Constitutive modeling refers to the development of equations describing the way that materials respond to various stimuli. In classical deformable body mechanics, a simple constitutive model might predict the stress required to induce a given strain; the canonical example is Hooke’s law of isotropic linear elasticity. More broadly, a constitutive model predicts increments in some macroscale state variables of interest (such as stress, entropy, polarization, etc.) that arise from changes in other macroscale state variables (strain, temperature, electric field, etc.).

Constitutive equations are ultimately implemented into a finite element code to close the set of equations required to solve problems of practical interest. This course describes a few common constitutive equations, explaining what features you would see in experimental data or structural behavior that would prompt you to select one constitutive model over another, how to use them in a code, how to test your understanding of the model, how to check if the code is applying the model as advertised in its user’s manual, and how to quantitatively assess the mathematical and physical believability of the solution.

The related – but fundamentally different – discipline of Materials Science aims to reveal the underlying microscale physical mechanisms (such as grain structure, dislocation density, etc.) that give rise to the relationships observed in the laboratory. Stated differently, constitutive equations predict what happens, whereas materials science explains why it happens. Materials Science plays an essential role in revealing appropriate definitions of and relationships between macroscale state variables. As such, even though this course focuses on the implementation and testing of the final equations themselves, the reasoning behind the equations (whether based on empirical observations or microscale or atomistic considerations) is essential to check the predictions and to assess if the equations are being used within their applicability domains.

Working from a premise that (aside from user input typos) the largest source of error in typical engineering finite-element simulations is modeling uncertainty in the constitutive equations, this course surveys a small selection of common constitutive models as a means of illustrating principles of verification (which is evidence that the equations are solved correctly) and validation (which is evidence that the equations are realistic).  Students will write their own stand-alone constitutive model driver, which will be compared with single-element testing of finite element models that purport to implement the same equations. Considerable emphasis is placed on exposing applicability limits of constitutive models.

 For the full course syllabus, click here.