Introduction

StressCheck is based on the displacement formulation of the finite element method. Therefore the basic information generated by StressCheck is an approximation to the displacement vector components. This approximation is characterized by the set of standard shape functions and their coefficients. Thus, in the case of two-dimensional elasticity, the displacement vector components on the k^th quadrilateral element are known in the form:

where n is the number of shape functions. The value of n depends on the polynomial degree p and whether the product or trunk space is used. N_i(ξ,η) (i=1,2,…,n) represent the standard shape functions defined in Introduction to Finite Element Analysis: Formulation, Verification and Validation by Barna Szabó and Ivo Babuška. All user-specified functions are computed from the displacement vector components. StressCheck computes a set of commonly used engineering functions, such as stresses, strains, etc. The available standard StressCheck functions, and corresponding symbols, are listed below for Planar, 3D and Axisymmetric references. In addition, any combination of the standard StressCheck functions can be computed through user-specified formulas or through the use of the StressCheck Calculator.

Planar/3D Elasticity Functions

Planar/3D Elasticity functions are described in terms of Cartesian (i.e. X, Y, Z) space.

Displacements

The following are symbols & standard functions for displacements in the Planar/3D reference:

Strains

The following are symbols & standard functions for strains in the Planar/3D reference:

By definition the normal strains are:

And the shear strains are:

The principal strains are the eigenvalues of the strain tensor. They are the strain values ε which satisfy the condition:

In two-dimensional problems, γ_yz = γ_xz = 0. The three roots are the principal strains, denoted by ε₁, ε₂, ε₃. In two-dimensional problems the principal strains are ordered such that ε₁ ≥ ε₂, and ε₃ = ε_z. In three dimensions ε₁ ≥ ε₂ ≥ ε₃. The normalized eigenvectors are the unit vectors which define the directions of the principal strains.

The equivalent strain is related to the von Mises theory of yield. For linear elastic isotropic materials:

And for elasto-plastic materials:

Stresses

The following are symbols & standard functions for stresses in the Planar/3D reference:

The sign convention for the stress tensor components is illustrated below. In two dimensions τ_xz = τ_zx = 0 and τ_yz = τ_zy = 0. The directional stresses, are computed multiplying the material stiffness matrix [E] by the strain tensor {ε}:

The principal stresses are the eigenvalues of the stress tensor. They are the stress values σ which satisfy the condition:

The three roots of the equation above are the principal stresses, denoted by σ₁, σ₂, σ₃. In two-dimensional problems the principal stresses are ordered such that σ₁ ≥ σ₂ and σ₃ = σ_z. In three dimensions σ₁ ≥ σ₂ ≥ σ₃. The corresponding normalized eigenvectors are the unit vectors in the direction of the principal stresses.

The equivalent stress σ_eq is by definition:

σ_eq is related to the von Mises yield criterion. The maximum shear stress is, by definition:

where σ₁ is the largest and σ₃ is the smallest principal stresses. τ_max is related to the Tresca yield criterion. In Planar Elasticity, the maximum shear stress is computed from σ₁ and σ₂.

Cylindrical System Extractions

If a local Cylindrical system is chosen for computation, the following conventions are used to relate the standard function symbols in Cartesian space (i.e. X, Y, Z) to their counterparts in Cylindrical space (i.e. R, T, Z):

X ⇒ R (radial)

Y ⇒ T (tangential/hoop)

Z ⇒ Z

For example, if a local Cylindrical system is selected for computation, and the standard function “Sx” is selected, the output will be the radial stress σ_r, and if the standard function “Sy” is selected, the output will be the tangential stress σ_t.

For an example of extracting results in cylindrical coordinates, refer to StressCheck Tutorial: Results in Cylindrical Coordinates (R, T, Z).

Axisymmetric (Axisym.) Elasticity Functions

Axisymmetric Elasticity functions are described in terms of (R, Z) space. It is assumed that there is no circumferential displacement (i.e. u_θ = 0).

Displacements

The following are symbols & standard functions for displacements in the Axisymmetric reference:

Strains

The following are symbols & standard functions for strains in the Axisymmetric reference:

In Axisymmetric Elasticity, the displacement vector components u_r(r, z) and u_z(r, z) are computed for each element. The strains are then computed as:

The principal strains (ε₁, ε₂) are computed in the r-z plane as follows:

Note that the third principal strain (ε₃) is equal to ε_t. The equivalent strain is:

Stress

The following are symbols & standard functions for stresses in the Axisymmetric reference:

The stress components are determined from the stress-strain relationships. For isotropic materials for example, we have:

The principal stresses (σ₁, σ₂) are computed in the r-z plane as follows:

Note that the third principal stress (σ₃) is equal to σ_t. The equivalent stress is:

And the maximum shear stress (τ_max) is:

Special Functions

Additionally, special functions may available for some output classes (e.g. Plot, Min/Max, Points). These special functions include the error indicator, user-specified formula, the StressCheck Calculator, fracture mechanics parameters and multi-body contact solution functions. Note: fracture mechanics parameters require a radius of integration to be entered in the “Rad.” field.

For more information on the special functions:

• Error Estimation Overview
• Formula & Calculator Options
• Fracture Mechanics Overview
• Multi-Body Contact Overview