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 have 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 [1]): \[ \sigma_{\mathrm{eq}}^{B}= \sqrt[a]{ \frac{1}{4}\left( \sum_{i=0}^{3} \sum_{j=0}^{3} {\left|s'_{i}-s''_{j}\right|}^{a} \right) } \]

where \(s'_{i}\) and \(s''_{i}\) are the eigenvalues of two transformed stresses \(\underline{s}'\) and \(\underline{s}''\) by two linear transformations \(\underline{\underline{\mathbf{L}}}'\) 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. They 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 parameters:

• 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 use 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 use coefficients \(c^{B}\) 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 provides an overload of the makeBarlatLinearTransformation whose template parameters are the modelling hypothesis and the orthotropic axis conventions. The purpose of this overload is to swap appropriate coefficients to get a consistent definition of the linear transformations for all the modelling hypotheses.

General functionalities

Computation of the inverse of the Langevin function

The inverse Langevin function is used in many statistically based network behaviours describing rubber-like materials.

The Langevin function \(\mathcal{L}\) is defined as follows: \[ \mathcal{L}\left(x\right)=\displaystyle\frac{\displaystyle 1}{\displaystyle \coth\left(x\right)}-\displaystyle\frac{\displaystyle 1}{\displaystyle x} \]

The complexity of the inverse Langevin function \(\mathcal{L}^{-1}\left(x\right)\) motivated the development of various approximations [2–4].

Figure 1 compares those approximations. The approximations of Bergström and Boyce [3] and Jedynak [4] are indistinguishable. See Jedynak for a quantitative discussion of the error generated by those approximations [4]. It is worth noting that all those approximations mostly differ near the pole of the inverse Langevin function \(\mathcal{L}^{-1}\left(x\right)\).

The InverseLangevinFunctionApproximations enumeration lists the approximations that have been implemented and that can be evaluated in a constexpr context:

• The value COHEN_1991 is associated with the approximation proposed by Cohen [2]: \[ \mathcal{L}^{-1}\left(x\right) \approx y\displaystyle\frac{\displaystyle 3-y^{2}}{\displaystyle 1-y^{2}}. \]
• The value JEDYNAK_2015 is associated with the approximation proposed by Jedynak [4]: \[ \mathcal{L}^{-1}\left(x\right) \approx y \, \displaystyle\frac{\displaystyle c_{0} + c_{1}\,y + c_{2}\,y^{2}}{\displaystyle 1 + d_{1}\,y + d_{2}\,y^{2}} \]
• KUHN_GRUN_1942 or MORCH_2022 are associated with a Taylor expansion of \(\mathcal{L}^{-1}\left(x\right)\) at \(0\): \[ \mathcal{L}^{-1}\left(x\right) \approx y\,P\left(y^{2}\right) \quad\text{with}\quad P\left(y^{2}\right)=\sum_{i=0}^{9}c_{i}\,\left(y^{2}\right)^{i} \] where \(P\) is a \(9\)th order polynomial. Hence, the Taylor expression is of order \(19\).

The computeApproximateInverseLangevinFunction computes one approximation of the inverse Langevin function and the computeApproximateInverseLangevinFunctionAndDerivative function computes an approximation of the inverse Langevin function and its derivative. These functions have two template parameters: the approximation selected and the numeric type to be used. By default, the JEDYNAK_2015 approximation is used and the numeric type can be deduced from the type of the argument.

The approximation proposed by Bergström and Boyce [3] is given by the following function: \[ \mathcal{L}^{-1}\left(x\right) \approx \left\{ \begin{aligned} c_{1} \tan\left(c_{2} \, x\right) + c_{3} \, x &\quad\text{if}\quad \left|x\right| \leq c_{0}\\ \displaystyle\frac{\displaystyle 1}{\displaystyle \mathop{sign}\left(x\right)-x}&\quad\text{if}\quad \left|x\right| > c_{0} \end{aligned} \right. \]

The computeBergstromBoyce1998ApproximateInverseLangevinFunction function computes this approximation and the computeBergstromBoyce1998ApproximateInverseLangevinFunctionAndDerivative function computes this function and its derivative. These functions can’t be declared constexpr because the tangent function is not constexpr. These functions have one template parameter, the numeric type to be used. This template parameter can be automatically deduced from the type of the argument.

Example of usage

using ApproximationFunctions = InverseLangevinFunctionApproximations;
// compute Cohen's approximation of the inverse Langevin function
const auto v = computeApproximateInverseLangevinFunction<
            ApproximationFunctions::COHEN_1991>(y)
// compute Jedynak's approximation of the inverse Langevin function and its derivative
const auto [f, df] = computeApproximateInverseLangevinFunctionAndDerivative(y)

\(\pi\)-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 pressure.

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 solvers can’t have a unique definition of the orthotropic axes valid 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 \(1D\) axisymmetric generalised plane stress.
• \(\left(rr,tt,zz,...\right)\) in \(2D\) plane stress, \(2D\) plane strain and \(2D\) generalized plane strain.

With those conventions, named Pipe in MFront, 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.

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.

If we were to model plates, an 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, named Plate in MFront is only valid for \(3D\), \(2D\) plane stress, \(2D\) plane strain and \(2D\) generalized plane strain modelling hypotheses.

Isotropic elastic moduli

Three data structures are defined to represent the isotropic moduli of an isotropic material:

• KGModuli (attributes: kappa,mu)
• YoungNuModuli (attributes: young,nu)
• LambdaMuModuli (attributes: lambda,mu)

It can be constructed as follows:

const auto KG = KGModuli<stress>(ka,mu);

And its attributes can be recovered as follows:

const auto K = KG.kappa;
const auto G = KG.mu;

It can be converted to the other structures describing isotropic moduli as follows:

const auto Enu = KG.ToYoungNu();
const auto LambdaMu = KG.ToLambdaMu();

Moreover, some useful functions allow to go from one to another:

const auto C = computeIsotropicStiffnessTensor<stress>(Enu);
const auto KG = computeKGModuli<stress>(C);

Note that computeKGModuli makes a projection on the fourth-order tensors \(\underline{\underline{\mathbf{J}}}\) and \(\underline{\underline{\mathbf{K}}}\) if \(\underline{\underline{\mathbf{C}}}\) is not isotropic. The projected moduli are given by

\(k=\dfrac13\underline{\underline{\mathbf{C}}}::\underline{\underline{\mathbf{J}}},\qquad\mu=\dfrac1{10}\underline{\underline{\mathbf{C}}}::\underline{\underline{\mathbf{K}}}\)

It can be checked that \(\underline{\underline{\mathbf{C}}}\) is isotropic by doing

const auto eps = 1e-6;
const bool = isIsotropic(C,eps);

Here, isIsotropic first projects \(\underline{\underline{\mathbf{C}}}\) on the isotropic basis, and constructs the isotropized version of \(\underline{\underline{\mathbf{C}}}\). Then, it computes the relative difference between \(\underline{\underline{\mathbf{C}}}\) and its isotropized, by using the \(L2\)-norm. This difference is compared to the tolerance eps.