This web-based class is an extension of my introductory polymer modeling training class. The class includes a more in depth review of the theory of different material models, and contains hands-on exercises designed to teach how to use the different material models to quickly and accurately solve real problems.

The class will take place on May 8 and 9, 2013. See the following page for more info.

Contact us soon if you want to register!

Model Theory

The Parallel Network (PN) model is an advanced material model for predicting the non-linear viscoplastic response of any polymer material. The rheological structure of the model can be represented using a number of parallel networks, as shown in the following figure.



The PN model is a generalization of both the Bergstrom-Boyce (BB) model, and the Three Network (TN) model. Depending on the selected model structure, the material model can predict almost any isotropic or anisotropic response.


Supported Model Components

The following is a summary of the different supported components and models that can be used with the Parallel Network (PN) model. The details of these different model options are presented in the PolyUMod User's Guide.

The model supports the following elastic components:

• Linear elastic
• Neo-Hookean hyperelastic model
• Mooney-Rivlin hyperelastic model
• Eight-chain hyperelastic model
• Yeoh hyperelastic model
• Gent hyperelastic model
• Bergstrom anisotropic eight-chain model
• Hyperfoam model
• Bischoff anisotropic eight-chain model
• Yeoh hyperelastic model with global stiffening
• Eight-chain model with rate-dependent stiffness
• Four parameter eight-chain hyperelastic model
• Ogden model
• Eight-chain hyperelastic model with small strain softening
• Eight-chain hyperelastic model with I2-dependence
• Holzapfel-Gasser-Ogden model
• Eight-chain model with different stiffness in tension and compression
• Hyperfoam model with different stiffness in tension and compression
• Yeoh hyperelastic model with I2-dependence and pressure dependent bulk modulus
• Hyperfoam model with small strain softening
• Alternative Neo-Hookean hyperelastic model
• Holzapfel-Gasser-Ogden-Bergstrom model

Each of these elastic components can have one of the following optional temperature dependence behaviors:

• Linear temperature dependence
• Exponential temperature dependence
• Power-law temperature dependence
• Piecewise linear temperature dependence

Each of the elastic components can also have one of the following optional thermal expansion behaviors:

• Linear thermal expansion
• Quadratic thermal expansion
• Linear thermal expansion with plasticity

Each of the elastic components can also have one of the following optional damage accumulation behaviors:

• Ogden-Roxburgh Mullins effect
• Enhanced Ogden-Roxburgh Mullins effect
• Linear damage from plastic strain
• Exponential damage growth after initiation
• Enhanced Ogden-Roxburgh Mullins effect with temperature dependence
• Damage evolution from plastic strain

Each of the elastic components can also have one or more of the following optional failure models:

• Max principal stress
• Max Mises stress
• Max principal strain
• Mises strain
• Molecular chain strain
• Molecular chain strain from viscous flow
• Rate of viscous flow
• Plastic Hill strain

In addition to an elastic component, each network can also have an optional flow component. The following flow components are supported by the Parallel Network (PN) model:

• Linear viscoelastic flow
• Power-law flow
• Bergstrom-Boyce (BB) flow
• Exponential energy activated flow
• Anisotropic BB-type flow
• Bergstrom-Boyce network-dependent flow
• Power-law flow with strain-dependence
• Power-law shear and volumetric flow
• Chaboche non-linear kinematic hardening plasticity
• Double power-law flow
• Sinh energy activation flow
• Bergstrom-Boyce flow with strain-dependent m
• Tsai-Wu power flow
• Anisotropic double powerlaw-type flow

Each of these flow types can be combined with one of the following optional temperature dependence models:

• Linear temperature dependence
• Exponential temperature dependence
• Power-law temperature dependence
• Piecewise linear temperature dependence
• Temperature raised to a power

Each of the flow types can be combined with one of the following optional pressure dependence models:

• Linear pressure dependence
• Truncated linear pressure dependence
• Different flow in tension and compression

Each of the flow models can also be combined with one of the following optional yield evolution models:

• Piecewise linear flow resistance
• Exponential evolution of the flow resistance
• Double exponential evolution of the flow resistance
• Increasing exponential evolution of the flow resinstance
• Anisotropic exponential evolution of the flow resistance
• Linear rate evolution with plastic strain
• Incremental double exponential evolution of the flow resistance

In addition to these elastic and flow networks, the PN-model supports any combination of the the following optional global failure conditions:

• Max principal true stress
• Mises true stress
• Max principal true strain
• Mises true strain
• True chain strain
• True Hill stress
• True Hill strain
• True stress-based damage accumulation
• True strain-based damage accumulation
• Anisotropic true strain-based damage accumulation
• 2D anisotropic true chain strain
• Bergstrom anisotropic eight-chain failure model
• Bischoff anisotropic eight-chain failure model
• Max true stress failure model
• Max true strain failure model
• Tsai-Hill anisotropic failure model (2D)
• Tsai-Wu anisotropic stress-based failure model (2D)
• Tsai-Wu anisotropic stress-based failure model (3D)
• Polynomial strain-based anisotropic failure model (2D)
• Polynomial strain-based anisotropic failure model (3D)
• Max true fiber strain
• Rate-dependent damage model 1
• Rate-dependent damage model 2
• True chain strain (for tensile stress)
• Max principal true strain (for tensile stress)
• Mises engineering stress

Each of these global failure models can also be combined with an optional damage growth model:

• Exponential growth
• Powerlaw growth

Each of the global failure models can also be combined with an optional temperature dependence model:

• Linear temperature dependence
• Exponential temperature dependence
• Power-law temperature dependence
• Piecewise linear temperature dependence

Each of the global failure models can also be combined with an optional rate-dependence model:

• Linear strain-rate dependence
• Exponential strain-rate dependence
• Piecewise linear strain-rate dependence
• Piecewise linear strain-rate dependence on logarithmic strain-rate

Each of the global failure models can also be combined with an optional output safety factor model:

• Output safety factor
• Output risk factor

The ability of the PN-model to combined all of these different components makes the material model extremely powerful and useful.


Needed Experimental Data

Since the PN-model is modular in structure, it is not possible to give specific information about what experimental data is necessary for a suitable material model calibration. The necessary experimental tests will depend on the specific PN-model structure is selected.


How To Calibrate

The PN-model can be calibrated using the MCalibration software.


Strength and Limitations of the PN-Model

The PN-model can provide exceptionally accurate material model predictions even for complicated anisotropic viscoplastic materials. The main limitation of the PN-model is that it can take some time for a new engineer to fully understand how to most appropriately structure a material model for a new material.


Additional Information

My polymer modeling classes include a nice review of the PN-model.

Model Theory

The Bergstrom-Boyce (BB) model is one of my favorite material models, partly because it was the first advanced material model that I developed, and partly because it actually works really well despite being a simple model.

The following figure shows a rheological representation of the BB-model.



The model consists of two parallel networks: Network A gives the equilibrium response of the material, and network B gives the viscoelastic contribution to the stress. This model structure is deceivingly similar to the rheological representation of linear viscoelasticity (LVE) theory. The BB-model, however, is significantly different than LVE. For example, when using linear viscoelasticity it is typically necessary to use many different parallel networks (where each network corresponds to one Prony series term), but due to the non-linear flow element (dashpot) that is used in the BB-model it is very rarely necessary to have more than two parallel networks.

Network A consists of a hyperelastic component, and Network B consists of a hyperelastic component in series with a non-linear viscoelastic component. The details of the model, and the individual components, are presented in the following papers [paper1, paper2, paper3, My PhD Thesis].

The BB-model is a non-linear viscoelastic material model that can be used to predict:

• strain-rate dependence
• stress relaxation
• creep
• energy loss during cyclic loading (hysteresis)
• Mullins damage during the first few load cycles

The following figure shows representative model predictions. In this case the predictions were obtained in uniaxial tension followed by unloading to zero stress. There are three predictions shown, each corresponding to a different applied strain rate (-0.01/s, -0.1/s, -1/).




Needed Experimental Data

The BB-model can be calibrated using uniaxial tension or compression data alone. Since the material model is non-linear viscoelastic, it is important to use experimental data with multiple strain-rates and/or stress relaxation or creep segments. Send me a message if you would like help with the experimental characterization or calibration of the BB-model.


How to Calibrate

The MCalibration software makes it very easy to calibrate the BB-model.


Strength and Limitations of the BB-Model

The BB-model is available as a built-in feature in Abaqus ans ANSYS, it is also available through the PolyUMod library.
The PolyUMod implementation of the BB-model has the advantage that it works with (virtually) all element types, it works in both implicit and explicit simulations, it works with thermomechanical loads, it includes Mullins damage considerations, and it has often better convergence properties than the built-in implementations.


Additional Information

My introductory polymer modeling classes include a nice derivation and review of the BB-model. The forum section also contains a lots of information about the BB-model.

Metal Plasticity Theory

Metal plasticity theory is a topic that is well known (and covered many text books). The plasticity models are typically to easy to use and can provide accurate results for many metals, but are also sometimes used to predict the mechanical response of solid polymers.


Types of Plasticity Models

Most FE software supports a number of different plasticity models. Here is a list of commonly available isotropic plasticity models:

• J2-plasticity with isotropic hardening
• Kinematic hardening plasticity
• Combined hardening plasticity
• Johnson-Cook plasticity
• Drucker-Prager plasticity
• Gurson porous plasticity

Note that all of these models can be calibrated and examined using the MCalibration software.


Material Classes

The different plasticity models are potentially useful for thermoplastics and thermoset polymers (as long as the temperature is not too high), but should not be used for elastomers or other soft materials.


Needed Experimental Data

Most plasticity models can be calibrated using uniaxial tension and/or compression data only. Unless the material is anisotropic there is usually no benefit to using data obtained in multiple loading modes.


How to Calibrate

The MCalibration software is an excellent tool for calibrating the different plasticity models. MCalibration can, for example, calibrate all material models in Abaqus and ANSYS.


Strengths of Metal Plasticity Models

• Easy to use and calibrate.
• Numerically efficient since the governing equations are simple.

Limitations of Metal Plasticity Models

Can give accurate results for monotonic loading of thermoplastics and thermosets, but is not accurate for elastomers or other soft materials, or any situation involving unloading or cyclic loading.


Additional Information

My introductory polymer modeling classes include a nice derivation and review of the equations behind plasticity models.