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{{Language_sel|LANG=ger|ARTIKEL=Mehrachsiger Spannungszustand}}
{{PSM_Infobox}}
Multiaxial stress state
__FORCETOC__

==Multiaxial Stress State==

In the general case of loading, where the stress vector (force vector per unit area) and the reference plane normals are neither parallel nor perpendicular to one another, it is possible to decompose the stress into a normal stress component ''σ''zz and two mutually perpendicular shear stress components ''τ''xz and ''τ''yz using the rules of vector calculus ('''Fig. 1''').

[[File:mehrachsigerspannungszustand1.jpg]]
{|
|- valign="top"
|width="50px"|'''Fig. 1''':
|width="600px" |Decomposition of the stress ''σ'' acting on the reference plane ABCD into the normal stress component ''σ''zz and the shear stress components ''τ''xz and ''τ''yz
|}

In the case of more complex loading scenarios, however, it is necessary to describe the space stress and strain state independently of any specific reference plane. This requires nine stress components acting at the cross-sectional faces of an infinitesimally small cubic volume element, as shown in '''Fig. 2'''. In accordance with the normal forces resulting from the equilibrium of forces, stresses of equal magnitude but opposite direction act on the opposite sides of the volume element.

The resulting stress components can be represented in the form of a matrix as elements of a symmetric second-order tensor ('''Fig. 2'''):

{|
|-
|width="20px"|
|width="500px" | <math> \sigma_{ij}=\begin{bmatrix}
\sigma_{xx} & \tau_{xy} & \tau_{xz}\\
\tau_{yx} & \sigma_{yy} & \tau_{yz}\\
\tau_{zx} & \tau_{zy} & \sigma_{zz}
\end{bmatrix}
</math>
|(1)
|}

[[File:mehrachsigerspannungszustand2.jpg]]
{|
|- valign="top"
|width="50px"|'''Fig. 2''':
|width="600px" |Space stress state of the volume element
|}

==The symmetry of the stress tensor==

Owing to the symmetrical properties of the tensor (''σ''ij = ''σ''ji), the number of mutually independent stress components is reduced to six.

By means of a coordinate transformation, it is possible to calculate the magnitude of the stress components with respect to differently oriented coordinate systems x, y, z. Of particular importance here is the coordinate system with respect to which all shear stress components of the stress tensor vanish (''τ''ij = 0 for all i ≠ j). The axes of this coordinate system are designated as principal axes 1, 2 and 3, and the remaining normal stresses (''σ''ij where i = j) are designated as principal stresses ''σ''1, ''σ''2 and ''σ''3. A description of the stress state that is independent of the choice of coordinate system is possible using the invariants ''I''1, ''I''2 and ''I''3 of the stress tensor:

{|
|-
|width="20px"|
|width="500px" | <math>I_{1}=\sigma_{xx}+\sigma_{yy}+\sigma_{zz} \!</math> <math>I_{2}=\sigma_{xx}\sigma_{yy}+\sigma_{yy}\sigma_{zz}+\sigma_{zz}\sigma_{xx}-\tau_{xy}^2-\tau_{yz}^2-\tau_{zx}^2</math> <math>I_{3}=\sigma_{xx}\sigma_{yy}\sigma_{zz}+2\tau_{xy}\tau_{yz}\tau_{zx}-\sigma_{xx}\tau_{yz}^2-\sigma_{yy}\tau_{zx}^2-\sigma_{zz}\tau_{xy}^2 </math>
|(2)
|}

With regard to the effects of stress, a distinction can be made between changes in volume and changes in shape. Accordingly, the stress tensor can be broken down into a hydrostatic component (dilatation component) ''p'' ('''Eq. (3)'''):

{|
|-
|width="20px"|
|width="500px" | <math>p=\frac{1}{3}(\sigma_{xx}+\sigma_{yy}+\sigma_{zz})=\frac{I_{1}}{3}</math>
|(3)
|}

and a deviatoric component (deformation component) ''σ''´ij ('''Eq. (4)'''):

{|
|-
|width="20px"|
|width="500px" | <math>\sigma^\prime_{ij}=\begin{bmatrix}
(\sigma_{xx}-p) & \tau_{xy} & \tau_{xz}\\
\tau_{yz} & (\sigma_{yy}-p) & \tau_{yz}\\
\tau_{zx} & \tau_{zy} & (\sigma_{zz}-p)
\end{bmatrix}</math>
|(4)
|}

In more complex loading cases, an exact analysis of the relative displacements of neighbouring mass points is required to describe the state of deformation. As a result of such an analysis, the state of deformation is described by a strain tensor, the components of which are arranged in the form of a symmetric matrix, analogous to the stress tensor ('''Eq. (5)'''):

{|
|-
|width="20px"|
|width="500px" | <math>\epsilon_{ij}=\begin{bmatrix}
\epsilon_{xx} & \gamma_{xy} & \gamma_{xz}\\
\gamma_{yz} & \epsilon_{yy} & \gamma_{yz}\\
\gamma_{zx} & \gamma_{zy} & \epsilon_{zz}
\end{bmatrix}
</math>
|(5)
|}

The relative changes in length of the system with respect to the x, y and z axes of the coordinate system are described by the strains ''ε''xx, ''ε''yy and ''ε''zz. In contrast, angular changes are expressed by the shear components ''γ''xy, ''γ''yz and ''γ''zx. The strain tensor has properties that are formally similar to those of the stress tensor. For example, a system of principal axes 1, 2, 3 can be specified, relative to which the shear strains vanish and only the principal strains ''ε''1, ''ε''2 and ''ε''3 exist. Furthermore, it is possible to determine three invariants and to distinguish between a hydrostatic volume change component and a deviatoric shape change component.

==Hooke’s generalised law==

The relationship between the mechanical stress parameters of stress and [[Deformation|deformation]] is determined by the material behaviour and described by constitutive equations (material laws). It is extremely diverse, depending on the structural composition of the [[Material & Werkstoff|material]] under consideration and the loading conditions. In the field of [[Plastics|plastics]] alone, the spectrum ranges from brittle, hard, glass-like amorphous [[Polymer|polymers]], through [[Ductility Plastics|ductile]] [[Crystallinity|semi-crystalline]] [[Thermoplastic Material|thermoplastics]] and soft, elastic [[Elastomers|rubber materials]], to liquid-like polymer melts. Due to the diversity of the phenomena observed, a uniform description is virtually impossible. Therefore, under simplifying assumptions, basic types of mechanical behaviour are defined which allow an approximate description of the stress–strain relationship (see: [[Tensile Test|tensile test]]) within narrow limits of validity.

In the general case of multiaxial loading, the [[Energy Elasticity|energy-elastic behaviour]] is described by the generalised [[HOOKE's Law|HOOKE’s law]]. This is based on the assumption that each of the six components of the stress tensor ''σ''ij depends linearly on the six components of the strain tensor ''ε''kl:

{|
|-
|width="20px"|
|width="500px" | <math>\sigma_{ij}=C_{ijkl}\cdot \epsilon_{kl}</math>
|(6)
|}

{|
|-
|width="20px"|
|width="500px" | <math>\epsilon_{ij}=D_{ijkl}\cdot \sigma_{kl}</math>.
|(7)
|}

The proportionality constants between the components of the stress and strain tensors form a fourth-order tensor known as the elasticity or stiffness tensor '''''C'''''ijkl or, respectively, the compliance tensor '''''D'''''ijkl. This tensor consists of 81 components, of which, however, only 21 are independent of one another in static equilibrium. Symmetry properties of the material can lead to a further reduction in the number of independent components. For an isotropic material, two components are required to fully describe the elasticity or compliance tensor. The relationship between the stress and strain states of the isotropic material can thus be expressed in vector notation as follows [1] ('''Eq. 8'''):

{|
|-
|width="20px"|
|width="500px" | <math>\begin{Bmatrix}
\sigma_{xx}\\
\sigma_{yy}\\
\sigma_{zz}\\
\tau_{xy}\\
\tau_{yz}\\
\tau_{zx}
\end{Bmatrix}=\begin{bmatrix}
C_{11} &C_{12} &C_{12} &0 &0 &0 \\
C_{12} &C_{11} &C_{12} &0 &0 &0 \\
C_{12} &C_{12} &C_{11} &0 &0 &0 \\
0 &0 &0 &\frac{C_{11}-C_{12}}{2} &0 &0 \\
0 &0 &0 &0 &\frac{C_{11}-C_{12}}{2} &0 \\
0 &0 &0 &0 &0 &\frac{C_{11}-C_{12}}{2}
\end{bmatrix}\cdot \begin{Bmatrix}
\epsilon_{xx}\\
\epsilon_{yy}\\
\epsilon_{zz}\\
\gamma_{xy}\\
\gamma_{yz}\\
\gamma_{zx}
\end{Bmatrix}</math>
|(8)
|}

==Correlation between the elastic parameters ''E'', ''v'', ''K'' and ''G''==

The elastic constants ''C''11 and ''C''12 are related to the [[Elastic Modulus|modulus of elasticity ''E'']] and the [[Poisson's Ratio|transverse coefficient of contraction]] (´Poisson`s ratio) ''ν'' of the isotropic material:

{|
|-
|width="20px"|
|width="500px" | <math>E=\frac{\sigma_{2}-\sigma_{1}}{\epsilon_{2}-\epsilon_{1}}=\frac{F_{2}-F_{1}}{0,002 \ A_{0}}</math>
|(9)
|}

{|
|-
|width="20px"|
|width="500px" | <math>\mu_{b}=\left | \frac{\epsilon_{qb}}{\epsilon} \right |=\left | \frac{\Delta b \ l_{0}}{\Delta l \ b_{0}} \right |</math>.
|(10)
|}

From the modulus of elasticity ''E'' and the Poisson`s ratio ''ν'', further [[Material Parameter|material properties]] such as the [[Shear Modulus|shear modulus ''G'']] and the compressive modulus ''K'' (see: [[Energy Elasticity|energy elasticity]]) can be calculated:

{|
|-
|width="20px"|
|width="500px" | <math>G=\frac{\tau}{\gamma}=\frac{E}{2(1+\nu)}=\frac{C_{11}-C_{12}}{2}</math>
|(11)
|}

{|
|-
|width="20px"|
|width="500px" | <math>K=\frac{p}{\frac{\Delta V}{V_{0}}}=\frac{E}{3(1-2\nu)}=\frac{C_{11}+2C_{12}}{3}</math>.
|(12)
|}

==See also==

* [[Stress]]
* [[Uniaxial Stress State|Uniaxial stress state]]
* [[Fracture Behaviour of Plastics Components|Fracture behaviour of plastics components]]
* [[Fracture Types|Fracture types]]
* [[Impact Loading Plastics|Impact loading plastics]]
* [[Strength]]

==References==

{|
|-valign="top"
|[1]
|Lüpke, T.: Fundamental principles of mechanical behaviour. In: [[Grellmann,_Wolfgang|Grellmann, W.]], [[Seidler,_Sabine|Seidler, S.]] (Eds.): Polymer Testing. Carl Hanser, Munich (2022) 3rd Edition, pp. 71–74 (ISBN 978-1-56990-806-8; E-Book: ISBN 978-1-56990-807-5; see [[AMK-Library]] under A 22)
|}

[[Category:Tensile Test]]

Multiaxial Stress State - Revision history