|}}=&{\frac {2\mathrm {d} {\vec {X}}\cdot \mathbf {E} \cdot \mathrm {d} {\vec {Y}}}{|\mathrm {d} {\vec {x}}||\mathrm {d} {\vec {y}}|}}={\frac {2\mathrm {d} {\vec {X}}\cdot \mathbf {E} \cdot \mathrm {d} {\vec {Y}}}{{\sqrt {2{\vec {e}}_{1}\cdot \mathbf {E} \cdot {\vec {e}}_{1}+1}}\;|\mathrm {d} {\vec {X}}|\;{\sqrt {2{\vec {e}}_{2}\cdot \mathbf {E} \cdot {\vec {e}}_{2}+1}}\;|\mathrm {d} {\vec {Y}}|}}\end{aligned}}}$

With then results for the shear γ : ${\displaystyle {\vec {e}}_{2}={\tfrac {\mathrm {d} {\vec {Y}}}{|\mathrm {d} {\vec {Y}}|}}}$

${\displaystyle \sin(\gamma ):={\frac {\mathrm {d} {\vec {x}}\cdot \mathrm {d} {\vec {y}}}{|\mathrm {d} {\vec {x}}||\mathrm {d} {\vec {y}}|}}={\frac {2{\vec {e}}_{1}\cdot \mathbf {E} \cdot {\vec {e}}_{2}}{{\sqrt {1+2{\vec {e}}_{1}\cdot \mathbf {E} \cdot {\vec {e}}_{1}}}{\sqrt {1+2{\vec {e}}_{2}\cdot \mathbf {E} \cdot {\vec {e}}_{2}}}}}}$

The Euler-Almansi strain tensor

The Euler-Almansi strain tensor

${\displaystyle \mathbf {e} ={\frac {1}{2}}(\mathbf {1} -\mathbf {F} ^{\top -1}\cdot \mathbf {F} ^{-1})}$

can be motivated analogously to the Green-Lagrange strain tensor from the comparison of two material line elements and in the point : ${\displaystyle \mathrm {d} {\vec {x}}}$ ${\displaystyle \mathrm {d} {\vec {y}}}$ ${\displaystyle {\vec {x}}}$

${\displaystyle \mathrm {d} {\vec {x}}\cdot \mathrm {d} {\vec {y}}-\mathrm {d} {\vec {X}}\cdot \mathrm {d} {\vec {Y}}=\mathrm {d} {\vec {x}}\cdot \mathrm {d} {\vec {y}}-(\mathbf {F} ^{-1}\cdot \mathrm {d} {\vec {x}})\cdot (\mathbf {F} ^{-1}\cdot \mathrm {d} {\vec {y}})=2\mathrm {d} {\vec {x}}\cdot \mathbf {e} \cdot \mathrm {d} {\vec {y}}\,.}$

For the stretching in one direction we get: ${\displaystyle \varepsilon }$ ${\displaystyle {\vec {e}}_{1}}$

${\displaystyle \varepsilon ={\frac {1}{\sqrt {1-2{\vec {e}}_{1}\cdot \mathbf {e} \cdot {\vec {e}}_{1}}}}-1}$ With ${\displaystyle {\vec {e}}_{1}={\tfrac {\mathrm {d} {\vec {x}}}{|\mathrm {d} {\vec {x}}|}}\,.}$

The Hencky strain tensor

The Hencky distortion tensor is calculated using the principal axis transformation of the right-hand distance tensor . Because this is symmetrical and positive definite, its spectral decomposition reads ${\displaystyle \mathbf {U} }$

${\displaystyle \mathbf {U} =\sum _{i=1}^{3}\lambda _{i}{\hat {v}}_{i}\otimes {\hat {v}}_{i}}$

where λ _i are all positive eigenvalues and the eigenvectors of normalized to one and orthogonal in pairs . Then the Hencky strain tensor is calculated ${\displaystyle {\hat {v}}_{i}}$ ${\displaystyle \mathbf {U} }$

${\displaystyle \mathbf {E} _{H}:=\ln(\mathbf {U} ):=\sum _{i=1}^{3}\ln(\lambda _{i}){\hat {v}}_{i}\otimes {\hat {v}}_{i}\,.}$

His trail is because of

${\displaystyle \operatorname {Sp} (\mathbf {E} _{H})=\sum _{i=1}^{3}\ln(\lambda _{i}){\hat {v}}_{i}\cdot {\hat {v}}_{i}=\ln(\lambda _{1}\lambda _{2}\lambda _{3})=\ln(\operatorname {det} (\mathbf {U} ))=\ln(\operatorname {det} (\mathbf {F} ))}$

a measure of the compression in place. If the distortion is small

${\displaystyle \ln(\operatorname {det} (\mathbf {F} ))\approx \operatorname {Sp} (\mathbf {H} )=\operatorname {Sp} ({\boldsymbol {\varepsilon }})}$

which is why the trace of the displacement gradient or the linearized strain tensor then takes on this role.

The Piola and Finger strain tensor

Stretching and shearing of the normals (red and blue) on material surfaces (gray) in the course of a deformation

The Piola strain tensor is motivated by comparing the normal vectors to material surfaces. A family of surfaces can be represented by a scalar function ${\displaystyle \mathbf {E} _{P}=-\mathbf {E} _{(-1)}}$ ${\displaystyle {\vec {N}}}$

${\displaystyle \Phi ({\vec {X}},t)=C}$

and an area parameter can be defined. The normal vectors to these surfaces are the gradients ${\displaystyle C}$

${\displaystyle {\vec {N}}:=\operatorname {GRAD} (\Phi )=\sum _{i=1}^{3}{\frac {\mathrm {d} \Phi }{\mathrm {d} X_{i}}}{\vec {e}}_{i}\,.}$

In the course of a deformation it becomes

${\displaystyle {\begin{aligned}{\vec {n}}=&\operatorname {grad} (\Phi )=\sum _{i=1}^{3}{\frac {\mathrm {d} \Phi }{\mathrm {d} x_{i}}}{\vec {e}}_{i}=\sum _{i,j=1}^{3}{\frac {\mathrm {d} \Phi }{\mathrm {d} X_{j}}}{\frac {\mathrm {d} X_{j}}{\mathrm {d} x_{i}}}{\vec {e}}_{i}\\=&\sum _{i,j=1}^{3}{\frac {\mathrm {d} X_{j}}{\mathrm {d} x_{i}}}{\vec {e}}_{i}\otimes {\vec {e}}_{j}\cdot {\frac {\mathrm {d} \Phi }{\mathrm {d} X_{k}}}{\vec {e}}_{k}=\mathbf {F} ^{\top -1}\cdot {\vec {N}}.\end{aligned}}}$

Another scalar function can be used to define another family of surfaces whose normal vectors or via ${\displaystyle \Psi ({\vec {X}},t)}$ ${\displaystyle {\vec {M}}}$ ${\displaystyle {\vec {m}}}$

${\displaystyle {\vec {m}}=\mathbf {F} ^{\top -1}\cdot {\vec {M}}}$

are related. The comparison of the scalar products of the normal vectors in the deformed and undeformed position in a material point leads to the Piola strain tensor ${\displaystyle {\vec {X}}}$

${\displaystyle {\begin{aligned}{\vec {m}}\cdot {\vec {n}}-{\vec {M}}\cdot {\vec {N}}=&(\mathbf {F} ^{\top -1}\cdot {\vec {M}})\cdot (\mathbf {F} ^{\top -1}\cdot {\vec {N}})-{\vec {M}}\cdot {\vec {N}}\\=&2{\vec {M}}\cdot {\frac {1}{2}}(\mathbf {F} ^{-1}\cdot \mathbf {F} ^{\top -1}-\mathbf {1} )\cdot {\vec {N}}=2{\vec {M}}\cdot \mathbf {E} _{P}\cdot {\vec {N}}\,,\end{aligned}}}$

which is therefore a measure for the deformation of the material surfaces. The Piola strain tensor operates in the initial configuration.

Its counterpart in the current configuration is the finger tensor

${\displaystyle \mathbf {e} _{F}:={\frac {1}{2}}(\mathbf {1} -\mathbf {F\cdot F} ^{\top })={\frac {1}{2}}(\mathbf {1} -\mathbf {b} )}$

for the

${\displaystyle {\begin{aligned}{\vec {m}}\cdot {\vec {n}}-{\vec {M}}\cdot {\vec {N}}=&{\vec {m}}\cdot {\vec {n}}-(\mathbf {F} ^{\top }\cdot {\vec {m}})\cdot (\mathbf {F} ^{\top }\cdot {\vec {n}})\\=&2{\vec {m}}\cdot {\frac {1}{2}}(\mathbf {1-F\cdot F} ^{\top })\cdot {\vec {n}}=2{\vec {m}}\cdot \mathbf {e} _{F}\cdot {\vec {n}}\,.\end{aligned}}}$

can be derived.

Distortion Rates

→ Main article : Velocity gradient

All real materials are more or less rate-dependent, i.e. their resistance to deformation depends on the speed at which this deformation is brought about. Distortion rates are used to describe such a relationship. The material behavior is observer invariant , but most time derivatives of the distortions are not. However, a number of distortion speeds have been defined that are observer-invariant.

The right stretch tensor is objectively related to the body , which means that it is not influenced by a change in the reference system. The same also applies to its material time derivative. Accordingly, all distortion speeds formed from this time derivative, e.g. B. ${\displaystyle \mathbf {U} }$ ${\displaystyle {\dot {\mathbf {U} }}\,.}$

${\displaystyle {\dot {\mathbf {E} }}={\frac {1}{2}}({\dot {\mathbf {U} }}\cdot \mathbf {U} +\mathbf {U} \cdot {\dot {\mathbf {U} }})\,,}$

objectively related to the body.

In the spatial description it can be shown that the left stretch tensor is objective, but its rate is not. The spatial velocity gradient is used for the formulation of objective rates of the spatial distortion tensors ${\displaystyle \mathbf {v} }$ ${\displaystyle {\dot {\mathbf {v} }}}$

${\displaystyle \mathbf {l} ={\dot {\mathbf {F} }}\cdot \mathbf {F} ^{-1}=\mathbf {d+w} }$

defined, its symmetrical part

${\displaystyle \mathbf {d} :={\frac {1}{2}}(\mathbf {l+l} ^{\top })}$

spatial strain velocity tensor and its unsymmetrical part

${\displaystyle \mathbf {w} :={\frac {1}{2}}(\mathbf {l-l} ^{\top })}$

Is called spin or vortex tensor . Then the (objective) covariant Oldroyd derivative of a tensor reads : ${\displaystyle \mathbf {a} }$

${\displaystyle {\stackrel {\Delta }{\mathbf {a} }}:={\dot {\mathbf {a} }}+\mathbf {a\cdot l+l^{\top }\cdot a} \,.}$

For the Euler-Almansi tensor e it holds in particular

${\displaystyle {\stackrel {\Delta }{\mathbf {e} }}=\mathbf {d} =\mathbf {F} ^{\top -1}\cdot {\dot {\mathbf {E} }}\cdot \mathbf {F} ^{-1}\,.}$

The contravariant Oldroyd derivative of a tensor is defined as: ${\displaystyle \mathbf {a} }$

${\displaystyle {\stackrel {\nabla }{\mathbf {a} }}:={\dot {\mathbf {a} }}-\mathbf {l\cdot a-a\cdot l} ^{\top }\,.}$

The rates of the covariant tensors are usually formed with the covariant Oldroyd derivative and those of the contravariant tensors with the contravariant Oldroyd derivative. The Zaremba-Jaumann rate of a tensor is also objective and is defined as: ${\displaystyle \mathbf {a} }$

${\displaystyle {\stackrel {\circ }{\mathbf {a} }}:={\dot {\mathbf {a} }}+\mathbf {a\cdot w-w\cdot a} \,.}$

See also

• Mohr's circle of tension
• Tensor
• Dyadic product
• Formula collection tensor algebra
• Tensoranalysis formula collection

literature

• H. Altenbach: Continuum Mechanics . Springer, 2012, ISBN 978-3-642-24118-5 .
• G. Holzapfel: Nonlinear Solid Mechanics: A Continuum Approach for Engineering . Wiley, 2000, ISBN 978-0-471-82319-3 .
• P. Haupt: Continuum Mechanics and Theory of Materials . Springer, 2000, ISBN 3-540-66114-X .
• A. Bertram: Elasticity and Plasticity of Large Deformations: An Introduction . Springer, 2012, ISBN 978-3-642-24614-2 .

Individual evidence

1. ↑ The Frechet derivative of a function according to the restricted linear operator is the - it if exists - in all directions the Gâteaux derivative corresponds, so ${\displaystyle f}$ ${\displaystyle x}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle h}$

   ${\displaystyle {\mathcal {A}}(h)=\left.{\frac {\mathrm {d} }{\mathrm {d} s}}f(x+sh)\right|_{s=0}=\lim _{s\rightarrow 0}{\frac {f(x+sh)-f(x)}{s}}\quad \forall \;h}$

   applies. In it is scalar, vector or tensor valued but and similar. Then will too ${\displaystyle s\in \mathbb {R} \,,f,x\,{\textsf {und}}\,h}$ ${\displaystyle x}$ ${\displaystyle h}$

   ${\displaystyle {\mathcal {A}}={\frac {\partial f}{\partial x}}}$

   written.
2. ↑ ZP Bazant, L. Cedolin: Stability of Structures. Elastic, Inelastic, Fracture and Damage Theories . Oxford Univ. Press, 2003, ISBN 0-486-42568-1 .
3. ↑ BR Seth from the Indian Institute of Technology in Kharagpur was the first to show that the Green-Lagrange and the Euler-Almansi strain tensor are special cases of this more general strain measure [a] [b]. The idea was further developed by Rodney Hill in [c].

   [a] BR Seth: Generalized strain measure with applications to physical problems . MRC Technical Summary Report # 248 of the Mathematics Research Center, United States Army, University of Wisconsin, 1961, pp. 1-18, AD0266913.pdf

   [b] BR Seth: Generalized strain measure with applications to physical problems . IUTAM Symposium on Second Order Effects in Elasticity, Plasticity and Fluid Mechanics, Haifa 1962.

   [c] R. Hill: On constitutive inequalities for simple materials-I . In: Journal of the Mechanics and Physics of Solids. 16, No. 4, 1968, pp. 229-242.

4. ↑ ^{a b} Bertram (2012)
5. ↑ ^{a b c} main (2000)
6. ↑ ^{a b} Altenbach (2012)