`TFEL`

, `MFront`

and `MTest`

The page declares the new functionalities of the 3.2 version of the `TFEL`

project.

The `TFEL`

project is a collaborative development of CEA and EDF dedicated to material knowledge manangement with special focus on mechanical behaviours. It provides a set of libraries (including `TFEL/Math`

and `TFEL/Material`

) and several executables, in particular `MFront`

and `MTest`

.

`TFEL`

is available on a wide variety of operating systems and compilers.

A new page dedicated to the `python`

bindings of the `TFEL`

libraries is available here.

`TFEL`

librariesThe `TFEL`

project provides several libraries. This paragraph is about updates made in those libraries.

`Evaluator`

classThe `Evaluator`

class is used to interpret textual formula, as follows:

`getValue`

methodIn the previous example, each variable value had to be set using the `setVariableValue`

method. The new overloaded version of the `getValue`

method can take a map as argument as follows:

`Operator()`

Two overloaded versions of the `Evaluator::operator()`

has been introduced as a synonyms for the `getValue`

method:

`getCxxFormula`

methodThe `getCxxFormula`

method returns a string representing the evaluation of the formula in standard `C++`

. This method takes a map as argument which describes how certain variables shall be represented. This method can be used, as follows:

The previous code displays:

This function is the basis of a new functionality of the `MFront`

code generator (inline material properties), see Section 3.1.1 for details.

The following new mathematical functions have been introduced:

`exp2`

: returns the base-2 exponential.`expm1`

: returns the base-e exponential minus one.`cbrt`

: returns the cubic root`log2`

: computes the base-2 logarithm of the argment.`log1p`

: computes the logarithm of the argment plus one.`acosh`

: computes the inverse of the hyperbolic cosine.`asinh`

: computes the inverse of the hyperbolic sine.`atanh`

: computes the inverse of the hyperbolic tangent.`erf`

: computes the error function.`erfc`

: computes the complementary error function.`tgamma`

: computes the Gamma function.`lgamma`

: compute the natural logarithm of the gamma function.`hypot`

: returns the hypotenuse of a right-angled triangle whose legs are x and y, i.e. computes (.`atan2`

: returns the principal value of the arc tangent of \(y/x\), expressed in radians.

Let \(\underline{\sigma}\) be a stress tensor. Its deviatoric part \(\underline{s}\) is:

\[ \underline{s}=\underline{\sigma}-\displaystyle\frac{\displaystyle 1}{\displaystyle 3}\,{\mathrm{tr}\left(\underline{\sigma}\right)}\,\underline{I} =\left(\underline{\underline{\mathbf{I}}}-\displaystyle\frac{\displaystyle 1}{\displaystyle 3}\,\underline{I}\,\otimes\,\underline{I}\right)\,\colon\,\underline{\sigma} \]

The deviator of a tensor can be computed using the `deviator`

function.

As it is a second order tensor, the stress deviator tensor also has a set of invariants, which can be obtained using the same procedure used to calculate the invariants of the stress tensor. It can be shown that the principal directions of the stress deviator tensor \(s_{ij}\) are the same as the principal directions of the stress tensor \(\sigma_{ij}\). Thus, the characteristic equation is

\[ \left| s_{ij}- \lambda\delta_{ij} \right| = -\lambda^3+J_1\lambda^2-J_2\lambda+J_3=0, \]

where \(J_1\), \(J_2\) and \(J_3\) are the first, second, and third *deviatoric stress invariants*, respectively. Their values are the same (invariant) regardless of the orientation of the coordinate system chosen. These deviatoric stress invariants can be expressed as a function of the components of \(s_{ij}\) or its principal values \(s_1\), \(s_2\), and \(s_3\), or alternatively, as a function of \(\sigma_{ij}\) or its principal values \(\sigma_1\), \(\sigma_2\), and \(\sigma_3\). Thus,

\[ \begin{aligned} J_1 &= s_{kk}=0,\, \\ J_2 &= \textstyle{\frac{1}{2}}s_{ij}s_{ji} = \displaystyle\frac{\displaystyle 1}{\displaystyle 2}{\mathrm{tr}\left(\underline{s}^2\right)}\\ &= \displaystyle\frac{\displaystyle 1}{\displaystyle 2}(s_1^2 + s_2^2 + s_3^2) \\ &= \displaystyle\frac{\displaystyle 1}{\displaystyle 6}\left[(\sigma_{11} - \sigma_{22})^2 + (\sigma_{22} - \sigma_{33})^2 + (\sigma_{33} - \sigma_{11})^2 \right ] + \sigma_{12}^2 + \sigma_{23}^2 + \sigma_{31}^2 \\ &= \displaystyle\frac{\displaystyle 1}{\displaystyle 6}\left[(\sigma_1 - \sigma_2)^2 + (\sigma_2 - \sigma_3)^2 + (\sigma_3 - \sigma_1)^2 \right ] \\ &= \displaystyle\frac{\displaystyle 1}{\displaystyle 3}I_1^2-I_2 = \frac{1}{2}\left[{\mathrm{tr}\left(\underline{\sigma}^2\right)} - \frac{1}{3}{\mathrm{tr}\left(\underline{\sigma}\right)}^2\right],\,\\ J_3 &= \det\left(\underline{s}\right) \\ &= \displaystyle\frac{\displaystyle 1}{\displaystyle 3}s_{ij}s_{jk}s_{ki} = \displaystyle\frac{\displaystyle 1}{\displaystyle 3} {\mathrm{tr}\left(\underline{s}^3\right)}\\ &= \displaystyle\frac{\displaystyle 1}{\displaystyle 3}(s_1^3 + s_2^3 + s_3^3) \\ &= s_1s_2s_3 \\ &= \displaystyle\frac{\displaystyle 2}{\displaystyle 27}I_1^3 - \displaystyle\frac{\displaystyle 1}{\displaystyle 3}I_1 I_2 + I_3 = \displaystyle\frac{\displaystyle 1}{\displaystyle 3}\left[{\mathrm{tr}\left(\underline{\sigma}^3\right)} - {\mathrm{tr}\left(\underline{\sigma}^2\right)}{\mathrm{tr}\left(\underline{\sigma}\right)} +\displaystyle\frac{\displaystyle 2}{\displaystyle 9}{\mathrm{tr}\left(\underline{\sigma}\right)}^3\right]. \end{aligned} \]

where \(I_{1}\), \(I_{2}\) and \(I_{3}\) are the invariants of \(\underline{\sigma}\).

\(J_{2}\) and \(J_{3}\) are building blocks for many isotropic yield critera. Classically, \(J_{2}\) is directly related to the von Mises stress \(\sigma_{\mathrm{eq}}\):

\[ \sigma_{\mathrm{eq}}=\sqrt{\displaystyle\frac{\displaystyle 3}{\displaystyle 2}\,\underline{s}\,\colon\,\underline{s}}=\sqrt{3\,J_{2}} \]

The first and second derivatives of \(J_{2}\) with respect to \(\sigma\) can be trivially implemented, as follows:

```
constexpr const auto id = stensor<N,real>::Id();
constexpr const auto id4 = st2tost2<N,real>::Id();
// first derivative of J2
const auto dJ2 = deviator(sig);
// second derivative of J2
const auto d2J2 = eval(id4-(id^id)/3);
```

In comparison, the computation of the first and second derivatives of \(J_{3}\) with respect to \(\sigma\) are more cumbersome. In previous versions `TFEL`

, one had to write:

```
constexpr const auto id = stensor<N,real>::Id();
constexpr const auto id4 = st2tost2<N,real>::Id();
const auto I1 = trace(sig);
const auto I2 = (I1*I1-trace(square(sig)))/2;
const auto dI2 = I1*id-sig;
const auto dI3 = computeDeterminantDerivative(sig);
const auto d2I2 = (id^id)-id4;
const auto d2I3 = computeDeterminantSecondDerivative(sig);
// first derivative of J3
const auto dJ3 = eval((2*I1*I1/9)*id-(I2*id+I1*dI2)/3+dI3);
// second derivative of J3
const auto d2J3 = eval((4*I1/9)*(id^id)-((id^dI2)+(dI2^id)+i1*d2I2)/3+d2I3);
```

More efficient implementations are now available using the `computeDeviatorDeterminantDerivative`

and `computeDeviatorDeterminantSecondDerivative`

functions:

```
// first derivative of J3
const auto dJ3 = computeDeviatorDeterminantDerivative(sig);
// second derivative of J3
const auto d2J3 = computeDeviatorDeterminantSecondDerivative(sig);
```

By definition, \(J_{2}\) and \(J_{3}\) are the second and third invariants of the deviatoric part \(\underline{s}\) of the stress tensor \(\underline{\sigma}\) (see also Section 2.1.2):

\[ \left\{ \begin{aligned} J_2 &= \displaystyle\frac{\displaystyle 1}{\displaystyle 2}{\mathrm{tr}\left(\underline{s}^2\right)}\\ J_3 &= \det(\underline{s}) \\ \end{aligned} \right. \]

The first and second derivatives of \(J_{2}\) with respect to the stress tensor \(\underline{\sigma}\) are trivially computed and implemented (see Section 2.1.2).

The first and second derivatives of \(J_{2}\) with respect to the stress tensor \(\underline{\sigma}\) can be computed respectively by:

- The
`computesJ3Derivative`

function, which is a synonym for the`computeDeviatorDeterminantDerivative`

function defined in the`tfel::math`

namespace (see Section 2.1.2 for details). - The
`computeJ3SecondDerivative`

function, which is a synonym for the`computeDeviatorDeterminantSecondDerivative`

function defined in the`tfel::math`

namespace (see Section 2.1.2 for details).

Within the framework of the theory of representation, generalizations to anisotropic conditions of the invariants of the deviatoric stress have been proposed by Cazacu and Barlat (see Cazacu and Barlat (2001)):

- The generalization of \(J_{2}\) is denoted \(J_{2}^{O}\). It is defined by: \[ J_{2}^{O}= a_6\,s_{yz}^2+a_5\,s_{xz}^2+a_4\,s_{xy}^2+\frac{a_2}{6}\,(s_{yy}-s_{zz})^2+\frac{a_3}{6}\,(s_{xx}-s_{zz})^2+\frac{a_1}{6}\,(s_{xx}-s_{yy})^2 \] where the \(\left.a_{i}\right|_{i\in[1:6]}\) are six coefficients describing the orthotropy of the material.
- The generalization of \(J_{3}\) is denoted \(J_{3}^{O}\). It is defined by: \[ \begin{aligned} J_{3}^{O}= &\frac{1}{27}\,(b_1+b_2)\,s_{xx}^3+\frac{1}{27}\,(b_3+b_4)\,s_{yy}^3+\frac{1}{27}\,(2\,(b_1+b_4)-b_2-b_3)\,s_{zz}^3\\ &-\frac{1}{9}\,(b_1\,s_{yy}+b_2s_{zz})\,s_{xx}^2\\ &-\frac{1}{9}\,(b_3\,s_{zz}+b_4\,s_{xx})\,s_{yy}^2\\ &-\frac{1}{9}\,((b_1-b_2+b_4)\,s_{xx}+(b_1-b3+b_4)\,s_{yy})\,s_{zz}^3\\ &+\frac{2}{9}\,(b_1+b_4)\,s_{xx}\,s_{yy}\,s_{zz}\\ &-\frac{s_{xz}^2}{3}\,(2\,b_9\,s_{yy}-b_8\,s_{zz}-(2\,b_9-b_8)\,s_{xx})\\ &-\frac{s_{xy}^2}{3}\,(2\,b_{10}\,s_{zz}-b_5\,s_{yy}-(2\,b_{10}-b_5)\,s_{xx})\\ &-\frac{s_{yz}^2}{3}\,((b_6+b_7)\,s_{xx}-b_6\,s_{yy}-b_7\,s_{zz})\\ &+2\,b_{11}\,s_{xy}\,s_{xz}\,s_{yz} \end{aligned} \] where the \(\left.b_{i}\right|_{i\in[1:11]}\) are eleven coefficients describing the orthotropy of the material.

Those invariants may be used to generalize isotropic yield criteria based on \(J_{2}\) and \(J_{3}\) invariants to orthotropy.

The following functions

\(J_{2}^{0}\), \(J_{3}^{0}\) and their first and second derivatives with respect to the stress tensor \(\underline{\sigma}\) can be computed by the following functions:

`computesJ2O`

,`computesJ2ODerivative`

and`computesJ2OSecondDerivative`

.`computesJ3O`

,`computesJ3ODerivative`

and`computesJ3OSecondDerivative`

.

Those functions take the stress tensor as first argument and each orthotropic coefficients. Each of those functions has an overload taking the stress tensor as its firs arguments and a tiny vector (`tfel::math::tvector`

) containing the orthotropic coefficients.

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.

`python`

bindings`tfel.math`

module`stensor`

classThe following operations are supported:

- addition of two symmetric tensors.
- substraction of two symmetric tensors.
- multiplication by scalar.
- in-place addition by a symmetric tensor.
- in-place substraction by a symmetric tensor.
- in-place multiplication by scalar.
- in-place division by scalar.

The following functions have been introduced:

`makeStensor1D`

: builds a \(1D\) symmetric tensor from a tuple of three values.`makeStensor2D`

: builds a \(2D\) symmetric tensor from a tuple of three values.`makeStensor3D`

: builds a \(3D\) symmetric tensor from a tuple of three values.

`tfel.material`

moduleThe following functions are available:

`buildFromPiPlane`

: returns a tuple containing the three eigenvalues of the stress corresponding to the given point in the \(\pi\)-plane.`projectOnPiPlane`

: projects a stress state, defined its three eigenvalues or by a symmetric tensor, on the \(\pi\)-plane.

The following script shows how to build an isosurface of the von Mises equivalent stress in the \(\pi\)-plane:

```
from math import pi,cos,sin
import tfel.math as tmath
import tfel.material as tmaterial
nmax = 100
for a in [pi*(-1.+(2.*i)/(nmax-1)) for i in range(0,nmax)]:
s = tmath.makeStensor1D(tmaterial.buildFromPiPlane(cos(a),sin(a)))
seq = tmath.sigmaeq(s);
s *= 1/seq;
s1,s2 = tmaterial.projectOnPiPlane(s);
print(s1,s2);
```

The `computeHosfordStress`

function, which compute the Hosford equivalent stress, is available.

The following script shows how to print an iso-surface of the Hosford equivalent stress in the \(\pi\)-plane:

```
from math import pi,cos,sin
import tfel.math as tmath
import tfel.material as tmaterial
nmax = 100
for a in [pi*(-1.+(2.*i)/(nmax-1)) for i in range(0,nmax)]:
s = tmath.makeStensor1D(tmaterial.buildFromPiPlane(cos(a),sin(a)))
seq = tmaterial.computeHosfordStress(s,8,1.e-12);
s /= seq;
s1,s2 = tmaterial.projectOnPiPlane(s);
print(s1,s2);
```

The following functions are available:

`makeBarlatLinearTransformation1D`

: builds a \(1D\) linear transformation of the stress tensor.`makeBarlatLinearTransformation2D`

: builds a \(2D\) linear transformation of the stress tensor.`makeBarlatLinearTransformation3D`

: builds a \(3D\) linear transformation of the stress tensor.`computeBarlatStress`

: computes the Barlat equivalent Barlat stress.

The following script shows how to print an iso-surface of the Barlat equivalent stress for the 2090-T3 aluminum alloy in the \(\pi\)-plane (see Barlat et al. (2005)):

```
from math import pi,cos,sin
import tfel.math as tmath
import tfel.material as tmaterial
nmax = 100
l1 = tmaterial.makeBarlatLinearTransformation1D(-0.069888,0.079143,0.936408,
0.524741,1.00306,1.36318,
0.954322,1.06906,1.02377);
l2 = tmaterial.makeBarlatLinearTransformation1D(0.981171,0.575316,0.476741,
1.14501,0.866827,-0.079294,
1.40462,1.1471,1.05166);
for a in [pi*(-1.+(2.*i)/(nmax-1)) for i in range(0,nmax)]:
s = tmath.makeStensor1D(tmaterial.buildFromPiPlane(cos(a),sin(a)))
seq = tmaterial.computeBarlatStress(s,l1,l2,8,1.e-12);
s *= 1/seq;
s1,s2 = tmaterial.projectOnPiPlane(s);
print(s1,s2);
```

`MFront`

Various keywords (such as `@ElasticMaterialProperties`

, `@ComputeThermalExpansion`

, `@HillTensor`

, etc.) expects one or more material properties. In previous versions, those material properties were constants or defined by an external `MFront`

.

This new version allows those material properties to be defined by formulae, as follows:

```
@Parameter E0 =2.1421e11,E1 = -3.8654e7,E2 = -3.1636e4;
@ElasticMaterialProperties {"E0+(T-273.15)*(E1+E2*(T-273.15))",0.3}
```

As for material properties defined in external `MFront`

files, the material properties evaluated by formulae will be computed for updated values of their parameters. For example, if the previous lines were used in the `Implicit`

DSL, two variables `young`

and `young_tdt`

will be automatically made available:

- the
`young`

variable will be computed for the temperature \(T+\theta\,\Delta\,T\). - the
`young_tdt`

variable will be computed for the temperature \(T+\Delta\,T\).

The following header files have been renamed:

`Hosford.hxx`

has been moved in`Hosford1972YieldCriterion.hxx`

.`Barlat.hxx`

has been moved in`Barlat2004YieldCriterion.hxx`

.

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.

Cazacu, Oana, and Frédéric Barlat. 2001. “Generalization of Drucker’s Yield Criterion to Orthotropy.” *Mathematics and Mechanics of Solids* 6 (6):613–30. https://doi.org/10.1177/108128650100600603.