Library: General characteristics

  • Formulation based on the finite strain kinematic framework
  • Written in C++ using a modular strategy: flexibility to combine multiple material features
  • Material modules designed to be independent of the finite element solver (implicit or explicit)
  • Available for Windows and Linux
  • No installation required: shared library automatically called by the FEM solver

Models: Main features

  • Isotropic and orthotropic linear elasticity
  • Isotropic non-linear elasticity for rubber-like behaviour based on chain non-Gaussian network theory by Arruda-Boyce
  • Rate-independent plasticity: Kinematic and isotropic hardening, Von Mises, Druker-Prager, Paraboloidal
  • Visco-elastic models (Maxwell, Kelvin-Voigt, Three-parameters model, Prony series, etc)
  • Nonlinear viscous constitutive models like Boyce-Parks-Argon, Norton-Hoff, Bergstrom-Boyce, etc
  • Unified amorphous–crystalline viscoplastic hardening law for non-isothermal modelling of thermoplastics and thermosets
  • Thermo-mechanical coupling for all inelastic behaviours (for implicit and explicit solvers)
  • Continuum damage model (based on traction-separation cohesive law, brittle, quasi-brittle, tension-compression, multiple softening models: linear, bi-linear, tri-linear, super-elliptic, exponential)
  • Anisotropization module for unidirectional and bidirectional transformation
  • Integration schemes: Euler-forward, Mid-point, Runge-Kutta Gear-3, Runge-Kutta Gear-4, Multipoint

Model family

Our library consists of a collection of rheological elements or modules that can be arranged in different forms.

Some modules have to be used in combination with others in order to provide physical sense (for example, modules representing viscosity or damage).

Five basic categories are distinguished: (i) elementary linear response, (ii) elementary nonlinear response, (iii) complementary modules, (iv) classical solids and (v) combined advanced material responses. Some examples are shown below. (Notes: Not all possible combinations are listed. Curves in column "Behaviour overview" are only for illustrative purposes).

Elementary linear modules

Module name Behaviour overview Rheological representation Description
CSpringLinear

This module is the most basic material response accounting for the linear elastic behavior given by Hooke's law. This module is designed to be easily coupled with any other module. Isotropic or orthotropic responses can be chosen.

CDashpotLinear

This is a linear viscous module commonly used to build up classical linear visco-elastic materials.

Elementary Nonlinear Modules

Module name Behaviour overview Rheological representation Description
CSpringLangevin

This module describes a non-linear hyper-elastic material that follows the 3-chain and/or the 8-chain non-Gaussian network theory developed by Arruda-Boyce.

Reference: J. Mech. Phys. Solids, vol 41, issue 2, pp 389-412, 1993
CDashpotPowerLaw

This module describes a commonly used rate-denpendent plasticity given by a Norton-Hoff type model (power-law). The curve shown in "Behaviour overview" is obtained by connecting this module to a linear spring in series.

Reference: Proc. R. Soc. Lond. A. 348, 101-127, 1976
CDashpotArgonBoyce

This module provides the response given by the physically-based model developed by Argon for predicting the rate- and temperature- dependent behavior of polymer below the glassy transition temperature. The curve shown in "Behaviour overview" is obtained by connecting this module to a linear spring in series.

Reference: Mechanics of Materials 7 (1988) 15-33
CDashpotGeneralized

This module is an extension of the original Boyce-Parks-Argon model (rate- and temperature- dependent). Enhanced with a phenomenological approach, the model is capable of capturing the pre- and post- yield behavior exhibited by some thermoplastics (PMMA, PC) and thermoset polymers (Epoxy). The curve shown in "Behaviour overview" is obtained by connecting this module to a linear spring in series.

Reference: International Journal of Plasticity 62 (2014) 138-161
CDashpotBergstromBoyce

This module provides the nonlinear viscosity model developed by Bergstrom-Boyce. By combining this module with hype-elastic modelus, this model is proved to properly capture the time-dependency response in Chloroprene rubbers (synthetic rubber also known with trade name 'Neoprene') The curve shown in "Behaviour overview" is obtained by connecting this module to a linear spring in series.

Reference: J. Mech. Phys. Solids 46, 931-954, 1998
CDamage

This module represents the softening part taking place in a damage process given by a traction-separation law. Coupling between damage under tension and compression can be chosen. Different criteria of damage initiation (principal strain, pressure-dependent parabolic function, stored energy, dilatation density energy or plastic strain) and different types of softening (linear, bilinear, trilinear, superelliptic, exponential) are available.

Reference: Comp. Part B: Eng. 263 (2023) 110812 (study of energy absorption capability of vehicles' anti-intrusion bar made of thermoplastic composites)
CHealing

This module represents the healing mechanism in a damage process given either by a classical traction-separation law or a rate-dependent visco-damage model. This healing model incorporates the effects of resting time (RT), loading rate and damage level (both instantaneous and accumulated) on the healing variable.

Reference: J. Int. Mech. Sciences 260 (2023) 108644 (multimaterial-aimed healing model incorporating dependency on resting time, strain rate and damage)

Complementary Modules

Module name Behaviour overview Rheological representation Description
CConstThermal

This module allows for incoporating basic thermal properties such as thermal conductivity and specific heat. This module is essential to perform heat transfer analysis in 2D and 3D cases.

C<Dashpot>Anisotropic

This special module can transform any linear and nonlinear viscous module into anisotropic. One or multiple fiber directions can be provided as input.

Reference: Mater. Des. 206 (2021) 109772 (Formulation and application for unidirectional and bidirectional composites)

Classical Modules

Module name Behaviour overview Rheological representation Description
CKelvinVoigt
CSpringLinear
CDashpotLinear

This module represents the well-known Kelvin-Voigt solid constructed from a linear viscous damper in parallel with a linear elastic spring.

CMaxwell
CSpringLinear
CDashpotLinear

This module represents the well-known visco-elastic Maxwell model constructed from a linear viscous damper in series with a linear elastic spring.

CThreeParameters
CKelvinVoigt
CSpringLinear

This module provides the behaviour of the so-called "Standard Linear Solid model" (under Kelvin representation) that consists of a Kelvin-Voigt module in series with a linear elastic spring.

CElastoPlastic
CSpringLinear
CYld_vonMises
CYld_DruckerPrager
CYld_Paraboloidal

This module combines a linear elastic spring with a slider element in series to provide rate-independent plastic behaviour. Different yield criterion are implemented (Von Mises, Druker-Prager, pressure-dependent Paraboloidal). Kinematic or Isotropic hardening can be chosen. This module admits any user-defined strain-hardening evolution in tabular form.

Reference1: J. Int. Mech. Sciences 217 (2022) 106988 (Application for efficient iteration-free implementation of pressure-dependent plasticity)
Reference2: Compos. Sci. Technol. 235 (2023) 109949 (Application for multiscale modeling of unidirectional and bidirectional fiber-reinforced composites)
Reference3: J. Int. Mech. Sciences 250 (2023) 108278 (Technique to reformulate constitutive macromolecular-based thermoviscoplasticity into yield-based formulae)
CElasticDamage
CSpringLinear
CDamage

The continuum isotropic damage model is connected in series to a linear elastic module. In the sample picture, linear and superelliptic softening are illustrated under tension and compression.

Reference1: J. Mech. Phys. Solids 149 (2021) 104318 (Application for dynamic fracture with pressure-dependent failure)
Reference2: Mater. Des. 213 (2022) 110299 (Application for investigating fracture process in concrete beam containing self-healing capsules)

Combined Advanced Modules

Module name Behaviour overview Rheological representation Description
CPronyParallel
CSpringLinear
n x CMaxwell

This combined module provides the response ruled by the so-called Generalized Maxwell model, consisting of a linear elastic module connected in parallel to several parallel Maxwell branches. This approach is commonly used to incorporate the rate dependency effect on the material stiffness. Different viscosities were assigned to the dashpot of each Maxwell component to introduce different time scales.

Reference: Constr. Build. Mater. 283 (2021) 122725 (Application for the identification of viscoelastic parameters of asphalt matrix)
CPronySeries
CSpringLinear
n x CMaxwell

This module is similar to the previous one except for the connection in series between the Maxwell branches and the linear elastic spring.

CMaxwellElastoPlastic
CElastoplastic
n x CMaxwell

This module incorporates rate-independent plasticity into the Generalized Maxwell model.

CMaxwellElastoPlasticSeries
CElastoplastic
n x CMaxwell

This module is obtained as a result of adding in series the classical elasto-plastic module to a collection or parallel-conected Maxwell branches.

CElastoPlasticDamage
CElastoplastic
CDamage

By adding damage in series, this combined module provides a handy representation of a rate-independent elasto-plastic-damageable solid.

CMaxwellDamage
CMaxwell
CDamage

Connection between a classical Maxwell branch in series with a traction-separation-based damage module.

CMaxwellElastoPlasticDamage
CMaxwellElastoPlastic
CDamage

This module incorporates damage to the plastic-enriched version of the Generalized Maxwell model. The resultant module is a visco-elastic-plastic-damageable solid. By using a pressure-dependent yield criterion for the elasto-plastic component, this combined module makes it possible to describe asphaltic-based materials.

CMaxwellDashpotGeneralizedDamage
CSpringLinear
CMaxwell
CDashpotGeneralized
CDamage

This module provides strain-rate dependency in stiffness and strength by combining non-linear visco-plasticity, the Generalized Maxwell model and energy-based damage. This combined module makes it possible to describe the response of polymers.

Reference1: "Composite micro-scale model accounting for debonding, strain-rate dependence and damage under impact using an explicit finite element solver", 17th European Conference on Composite Materials
Reference2: "Modelling nonlinear material response of polymer matrices used in fiber-reinforced composites", Book chapter (ISBN: 9780128189856)
CAsphaltMatrix
CSpringLinear
CDashpotBergstromBoyce
CViscoDamage

This module provides strain-rate dependency by combining liner-elastiticy, the non-linear Bergstrom-Boyce viscous dashpot and a rate-dependent version of the damage model derived from CDamage. This combined module makes it possible to describe the response of the asphalt matrix.

Reference1: J. Int. Mech. Sciences 224 (2022) 107309 (rate- and pressure-dependent damageable model to capture compression–tension asymmetry in asphalt-based composites)
Reference2: 9th International Conference on Mechanics and Materials in Design (M2D2022), Funchal/Portugal, June 2022, ISBN:978-989-54756-3-6
Reference3: 10th International Conference on Mechanics and Materials in Design (M2D2024), Nagoya/Japan, September 2024, ISBN:TBD
CConstThermalDashpotGeneralized
CConstThermal
CSpringLinear
CDashpotGeneralized
CSpringLangevin

This advanded combined module integrates the thermal complementary module to achieve full thermo-mechanical coupling. With this thermal component, heat transfer analysis can be conducted by considering heat generated from dissipation energy of the mechanical part. Thermal softening effect under high strain rates can be captured accordingly. This module constitutive model captures the response of thermosets polymers such as Epoxy.

CExplicitSCP
CConstThermal
CSpringLinear
CDashpotGeneralized
CSpringLangevin
CElastoPlastic

This advanced module combination describes the thermo-mechancial response of Semi-Crystalline Polymers (SCP) such as PA6, nylon 101 or PE. The crystalline phase is explicitely described as an equivalent elastoplastic response that is activated according to the yield kinetics. Apart from self-heating and thermal softening, this model captures very accurately the "Double Yield" phenomenon.

Reference: Int. J. Plast. 153 (2022) 103233 (General formulation, validation and applicability)
CUnifiedSCP
CConstThermal
CSpringLinear
CDashpotGeneralizedUnified
CSpringLangevin

This module relies on a unified formulation to capture the thermo-mechanical response of a wide variety of polymers. It uses a single viscoplastic law integrating both amorphous and crystalline phases under non-isothermal conditions useful to model either thermoplastics or thermosets. Dependencies of temperature, strain rate and crystallinity on the material response can be studied with this module. Apart from its accurary and applicability range, this module needs only three crystalline-related parameters with clear interpretation from stress–strain curves. This unified model predicts the experimental results of plastics such as epoxy, PEEK, nylon 101, PA6, LDPE, HDPE, PP and polymer blends.

Reference1: Int. J. Plast. 159 (2022) 103469 (General formulation, validation and applicability)
Reference2: Mater. Des. 224 (2022) 111367 (Validation and applicability in quasistatic-to-dynamic regime)
Reference3: Polymers 15(5) (2023) (Application to multiscale modelling of UD composites under high speed compression)
Reference4: Polymers 16(8) (2024) (Application to charactrization and modeling of recycled polyolefin blends)
Reference5: "Advanced characterization and thermomechanical modeling of PP/PE blends for innovations in recycled composites", 21st European Conference on Composite Materials (vol.2, p.576-582, 2024, ISBN:9782912985019)

Applicability

Nowadays, Multiscale modeling is commonly used to develop new advanced composite materials, where three basic scales are covered. Industrial sectors like automotive, aerospace, construction and road engineering, energy or medicine are introducing the multiscale approach in their cycles of design, because the complexity of composites is high and intermediate description levels are necessary to introduce details of their internal architecture and material features. Having suitable constitutive material models for each scale is essential to get meaningful results from the multiscale models. Micro-scale models require relatively simple geometries but detailed material laws dedicated to each constituent. This scale brings great benefits in terms of understanding but demands more computational effort. An adequate material model is the key point to establish which phenomena are truly relevant in order to capture the behavior of the composite material at higher description levels.

Micro-scale modeling

The microscale level typically consists of three basic constituents: matrix, inclusions and matrix-inclusion interface. This level provides rich details on how deformation and damage occur at micrometer scales when an external macroscopic load is applied. This approach is suitable to any composite-based material in very different applications, like for instance, design of fiber-reinforced polymers in material engineering. New polymer-based lightweight structures (bike, car, airplane, medical orthosis,...) consist in glass or carbon fibers embedded in the polymer resin. The companion picture shows an example where we used our library: (left) microscale simulation of carbon fibers embedded in a viscoplastic matrix under tranversal load and (right) SEM image of fibre-matrix failure. (Research developed within the context of the M3Strength project).

Meso-scale modeling

In road engineering, asphalt concrete used in pavement design consists in stone aggregates embedded in an asphaltic bituminous matrix. The properties of the constituent materials used in this case share many aspects with fiber-reinforced polymers (see above): very stiff and brittle reinforcement inclusions (aggregates or fibers) embedded in a visco-elasto-plastic-damageable matrix (asphalt or polymer), which is greatly affected by temperature and loading conditions. The companion figure shows damage by compression in cylindric specimen using a constitutive model of our library to simulate the asphalt matrix. This material model is a combination of asphalt matrix and aggregates where the effect of small aggregates and voids have been homogenized in the overall response of the matrix. (Ongoing research developed by Ziwei DAI in her joint-PhD UGent-UTongji).

Macro-scale modeling

In practical and "human-size" applications, as those carried out by manufacturers of structural components (e.g., car anti-intrusion door bar, see companion figure), micro- or meso-scales cannot be used, as these simulations are not feasible due to excessive computation time. Nevertheless, the macroscopic description gets the benefits of being enriched with all the information obtained from the lower scales. Non-linear and rate-dependent effects described by some of the material models in our library can be transformed into their anisotropic version in an efficient manner. The effect of the fibers can be incorporated by kinematically restricting the nonlinearities along the desired directions. With this homogenization-like approach, the constitutive polymer model used in previous scales still preserves its original shape and facilitates a more clear correlation between results from different scales. (Research developed within the context of the RACE-TP project).

Contact

Address:
Mechanics of Materials and Structures research group
Department of Materials, Textiles and Chemical Engineering
Ghent University
Tech Lane Ghent Science Park – Campus A Technologiepark Building 46
B-9052 Zwijnaarde (Ghent)
BELGIUM

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