Uncertainty quantification of elastic material responses: testing, stochastic calibration and Bayesian model selection
- PDF / 3,132,437 Bytes
- 18 Pages / 595.224 x 790.955 pts Page_size
- 23 Downloads / 189 Views
ORIGINAL PAPER
Uncertainty quantification of elastic material responses: testing, stochastic calibration and Bayesian model selection Danielle Fitt1 · Hayley Wyatt2 · Thomas E. Woolley1 · L. Angela Mihai2 Received: 3 September 2019 / Accepted: 9 October 2019 © The Author(s) 2019
Abstract Motivated by the need to quantify uncertainties in the mechanical behaviour of solid materials, we perform simple uniaxial tensile tests on a manufactured rubber-like material that provide critical information regarding the variability in the constitutive responses between different specimens. Based on the experimental data, we construct stochastic homogeneous hyperelastic models where the parameters are described by spatially independent probability density functions at a macroscopic level. As more than one parametrised model is capable of capturing the observed material behaviour, we apply Bayes’ theorem to select the model that is most likely to reproduce the data. Our analysis is fully tractable mathematically and builds directly on knowledge from deterministic finite elasticity. The proposed stochastic calibration and Bayesian model selection are generally applicable to more complex tests and materials. Keywords Stochastic elasticity · Finite strain analysis · Hyperelastic material · Bayes’ factor · Experiments · Probabilities “This task is made more difficult than it otherwise would be by the fact that some of the test-pieces used have to be moulded individually, and it is difficult to make two rubber specimens having identical properties even if nominally identical procedures are followed in preparing them.” - R.S. Rivlin and D.W. Saunders [55]
1 Introduction The study of material elastic properties has traditionally used deterministic approaches, based on ensemble averages, to quantify constitutive parameters [36]. In practice, these parameters can meaningfully take on different values corresponding to possible outcomes of the experiments. The art and challenge of experimental setup is to get as close as possible to the ideal situations that can be analysed mathematically. From the mathematical modelling point of view, stochastic representations accounting for data dispersion are needed to improve assessment and predictions [9, 14, 18, 26, 45, 50, 61, 71]. Recently, stochastic models (described by a strain-energy density) were proposed for nonlinear elastic materials, where the parameters are characterised by probability distributions at a continuum level [37, 65–69]. These are advanced L. Angela Mihai
[email protected] Danielle Fitt [email protected] Hayley Wyatt [email protected] Thomas E. Woolley [email protected] 1
School of Mathematics, Cardiff University, Senghennydd Road, Cardiff, CF24 4AG, UK
2
School of Engineering, Cardiff University, The Parade, Cardiff, CF24 3AA, UK
D. Fitt et al.
phenomenological models that rely on the finite elasticity theory [15, 48, 76] and on the maximum entropy principle to enable the propagation of uncertainties from input data to output quantities [62]. The pr
Data Loading...
