# A Tutorial on Bayesian Inference to Identify Material Parameters in Solid Mechanics

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## Abstract

The aim of this contribution is to explain in a straightforward manner how Bayesian inference can be used to identify material parameters of material models for solids. Bayesian approaches have already been used for this purpose, but most of the literature is not necessarily easy to understand for those new to the field. The reason for this is that most literature focuses either on complex statistical and machine learning concepts and/or on relatively complex mechanical models. In order to introduce the approach as gently as possible, we only focus on stress–strain measurements coming from uniaxial tensile tests and we only treat elastic and elastoplastic material models. Furthermore, the stress–strain measurements are created artificially in order to allow a one-to-one comparison between the true parameter values and the identified parameter distributions.

## Glossary

- Conditional probability distribution
Conditional probability distribution \(\pi (y|x)\) provides the plausibility of proposition

*y*, given proposition*x*- Correlation
A general term for the dependence between pairs of random variables

- Correlation coefficient
A measure for the strength of the dependence between pairs of random variables

- Covariance
A measure that shows how two random variables depend on each other

- Covariance matrix
A symmetric matrix in which the off-diagonal elements are covariances of pairs of random variables and the diagonal elements are variances of random variables

- Credible interval (region)
An interval (or a region in the multivariate case) of a distribution in which it is believed that one or more random variables (parameters in this study) lie with a certain probability

- Dependence and independence
Two events are statistically independent if the occurrence of one has no influence on the probability of the occurrence of the other one (i.e. \(\pi (x)=\pi (x|y)\)). They are dependent if the occurrence of one has an influence on the probability of the occurrence of the other one (i.e. \(\pi (x)\ne \pi (x|y)\))

- Event
A set of outcomes of an experiment

- Joint distribution
A multivariate distribution

- Laplace approximation
An approximation of a distribution with a Gaussian distribution centred at the MAP

- Likelihood function
If the conditional probability distribution \(\pi (y|x)\) is regarded as a function of

*x*for given fixed*y*, the function is called a likelihood function. The likelihood describes the plausibility of a parameter, given observations- Marginal distribution
A probability distribution as a function of a single variable or a combination of subsets of variables associated with a multivariate distribution (e.g. \(\pi (x)\), \(\pi (y)\), \(\pi (x,y)\), \(\pi (x,z)\) and \( \pi (y,z)\), for joint distribution \(\pi (x,y,z)\)). A marginal distribution is obtained by integrating a multivariate distribution over one or more (but not all) other variables

- Markov chain
A stochastic model to describe a sequence of events in which the probability of each event only depends on the previous event

- Markov chain Monte Carlo (MCMC) methods
A set of techniques to draw samples (i.e. simulate observations) from probability distributions by the construction of a Markov chain

- Maximum a posteriori probability (MAP) point
A point at which the posterior distribution is (globally) maximum

- Mean (expected value)
A measure for the central value of the underlying distribution

- Multivariate distribution
A probability distribution of two or more random variables

- Point estimate
A scalar that measures a feature of a population, e.g. the mean value, the MAP point

- Population
The total set of all possible observations that can be made

- Posterior distribution (posterior)
The probability distribution that describes one’s knowledge about a random variable (parameter in this study) after obtaining new measurements

- Posterior predictive distribution (PPD)
The distribution of unobserved measurements (observations), given the measured (observed) data

- Prior distribution (prior)
The probability distribution that describes one’s a-priori knowledge about a random variable (parameter in this study)

- Probability
The likelihood (or plausibility) that a certain event occurs

- Probability density function (PDF)
The equation that describes a continuous probability distribution

- Probability distribution
A function that provides the probabilities of the occurrence of the possible outcomes of an experiment

- Random sample
A randomly chosen sample

- Random variable
A variable of which the value depends on the outcome of a random experiment

- Realisation
The value that a random variable takes or the outcome of an experiment after its occurrence

- Sample
A set of observations from a population with the purpose of investigating particular properties of the population

- Standard deviation
A measure for the possible deviation of a random variable from its mean. Large standard deviations indicate large possible differences; and vice versa

- Validation point
A measurement (observation) used to assess the quality of a prediction based on the identified parameters, that is not used for the identification itself

- Variance
The standard deviation squared

## Notes

### Acknowledgements

Hussein Rappel, Lars A.A. Beex and Stéphane P.A. Bordas would like to acknowledge the financial support from the University of Luxembourg. Hussein Rappel and Lars A.A. Beex are also grateful for the support of the Fonds National de la Recherche Luxembourg DFG-FNR grant INTER/DFG/16/1150192. Stéphane P.A. Bordas and Lars A.A. Beex are also grateful for the support of the Fonds National de la Recherche Luxembourg FNRS-FNR grant INTER/FNRS/15/11019432/EnLightenIt/Bordas. Jack S. Hale received funding from the National Research Fund, Luxembourg, and cofunded under the Marie Curie Actions of the European Commission (FP7-COFUND Grant No. 6693582).

### Compliance with Ethical Standards

### Conflict of interest

The authors declare that they have no conflict of interest.

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