Introduction

MFront version 3.1 provides an interface for the ANSYS Mechanical APDL (MAPDL) finite element solver. This interface is fairly features complete:

• Isotropic and orthotropic materials are supported.
• Small and finite strain behaviours are supported.

It shall be pointed out that the ANSYS solvers has a long history. In particular, the design choices made for the USERMAT interface were meant to allow the users to easily write finite-strain behaviours in rate-form.

MFront strives to provides behaviours that can be used “just-like” other USERMAT subroutines, but

Limitations

The USERMAT subroutines has a number of shortcomings compared with other interfaces:

• External state variables are not supported.
• Internal state variables can not be initialized to values different from zero.
• There is no standard way of defining the orthotropic axes for orthotropic behaviours.
• There is no way of controlling the increase/decrease of the time step.

The choice made to defined the orthotropic axes for orthotropic behaviours is detailed below.

Pitfalls

There are also cases of misuses of the generated libraries that can not be prevented by MFront. The most important ones are the following:

• The modelling hypothesis is not passed to the mechanical behaviours. As a consequence, there is not way of distinguishing between the plane strain hypothesis and the axisymmetrical hypothesis, which is mandatory for consistent orthotropic axes management. Therefore, the user has to explicitly call the implementation of the behaviour corresponding to the modelling hypothesis he wants to use, but there is not way to ensure the consistency of this choice and the choice of the modelling hypothesis make for the mechanical computation.
• Whether the computation takes into account finite strain transformation (non linear geometric effects according to the ANSYS wording) is not known to the behaviour, so some misuses of MFront behaviours can’t be prevented. For example, small strain behaviours can be used in finite strain transformation using the build-in Jauman corotational framework, which is only valid for isotropic behaviours. Using this framework with orthotropic behaviours can’t be prevented.
• The user must respect the order of definition of the material properties, some of them being implicitly defined by MFront. For example, we made the choice to pass the orthotropic axes as material properties. Those are automatically inserted at the beginning of the material properties list, after thermo-elastic properties, if the latter are requested. As a consequence, the number of material properties depends on the modelling hypothesis. The user is thus strongly advised to look at the input file example generated by MFront to have a consistent definition of the material properties.

Current status

The ansys interface is still in its early stage of development.

How the use MFront behaviours in ANSYS

When compiling mechanical behaviours with the ansys interface, MFront generates:

• Shared libraries containing one implementation of the considered behaviours per supported modelling hypotheses.
• Examples of input files in the ansys directory.
• A copy of a generic usermat.cpp file for the ANSYS solver.

Those various files and their usage are now described.

Notes

• The MFront libraries are only generated once, only the generic files needs to be recompiled at each run. This is very handy since compiling the MFront libraries can be time-consuming. Those libraries can be shared between computations and/or between users when placed in a shared folders.
• The fact that MFront generates one implementation per modelling hypothesis allows the distinction between the plane strain hypothesis and the axisymmetrical hypothesis. This is mandatory to consistently handle orthotropy.
• The name of the library can be changed by renaming it after the compilation or by using the @Library keyword.

Note on libraries locations

As explained above, MFront libraries will be loaded at the runtime time. This means that the libraries must be found by the dynamic loader of the operating system.

Under Linux

Under Linux, the search path for dynamic libraries are specified using the LD_LIBRARY_PATH variable environment. This variable defines a colon-separated set of directories where libraries should be searched for first, before the standard set of directories.

Depending on the configuration of the system, the current directory can be considered by default.

Under Windows

Under Windows, the dynamic libraries are searched:

• in the current directory
• in the directories listed in the PATH environment. This variable defines a semicolon-separated set of directories.

Generated input files

Here is an extract of the generated input file for a MFront behaviour named Plasticitiy for the plane strain modelling hypothesis for the ANSYS solver:

/com, Example for the 'PlaneStrain' modelling hypothesis
/com, List of material properties
/com, -YoungModulus
/com, -PoissonRatio
/com, -H
/com, -s0
tb,user,<mat_id>,<numer of temperatures>,4
/com, you shall now declare your material properties
/com, using the tbtemp an tbdata instructions.
/com, See the Ansys "USER Material Subroutine" guide for details.
/com, Declaration of state variables
tb,state,<mat_id>,,5

The generic usermat.cpp file

When generating the sources of an MFront file with the ansys interface, a subdirectory called ansys is created. This subdirectory contains the following files:

• Examples of MFront’ generated behaviours usage
• usermat.cpp: the generic usermat.cpp used to generate the usermatLib library, described below.
• test-usermat.cxx: the source of an executable which can be used to test an mfront-usermat.dat file. Its usage is described below.
• CMakeLists.txt: a cmake project file that can be used to generate the usermatLib library and the test-usermat executable.

The generic usermat.cpp file provided by MFront has the following role:

• Reading a file called mfront-usermat.dat that has to be present in the current directory. This file relates an ANSYS material identifier to a MFront behaviour.
• Loading the shared library and the behaviours implementation generated by MFront.
• Calling the MFront behaviour.

Warnings and errors are written in a file called mfront-usermat.log.

Compilation of the usermatLib library and the test-executable executable

This paragraph describe how to use the cmake to compile the usermatLib library and the test-executable executable.

Using Visual Studio 2017

In the ansys subdirectory generated by MFront, type:

$ cmake . -G "Visual Studio 15 2017 Win64"
$ cmake --build . --config Release

The usermatLib.dll must be copied in a directory pointed by the ANS_USER_PATH variable.

Under Unix

In the ansys subdirectory generated by MFront, type:

$ cmake . -DCMAKE_BUILD_TYPE=Release
$ cmake --build .

The usermatLib.dll must be copied in a directory pointed by the ANS_USER_PATH variable.

Detailed procedure

This example shows how to compile a file called ImplicitNorton.mfront and the usermatLib library under LiNuX.

1. Compile the MFront behaviour and generate the ansys subdirectory

$ mfront --obuild --interface=ansys ImplicitNorton.mfront 
Treating target : all
The following library has been built :
- libAnsysBehaviour.so :  ImplicitNorton_axis ImplicitNorton_pstress ImplicitNorton_pstrain ImplicitNorton_3D

2. Change the current directory

$ cd ansys/

3. configure the cmake project

$ cmake . -DCMAKE_BUILD_TYPE=Release
-- The C compiler identification is GNU 6.3.0
-- The CXX compiler identification is GNU 6.3.0
-- Check for working C compiler: /usr/bin/cc
-- Check for working C compiler: /usr/bin/cc -- works
-- Detecting C compiler ABI info
-- Detecting C compiler ABI info - done
-- Detecting C compile features
-- Detecting C compile features - done
-- Check for working CXX compiler: /usr/bin/c++
-- Check for working CXX compiler: /usr/bin/c++ -- works
-- Detecting CXX compiler ABI info
-- Detecting CXX compiler ABI info - done
-- Detecting CXX compile features
-- Detecting CXX compile features - done
CMake Warning (dev) in CMakeLists.txt:
  No cmake_minimum_required command is present.  A line of code such as

    cmake_minimum_required(VERSION 3.7)

  should be added at the top of the file.  The version specified may be lower
  if you wish to support older CMake versions for this project.  For more
  information run "cmake --help-policy CMP0000".
This warning is for project developers.  Use -Wno-dev to suppress it.

-- Configuring done
-- Generating done
-- Build files have been written to: /tmp/ansys

4. Compile the usermatLib library and the test-usermat executable

$ cmake --build .
Scanning dependencies of target test-usermat
[ 25%] Building CXX object CMakeFiles/test-usermat.dir/test-usermat.o
[ 50%] Linking CXX executable test-usermat
[ 50%] Built target test-usermat
Scanning dependencies of target usermatLib
[ 75%] Building CXX object CMakeFiles/usermatLib.dir/usermat.o
[100%] Linking CXX shared library libusermatLib.so
[100%] Built target usermatLib

At the end of those four steps, the ansys subdirectory contains:

• The libusermat.so library.
• The test-usermat executable.

Relating material identifier and MFront behaviours

The mfront-usermat.dat file gives a list of commands aimed to associate an ANSYS material identifier to a MFront behaviour. The syntax used closely follows ANSY APDL syntax.

Only two commands are actually supported:

• /com, which is used to introduce a comment line.
• tb,mfront which is used to associate an ANSYS material identifier to a MFront behaviour.

Here is an example of such a file:

/com, Associate the material id 2 to the Chaboche_3D behaviour
/com, implemented in the Zircaloy4Behaviours shared library.
tb,mfront,2,Zircaloy4Behaviours,Chaboche_3D
/com, Associate the material id 3 to the Creep_3D behaviour
/com, implemented in the Zircaloy4Behaviours shared library.
tb,mfront,3,UO2Behaviours,Creep_3D

New commands will eventually be introduced as needed to circumvent the various shortcomings of the USERMAT interface, notably:

• Adding the ability to give initial values to state variables
• Adding the ability to override the default values of parameters.
• etc.

Library name

For portability reasons, the library name can be stripped from the standard prefix (lib under UNIX) and from the file extension (.dll under Windows, .dylib under Mac Os, .so under LiNuX). The usermat function delivered with MFront will try every combinaison until a suitable on is found.

Testing an mfront-usermat.dat using test-usermat

The declarations in the mfront-usermat.dat file can lead to the following errors:

• The material libraries can not be found.
• The TFEL libraries can not be found.
• The functions implementing the behaviours can not be found.

Although every warning and errors are redirected in the mfront-usermat.log file, those errors can be painful to analyse in Ansys because the generic usermat function closes Ansys on error.

Thus, a simple executable called test-usermat is also provided. This executable reads the mfront-usermat.dat in the current directory. If its exists normally, then everything is ok.

Main features of the ANSYS interface

The ANSYS solver provides the USERMAT interface. In this case, the behaviour shall compute:

• The evolution of the state variables.
• The value the Cauchy stress at the end of the time step.
• The consistent tangent operator. The definition of the consistent tangent operator is given below.

For finite strain analyses, small strain behaviours can be written in rate form. The behaviour in integrated in the Jauman framework, which is only suitable for isotropic behaviours.

Supported behaviours

Isotropic and orthotropic behaviours are both supported.

Small and finite strain behaviours are supported.

Modelling hypotheses

The following modelling hypotheses are supported:

• tridimensional (3D)
• plane strain (pstrain)
• plane stress (including shell elements) (pstress)
• axisymmetrical (axis)

The generalised plane strain hypothesis is currently not supported.

Orthotropic behaviours

By nature, orthotropic behaviours are expressed in a preferential frame defined by the material orthotropic axes. There is no standard way of defining the orthotropic axes in ANSYS.

We choose to add the definition of those axes to the list of material properties. \(2\) additional material properties are required in \(2D\), and \(6\) additional material properties in \(3D\):

• In \(2D\), the two additional material properties are the two components of the vector defining the first axis of the material frame. This vector is thus supposed to be contained in the \(2D\) plane. The second axis is also assumed to be contained in this plane and perpendicular to the first one. With those assumptions, this vector can be deduced from the first one and does not have to be defined. The third direction is required to follow the out-of-plane direction.
• In \(3D\), the six additional material properties defines two vectors defining respectively the first and second axes of the material frame. The components of the first vectors are given before the components of the second vector. Those vectors are supposed to be orthogonal. The third vector, associated to the third axis, is implicitly given by the cross-product of the two first axes and does not have to be defined.

The user shall use the input file example generated by MFront to see their relative positions of the material properties associated to the definition of the orthotropic axes.

Those definitions are only meaningful if the direction of orthotropy are constants.

Finite strain strategies

Engineers are used to write behaviours based on an additive split of strains, as usual in small strain behaviours. Different strategies exist to:

• write finite strain behaviours that preserve this property.
• guarantee some desirable properties such as energetic consistency and objectivity.

Through the @ANSYSFiniteStrainStrategy, the user can select on of various finite strain strategies supported by MFront, which are described in this paragraph.

Note

The usage of the @ANSYSFiniteStrainStrategy keyword is mostly deprecated since MFront 3.1: see the @StrainMeasure keyword.

The Native finite strain strategy

Among them is the Native finite strain strategy which relies on build-in ANSYS facilities to integrate the behaviours written in rate form. The Native finite strain strategy will use the Jauman rate.

This strategy has some theoretical drawbacks (hypoelasticity, restriction to isotropic behaviours, etc…) and is not portable from one code to another.

Recommended finite strain strategies

Two other finite strain strategies are available in MFront for the ansys interface (see the @ANSYSFiniteStrainStrategy keyword):

• FiniteRotationSmallStrain: this finite strain strategy is fully described in [1],[3]
• MieheApelLambrechtLogarithmicStrain: this finite strain strategy is fully described in [4] and
  1. This finite strain strategy is yet to be implemented.

Those two strategies use lagrangian tensors, which automatically ensures the objectivity of the behaviour.

Each of these two strategies define an energetic conjugate pair of strain or stress tensors:

• For the FiniteRotationSmallStrain case, the strain tensor is the Green-Lagrange strain and the conjugated stress is the second Piola-Kirchhoff stress.
• For the MieheApelLambrechtLogarithmicStrain, the strain tensor is the langrangian Hencky strain tensor, i.e. the logarithm of the stretch tensor.

The first strategy is suited for reusing behaviours that were identified under the small strain assumptions in a finite rotation context. The usage of this behaviour is still limited to the small strain assumptions.

The second strategy is particularly suited for metals, as incompressible flows are characterized by a deviatoric logarithmic strain tensor, which is the exact transposition of the property used in small strain behaviours to handle plastic incompressibility. This means that all valid consistutive equations for small strain behaviours can be automatically reused in finite strain analysis. This does not mean that a behaviour identified under the small strain assumptions can be directly used in a finite strain analysis: the identification would not be consistent.

Those two finite strain strategies are fairly portable and are available (natively or via MFront) in Cast3M, Code_Aster, Europlexus, Abaqus/Standard and Abaqus/Explicit and Zebulon, etc…

Consistent tangent operator for finite strain behaviours

The “ANSYS User Subroutines Reference Guide” gives indicates that the tangent moduli required by ANSYS \(\underline{\underline{\mathbf{C}}}^{MJ}\) is closely related to \(\underline{\underline{\mathbf{C}}}^{\tau\,J}\), the moduli associated to the Jauman rate of the Kirchhoff stress :

\[ J\,\underline{\underline{\mathbf{C}}}^{MJ}=\underline{\underline{\mathbf{C}}}^{\tau\,J} \]

where \(J\) is the derterminant of the deformation gradient \({\underset{\tilde{}}{\mathbf{F}}}\).

By definition, \(\underline{\underline{\mathbf{C}}}^{\tau\,J}\) satisfies: \[ \overset{\circ}{\underline{\tau}}^{J}=\underline{\underline{\mathbf{C}}}^{\tau\,J}\,\colon\underline{D} \] where \(\underline{D}\) is the rate of deformation.

Appendix

Most information reported here are extracted from the book of Belytschko ([5]).

Relations between tangent operator

Relation with the moduli associated to the Truesdell rate of the Cauchy Stress \(\underline{\underline{\mathbf{C}}}^{\sigma\,T}\)

The moduli associated to the Truesdell rate of the Cauchy Stress \(\underline{\underline{\mathbf{C}}}^{\sigma\,T}\) is related to \(\underline{\underline{\mathbf{C}}}^{\tau\,J}\) by the following relationship:

\[ \underline{\underline{\mathbf{C}}}^{\tau\,J}=J\,\left(\underline{\underline{\mathbf{C}}}^{\sigma\,T}+\underline{\underline{\mathbf{C}}}^{\prime}\right)\quad\text{with}\quad\underline{\underline{\mathbf{C}}}^{\prime}\colon\underline{D}=\underline{\sigma}\,.\,\underline{D}+\underline{D}\,.\,\underline{\sigma} \]

Thus,

\[ \underline{\underline{\mathbf{C}}}^{MJ}=\underline{\underline{\mathbf{C}}}^{\sigma\,T}+\underline{\underline{\mathbf{C}}}^{\prime} \]

Relation with the spatial moduli \(\underline{\underline{\mathbf{C}}}^{s}\)

The spatial moduli \(\underline{\underline{\mathbf{C}}}^{s}\) is associated to the Lie derivative of the Kirchhoff stress \(\mathcal{L}\underline{\tau}\) , which is also called the convected rate or the Oldroyd rate:

\[ \mathcal{L}\underline{\tau}=\underline{\underline{\mathbf{C}}}^{s}\,\colon\,\underline{D} \]

The spatial moduli is related to the moduli associated to Truesdell rate of the Cauchy stress \(\underline{\underline{\mathbf{C}}}^{\sigma\,T}\):

\[ \underline{\underline{\mathbf{C}}}^{\sigma\,T}=J^{-1}\,\underline{\underline{\mathbf{C}}}^{s} \]

Thus, we have: \[ \underline{\underline{\mathbf{C}}}^{MJ}= J^{-1}\underline{\underline{\mathbf{C}}}^{s}+\underline{\underline{\mathbf{C}}}^{\prime} = J^{-1}\left(\underline{\underline{\mathbf{C}}}^{s}+\underline{\underline{\mathbf{C}}}^{\prime\prime}\right)\quad\text{with}\quad\underline{\underline{\mathbf{C}}}^{\prime\prime}\colon\underline{D}=\underline{\tau}\,.\,\underline{D}+\underline{D}\,.\,\underline{\tau} \]

Relation with \(\underline{\underline{\mathbf{C}}}^{\mathrm{SE}}\)

The \(\underline{\underline{\mathbf{C}}}^{\mathrm{SE}}\) relates the rate of the second Piola-Kirchhoff stress \(\underline{S}\) and the Green-Lagrange strain rate \(\underline{\varepsilon}^{\mathrm{GL}}\):

\[ \underline{\dot{S}}=\underline{\underline{\mathbf{C}}}^{\mathrm{SE}}\,\colon\,\underline{\dot{\varepsilon}}^{\mathrm{GL}} \]

As the Lie derivative of the Kirchhoff stress \(\mathcal{L}\underline{\tau}\) is the push-forward of the second Piola-Kirchhoff stress rate \(\underline{\dot{S}}\) and the rate of deformation \(\underline{D}\) is push-forward of the Green-Lagrange strain rate \(\underline{\dot{\varepsilon}}^{\mathrm{GL}}\), \(\underline{\underline{\mathbf{C}}}^{s}\) is the push-forward of \(\underline{\underline{\mathbf{C}}}^{\mathrm{SE}}\):

\[ C^{c}_{ijkl}=F_{im}F_{jn}F_{kp}F_{lq}C^{\mathrm{SE}}_{mnpq} \]

Link with \(\displaystyle\frac{\displaystyle \partial \underline{\sigma}}{\displaystyle \partial {\underset{\tilde{}}{\mathbf{F}}}}\)

For all variation of the deformation gradient \(\delta\,{\underset{\tilde{}}{\mathbf{F}}}\), the Jauman rate of the Kirchhoff stress satisfies: \[ \underline{\underline{\mathbf{C}}}^{\tau\,J}\,\colon\delta\underline{D}=\delta\underline{\tau}-\delta{\underset{\tilde{}}{\mathbf{W}}}.\underline{\tau}+\underline{\tau}.\delta{\underset{\tilde{}}{\mathbf{W}}} \]

with:

• \(\delta{\underset{\tilde{}}{\mathbf{L}}}= \delta{\underset{\tilde{}}{\mathbf{F}}}\,.\,{\underset{\tilde{}}{\mathbf{F}}}^{-1}\)
• \(\delta{\underset{\tilde{}}{\mathbf{W}}}= \displaystyle\frac{\displaystyle 1}{\displaystyle 2}\left(\delta{\underset{\tilde{}}{\mathbf{L}}}-{\delta{\underset{\tilde{}}{\mathbf{L}}}^{\mathrm{T}}}\right)\)
• \(\delta{\underset{\tilde{}}{\mathbf{D}}}= \displaystyle\frac{\displaystyle 1}{\displaystyle 2}\left(\delta{\underset{\tilde{}}{\mathbf{L}}}+{\delta{\underset{\tilde{}}{\mathbf{L}}}^{\mathrm{T}}}\right)\)

Thus, the derivative of the Kirchhoff stress with respect to the deformation gradient yields: \[ \displaystyle\frac{\displaystyle \partial \underline{\tau}}{\displaystyle \partial {\underset{\tilde{}}{\mathbf{F}}}}=\underline{\underline{\mathbf{C}}}^{\tau\,J}\,.\,\displaystyle\frac{\displaystyle \partial \underline{D}}{\displaystyle \partial {\underset{\tilde{}}{\mathbf{F}}}}+\left(\partial^{\star}_{r}\left(\tau\right)-\partial^{\star}_{l}\left(\tau\right)\right)\,.\,\displaystyle\frac{\displaystyle \partial {\underset{\tilde{}}{\mathbf{W}}}}{\displaystyle \partial {\underset{\tilde{}}{\mathbf{F}}}} \]

with \(\delta\,\underline{D}=\displaystyle\frac{\displaystyle \partial \underline{D}}{\displaystyle \partial {\underset{\tilde{}}{\mathbf{F}}}}\,\colon\,\delta\,{\underset{\tilde{}}{\mathbf{F}}}\) and \(\delta\,{\underset{\tilde{}}{\mathbf{W}}}=\displaystyle\frac{\displaystyle \partial {\underset{\tilde{}}{\mathbf{W}}}}{\displaystyle \partial {\underset{\tilde{}}{\mathbf{F}}}}\,\colon\,\delta\,{\underset{\tilde{}}{\mathbf{F}}}\)

\[ \displaystyle\frac{\displaystyle \partial \underline{\sigma}}{\displaystyle \partial {\underset{\tilde{}}{\mathbf{F}}}}=\displaystyle\frac{\displaystyle 1}{\displaystyle J}\left(\displaystyle\frac{\displaystyle \partial \underline{\tau}}{\displaystyle \partial {\underset{\tilde{}}{\mathbf{F}}}}-\underline{\sigma}\,\otimes\,\displaystyle\frac{\displaystyle \partial J}{\displaystyle \partial {\underset{\tilde{}}{\mathbf{F}}}}\right) \]

Numerical approximation of \(\underline{\underline{\mathbf{C}}}^{MJ}\)

Following [6], an numerical approximation of \(\underline{\underline{\mathbf{C}}}^{MJ}\) is given by: \[ \underline{\underline{\mathbf{C}}}^{MJ}_{ijkl}\approx\displaystyle\frac{\displaystyle 1}{\displaystyle J\,\varepsilon}\left(\underline{\tau}_{ij}\left({\underset{\tilde{}}{\mathbf{F}}}+{\underset{\tilde{}}{\mathbf{\delta F}}}^{kl}\right)-\underline{\tau}_{ij}\left({\underset{\tilde{}}{\mathbf{F}}}\right)\right) \]

where the perturbation \({\underset{\tilde{}}{\mathbf{\delta F}}}^{kl}\) is given by:

\[ {\underset{\tilde{}}{\mathbf{\delta F}}}^{kl}=\displaystyle\frac{\displaystyle \varepsilon}{\displaystyle 2}\left(\vec{e}_{k}\otimes\vec{e}_{l}+\vec{e}_{l}\otimes\vec{e}_{k}\right)\,.\,{\underset{\tilde{}}{\mathbf{F}}} \]

Such perturbation leads to the following rate of deformation: \[ \delta\,\underline{D}=\left({\underset{\tilde{}}{\mathbf{\delta F}}}^{kl}\right)\,{\underset{\tilde{}}{\mathbf{F}}}^{-1}=\displaystyle\frac{\displaystyle \varepsilon}{\displaystyle 2}\left(\vec{e}_{k}\otimes\vec{e}_{l}+\vec{e}_{l}\otimes\vec{e}_{k}\right) \]

The spin rate \(\delta\,\underline{W}\) associated with \({\underset{\tilde{}}{\mathbf{\delta F}}}^{kl}\) is null.

Relation with other moduli

The previous relation can be used to relate to other moduli. See the section describing the isotropic case for details.

Biblography