Function Objects

Prepared by Ivan Batistić

Section Aims

This document describes solids4foam function objects which are not available within the standard OpenFOAM package;
Function objects are various post-processing functionalities that are executing during simulation run time;

Function objects are placed at the bottom of the system/controlDict file; for example:

functions
{
    forceDisp
    {
        type          solidForcesDisplacements;
        historyPatch  cylinderFixed;
    }
    patchForce
    {
        type    			solidForces;
        historyPatch  top;
    }
}
  

`cantileverAnalyticalSolution`

Function object purpose
To generate the analytical solution fields for the bending slender cantilever problem. The analytical solution is taken from [C.E. Augarde, A.J. Deeks, The use of Timoshenko's exact solution for a cantilever beam in adaptive analysis. Finite Elements in Analysis and Design, 44, 2008]:

\[\sigma_{xx} = \frac{P(L-x)y}{I}, \qquad \sigma_{yy}=0, \qquad \sigma_{xy}=-\frac{P}{2I}\left( \frac{D^2}{4}-y^2\right),\] \[u_x = \frac{Py}{6EI} \left((6L-3x)x+(2+\nu)\left(y^2-\frac{D^2}{4}\right) \right),\] \[u_y = \frac{P}{6EI} \left(3\nu y^2(L-x)+(4+5\nu)\frac{D^2x}{4}+(3L-x)x^2 \right),\]

where $\nu$ is Poisson's ratio, $E$ is Young modulus, $I$ is the second moment of are of the cross-section, $P$ is applied load and $L$ is length of the beam.

Note

Above analytical solution can be used only if the shearing forces on the ends are distributed according to the same parabolic law as the shearing stress $\tau_{xy}$ and the intensity of the normal forces at the built-in end is proportional to $y$.

Warning

The current version of the code assumes a rectangular cross-section with unit width and automatically calculates the second moment of inertia!

Example of usage

functions
{
    cantileverSolution
    {
        type    cantileverAnalyticalSolution;
  
        E      00e6;
        nu      0.3;
        L 	     10;
        D		0.1;
        P		1e5;
          
        //Optional
        cellDisplacement true;
        pointDisplacement true;
        cellStress true;
        pointStress true;
    }
}
  

Arguments
- P load applied in the minus $y$ direction at the other end of the beam;
- L length of the beam;
- D depth of the beam;
- E Young's modulus;
- nu Poisson's ratio.
Optional arguments
- cellDisplacement write analytical solution for cell-centred displacement field; default is true;
- pointDisplacement write analytical solution for vertex-centred displacement field; default is true;
- cellStress write analytical solution for cell-centred stress field; default is true;
- pointStress write analytical solution for vertex-centred stress field; default is true;
Outputs
- Analytical solution for the stress tensor field analyticalStress in time directories;
- Analytical solution for the displacement field analyticalD in time directories.
- cellStressDifference field; difference between analytical stress and calculated one: analyticalStress-sigma;
- DDiference field; difference between analytical displacement and calculated one: analyticalD-D.
- Log at the end of each time-step: Component: 1
  Norms: mean L1, mean L2, LInfL: 0.12 0.2 0.5
  ...
Tutorial case in which it is used: :
solids/linearElasticity/cantilever2d/vertexCentredCantilever2d

`contactPatchTestAnalyticalSolution`

Function object purpose
To generate the analytical solution for contact patch test. The analytical solution for the stress field is taken from [ Crisfield MA. Re-visiting the contact patch test. Int J Numer Methods Eng. 2000]:

\[\sigma_{x} = \tau_{xy}=0\qquad \sigma_{y} = \dfrac{E}{1-\nu^2}\Delta \qquad \sigma_z = \nu \sigma_y.\]

where $E$ is Young's modulus, $\nu$ Poisson's ratio and $\Delta$ prescribed displacement of upper block top surface.

Note

To use this analytical solution, the bottom surface of the lower block must freely deform in the tangential direction. It can be fixed only in the case of zero Poisson’s ratio.

Example of usage

functions
{
	analyticalSolution
    {
        type    contactPatchTestAnalyticalSolution;
  
        displacement   0.01;
  
        E       1e6;
        nu      1e-15;
    }
}
  

Arguments
- displacement upper block prescribed vertical displacement;
- E Young's modulus;
- nu Poisson's ratio.
Optional arguments
- None.
Outputs
- Analytical solution for stress tensor field analyticalStress in time directories.
- Scalar field of relative error named relativeError and defined as: $e(\%)=\dfrac{\left| \sigma_y - \sigma_y^{analytical} \right|}{\left|\sigma_y^{analytical}\right|} \cdot 100.$
Tutorial case in which it is used:
solids/linearElasticity/contactPatchTest

`curvedCantileverAnalyticalSolution`

Function object purpose
To generate the analytical solution for a curved cantilever beam with end loading and traction-free inner and outer surface. The analytical solution for the stress field is taken from [ Sadd MH. Elasticity: Theory, Applications, and Numerics. Elsevier 2009]:

\[N = {a}^2 - {b}^2 + ({a}^2+{b}^2)\;\text{ln}\left(\frac{b}{a}\right)\] \[\sigma_{r} = \frac{P}{N}\left(r+\frac{a^2b^2}{r^3}-\frac{a^2+b^2}{r}\right)\sin (\theta),\] \[\sigma_{\theta} = \frac{P}{N}\left(3r-\frac{a^2b^2}{r^3}-\frac{a^2+b^2}{r}\right)\sin (\theta),\] \[\tau_{r\theta} = \tau_{\theta r} =-\frac{P}{N}\left(r+\frac{a^2b^2}{r^3}-\frac{a^2+b^2}{r}\right)\cos (\theta),\]

where $a$ is beam inner radius, $b$ is beam outer radius and $P$ is applied shear force.

Example of usage

functions
{
    analyticalSolution
    {
        type    curvedCantileverAnalyticalSolution;
    
        rInner  0.31;
        rOuter  0.33;
   
        force   4;
        E       100;
        nu      0.3;
    }
}
  

Arguments
- rInner inner beam radius;
- rOuter outer beam radius;
- force applied force in (N/m) at beam free end;
- E Young's modulus;
- nu Poisson's ratio.
Optional arguments
- None.
Outputs
- Analytical solution for stress tensor field analyticalStress in time directories.
Tutorial case in which it is used:
solids/linearElasticity/curvedCantilever

`hotCylinderAnalyticalSolution`

Function object purpose
To generate the analytical solution (temperature and stress) fields for the case of a thermally-stressed pipe/cylinder. The analytical solution is taken from [ Timoshenko, Stephen. Theory of elasticity. Oxford, 1951.]:

\[\sigma_r = \frac{\alpha E \Delta T}{2(1-\nu)\ln\frac{b}{a}}\left( -\ln \frac{b}{r} - \frac{a^2}{(b^2-a^2)}\left( 1-\frac{b^2}{r^2} \right) \ln \frac{b}{a} \right),\] \[\sigma_{\theta} = \frac{\alpha E \Delta T}{2(1-\nu)\ln\frac{b}{a}}\left( 1-\ln \frac{b}{r} - \frac{a^2}{(b^2-a^2)}\left( 1+\frac{b^2}{r^2} \right) \ln \frac{b}{a} \right),\] \[T = \displaystyle{\frac{\Delta T}{\ln \frac{b}{a}} \ln \frac{b}{r}},\]

where $a$ is pipe inner radius, $b$ is pipe outer radius, $\nu$ is Poisson's ratio, $E$ is Young modulus, $\alpha$ is coefficient of linear thermal expansion and $\Delta T$ is temperature difference between inner and outer pipe surface.

Example of usage

functions
{
    analyticalHotCylinder
    {
        type    hotCylinderAnalyticalSolution;
  
        rInner  0.5;
        rOuter  0.7;
  
        TInner  100;
        TOuter  0;
  
        E       200e9;
        nu      0.3;
        alpha   1e-5;
    }
}
  

Arguments
- rInner inner pipe radius;
- rOuter outer pipe radius;
- TInner temperature on the inner pipe surface;
- TOuter temperature on the outer pipe surface;
- E Young's modulus;
- nu Poisson's ratio;
- alpha coefficient of linear thermal expansion.
Optional arguments
- None.
Outputs
- Analytical solution for hoop stress field analyticalHoopStress in time directories.
- Analytical solution for radial stress field analyticalRadialStress in time directories.
- Analytical solution for temperature field analyticalT in time directories.
Tutorial case in which it is used:
solids/thermoelasticity/hotCylinder/hotCylinder

`plateHoleAnalyticalSolution`

Function object purpose
To generate the analytical solution fields for the "hole in a plate" case. The analytical solution for the stress field is taken from [ Timoshenko, Stephen. Theory of elasticity. Oxford, 1951.]:

\[\sigma_r = \frac{T}{2}\left( 1-\frac{a^2}{r^2}\right) + \frac{T}{2} \left( 1+\frac{3a^4}{r^4} - \frac{4a^2}{r^2} \right)cos(2\theta),\] \[\sigma_{\theta} = \frac{T}{2}\left( 1+\frac{a^2}{r^2}\right) - \frac{T}{2} \left( 1+\frac{3a^4}{r^4} \right)\cos(2\theta),\] \[\sigma_{r\theta} = - \frac{T}{2} \left( 1-\frac{3a^4}{r^4} + \frac{2a^2}{r^2} \right)\sin(2\theta),\]

same in cartesian coordinates:

\[\sigma_{xx} = T \left( 1-\frac{a^2}{r^2}\left(\frac{3}{2}\cos(2\theta)+\cos(4\theta) \right) + \frac{3}{2}\frac{a^4}{r^4}\cos(4\theta) \right),\] \[\sigma_{yy} = T \left( -\frac{a^2}{r^2}\left(\frac{1}{2}\cos(2\theta)-\cos(4\theta) \right) - \frac{3}{2}\frac{a^4}{r^4}\cos(4\theta) \right),\] \[\sigma_{xy} = T \left( -\frac{a^2}{r^2}\left(\frac{1}{2}\cos(2\theta)+\sin(4\theta) \right) + \frac{3}{2}\frac{a^4}{r^4}\sin(4\theta) \right).\]

Displacement field in cartesian coordinates:

\[u_x = \frac{Ta}{8\mu}\left( \frac{r}{a}(\kappa+1)\cos\theta+\frac{2a}{r}\left((1+\kappa)\cos(\theta)+\cos (3\theta)\right)-\frac{2a^3}{r^3}\cos(3\theta) \right),\] \[u_y = \frac{Ta}{8\mu}\left( \frac{r}{a}(\kappa-3)\sin\theta+\frac{2a}{r}\left((1-\kappa)\sin(\theta)+\sin (3\theta)\right)-\frac{2a^3}{r^3}\sin(3\theta) \right),\]

where $a$ is hole radius, $T$ is far field traction in $x$ direction, $\nu$ is Poisson's ratio, $\mu$ is shear modulus and $\kappa$ parameter is equal to $3-4\nu$.

Example of usage

functions
{
    plateHoleSolution
    {
        type    plateHoleAnalyticalSolution;
    
        holeRadius  1;
        farFieldTractionX  1e6;
   
        E       100;
        nu      0.3;
          
        //Optional
        cellDisplacement true;
        pointDisplacement true;
        cellStress true;
        pointStress true;
    }
}
  

Arguments
- holeRadius radius of the hole centred on the origin;
- farFieldTractionX far-field traction in the $x$ direction;
- E Young's modulus;
- nu Poisson's ratio.
Optional arguments
- cellDisplacement write analytical solution for cell-centred displacement field; default is true;
- pointDisplacement write analytical solution for vertex-centred displacement field; default is true;
- cellStress write analytical solution for cell-centred stress field; default is true;
- pointStress write analytical solution for vertex-centred stress field; default is true;
Outputs
- Analytical solution for stress tensor field analyticalStress in time directories;
- Analytical solution for the displacement field analyticalD in time directories.
Tutorial case in which it is used:
None.

`fsiConvergenceData`

Function object purpose
Reports the number of outer correctors required at each time-step to reach convergence of the FSI coupling.

Example of usage

functions
{
    fsiConvData
    {
        type fsiConvergenceData;     
        //Optional
        region fluid;
    }
}
  

Arguments
- None
Optional arguments
- region name; the default value is set to region0.
Outputs
- Output file: postProcessing/0/fsiConvergenceData.dat ;
- Output file format:
```
# Time nFsiCorrectors
0 15
1 12
...
```
Tutorial case in which it is used:
fluidSolidInteraction/heatTransfer/flowOverHeatedPlate
fluidSolidInteraction/heatTransfer/thermalCavity

`hydrostaticPressure`

Function object purpose
Outputs the hydrostatic component of the stress tensor field
\[\sigma_h= -\frac{1}{3}\text{tr}( \mathbf{\sigma}).\]
where $\mathbf{\sigma}$ is stress tensor.

Example of usage

functions
{
    meanStress
    {
        type    hydrostaticPressure;
    }
}
  

Arguments
- None.
Optional arguments
- None.
Outputs
- Scalar field hydrostaticPressure in time directories;
- Log at the end of each time-step:
```
Hydrostatic pressure: min = 150, max = 500
```
Tutorial case in which it is used:
None

`principalStresses`

Function object purpose
Calculate and write principal stress fields. It assumed that the stress tensor is called sigma or sigmaCauchy. Three vector fields are created: sigmaMax, sigmaMid, sigmaMin. sigmaMax is the most positive/tensile principal stress multiplied by the corresponding principal direction; sigmaMid is the middle principal stress multiplied by the corresponding principal direction; sigmaMin is the most negative/compressive principal stress multiplied by the corresponding principal direction.

Example of usage

functions
{
    principalStresses1
    {
        type principalStresses;
  
        // Optional
        compressionPositive   true;
        region    region0;
    }
}
  

Arguments
- None.
Optional arguments
- region name; the default value is set to region0.
- compressionPositive specify if compression is considered positive; default is false.
Outputs
- sigmaMinDir vector field in time directories;
- sigmaMin scalar field in time directories;
- sigmaMaxDir vector field in time directories;
- sigmaMax scalar field in time directories;
- sigmaMidDir vector field in time directories;
- sigmaMid scalar field in time directories;
- sigmaDIff field; difference between sigmaMax and sigmaMin fields.
Tutorial case in which it is used:
None

`solidDisplacements`

Function object purpose
Reports the minimum and maximum values of displacement components together with the arithmetic average value of the displacement.

Example of usage

functions
{
    patchDisplacements
    {
        type    solidDisplacements;
        historyPatch     top;
    }
}
  

Arguments
- historyPatch is the name of the patch.
  
  Note
  
  The non-existing patch name will not stop the simulation.
Optional arguments
- None.

Outputs

Output file: postProcessing/0/solidDisplacements<historyPatch>.dat ;

Output file format:

# Time minX minY minZ maxX maxY maxZ avX avY avZ
1 1 1 1 3 4 5 1.2 2 4
2 1 1 1 3 3 2 1 3 5
...
  

Tutorial case in which it is used:
solids/hyperelasticity/longWall
solids/elastoplasticity/neckingBar

`solidForces`

Function object purpose
Reports the overall force $\mathbf{f}$ and normal force $f_n$ for specified patch:
\[\mathbf{f} = \sum_{N_f} \mathbf{n}_f \cdot \boldsymbol{\sigma}_f, \qquad {f}_n = \mathbf{f} \cdot \mathbf{n}_f,\]
where $\mathbf{n}_f$ is outward unit normal vector, $\boldsymbol{\sigma}$ is Cauchy stress and $N_f$ is number of faces on specified patch. Subscript $f$ is used to denote face centre value. In the case of TL formulation, the current boundary unit normal vector $\mathbf{n}_f$ is calculated using total deformation gradient and its Jacobian $J \: \mathbf{F}_{inv}^T \cdot \mathbf{n}_f$.

Example of usage

functions
{
    patchForce
    {
        type    solidForces;
        historyPatch     top;
    }
}
  

Arguments
- historyPatch is the name of the patch.
  
  Note
  
  The non-existing patch name will not stop the simulation.
Optional arguments
- None.
Outputs
- Output file: postProcessing/0/solidForces<historyPatch>.dat;
- Output file format:
```
# Time forceX forceY forceZ normalForce
1 40 40 50 48
2 40 60 70 45
...
```
Tutorial case in which it is used:
solids/linearElasticity/punch
solids/linearElasticity/plateHole
solids/elastoplasticity/pipeCrush
solids/elastoplasticity/uniaxialTension
solids/elastoplasticity/impactBar
solids/elastoplasticity/simpleShear
solids/elastoplasticity/perforatedPlate
solids/elastoplasticity/cylinderCrush
solids/abaqusUMATs/plateHoleTotalDispUMAT
solids/hyperelasticity/plateHoleTotalLag
solids/hyperelasticity/cylinderCrush
fluidSolidInteraction/3dTube
fluidSolidInteraction/3dTubeRobin
fluidSolidInteraction-preCICE/3dTube

`solidForcesDisplacements`

Function object purpose
Reports the overall force $f$ vs arithmetic average displacement $\bar{\mathbf{u}}$ for specified patch:
\[\mathbf{f} = \sum_{N_f} \mathbf{n}_f \cdot \boldsymbol{\sigma}_f,\] \[\bar{\mathbf{u}} = \frac{1}{N_f} \left( \sum_{N_f} \mathbf{u}_f \right),\]
where $\mathbf{n}_f$ is outward unit normal vector, $\boldsymbol{\sigma}$ is Cauchy stress, $\mathbf{u}$ is displacement vector and $N_f$ is number of faces on specified patch. Subscript $f$ is used to denote face centre value. In the case of TL formulation, the current boundary unit normal vector $\mathbf{n}_f$ is calculated using total deformation gradient and its Jacobian $J \: \mathbf{F}_{inv}^T \cdot \mathbf{n}_f$.

Example of usage

functions
{
    patchForceDisplacements
    {
        type    solidForcesDisplacements;
        historyPatch     top;
    }
}
  

Arguments
- historyPatch is the name of the patch.
  
  Note
  
  The non-existing patch name will not stop the simulation.
Optional arguments
- None.

Outputs

Output file: postProcessing/0/solidForcesDisplacements<historyPatch>.dat;

Output file format:

# Time dispX dispY dispZ forceX forceY forceZ
1 0.1 0.1 0.1 40 40 50
2 0.1 0.2 0.2 40 60 70
...
  

Tutorial case in which it is used:
solids/linearElasticity/punch
solids/linearElasticity/plateHole
solids/elastoplasticity/pipeCrush
solids/elastoplasticity/perforatedPlate
solids/elastoplasticity/cylinderCrush
solids/abaqusUMATs/plateHoleTotalDispUMAT
solids/hyperelasticity/plateHoleTotalLag

`solidKineticEnergy`

Function object purpose
Reports the kinetic energy $E_k$ of a solid:
\[E_k = \displaystyle{\frac{1}{2} \sum_{N_P} \rho_P \;( \mathbf{v}_P \cdot \mathbf{v}_P) \; V_p},\]
where $\rho$ is density, $\mathbf{v}$ is velocity, $N_P$ is number of cells and $V$ is volume. Subscript $P$ is used to denote cell centre value.

Example of usage

functions
{
    kineticEnergy
    {
        type    solidKineticEnergy;
    }
}
  

Arguments
- None.
Optional arguments
- None.
Outputs
- Output file: postProcessing/0/solidKineticEnergy.dat ;
- Output file format:
```
# Time kineticEnergy
1 0.10
2 0.11
...
```
Tutorial case in which it is used:
None

`solidPointDisplacement`

Function object purpose
Reports displacement vector value at closest mesh point to the specified point. The closest mesh point is determined using Euclidean distance. The displacement field (defined at cell centres) is interpolated to mesh points using the least squares interpolation.

Example of usage

functions
{
    pointDisp
    {
        type    solidPointDisplacement;
        point   (105e-6 50e-6 0);
        // Optional
        region  "solid";
    }
}
  

Arguments
- point - monitoring point vector.
Optional arguments
- region name; in the case of structural analysis, the default value is set to solid otherwise it is region0.
Outputs
- Output file: postProcessing/0/solidPointDisplacement_<functionObjectName>.dat ;
- Output file format:
```
# Time Dx Dy Dz magD
1 0.10 0.20 0.10 0.24494897
2 0.11 0.20 0.22 0.26551836
...
```
Tutorial case in which it is used:
solids/hyperelasticity/cylinderCrush
solids/hyperelasticity/cylindricalPressureVessel
solids/abaqusUMATs/plateHoleTotalDispUMAT
solids/elastoplasticity/perforatedPlate
solids/elastoplasticity/cooksMembrane
solids/viscoelasticity/viscoTube
solids/linearElasticity/cooksMembrane
solids/abaqusUMATs/plateHoleTotalDispUMAT
solids/linearElasticity/wobblyNewton solids/linearElasticity/plateHole
fluidSolidInteraction/beamInCrossFlow fluidSolidInteraction/HronTurekFsi3
fluidSolidInteraction/3dTubeRobin

`solidPointDisplacementAlongLine`

Function object purpose
Reports reports displacement value along a line specified by the user. The displacement field (defined at cell centres) is interpolated to mesh points using the least squares interpolation.

Example of usage

functions
{
    pointStress
    {
        type    solidPointDisplacementAlongLine;
        startPoint   (0 0 0);
        endPoint	 (2 0 0);
  
        // Optional
        minDist  1e-5;
        region  "solid";
    }
}
  

Warning

This function object is currently only implemented for serial run!

Arguments
- startPoint line start point;
- endPoint line end point.
Optional arguments
- minDist maximum distance at which mesh point will be included in line plot;
- region name; in the case of structural analysis, the default value is set to solid otherwise it is region0.

Outputs

Output file: postProcessing/0/solidPointDisplacementAlongLine<functionObjectName>.dat ;

Output file format:

# PointID PointCoord Dx Dy Dz mag
152 (0.2 0.3 0.5) 0.1 0.2 0.1 0.2449
258 (0.4 0.3 0.5) 0.1 0.2 0.2 0.3
...
  

Tutorial case in which it is used:
None.

`solidPointStress`

Function object purpose
Reports stress value at closest mesh point to the specified point. The closest mesh point is determined using Euclidean distance. The stress field (defined at cell centres) is interpolated to mesh points using the least squares interpolation.

Example of usage

functions
{
    pointStress
    {
        type    solidPointStress;
        point   (0.0075 0 0);
        // Optional
        region  "solid";
    }
}
  

Arguments
- point - monitoring point vector.
Optional arguments
- region name; in the case of structural analysis, the default value is set to solid otherwise it is region0.

Outputs

Output file: postProcessing/0/solidPointStress_<functionObjectName>.dat ;

Output file format:

# Time XX XY XZ YY YZ ZZ
1 1e6 1e6 1e6 1e6 2e6 3e6 
2 1e6 3e6 2e6 2e6 2e6 3e6
...
  

Tutorial case in which it is used:
solids/viscoelasticity/viscoTube

`solidPointTemperature`

Function object purpose
Reports temperature value at closest mesh point to the specified point. The closest mesh point is determined using Euclidean distance. The temperature field (defined at cell centres) is interpolated to mesh points using the least squares interpolation.

Example of usage

functions
{
    pointTemp
    {
        type    solidPointTemperature;
        point   (105e-6 50e-6 0);
        // Optional
        region  "solid";
    }
}
  

Arguments
- point - monitoring point vector.
Optional arguments
- region name; in the case of structural analysis, the default value is set to solid otherwise it is region0.
Outputs
- Output file: postProcessing/solidPointTemperature_<functionObjectName>.dat ;
- Output file format:
```
# Time value
1 225
2 226
...
```
Tutorial case in which it is used:
None

`solidPotentialEnergy`

Function object purpose
Reports the potential energy of a solid:
\[h_P = \frac{\mathbf{g}}{|\mathbf{g}|} \cdot (\mathbf{r}_P+\mathbf{u}_P-\mathbf{r}_{ref}),\] \[E_p = \sum_{N_P} \rho_P \: |\mathbf{g}| \: h_P V_P ,\]
where $\rho$ is density, $\mathbf{u}$ is displacement, $N_P$ is number of cells, $g$ is gravity, $V$ is volume, $\mathbf{r}_{ref}$ is reference point with zero potential energy and $\mathbf{r}_P$ is positional vector of cell centroid. Subscript $P$ is used to denote cell centre value.

Example of usage

functions
{
    potentialEnergy
    {
        type    solidPotentialEnergy;
        referencePoint   (10 50 0);
    }
}
  

Arguments
- referencePoint is a coordinate at which the potential energy is zero.
  
  Note
  
  The value for the uniform gravity field $g$ is specified at constant/g!
Optional arguments
- None.
Outputs
- Output file: postProcessing/0/solidPotentialEnergy.dat ;
- Output file format:
```
# Time potentialEnergy
1 500
2 520
...
```
Tutorial case in which it is used:
None

Warning

solidPotentialEnergy is currently implemented only for linear geometry solid models!

`solidStresses`

Function object purpose
Reports the arithmetic average stress on the patch of a solid:
\[\bar{\boldsymbol{\sigma}} = \frac{1}{N_f} \left( \sum_{N_f} \boldsymbol{\sigma}_f \right)\]
where subscript $f$ is used to denote patch face centre value, $\boldsymbol{\sigma}$ is Cauchy stress tensor and $N_f$ is overall number of patch faces

Example of usage

functions
{
    topStress
    {
       type         solidStresses;
       historyPatch top;
    }
}
  

Arguments
- historyPatch is the name of the patch.
  
  Note
  
  The non-existing patch name will not stop the simulation.
Optional arguments
- None.

Outputs

Output file: postProcessing/0/solidStresses<historyPatch>.dat ;

Output file format:

# Time XX XY XZ YY YZ ZZ
1 1e6 1e6 1e6 1e6 2e6 3e6 
2 1e6 3e6 2e6 2e6 2e6 3e6
...
  

Tutorial case in which it is used:
solids/elastoplasticity/cylinderExpansion solids/hyperelasticity/longWall

`solidTorque`

Function object purpose
Reports torque on the specified patch about the given axis:
\[\mathbf{r}_m = (\mathbf{r}_f - \mathbf{r}_{pa}) - \mathbf{a}(\mathbf{a} \cdot (\mathbf{r}_f - \mathbf{r}_{pa})),\] \[\text{Torque} = \sum_{N_f} \mathbf{a} \cdot (\mathbf{r}_m \times (\mathbf{S}_f \cdot \boldsymbol{\sigma})),\]
where $\mathbf{r}_{pa}$ is point on axis, $\mathbf{a}$ is axis direction, $\mathbf{r}_f$ is face centre vector, $\mathbf{S}_f$ boundary face area vector, $\boldsymbol{\sigma}_f$ Cauchy stress vector and $N_f$ number of boundary patch faces. In the case of TL formulation, the current boundary face area vector $\mathbf{S}_f$ is calculated using total deformation gradient and its Jacobian $J \: \mathbf{F}_{inv}^T \cdot \mathbf{S}_f$.

Example of usage

functions
{
    patchTorque
    {
        type    solidTorque;
          
        historyPatch     right;
        pointOnAxis      (0 0 0);
        axisDirection    (0 0 1);
    }
}
  

Arguments
- pointOnAxis - point on axis;
- axisDirestion - axis vector, does not require to be normalised;
- historyPatch is the name of the patch.
  
  Note
  
  The non-existing patch name will not stop the simulation.
Optional arguments
- None
Outputs
- Output file: postProcessing/0/solidTorque<historyPatch>.dat ;
- Output file format:
```
# Time torqueX torqueY torqueZ
1 10 12 13
2 12 13 13
...
```
Tutorial case in which it is used:
None

`solidTractions`

Function object purpose
Writes boundary traction as a vector field:
\[\text{Updated Lagrangian (UL):} \qquad \mathbf{t}_b = \mathbf{n}_b \cdot \boldsymbol{\sigma}\] \[\text{Total Lagrangian (TL):} \qquad \mathbf{t}_b = \frac{\textbf{F}_{inv}^T \cdot \textbf{n}_b}{|\textbf{F}_{inv}^T \cdot \textbf{n}_b|} \cdot \boldsymbol{\sigma}\] \[\text{Small strain approach:} \qquad \mathbf{t}_b = \mathbf{n}_b \cdot \boldsymbol{\sigma}\]
where $\boldsymbol{\sigma}$ is Cauchy stress tensor, $\mathbf{n}_b$ boundary outward unit vector and $\mathbf{F}_{inv}$ inverse of total deformation gradient. Note that in case of UL formulation boundary normal $\mathbf{n}_b$ is in current configuration while in the case of TL approach it is in initial configuration.

Example of usage

functions
{
    patchTractions
    {
        type    solidTractions;
    }
}
  

Arguments
- None.
Optional arguments
- None.
Outputs
- Vector field traction in time directories;
Tutorial case in which it is used:
None

`stressTriaxiality`

Function object purpose
Outputs the stress triaxiality field (mean i.e. hydrostatic stress divided by equivalent stress):
\[\sigma_h= -\frac{1}{3}\text{tr}( \mathbf{\boldsymbol{\sigma}}).\] \[T.F. = \frac{-\sigma_h}{\sigma_{eq}}\]
where $T.F$ is triaxiality factor, $\boldsymbol{\sigma}$ is Cauchy stress and $\mathbf{\sigma}_{eq}$ is Von Mises equivalent stress.

Example of usage

functions
{
    triaxility
    {
        type    stressTriaxility;
        // Optional
        region  "region0";
    }
}
  

Arguments
- None.
Optional arguments
- region name; the default value is set to region0.
Outputs
- Scalar field stressTriaxiality in time directories;
- Log at the end of each time-step:
```
Stress triaxiality: min = 0.2, max = 0.4
```
Tutorial case in which it is used:
None

`transformStressToCylindrical`

Function object purpose
Transform stress tensor to cylindrical coordinate system: $\sigma_{transformed} = \mathbf{R} \cdot \sigma \cdot \mathbf{R}^{T},$

where $\mathbf{R}$ is rotation tensor.

Example of usage

functions
{
    transformStressToCylindrical
    {
        type        transformStressToCylindrical;
  
        origin      (0 0 0);
        axis        (0 0 1);
    }
}
  

Arguments
- origin - origin point;
- axis - axis vector, does not require to be normalised;
Optional arguments
- None.
Outputs
- Transformed sigma stress field named sigma:Transformed in time directories;
  
  When visualizing in Paraview, keep in mind that the stress components in the cylindrical coordinate system will have the same names as the ones from the Cartesian coordinate system.
  \[\boldsymbol{\sigma} = \left[\begin{array}{ccc}\sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\ & \sigma_{yy} & \sigma_{yx} \\ & & \sigma_{zz}\end{array}\right]\equiv\left[\begin{array}{ccc}\sigma_{R R} & \sigma_{R \theta} & \sigma_{R \phi} \\ & \sigma_{\theta \theta} & \sigma_{\theta \phi} \\ & & \sigma_{\phi \phi}\end{array}\right]\]
Tutorial case in which it is used:
solids/linearElasticity/pressurisedCylinder
solids/thermoelasticity/hotCylinder/hotCylinder
solids/multiMaterial/layeredPipe