Multiaxial Stress State

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Multiaxial stress state

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).

Fig. 1:

Decomposition of the stress σ acting on the reference plane ABCD into the normal stress component σzz and the shear stress components τxz and τyz

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):

$\sigma _{ij}={\begin{bmatrix}\sigma _{xx}&\tau _{xy}&\tau _{xz}\\\tau _{yx}&\sigma _{yy}&\tau _{yz}\\\tau _{zx}&\tau _{zy}&\sigma _{zz}\end{bmatrix}}$

(1)

Fig. 2:

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 σ₁, σ₂ and σ₃. A description of the stress state that is independent of the choice of coordinate system is possible using the invariants I₁, I₂ and I₃ of the stress tensor:

$I_{1}=\sigma _{xx}+\sigma _{yy}+\sigma _{zz}\!$ $I_{2}=\sigma _{xx}\sigma _{yy}+\sigma _{yy}\sigma _{zz}+\sigma _{zz}\sigma _{xx}-\tau _{xy}^{2}-\tau _{yz}^{2}-\tau _{zx}^{2}$ $I_{3}=\sigma _{xx}\sigma _{yy}\sigma _{zz}+2\tau _{xy}\tau _{yz}\tau _{zx}-\sigma _{xx}\tau _{yz}^{2}-\sigma _{yy}\tau _{zx}^{2}-\sigma _{zz}\tau _{xy}^{2}$

(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)):

$p={\frac {1}{3}}(\sigma _{xx}+\sigma _{yy}+\sigma _{zz})={\frac {I_{1}}{3}}$

(3)

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

$\sigma _{ij}^{\prime }={\begin{bmatrix}(\sigma _{xx}-p)&\tau _{xy}&\tau _{xz}\\\tau _{yz}&(\sigma _{yy}-p)&\tau _{yz}\\\tau _{zx}&\tau _{zy}&(\sigma _{zz}-p)\end{bmatrix}}$

(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)):

$\epsilon _{ij}={\begin{bmatrix}\epsilon _{xx}&\gamma _{xy}&\gamma _{xz}\\\gamma _{yz}&\epsilon _{yy}&\gamma _{yz}\\\gamma _{zx}&\gamma _{zy}&\epsilon _{zz}\end{bmatrix}}$

(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 ε₁, ε₂ and ε₃ 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 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 under consideration and the loading conditions. In the field of plastics alone, the spectrum ranges from brittle, hard, glass-like amorphous polymers, through ductile semi-crystalline thermoplastics and soft, elastic 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) within narrow limits of validity.

In the general case of multiaxial loading, the energy-elastic behaviour is described by the generalised 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:

$\sigma _{ij}=C_{ijkl}\cdot \epsilon _{kl}$

(6)

$\epsilon _{ij}=D_{ijkl}\cdot \sigma _{kl}$.

(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):

${\begin{Bmatrix}\sigma _{xx}\\\sigma _{yy}\\\sigma _{zz}\\\tau _{xy}\\\tau _{yz}\\\tau _{zx}\end{Bmatrix}}={\begin{bmatrix}C_{11}&C_{12}&C_{12}&0&0&0\\C_{12}&C_{11}&C_{12}&0&0&0\\C_{12}&C_{12}&C_{11}&0&0&0\\0&0&0&{\frac {C_{11}-C_{12}}{2}}&0&0\\0&0&0&0&{\frac {C_{11}-C_{12}}{2}}&0\\0&0&0&0&0&{\frac {C_{11}-C_{12}}{2}}\end{bmatrix}}\cdot {\begin{Bmatrix}\epsilon _{xx}\\\epsilon _{yy}\\\epsilon _{zz}\\\gamma _{xy}\\\gamma _{yz}\\\gamma _{zx}\end{Bmatrix}}$

(8)

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

The elastic constants C₁₁ and C₁₂ are related to the modulus of elasticity E and the transverse coefficient of contraction (´Poisson`s ratio) ν of the isotropic material:

$E={\frac {\sigma _{2}-\sigma _{1}}{\epsilon _{2}-\epsilon _{1}}}={\frac {F_{2}-F_{1}}{0,002\ A_{0}}}$

(9)

$\mu _{b}=\left|{\frac {\epsilon _{qb}}{\epsilon }}\right|=\left|{\frac {\Delta b\ l_{0}}{\Delta l\ b_{0}}}\right|$.

(10)

From the modulus of elasticity E and the Poisson`s ratio ν, further material properties such as the shear modulus G and the compressive modulus K (see: energy elasticity) can be calculated:

$G={\frac {\tau }{\gamma }}={\frac {E}{2(1+\nu )}}={\frac {C_{11}-C_{12}}{2}}$

(11)

$K={\frac {p}{\frac {\Delta V}{V_{0}}}}={\frac {E}{3(1-2\nu )}}={\frac {C_{11}+2C_{12}}{3}}$.

(12)

See also

• Stress
• Uniaxial stress state
• Fracture behaviour of plastics components
• Fracture types
• Impact loading plastics
• Strength

References

[1]	Lüpke, T.: Fundamental principles of mechanical behaviour. In: Grellmann, W., 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)