Formulations in Stress analysis

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Within QuickField package, the plane stress, the plane strain and the axisymmetric stress models are available with both isotropic and orthotropic materials.
The plane stress model is suitable for analyzing structures that are thin in the out-of-plane direction, e.g., thin plates subject to in-plane loading. Out-of-plane direct stress and shear stresses are assumed to be negligible.
The plane strain model is formulated by assuming that out-of-plane strains are negligible. This model is suitable for structures that are thick in the out-of-plane direction.

Displacement, Strain and Stress

The displacement field is assumed to be completely defined by the two components of the displacement vector δ in each point:

Plane problems: \[ \{δ\} = \begin{Bmatrix} δ_x\\ δ_y \end{Bmatrix} \]

Axisymmetric problems: \[ \{δ\} = \begin{Bmatrix} δ_z\\ δ_r \end{Bmatrix} \]

Only three components of strain and stress tensors are independent in both plane stress and plane strain cases. The strain-displacement relationship is defined as: \[ \{ε\} = \begin{Bmatrix} ε_x\\ ε_y \\ γ_{xy} \\ \end{Bmatrix} = \begin{Bmatrix} \frac{\partial δ_x}{\partial x}\\ \frac{\partial δ_y}{\partial y} \\ \frac{\partial δ_x}{\partial y} + \frac{\partial δ_y}{\partial x} \\ \end{Bmatrix} \]

The corresponding stress components: \[ \{σ\} = \begin{Bmatrix} σ_x\\ σ_y \\ τ_{xy} \\ \end{Bmatrix} \]

The axisymmetric problem formulation also includes the out-of-plane direct strain ε_θ, caused by the radial deformation. The strain-displacement relationship is defined as: \[ \{ε\} = \begin{Bmatrix} ε_z\\ ε_r \\ ε_θ \\ γ_{rz} \\ \end{Bmatrix} = \begin{Bmatrix} \frac{\partial δ_z}{\partial z}\\ \frac{\partial δ_r}{\partial r} \\ \frac{δ_r}{r}\\ \frac{\partial δ_z}{\partial r} + \frac{\partial δ_r}{\partial z} \\ \end{Bmatrix} \]

The corresponding stress components: \[ \{σ\} = \begin{Bmatrix} σ_z\\ σ_r\\ σ_θ\\ τ_{rz} \\ \end{Bmatrix} \]

The equilibrium equations for the plane problems are: \[ \left \lbrace \begin{matrix} \frac{\partial σ_x}{\partial x} + \frac{\partial τ_{xy}}{\partial y} = -f_x\\ \frac{\partial τ_{xy}}{\partial x} + \frac{\partial σ_{y}}{\partial y} = -f_y \end{matrix} \right ., \]

and for the axisymmetric problems are: \[ \left \lbrace \begin{matrix} \frac{1}{r}\frac{\partial (rσ_r)}{\partial r} + \frac{\partial τ_{rz}}{\partial z} = -f_r\\ \frac{1}{r}\frac{\partial (rτ_{rz})}{\partial r} + \frac{\partial σ_{z}}{\partial z} = -f_z \end{matrix} \right ., \]

where f_x, f_y and f_z, f_r are components of the volume force vector.

For linear elasticity, the stresses are related to the strains using relationship of the form \[\{σ\} = [D](\{ε\} - \{ε_0\}), \] where D is a matrix of elastic constants, and ε₀ is the initial thermal strain. The specific form of the matrix depends on a particular problem formulation.

For plane stress and isotropic material: \[ [D] = \frac{E}{1 - ν^2} \left [ \begin{matrix} 1 && ν && 0\\ ν && 1 && 0\\ 0 && 0 &&\frac{1 - ν}{2} \end{matrix} \right ] \]

For plane stress and orthotropic material: \[ [D] = \left [ \begin{matrix} \frac{1}{E_x} && -\frac{ν_{xy}}{E_y} && 0\\ -\frac{ν_{xy}}{E_y} && \frac{1}{E_y} && 0\\ 0 && 0 && \frac{1}{G_{xy}} \end{matrix} \right ]^{-1} \]

For plane strain and isotropic material: \[ [D] = \frac{E(1-ν)}{(1+ν)(1-2ν)} \left [ \begin{matrix} 1 && \frac{ν}{1-ν} && 0\\ \frac{ν}{1-ν} && 1 && 0\\ 0 && 0 && \frac{1-2ν}{2(1-ν)} \end{matrix} \right ] \]

For plane strain and orthotropic material: \[ [D] = \left [ \begin{matrix} \frac{1}{E_x} - \frac{ν_{xz}^2}{E_z} && -\frac{ν_{xy}}{E_y} -\frac{ν_{xz}ν_{yz}}{E_z} && 0\\ -\frac{ν_{xy}}{E_y} -\frac{ν_{xz}ν_{yz}}{E_z} && \frac{1}{E_y} - \frac{ν_{yz}^2}{E_z} && 0\\ 0 && 0 && \frac{1}{G_{xy}} \end{matrix} \right ]^{-1} \]

For axisymmetric problem and isotropic material: \[ [D] = \frac{E(1-ν)}{(1+ν)(1-2ν)} \left [ \begin{matrix} 1 && \frac{ν}{1-ν} && \frac{ν}{1-ν} && 0\\ \frac{ν}{1-ν} && 1 && \frac{ν}{1-ν} && 0\\ \frac{ν}{1-ν} && \frac{ν}{1-ν} && 1 && 0\\ 0 && 0 && 0 && \frac{1-2ν}{2(1-ν)} \end{matrix} \right ] \]

For axisymmetric problem and orthotropic material: \[ [D] = \left [ \begin{matrix} \frac{1}{E_z} && -\frac{ν_{zr}}{E_r} && -\frac{ν_{zθ}}{E_θ} && 0\\ -\frac{ν_{zr}}{E_r} && \frac{1}{E_r} && - \frac{ν_{rθ}}{E_θ} && 0\\ -\frac{ν_{zθ}}{E_θ} && - \frac{ν_{rθ}}{E_θ} && \frac{1}{E_θ} && 0\\ 0 && 0 && 0 && \frac{1}{G_{zr}} \end{matrix} \right ]^{-1} \]

In all these equations E denotes Young's modulus of the isotropic material; E_x, E_y, E_z, E_r, and E_θ are the Young's moduli of the orthotropic material along the corresponding axes;
ν is a Poisson's ratio for isotropic material; ν_yx, ν_zx, ν_zy, νrz, ν_θz, ν_θr are the Poisson's ratios for orthotropic material;
G_xy and G_zr are the shear moduli.