Mathematicians propose new model to quantify materials uncertainties

Researchers have developed a novel set of tools for modelling uncertainties in materials, which may provide a more accurate representation of material behaviour in the real world.

During development, our bodies have the amazing ability to change the density of our heart muscle to accommodate differences. Even in the carefully-controlled manufacture of human-made materials, variations in product output exist. No matter if it is coronary, metal or rubber, variability in a material can lead to unexpected outcomes.

Nonlinear elasticity, the study of material elastic properties, has traditionally used deterministic approaches based on average values to quantify constitutive material parameters. In practice, these parameters can meaningfully take on different values, corresponding to possible outcomes of the experimental tests.

Recent work from L. Angela Mihai (Cardiff Mathematics) together with Thomas E.Woolley (Cardiff Mathematics) and Alain Goriely (Oxford Mathematics) suggests that stochastic representations, accounting for data dispersion, are needed to improve assessment and predictions.

Their new research has developed stochastic hyperelastic material models, described by a strain-energy density where the parameters are random variables characterised by probability density functions. Constructed through a Bayesian identification procedure, these models rely on the notion of entropy (or uncertainty) and the maximum entropy principle and enable the propagation of uncertainties from input data to output quantities of interest.

The team investigated the effect of probabilistic model parameters on predicted mechanical responses analytically, for different bodies with simple geometries at finite strain deformations.

Their published case studies, so far, include:

• the classical problem of the Rivlin cube
• the inflation of pressurised spherical and cylindrical shells
• the cavitation of a sphere under uniform tensile dead load

Each of these structures can be used to model and understand, for example, the structural mechanics of different parts of the heart, such as atria, ventricles, or arteries.

By combining knowledge from elasticity, statistics and probability theories, the research offers a richer set of tools to model natural and synthetic materials.

Further research in this area may lead to more accurate representations of material behaviour in the future, which can be used to improve assessment and prediction in areas as diverse as science, manufacturing, infrastructure and medical research.

Recently, L. Angela Mihai has been awarded a new EPSRC grant to support her research in stochastic finite strain analysis. This will enable her to address some important challenges associated with the consideration and quantification of uncertainties in material responses. Her long-term goal is to develop stochastic elasticity methods as powerful and effective tools available for future applications.