Elasticity tensor

The elasticity tensor maps the distortions to the stresses in the linear elasticity . It can be used to map anisotropic material behavior. The branch of physics that deals with elastic deformations is called the theory of elasticity . It is part of continuum mechanics and is characterized by the fact that elastic deformations are reversible: after the external force has ceased, the material returns to its original shape. This is no longer the case when it comes to fractures or plastic flow - the latter case is dealt with by the plasticity theory .

Elasticity tensor

Designation of the normal and shear stresses on a 3-dimensional material volume

Mechanical stresses are considered for the calculation as a force on a (sectional) surface of a body. For the calculation, a force is divided into the component normal stresses ( ), perpendicular to the selected plane, and shear stresses ( ), in the plane. These different stresses are summarized in the stress tensor : ${\displaystyle \sigma }$ ${\displaystyle \tau }$

${\displaystyle {\overline {\overline {\sigma }}}={\begin{bmatrix}\sigma _{11}&\tau _{12}&\tau _{13}\\\tau _{21}&\sigma _{22}&\tau _{23}\\\tau _{31}&\tau _{32}&\sigma _{33}\end{bmatrix}}}$

Accordingly, the deformations are summarized in the strain tensor :

${\displaystyle {\overline {\overline {\varepsilon }}}={\begin{bmatrix}\varepsilon _{11}&\varepsilon _{12}&\varepsilon _{13}\\\varepsilon _{21}&\varepsilon _{22}&\varepsilon _{23}\\\varepsilon _{31}&\varepsilon _{32}&\varepsilon _{33}\end{bmatrix}}={\begin{bmatrix}\varepsilon _{11}&\gamma _{12}/2&\gamma _{13}/2\\\gamma _{12}/2&\varepsilon _{22}&\gamma _{23}/2\\\gamma _{13}/2&\gamma _{23}/2&\varepsilon _{33}\end{bmatrix}}}$

With a simple eraser you can see that pulling along the x-axis not only causes a deformation in the x-direction, but also makes the eraser thinner on the side ( transverse contraction ), i.e. i.e., is also linearly related to the lateral displacements and . ${\displaystyle \sigma _{11}}$ ${\displaystyle \varepsilon _{22}}$ ${\displaystyle \varepsilon _{33}}$

In the following, small deflections are assumed. This allows the generalized Hooke's law to be used, which establishes a linear relationship between stress and distortion

${\displaystyle \sigma =C\varepsilon }$.

Here the elasticity tensor is a fourth order tensor, with 3 × 3 × 3 × 3 = 81 components. The relationship is by component ${\displaystyle C}$

${\displaystyle \sigma _{ij}=\sum _{k=1}^{3}\sum _{l=1}^{3}C_{ijkl}\varepsilon _{kl}}$

or with Einstein's summation convention

${\displaystyle \sigma _{ij}=C_{ijkl}\varepsilon _{kl}}$.

To simplify the representation, symmetries of the tensors involved can be used under certain conditions :

• The stress tensor is due to the angular momentum balance symmetrical: . Here - as in the vast majority of applications - a ( quasi -) static approach is implicitly assumed. ${\displaystyle \sigma }$ ${\displaystyle \sigma _{ij}=\sigma _{ji}}$
• Under the assumption of small deflections, the strain tensor is usually assumed to be a linearized strain tensor . In this case, it is symmetrical by definition; H. with the indices from the above formula applies . ${\displaystyle \varepsilon }$ ${\displaystyle \varepsilon _{kl}=\varepsilon _{lk}}$

These two conditions result in the two secondary symmetries

${\displaystyle C_{ijkl}=C_{jikl}\quad }$ and ${\displaystyle \quad C_{ijkl}=C_{ijlk}}$

of the elasticity tensor. It also means that the matrices and each have only 6 independent components. The number of independent components in the elasticity tensor has been reduced to 6 × 6 = 36. ${\displaystyle {\overline {\overline {\sigma }}}}$ ${\displaystyle {\overline {\overline {\varepsilon }}}}$

The main symmetry

${\displaystyle C_{ijkl}=C_{klij}}$

follows from the hyperelasticity , which contains Hooke's law assumed here as a special case, taking Schwarz's theorem into account . This reduces the number of independent elasticity components to 21.

With the help of Voigt's notation , the components of the stress and strain matrices are each summarized in a column vector. This allows Hooke's law in the Kelvin-Voigt notation

${\displaystyle {\begin{bmatrix}\sigma _{11}\\\sigma _{22}\\\sigma _{33}\\\tau _{23}\\\tau _{13}\\\tau _{12}\end{bmatrix}}={\begin{bmatrix}C_{11}&C_{12}&C_{13}&C_{14}&C_{15}&C_{16}\\C_{21}&C_{22}&C_{23}&C_{24}&C_{25}&C_{26}\\C_{31}&C_{32}&C_{33}&C_{34}&C_{35}&C_{36}\\C_{41}&C_{42}&C_{43}&C_{44}&C_{45}&C_{46}\\C_{51}&C_{52}&C_{53}&C_{54}&C_{55}&C_{56}\\C_{61}&C_{62}&C_{63}&C_{64}&C_{65}&C_{66}\\\end{bmatrix}}{\begin{bmatrix}\varepsilon _{1}\\\varepsilon _{2}\\\varepsilon _{3}\\\gamma _{23}\\\gamma _{13}\\\gamma _{12}\end{bmatrix}}}$,

or in the Kelvin-Mandel notation

${\displaystyle {\begin{bmatrix}\sigma _{11}\\\sigma _{22}\\\sigma _{33}\\{\sqrt {2}}\tau _{23}\\{\sqrt {2}}\tau _{13}\\{\sqrt {2}}\tau _{12}\end{bmatrix}}={\begin{bmatrix}C_{11}&C_{12}&C_{13}&C_{14}&C_{15}&C_{16}\\C_{21}&C_{22}&C_{23}&C_{24}&C_{25}&C_{26}\\C_{31}&C_{32}&C_{33}&C_{34}&C_{35}&C_{36}\\C_{41}&C_{42}&C_{43}&C_{44}&C_{45}&C_{46}\\C_{51}&C_{52}&C_{53}&C_{54}&C_{55}&C_{56}\\C_{61}&C_{62}&C_{63}&C_{64}&C_{65}&C_{66}\\\end{bmatrix}}{\begin{bmatrix}\varepsilon _{11}\\\varepsilon _{22}\\\varepsilon _{33}\\{\sqrt {2}}\varepsilon _{23}\\{\sqrt {2}}\varepsilon _{13}\\{\sqrt {2}}\varepsilon _{12}\end{bmatrix}}}$,

in this notation due to the main symmetry . Depending on the material and its symmetry properties, further components can be eliminated, as will become clear below. ${\displaystyle C_{ij}=C_{ji}}$

In the case of a quadratic non-linear material, the relationship arises

${\displaystyle \sigma _{ij}=C_{ij}+C_{ijkl}\varepsilon _{kl}+C_{ijklmn}\varepsilon _{kl}\varepsilon _{mn}}$

between stress tensor and strain tensor. Here, too, the above-mentioned symmetries can be used and the matrix-vector notation can be introduced.

Special elasticity laws

Complete anisotropy

→ Main article : anisotropy

The complete ( triclinic ) anisotropy is the most general form of a law of elasticity. It is characterized by the following properties:

• no planes of symmetry in the material
• 21 independent elastic constants describe the law
• The modulus of elasticity is directional
• all couplings are available
• Stiffness matrix is ​​fully occupied

Many fiber-plastic composites are anisotropic. Engineers try to use the effects resulting from complete anisotropy.

Monoclinic anisotropy

The monoclinic anisotropy is of little importance for construction materials. The following properties characterize the monoclinic anisotropy:

• 1 plane of symmetry in the material
• 13 independent elastic constants describe the law
• The modulus of elasticity is directional
• Couplings available

Rhombic anisotropy (orthotropy)

→ Main article : Orthotropy

Many construction materials are orthotropic , e.g. B. technical wood , fabric , many fiber-plastic composites, rolled sheets with texture, etc. The orthotropy must not be confused with the anisotropy. The mere direction-dependent modulus of elasticity is not yet an indication of the anisotropy. Orthotropy is a special case of a completely anisotropic law of elasticity. Orthotropy is characterized by the following properties:

• 3 levels of symmetry in the material
• 9 independent elastic constants describe the law
• The modulus of elasticity is directional
• no expansion-displacement coupling present

So orthotropic materials do not produce shear distortion when stretched. This makes them easy to handle for the designer. Therefore, in fiber composite technology, orthotropic layers such as the balanced angle composite are used specifically . Laminated wood is constructed in such a way that it has orthotropic properties.

${\displaystyle C^{-1}={\begin{bmatrix}{\frac {1}{E_{1}}}&-{\frac {\nu _{21}}{E_{2}}}&-{\frac {\nu _{31}}{E_{3}}}&0&0&0\\-{\frac {\nu _{12}}{E_{1}}}&{\frac {1}{E_{2}}}&-{\frac {\nu _{32}}{E_{3}}}&0&0&0\\-{\frac {\nu _{13}}{E_{1}}}&-{\frac {\nu _{23}}{E_{2}}}&{\frac {1}{E_{3}}}&0&0&0\\0&0&0&{\frac {1}{G_{23}}}&0&0\\0&0&0&0&{\frac {1}{G_{31}}}&0\\0&0&0&0&0&{\frac {1}{G_{12}}}\\\end{bmatrix}}}$

Remarks:

• The matrix and thus its inverse are symmetrical. In general, however, the constants used in the illustration for which , and applies are not symmetrical . ${\displaystyle C}$ ${\displaystyle \nu _{ij}}$ ${\displaystyle \nu _{12}={\frac {E_{1}}{E_{2}}}\nu _{21}}$ ${\displaystyle \nu _{13}={\frac {E_{1}}{E_{3}}}\nu _{31}}$ ${\displaystyle \nu _{23}={\frac {E_{2}}{E_{3}}}\nu _{32}}$
• In the above matrix, the following convention is used to calculate the transverse contraction, ${\displaystyle \nu _{xy}=-{\frac {\varepsilon _{y}}{\varepsilon _{x}}}}$

Transverse isotropy

→ Main article : Transverse isotropy

The transverse isotropy is characterized by the fact that the law of elasticity can be rotated around an axis without it changing. So it is invariant to the rotation. An example of a transversely isotropic material is a round wood or a unidirectional layer. The elastic properties of the round timber do not change when it is turned around its longitudinal axis. Nevertheless, the wood has different modules along and across the grain. The transverse isotropy is characterized by the following properties:

• 3 levels of symmetry in the material
• 5 independent elastic constants describe the law, is a possible choice , , , and ${\displaystyle E_{1}}$ ${\displaystyle E_{2}}$ ${\displaystyle \nu _{12}}$ ${\displaystyle \nu _{23}}$ ${\displaystyle G_{12}}$
  • Because it applies and the other quantities in the matrix result from the relationship . ${\displaystyle \nu _{32}=\nu _{23}}$ ${\displaystyle {\frac {\nu _{12}}{E_{1}}}={\frac {\nu _{21}}{E_{2}}}}$
• The modulus of elasticity is direction-dependent, two modules are identical
• no expansion-displacement coupling present

The transverse isotropy is a special case of the general orthotropy.

${\displaystyle C^{-1}={\begin{bmatrix}{\frac {1}{E_{1}}}&-{\frac {\nu _{21}}{E_{2}}}&-{\frac {\nu _{21}}{E_{2}}}&0&0&0\\-{\frac {\nu _{12}}{E_{1}}}&{\frac {1}{E_{2}}}&-{\frac {\nu _{32}}{E_{2}}}&0&0&0\\-{\frac {\nu _{12}}{E_{1}}}&-{\frac {\nu _{23}}{E_{2}}}&{\frac {1}{E_{2}}}&0&0&0\\0&0&0&{\frac {2(1+\nu _{23})}{E_{2}}}&0&0\\0&0&0&0&{\frac {1}{G_{31}}}&0\\0&0&0&0&0&{\frac {1}{G_{21}}}\\\end{bmatrix}}}$

Isotropy

The isotropic law is the best known and most important law of elasticity. Almost all metals and unreinforced plastics can be written with it. Even short-fiber reinforced plastics can be isotropic if the reinforcing fibers are statistically distributed (see: Fiber-matrix semi-finished products ). For the designer, the isotropic law of elasticity is mainly characterized by its invariance with respect to rotation. In a construction it is therefore irrelevant how the isotropic material is oriented. Rolled metallic sheets can have a weak anisotropy.

• infinitely many levels of symmetry in the material
• 2 independent elastic constants describe the law
• The modulus of elasticity is not direction-dependent, two modules are identical
• there is no displacement-elongation coupling

Kelvin Voigt notation

${\displaystyle C^{-1}={\begin{bmatrix}{\frac {1}{E}}&-{\frac {\nu }{E}}&-{\frac {\nu }{E}}&0&0&0\\-{\frac {\nu }{E}}&{\frac {1}{E}}&-{\frac {\nu }{E}}&0&0&0\\-{\frac {\nu }{E}}&-{\frac {\nu }{E}}&{\frac {1}{E}}&0&0&0\\0&0&0&{\frac {2(1+\nu )}{E}}&0&0\\0&0&0&0&{\frac {2(1+\nu )}{E}}&0\\0&0&0&0&0&{\frac {2(1+\nu )}{E}}\\\end{bmatrix}}}$; ${\displaystyle C={\begin{bmatrix}\lambda +2G&\lambda &\lambda &0&0&0\\\lambda &\lambda +2G&\lambda &0&0&0\\\lambda &\lambda &\lambda +2G&0&0&0\\0&0&0&G&0&0\\0&0&0&0&G&0\\0&0&0&0&0&G\\\end{bmatrix}}}$

Kelvin-Mandel spelling

${\displaystyle C^{-1}={\begin{bmatrix}{\frac {1}{E}}&-{\frac {\nu }{E}}&-{\frac {\nu }{E}}&0&0&0\\-{\frac {\nu }{E}}&{\frac {1}{E}}&-{\frac {\nu }{E}}&0&0&0\\-{\frac {\nu }{E}}&-{\frac {\nu }{E}}&{\frac {1}{E}}&0&0&0\\0&0&0&{\frac {(1+\nu )}{E}}&0&0\\0&0&0&0&{\frac {(1+\nu )}{E}}&0\\0&0&0&0&0&{\frac {(1+\nu )}{E}}\\\end{bmatrix}}}$; ${\displaystyle C={\begin{bmatrix}\lambda +2G&\lambda &\lambda &0&0&0\\\lambda &\lambda +2G&\lambda &0&0&0\\\lambda &\lambda &\lambda +2G&0&0&0\\0&0&0&2G&0&0\\0&0&0&0&2G&0\\0&0&0&0&0&2G\\\end{bmatrix}}}$

See also: Lamé constants .

Couplings

The different elasticity laws are characterized by their couplings. A coupling describes the effect that the material reacts with a deformation outside the effective direction of the load.

Expansion-transverse expansion coupling

Expansion-transverse expansion coupling

This is the best known coupling. It is also known as transverse contraction coupling. The coupling causes the material to constrict when it is pulled or to widen when it is pushed. Engineers have learned to deal with the strain coupling and apply it specifically, e.g. B. riveting. Practically all construction materials have this coupling.

• responsible terms: ${\displaystyle a_{1\dots 3{,}1\dots 3},i\neq j}$

Expansion-displacement coupling

Expansion-displacement coupling

This coupling occurs particularly with anisotropic materials. They do not have orthotropic materials. The stretch-displacement coupling creates a displacement when the material is stretched. Colloquially, this is also referred to as delay. With the help of the classical laminate theory it can be investigated whether a material has an elongation-displacement coupling.

• responsible terms: as well as ${\displaystyle a_{1\dots 3{,}4\dots 6}}$ ${\displaystyle a_{4\dots 6{,}1\dots 3}}$

Shift-shift coupling

The shift-shift coupling occurs only with anisotropic materials. A shift in the plane also creates a shift out of the plane.

• responsible terms: ${\displaystyle a_{4\dots 6{,}4\dots 6},i\neq j}$

Individual evidence

1. ↑ Tribikram Kundu: Ultrasonic and Electromagnetic NDE for Structure and Material Characterization . CRC Press, 2012, p. 19th ff . ( limited preview in Google Book search).
2. ↑ Helmut Schürmann: Constructing with fiber-plastic composites. 2nd edition. Springer 2008, ISBN 978-3-540-72189-5 , page 183.