# Equivalent stress

## Hosford equivalent stress

The header TFEL/Material/Hosford1972YieldCriterion.hxx introduces three functions which are meant to compute the Hosford equivalent stress and its first and second derivatives. This header is automatically included by MFront

The Hosford equivalent stress is defined by: $\sigma_{\mathrm{eq}}^{H}=\sqrt[a]{\displaystyle\frac{\displaystyle 1}{\displaystyle 2}\left({\left|\sigma_{1}-\sigma_{2}\right|}^{a}+{\left|\sigma_{1}-\sigma_{3}\right|}^{a}+{\left|\sigma_{2}-\sigma_{3}\right|}^{a}\right)}$ where $$s_{1}$$, $$s_{2}$$ and $$s_{3}$$ are the eigenvalues of the stress.

Therefore, when $$a$$ goes to infinity, the Hosford stress reduces to the Tresca stress. When $$n = 2$$ the Hosford stress reduces to the von Mises stress.

The following functions has been implemented:

• computeHosfordStress: return the Hosford equivalent stress
• computeHosfordStressNormal: return a tuple containing the Hosford equivalent stress and its first derivative (the normal)
• computeHosfordStressSecondDerivative: return a tuple containing the Hosford equivalent stress, its first derivative (the normal) and the second derivative.

#### Example

The following example computes the Hosford equivalent stress, its normal and second derivative:

stress seq;
Stensor  n;
Stensor4 dn;
std::tie(seq,n,dn) = computeHosfordStressSecondDerivative(s,a,seps);

In this example, s is the stress tensor, a is the Hosford exponent, seps is a numerical parameter used to detect when two eigenvalues are equal.

If C++-17 is available, the previous code can be made much more readable:

const auto [seq,n,dn] = computeHosfordStressSecondDerivative(s,a,seps);

## Barlat equivalent stress

The header TFEL/Material/Barlat2004YieldCriterion.hxx introduces various functions which are meant to compute the Barlat equivalent stress and its first and second derivatives. This header is automatically included by MFront for orthotropic behaviours.

The Barlat equivalent stress is defined as follows (see Barlat et al. (2005)): $\sigma_{\mathrm{eq}}^{B}= \sqrt[a]{ \frac{1}{4}\left( \sum_{i=0}^{3} \sum_{j=0}^{3} {\left|s'_{i}-s''_{j}\right|}^{a} \right) }$

where $$s'_{i}$$ and $$s''_{i}$$ are the eigenvalues of two transformed stresses $$\underline{s}'$$ and $$\underline{s}''$$ by two linear transformation $$\underline{\underline{\mathbf{L}}}'$$ and $$\underline{\underline{\mathbf{L}}}''$$: \left\{ \begin{aligned} \underline{s}' &= \underline{\underline{\mathbf{L'}}} \,\colon\,\underline{\sigma}\\ \underline{s}'' &= \underline{\underline{\mathbf{L''}}}\,\colon\,\underline{\sigma}\\ \end{aligned} \right.

The linear transformations $$\underline{\underline{\mathbf{L}}}'$$ and $$\underline{\underline{\mathbf{L}}}''$$ are defined by $$9$$ coefficients (each) which describe the material orthotropy. There are defined through auxiliary linear transformations $$\underline{\underline{\mathbf{C}}}'$$ and $$\underline{\underline{\mathbf{C}}}''$$ as follows: \begin{aligned} \underline{\underline{\mathbf{L}}}' &=\underline{\underline{\mathbf{C}}}'\,\colon\,\underline{\underline{\mathbf{M}}} \\ \underline{\underline{\mathbf{L}}}''&=\underline{\underline{\mathbf{C}}}''\,\colon\,\underline{\underline{\mathbf{M}}} \end{aligned} where $$\underline{\underline{\mathbf{M}}}$$ is the transformation of the stress to its deviator: $\underline{\underline{\mathbf{M}}}=\underline{\underline{\mathbf{I}}}-\displaystyle\frac{\displaystyle 1}{\displaystyle 3}\underline{I}\,\otimes\,\underline{I}$

The linear transformations $$\underline{\underline{\mathbf{C}}}'$$ and $$\underline{\underline{\mathbf{C}}}''$$ of the deviator stress are defined as follows: $\underline{\underline{\mathbf{C}}}'= \begin{pmatrix} 0 & -c'_{12} & -c'_{13} & 0 & 0 & 0 \\ -c'_{21} & 0 & -c'_{23} & 0 & 0 & 0 \\ -c'_{31} & -c'_{32} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & c'_{44} & 0 & 0 \\ 0 & 0 & 0 & 0 & c'_{55} & 0 \\ 0 & 0 & 0 & 0 & 0 & c'_{66} \\ \end{pmatrix} \quad \text{and} \quad \underline{\underline{\mathbf{C}}}''= \begin{pmatrix} 0 & -c''_{12} & -c''_{13} & 0 & 0 & 0 \\ -c''_{21} & 0 & -c''_{23} & 0 & 0 & 0 \\ -c''_{31} & -c''_{32} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & c''_{44} & 0 & 0 \\ 0 & 0 & 0 & 0 & c''_{55} & 0 \\ 0 & 0 & 0 & 0 & 0 & c''_{66} \\ \end{pmatrix}$

The following functions have been implemented:

• computeBarlatStress: return the Barlat equivalent stress
• computeBarlatStressNormal: return a tuple containing the Barlat equivalent stress and its first derivative (the normal)
• computeBarlatStressSecondDerivative: return a tuple containing the Barlat equivalent stress, its first derivative (the normal) and the second derivative.

### Linear transformations

To define the linear transformations, the makeBarlatLinearTransformation function has been introduced. This function takes two template parameter:

• the space dimension ($$1$$, $$2$$, and $$3$$)
• the numeric type used (automatically deduced)

This functions takes the $$9$$ coefficients as arguments, as follows:

const auto l1 = makeBarlatLinearTransformation<3>(c_12,c_21,c_13,c_31,
c_23,c_32,c_44,c_55,c_66);

Note In his paper, Barlat and coworkers uses the following convention for storing symmetric tensors:

$\begin{pmatrix} xx & yy & zz & yz & zx & xy \end{pmatrix}$

which is not consistent with the TFEL/Cast3M/Abaqus/Ansys conventions:

$\begin{pmatrix} xx & yy & zz & xy & xz & yz \end{pmatrix}$

Therefore, if one wants to uses coeficients $$c^{B}$$ given by Barlat, one shall call this function as follows:

const auto l1 = makeBarlatLinearTransformation<3>(cB_12,cB_21,cB_13,cB_31,
cB_23,cB_32,cB_66,cBB_55,cBB_44);

The TFEL/Material library also provide an overload of the makeBarlatLinearTransformation which template parameters are the modelling hypothesis and the orthotropic axis conventions. The purpose of this overload is to swap appriopriate coefficients to get a consistent definition of the linear transforamtions for all the modelling hypotheses.

# General functionalities

## $$\pi$$-plane

The $$\pi$$-plane is defined in the space defined by the three eigenvalues $$S_{0}$$, $$S_{1}$$ and $$S_{2}$$ of the stress by the following equations: $S_{0}+S_{1}+S_{2}=0$

This plane contains deviatoric stress states and is perpendicular to the hydrostatic axis. A basis of this plane is given by the following vectors: $\vec{n}_{0}= \frac{1}{\sqrt{2}}\, \begin{pmatrix} 1 \\ -1 \\ 0 \end{pmatrix} \quad\text{and}\quad \vec{n}_{1}= \frac{1}{\sqrt{6}}\, \begin{pmatrix} -1 \\ -1 \\ 2 \end{pmatrix}$

This plane is used to characterize the iso-values of equivalent stresses which are not sensitive to the hydrostatic pression.

Various functions are available:

• projectOnPiPlane: this function projects a stress state on the $$\pi$$-plane.
• buildFromPiPlane: this function builds a stress state, defined by its three eigenvalues, from its coordinate in the $$\pi$$-plane.

# Orthotropic axes convention

Most finite element solver can’t have a uniq definition of the orthotropic axes for all the modelling hypotheses.

For example, one can define a pipe using the following axes definition:

• $$\left(rr,zz,tt,...\right)$$ in $$3D$$, $$2D$$ axysymmetric, $$1D$$ axisymmetric generalised plane strain or generalised plane stress.
• $$\left(rr,tt,zz,...\right)$$ in $$2D$$ plane stress, strain, generalized plane strain.

With those conventions, the axial direction is either the second or the third material axis, a fact that must be taken into account when defining the stiffness tensor, the Hill tensor(s), the thermal expansion, etc.

If we were to model plates, a appropriate convention is the following:

• The first material axis is the rolling direction
• The second material axis is the in plane direction perpendicular to the rolling direction (transverse direction).
• The third material axis is the normal to the plate.

This convention is only valid for $$3D$$, $$2D$$ axysymmetric, $$1D$$ axisymmetric generalised plane strain or generalised plane stress. - $$\left(rr,tt,zz,...\right)$$ in $$2D$$ plane stress, strain, generalized plane strain.

With those conventions, the axial direction is either the second or the third material axis, a fact that must be taken into account when defining the stiffness tensor, the Hill tensor(s), the thermal expansion, etc.

If we were to model plates, a appropriate convention is the following:

• The first material axis is the rolling direction
• The second material axis is the in plane direction perpendicular to the rolling direction (transverse direction).
• The third material axis is the normal to the plate.

This convention is only valid for $$3D$$, $$2D$$ axysymmetric, $$1D$$ axisymmetric generalised plane strain or generalised plane stress. - $$\left(rr,tt,zz,...\right)$$ in $$2D$$ plane stress, strain, generalized plane strain.

With those conventions, the axial direction is either the second or the third material axis, a fact that must be taken into account when defining the stiffness tensor, the Hill tensor(s), the thermal expansion, etc.

If we were to model plates, a appropriate convention is the following:

• The first material axis is the rolling direction
• The second material axis is the in plane direction perpendicular to the rolling direction (transverse direction).
• The third material axis is the normal to the plate.

By definition, this convention is only valid for $$3D$$, $$2D$$ plane stress, strain and generalized plane strain modelling hypotheses.

Barlat, F., H. Aretz, J. W. Yoon, M. E. Karabin, J. C. Brem, and R. E. Dick. 2005. “Linear Transfomation-Based Anisotropic Yield Functions.” International Journal of Plasticity 21 (5):1009–39. https://doi.org/10.1016/j.ijplas.2004.06.004.