Introductory Concepts

Constitutive Models and Calibration

In a numerical simulation a material definition, or constitutive model, is a mathematical representation that is used to approximate the response of a real material subjected to external loadings (mechanical, thermal, etc., ). Constitutive models for mechanical behaviors define relationships between components of stress and strain. For example, in a linear elastic constitutive model, the material parameter Young's modulus ( $\mathit{E}$ ) is used to define a linear relationship between the uniaxial stress ( $\mathit{\sigma }$ ) and strain ( $\mathit{\epsilon }$ ) in a standard tension test as, $\mathit{\sigma }\mathit{=}\mathit{E}\mathit{\epsilon }$ . A simple elastic-plastic constitutive model for a metal, such as steel, might include a temperature independent form of the Johnson-Cook plastic hardening model that assumes that the yield stress ( ${\mathit{\sigma }}^{\mathit{0}}$ ) can be written as ${\mathit{\sigma }}^{\mathit{0}}\mathit{=}\mathit{A}\mathit{+}\mathit{B}{\mathit{\left(}{\overline{\mathit{\epsilon }}}^{\mathit{p}\mathit{l}}\mathit{\right)}}^{\mathit{n}}$ , where $\mathit{A}\mathit{,}\mathit{B}\mathit{,}\mathit{n}$ are material parameters and ${\overline{\mathit{\epsilon }}}^{\mathit{p}\mathit{l}}$ is the equivalent plastic strain.

For a constitutive model to be useful in a numerical simulation it must be an appropriate material model for the intended applications and the material parameters must be calibrated in order to reproduce the required material response. The basic steps for creating a calibrated constitutive model are:

• Collect and import appropriate test data.
• Create or select numerical simulations that accurately model the physical tests that were used to create the imported test data.
• Select a proper constitutive model.
• Set up and run a calibration.

Selecting Test Data for a Calibration

To begin you will need to collect an appropriate set of test data. Ideally, the test data should cover the range of expected in-service loading conditions that the material will experience. In-service loading conditions can include expected: strain levels, strain rates, deformation modes and, environmental conditions such as temperature, etc. It is important that you also understand the requirements of your constitutive model and provide enough test data to properly calibrate all of the behaviors.

For a simple example, suppose you want to calibrate a linear isotropic elastic material model that has Young's modulus $\left(\mathit{E}\right)$ and Poisson's ratio $\left(\mathit{\nu }\right)$ as material parameters. If you have a single set of uniaxial test data that only includes the uniaxial stress data as a function of uniaxial strain you will be able to calibrate $\mathit{E}$ but not $\mathit{\nu }$ . To calibrate $\mathit{\nu }$ you will need another set of test data, such as lateral strain as a function of uniaxial strain or perhaps pressure as a function of volumetric compression. If the required test data is not available you will need to provide an estimate for $\mathit{\nu }$ and only calibrate $\mathit{E}$ . In another example, if you want to calibrate the plastic behavior of an elastic-plastic constitutive model your test data needs to include stress and strain values in the material's plastic regime. It might be perfectly acceptable to only calibrate some of the material parameters in your constitutive model. In the plasticity example, it might be that you already have good estimates for $\mathit{E}$ and $\mathit{\nu }$ in which case you can directly specify these values and only calibrate the plastic behavior.

The Material Calibration app provides a complete set of tools to import, create and, preprocess test data sets (see Test Data and Built-In Models, Test Data and Imported Finite Element Models, and Test Data Processing).

Numerical Simulation

The calibration process is an optimization exercise in which one of more numerical simulations are used to compute response data that correlates with the test data. This requires that you select or provide numerical simulation models designed to approximate the responses of a physical test. In addition you will need to identify matched pairs of measured test data responses and computed responses from the numerical simulation that can be used to quantify the accuracy of the simulation response.

Consider the calibration of a steel alloy that will be used in a thin sheet stamping operation in which the only test data that you have access to is from a single uniaxial test where the coupon was pulled past plastic yield. Since the material is steel, and the loading is primarily monotonic with plastic yield, a reasonable material model for the numerical simulation would be an isotropic elastic-plastic material with Johnson-Cook plastic hardening. In the actual physical test the coupon is clamped into a testing machine and monotonically stretched to some prescribed value of displacement. Test data that would be commonly collected as a function of time includes: clamp displacement, gauge length stretch, uniaxial force response, and perhaps temperature. A reasonable numerical simulation of such a test would most likely be driven by prescribed uniaxial displacement boundary conditions that replicate the uniaxial stretch in the coupon as a function of time. It would also enforce a uniaxial state of stress. If necking of the coupon does not occur during the test, a numerical simulation with a homogeneous deformation gradient would be a reasonable engineering assumption. If necking did occur a simulation model that supports non-homogenous deformations would be needed, or the test data beyond necking would need to be removed from the test data set. The fidelity of the simulation model would be quantified by a computed error norm between the computed uniaxial force/stress in the simulation model and the measured test data. Note that including the difference between the prescribed uniaxial displacements in the test data and the simulation as part of the error function does not make sense because by definition these displacements are identical and would therefore contribute nothing to the error calculation..

The simple example discussed above highlights the following key aspects of a simulation model:

• Time Data: A strictly monotonic scalar parametrization for the simulation.
• Prescribed Conditions: Conditions that are explicitly enforced as a function of position and time in the simulation model. In a displacement-controlled scenario the displacement at a given point/region is assumed to be a known function of time, the resultant (conjugate) force/pressure is a response and needs to be computed. In a load-controlled scenario the load is assumed to be a known function of time, and the resultant displacement is a response and needs to be computed. Environmental conditions such as temperature can also be prescribed.
• Response Data: Non-prescribed responses to prescribed loadings and displacement boundary conditions.
• Response Data Matching: Matching one or more simulation responses to collected test data that can be used in an error calculation.

The Material Calibration app comes with a set of easy to use built-in simulation models that cover the most common types of material testing procedures. It also allows you to import your own finite element simulation models (see About Execution Modes and Simulation Models ).

Selecting a Constitutive Model

To select an appropriate constitutive model it is critical that you understand and explicitly define the requirements of your numerical simulations and the range of needed material responses. For example, suppose you need a material model for an NVH simulation of a steel automobile chassis. Since the simulation is primarily linear in nature, it might be sufficient to calibrate a simple linear elastic material from uniaxial test data in the elastic regime. Suppose, however, that you need to perform a crash simulation of the chassis. In this case you will need a material model that can accurately predict the high-speed plastic response, and perhaps failure, of your steel. So depending on your simulation needs, you might choose to define and calibrate different constitutive models for the same material. The interactive nature of the Material Calibration app can help you in this decision process by allowing you to easily test and compare the responses of a wide range of constitutive models (see About Material Models and Execution Modes).

Running a Calibration

Once an appropriate numerical simulation is available a calibration can be performed by running the numerical simulation with a set of material parameters and computing an error norm of the difference between the measured force in the test data and the computed force from the simulation model. If the measure of the error norm is small enough, small being determined by you, the calibration is complete and the current material model and set of material parameters represent the calibrated model. If the error norm is too large, the current set of material parameters are automatically modified in an attempt to reduce the error norm and the process is repeated until an acceptable error norm is achieved.

The Material Calibration app lets you easily setup and run calibrations (see Running a Calibration Job) and offers a wide range of powerful optimization options and controls (see Optimization Controls).