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# Standard Engineering Functions

## 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:

Ux | Displacement component in the x-direction u_{x} |

Uy | Displacement component in the y-direction u_{y} |

Uz | Displacement component in the z-direction u_{z} |

|U| | Total displacement |u| |

### Strains

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

Ex | Normal strain ε_{x} |

Ey | Normal strain ε_{y} |

Ez | Normal strain ε_{z} |

Gxy | Shear strain γ_{xy} |

Gyz | Shear strain γ_{yz} |

Gxz | Shear strain γ_{xz} |

E1 | 1st Principal strain ε_{1} |

E2 | 2nd Principal strain ε_{2} |

E3 | 3rd Principal strain ε_{3} |

Eeq | von Mises strain ε_{eq} |

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 ε_{1}, ε_{2}, ε_{3}. In two-dimensional problems the principal strains are ordered such that ε_{1 }≥ ε_{2}, and ε_{3} = ε_{z}. In three dimensions ε_{1 }≥ ε_{2 }≥ ε_{3}. 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:

Sx | Normal stress σ_{x} |

Sy | Normal stress σ_{y} |

Sz | Normal stress σ_{z} |

Txy | Shear stress τ_{xy} |

Tyz | Shear stress τ_{yz} |

Txz | Shear stress τ_{xz} |

S1 | 1st Principal stress σ_{1} |

S2 | 2nd Principal stress σ_{2} |

S3 | 3rd Principal stress σ_{3} |

Seq | von Mises stress σ_{eq} |

Tmax | Maximum shear stress τ_{max} |

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 σ_{1}, σ_{2}, σ_{3}. In two-dimensional problems the principal stresses are ordered such that σ_{1 }≥ σ_{2 }and σ_{3 }= σ_{z}. In three dimensions σ_{1 }≥ σ_{2 }≥ σ_{3}. 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 σ_{1} is the largest and σ_{3} is the smallest principal stresses. τ_{max} is related to the Tresca yield criterion. In Planar Elasticity, the maximum shear stress is computed from σ_{1} and σ_{2}.

### 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:

Ur | Displacement component in the R-direction u_{r} |

Uz | Displacement component in the Z-direction u_{z} |

|U| | Total displacement |u| |

### Strains

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

Er | Normal strain ε_{r }in the radial direction |

Ez | Normal strain ε_{z }in the Z-direction |

Et | Normal strain ε_{t }in the circumferential direction |

Grz | Shear strain γ_{rz} |

E1 | Principal strain ε_{1 }in the rz-plane |

E2 | Principal strain ε_{2 }in the rz-plane |

Eeq | Equivalent strain ε_{eq} |

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 (ε_{1}, ε_{2}) are computed in the r-z plane as follows:

Note that the third principal strain (ε_{3}) is equal to ε_{t}. The equivalent strain is:

### Stress

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

Sr | Normal stress σ_{r} |

Sz | Normal stress σ_{z} |

St | Normal stress σ_{t} |

Trz | Shear stress τ_{rz} |

S1 | Principal stress σ_{1 }in the rz-plane |

S2 | Principal stress σ_{2 }in the rz-plane |

Seq | Equivalent stress σ_{eq} |

Tmax | Maximum shear stress τ_{max }in the rz-plane |

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

The principal stresses (σ_{1}, σ_{2}) are computed in the r-z plane as follows:

Note that the third principal stress (σ_{3}) 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.

Error | Error Indicator. Available in the Plot output class only. |

Fmla | Formula. Using this option, any mathematical expression containing the standard functions can be computed for a given solution. Available in the Plot, Min/Max and Points output classes. |

Calc | Calculator. Using this option, any mathematical expression containing standard functions can be computed for any arbitrary combination of solutions. Available in the Plot, Min/Max and Points output classes. |

K1 | Mode I stress intensity factor (SIF) via the Contour Integral Method (CIM). Available in the Points output class only. |

K2 | Mode II stress intensity factor (SIF) via the Contour Integral Method (CIM). Available in the Points output class only. |

T-str | T-stress. Available in the Points output class only. |

J1p | Mode I energy release rate (ERR) via the path J-integral (J-path). Available in the Points output class only. |

J2p | Mode II energy release rate (ERR) via the path J-integral (J-path). Available in the Points output class only. |

J3p | Mode III energy release rate (ERR) via the path J-integral (J-path). Available in the Points output class only. |

Initial Gap | Pre-solution (initial) gap measurement between contact pairs. Available in the Plot output class for the case of 3D multi-body contact solutions only. |

Final Gap | Post-solution (final) gap measurement between contact pairs. Available in the Plot output class for the case of 3D multi-body contact solutions only. |

Contact Press. | Contact pressures (surface tractions) developed between contact pairs. Available in the Plot output class for the case of 3D multi-body contact solutions only. |

For more information on the special functions: