Time Domain Deformation Modes and Test Data

Deformation Modes in Time Domain

In the time domain there are five available deformation modes for you to choose from: Uniaxial, Biaxial, Planar, Simple Shear and, Volumetric. When you select a deformation mode, you are choosing a built-in simulation model that will be available for calibration. Each deformation mode has a minimal set of required test data that includes: time, a prescribed driving strain, and a primary stress response. In each case, the primary stress response is the conjugate to the driving strain. Some of the deformation modes also support optional secondary strain and stress responses that you can use in a calibration. The term secondary in this context means they are not conjugate to the driving strain. The available strain and stress labels for each deformation mode appear in the table below.

Table 1. Test Data Labels

Deformation Mode	Prescribed Driving Strain	Primary Stress Response	Secondary Strain Response	Secondary Stress Response
Uniaxial	Nominal Uniaxial Strain $\mathbf{\left(}{\mathit{ϵ}}_{\mathit{U}}\mathbf{\right)}$ or True Uniaxial Strain $\mathbf{\left(}{\mathit{\epsilon }}_{\mathit{U}}\mathbf{\right)}$	Nominal Uniaxial Stress $\mathbf{\left(}{\mathit{T}}_{\mathit{U}}\mathbf{\right)}$ or True Uniaxial Stress $\mathbf{\left(}{\mathit{\sigma }}_{\mathit{U}}\mathbf{\right)}$	Nominal Lateral Strain $\mathbf{\left(}{\mathit{ϵ}}_{\mathit{L}}\mathbf{\right)}$ or Nominal Transverse Strain $\mathbf{\left(}{\mathit{ϵ}}_T\mathbf{\right)}$	Not Supported
Biaxial	Nominal Biaxial Strain $\mathbf{\left(}{\mathit{ϵ}}_{\mathit{B}}\mathbf{\right)}$ or True Biaxial Strain $\mathbf{\left(}{\mathit{\epsilon }}_{\mathit{B}}\mathbf{\right)}$	Nominal Biaxial Stress $\mathbf{\left(}{\mathit{T}}_{\mathit{B}}\mathbf{\right)}$ or True Biaxial Stress $\mathbf{\left(}{\mathit{\sigma }}_{\mathit{B}}\mathbf{\right)}$	Nominal Lateral Strain $\mathbf{\left(}{\mathit{ϵ}}_{\mathit{L}}\mathbf{\right)}$	Not Supported
Planar	Nominal Planar Strain $\mathbf{\left(}{\mathit{ϵ}}_{\mathit{P}}\mathbf{\right)}$ or True Planar Strain $\mathbf{\left(}{\mathit{\epsilon }}_{\mathit{P}}\mathbf{\right)}$	Nominal Planar Stress $\mathbf{\left(}{\mathit{T}}_{\mathit{P}}\mathbf{\right)}$ or True Planar Stress $\mathbf{\left(}{\mathit{\sigma }}_{\mathit{P}}\mathbf{\right)}$	Nominal Lateral Strain $\mathbf{\left(}{\mathit{ϵ}}_{\mathit{L}}\mathbf{\right)}$	Not Supported
Simple Shear	Nominal Shear Strain $\mathbf{\left(}\mathit{\gamma }\mathbf{\right)}$	Nominal Shear Stress $\mathbf{\left(}{\mathit{T}}_{\mathit{S}}\mathbf{\right)}$	Not Supported	Nominal Lateral Stress $\mathbf{\left(}{\mathit{T}}_{\mathit{L}}\mathbf{\right)}$
Volumetric	Volume Ratio $\mathbf{\left(}\mathit{J}\mathbf{\right)}$	Pressure $\mathbf{\left(}\mathit{p}\mathbf{\right)}$	Not Supported	Not Supported

From a robustness and accuracy standpoint, it is recommended that the primary stress responses are always included in the objective function during a calibration. Nominal lateral strain should only be included if the material has a reasonable amount of compressibility, such as in the case of foam. For nearly incompressible materials, such as rubber, the objective function calculations can be very sensitive to small errors in lateral strain test data. Therefore, you should use primary stress responses only for these materials.

The simulations for all the deformation modes are based on a homogenous deformation. This is typically a good assumption for isotropic materials or orthotropic materials whose primary directions align with the deformation field with a reasonable amount of deformation. For test data that includes inhomogeneous deformation, such as plastic necking, a FE-based model might be more appropriate. Otherwise, you should remove the test data past necking from the set before running a calibration in non-FE-based mode.

The Uniaxial, Biaxial, and Planar deformation modes support both nominal and true strain and stress test data. However, you cannot mix nominal and true data in the same test data set. For example, if for a given set of test data you import nominal strain data the set must include only nominal stress values, and vice versa. Note that test data sets that use nominal test data can be included in the same calibration run that includes test data sets that contain true data.

Converting True Stress and Strain into Nominal Stress and Strain

During a simulation, the analytical execution mode (see About Material Models and Execution Modes) only uses nominal stress and strain values and therefore it is generally recommended that you import nominal stress and strain values. If you import true stress and strain test data, the app internally converts these values to nominal stress and strain for the analytic kernel using the following formulas:

• ${\sigma }_{nom}=\frac{{\sigma }_{true}}{1+{ϵ}_{nom}}$
• ${ϵ}_{nom}=e^{{ϵ}_{true}}-1$

The conversion formulas are based on the assumption that the material is incompressible. This assumption is typically acceptable for rubber materials and metal plasticity. For other materials such as foam, which is typically highly compressible, the conversion formulas might introduce unacceptable errors. Therefore, you should import test data as nominal stress and strain.

Simulations that are run in numerical execution mode can directly use either true or nominal values.

Time Data

Time data must be included with every imported test data set. You can explicitly supply time data, or you can specify a primary driving strain rate, $\dot{\epsilon }$ , or total time of the test, $T$ , and the app computes the time data for you. If you specify a strain rate, the app computes the time data starting at zero and uses a time increment of $\mathit{▵}{\mathit{t}}_{\mathit{i}}\mathit{=}\frac{\mathit{|}\mathit{▵}{\mathit{\epsilon }}_{\mathit{i}}\mathit{|}}{\dot{\mathit{\epsilon }}}$ for the $i^{th}$ data point $(i>1)$ , where $\mathit{▵}{\mathit{\epsilon }}_{\mathit{i}}\mathit{=}{\mathit{\epsilon }}_{\mathit{i}}\mathit{-}{\mathit{\epsilon }}_{\mathit{i}\mathit{-}\mathit{1}}$ . If you specify the total time of the test, the app computes a strain rate as $\dot{\mathit{\epsilon }}\mathit{=}\frac{\mathit{\sum }_{\mathit{i}\mathit{=}\mathit{2}}^{\mathit{N}}\mathit{|}\mathit{▵}{\mathit{\epsilon }}_{\mathit{i}}\mathit{|}}{\mathit{T}}\mathit{,}$ where $N$ is the number of data points.

The time data must begin at zero and be strictly monotonically increasing. If your imported test data does not provide time data explicitly and instead specifies a strain rate or total time for the test, you must make sure that the primary driving strain data does not have any sequential rows with the same strain values. If you have sequential rows with the same strain values in this case, one or more of the strain increment magnitudes $\mathit{|}\mathit{▵}{\mathit{\epsilon }}_{\mathit{i}}\mathit{|}$ will be zero, which will lead to incorrect time data.

Temperature

You can explicitly specify varying temperatures as a column of test data, or you can specify a single test temperature.

Test Data Weights

For some test data sets imported for calibration, you might want to put more emphasis on some parts of the data than on others. In this case, you can define varying relative weights as a column of test data—larger weights put more emphasis on a row of data, smaller weights put less emphasis. Because the scaling is relative, the app internally normalizes the defined weights.

Minimal Required Test Data for Deformation Modes

The Uniaxial deformation mode simulates a standard uniaxial test. The minimal required test data are: time, uniaxial strain ( ${\mathit{ϵ}}_{\mathit{U}}$ or ${\mathit{\epsilon }}_{\mathit{U}}$ ), and uniaxial stress ( ${\mathit{T}}_{\mathit{U}}$ or ${\mathit{\sigma }}_{\mathit{U}}$ ). Nominal lateral strain ( ${\mathit{ϵ}}_{\mathit{L}}$ ) can also be imported. Note that since you cannot mix nominal and true data in the same test data set if you import nominal lateral strain, you must import nominal uniaxial strain ( ${\mathit{ϵ}}_{\mathit{U}}$ ) and nominal uniaxial stress ( ${\mathit{T}}_{\mathit{U}}$ ).

The Biaxial deformation mode simulates a standard biaxial test. The minimal required test data are: time, biaxial strain ( ${\mathit{ϵ}}_{\mathit{B}}$ or ${\mathit{\epsilon }}_{\mathit{B}}$ ), and biaxial stress ( ${\mathit{T}}_{\mathit{B}}$ or ${\mathit{\sigma }}_{\mathit{B}}$ ). Nominal lateral strain ( ${\mathit{ϵ}}_{\mathit{L}}$ ) can also be imported. Note that since you cannot mix nominal and true data in the same test data set if you import nominal lateral strain, you must import nominal biaxial strain ( ${\mathit{ϵ}}_{\mathit{B}}$ ) and nominal biaxial stress ( ${\mathit{T}}_{\mathit{B}}$ ).

The Planar deformation mode simulates a standard planar test. The minimal required test data are: time, planar strain ( ${\mathit{ϵ}}_{\mathit{P}}$ or ${\mathit{\epsilon }}_{\mathit{P}}$ ), and planar stress ( ${\mathit{T}}_{\mathit{P}}$ or ${\mathit{\sigma }}_{\mathit{P}}$ ). Nominal lateral strain ( ${\mathit{ϵ}}_{\mathit{L}}$ ) can also be imported. Because you cannot mix nominal and true data in the same test data set if you import nominal lateral strain, you must import nominal planar strain ( ${\mathit{ϵ}}_{\mathit{P}}$ ) and nominal planar stress ( ${\mathit{T}}_{\mathit{P}}$ ).

The Simple Shear deformation mode simulates a standard simple shear test. The minimal required test data are: time, nominal shear strain ( $\mathit{\gamma }$ ), and nominal shear stress ( ${\mathit{T}}_{\mathit{S}}$ ). Nominal lateral stress ( ${\mathit{T}}_{\mathit{L}}$ ) can also be imported.

The Volumetric deformation mode simulates a standard volumetric test. The minimal required test data are: time, volume ratio ( $\mathit{J}\mathbf{=}{\mathit{\lambda }}_{\mathit{V}}^{\mathit{3}}$ ), and pressure ( $\mathit{p}\mathbf{=}\mathbf{-}\frac{\mathbf{1}}{\mathbf{3}}\mathbf{(}{\mathit{\sigma }}_{\mathit{11}}\mathbf{+}{\mathit{\sigma }}_{\mathit{22}}\mathbf{+}{\mathit{\sigma }}_{\mathit{33}}\mathbf{)}$ ).