SofaViscoElastic

Table of contents

• Introduction
• Visco-Hyperelasticity
• Installation
• Python Functions and Bindings
• Algorithm
• Applications

Introduction

SofaViscoElastic is a Software Open Architecture Framework (SOFA) plugin, which implements the fundamental linear viscoelastic and visco-hyperelastic constitutive laws applied to tetrahedral meshes. Viscoelasticity is a property of elastomeric materials that influences their mechanical behavior under dynamic conditions. Viscoelastic constitutive equations are dependent on the stress/strain rate. At low stress/strain rates, a viscoelastic material behaves like a viscous liquid-like material, while at high stress/strain rates, the same material behaves like a Hookean solid. The simplest viscoelastic models are:

These two models represent the basic unit that constitutes the viscoelastic materials. They are composed of an elastic part described by the spring symbol and a viscous one represented by the dashpot. The Maxwell and Kelvin-Voigt models describe the behavior of a few kinds of materials, like silly-putty and gels. Furthermore, they are unstable theoretical models under creep (Maxwell) or stress relaxations (Kelvin-Voigt) conditions. Elastomers and rubbers are polymeric materials in nature but are also used in several industrial applications. Many research fields are involved in developing and using new elastomeric materials and rubbers, such as soft robotics and surgical applications. For this reason, this plugin is indicated for users who want a realistic mechanical simulation of these materials afflicted by the viscoelastic effect. To describe their viscoelastic properties, different viscoelastic models have to be used, like the Standard Linear Solid (SLS) Maxwell/Kelvin-Voigt representation:

They add another spring in parallel (Maxwell representation) or in series (Kelvin-Voigt representation) to make the model stable under creep and stress relaxation. They are excellent for describing viscoelastic polymer rheology. The SofaViscoElastic plugin presents nine different viscoelastic models. For more theoretical information, users can refer to the paper "Considering the viscoelastic effects in soft robotic modeling" by Ferrentino et al., submitted in the Soft Robotic Journal (SORO).

Visco-Hyperelasticity

The visco-hyperelastic describes the mechanical behavior of the elastomers in large deformations. In this regime, the hyperelasticity of the material is shown in parallel to the viscous effects, in particular, for this plugin:

Hence, it combines the hyperelastic models already implemented in Sofa with Maxwell Branches. The user can choose for each visco-hyperelastic model until the second order (two Maxwell branches in parallel).

Hysteresis

The material models don't include hysteresis modeling, which will be added in future works.

Installation

This plugin is available for Ubuntu/Linux, Macintosh, and Windows. The only dependency is the SOFA plugin "SofaPython3" (make sure it is installed). To install this plugin from the source, the user has to download this folder and place it in:

 $ /home/adminName/sofa/src/applications/plugins

Then, the user has to write this in the CMakeLists.txt present in the previous destination:

$ sofa_add_subdirectory(plugin SofaViscoElastic SofaViscoElastic)

then recompile SOFA and it should start its installation. Enjoy! If you have any problems, please get in touch with the author at [email protected].

Python Functions and Bindings

The principal function of this plugin is the so-called TethrahedronViscoelasticityFEMForceField, which applies the viscoelastic constitutive law to the tetrahedral mesh uploaded in SOFA, the syntax in Python is the following :

The additional fields to fill in are:

• template: related to the DOF expressed in the Mechanical Object in SOFA.
• name: The name chosen by the user for the function (will appear in the SOFA simulation Graph)
• materialname: The name of the viscoelastic model that the user wants to use, he can choose between: (Linear Viscoelasticity)
  • MaxwellFirstOrder
  • KelvinVoigtFirstOrder
  • SLSMaxwellFirstOrder
  • SLSKelvinVoigtFirstOrder
  • Burgers
  • MaxwellSecondOrder
  • KelvinVoigtSecondOrder
  • SLSMaxwellSecondOrder
  • SLSKelvinVoigtSecondOrder (Visco-Hyperelasticity)
  • SLSNeohookeanFirstOrder
  • SLSNeohookeanSecondOrder
  • SLSMooneyRivlinFirstOrder
  • SLSMooneyRivlinSecondOrder
  • SLSOgdenFirstOrder
  • SLSOgdenSecondOrder
  • SLSStableNeoHookeanFirstOrder
  • SLSStableNeoHookeanSecondOrder
• ParameterSet: the lists of the material constants proper of the viscoelastic model chosen by the user. In particular, the user has to define the Young Moduli ($E_i$) and the relaxation times ($τ_i$) defined as the ratio between the viscosity ($η_i$) and the relative Young modulus. Ultimately, the user must specify the Poisson Ratio (ν).

For the simulation, it is strictly recommended to set the Rayleigh coefficient of the Solver to 0:

The simulation results are strictly dependent on the time step (dt). The author advises to use this range of time steps:

$$ dt \leq {τ_i \over 100} $$

Furthermore, in the plugin are integrated some Python Bindings to export some internal parameters of specific tetrahedrons:

The user can choose these quantities:

• getShapeVector(): Get the shape Vector of the specific tetrahedron (specified in the brackets).
• getFiberDirection(): get the fiber direction of the tetrahedron.
• getVolume(): get the volume of the tetrahedron.
• getRestVolume(): get the rest volume of the tetrahedron.
• getVolScale(): get the volume scale of the tetrahedron.
• getF(): get the deformation gradient (F) of the tetrahedron.
• getSPKStress(): get the stresses of the Second-Piola Kirchhoff tensor.
• getCauchyStress(): get the stresses Cauchy tensor.
• getVonMisesStress(): get the Von Mises stresses.

P.S. The stresses are per Element not per Node.

Algorithm

In Figure 4 of the paper, the SLS-Maxwell model of first order is used as an example to understand the algorithm used in this plugin. This section explains the principal steps of the algorithm. First, the plugin aims to calculate the deviatoric and hydrostatic parts of the stress tensor. The deviatoric part calculation derives from the stress balance on the springs-dashpot 3D model:

Where $σ^{dev}$ is the deviatoric stress tensor, $ε$ is the strain tensor, while $ε^{· v}$ is the strain rate which acts in the dashpot of the model. Applying the discretization of this equation, in particular an Euler backward scheme on the strain rate formulation, we obtain:

The terms with the exponent "n" refer to the quantity calculated at the current time step, while the ones with the exponent "n-1" refer to the quantity calculated at the previous time step. Hence, in viscoelastic materials, the actual strain stress status depends on the previous strain stress conditions, which means that the material has the "memory" of its previous internal stress/strain state.
Instead, the hydrostatic part of the stress tensor is calculated using this general formula for the $4^{th}$ order elasticity tensor for visco-elastic materials:

$E^{*}(t)$ is the so-called Prony series and depends on the particular model of viscoelasticity. For more details on the algorithm, please consult Section 3 of the paper.

Future works

Visco-Hyperelastixcity will be added with Hysteresis modeling/parametrization. Stay updated!

Applications

The plugin develops different example scenes in Python3. The examples show a cylindrical beam considering viscoelastic constitutive models and undergoing a creep test and a stress relaxation test. The plugin was developed in collaboration between the Brubotics lab of the VUB (Vrije Universiteit Brussel) University and the DEFROST team of the INRIA Institute in Lille. The authors of this plugin are looking for future collaborations for further developments. For other info please contact: [email protected].