# Limit analysis of debonding states in multi-body systems of stochastic hyperelastic material

Lead Research Organisation:
Cardiff University

Department Name: Sch of Mathematics

### Abstract

The aim of this project is to establish effective mathematical formulations and construct reliable numerical solution procedures for the debonding analysis of multi-body systems of stochastic hyperelastic material subject to large strain deformations. The theoretical and computational challenges raised by these systems range from the large deformation of individual bodies, to the detection of contact and openings between them, to the estimation of the probability distribution for the critical load such that debonding through loss of contact can or cannot occur. Even though debonding through loss of contact is a mechanism for damage initiation and crack propagation in many natural and engineered materials, it has been insufficiently investigated. For these materials, deterministic approaches, which are based on average data values, can greatly underestimate or overestimate the damage, and stochastic representations accounting also for data dispersion are needed.

In recent years, there has been a growing interest in stochastic modelling techniques for engineering and biomedical applications, where uncertainties in the material parameters calibrated to sparse and approximate observational data cannot be ignored. In the quest for estimating material uncertainties, stochastic finite elasticity introduces stochastic features into the finite elasticity theory in order to characterise the variability in the elastic responses of materials, which are rarely deterministic. Within this framework, stochastic hyperelastic materials are advanced phenomenological models described by a strain-energy function where the parameters are random variables characterised by probability density functions. These models rely on the notion of entropy (or uncertainty) and on the maximum entropy principle for a discrete probability distribution, and are able to propagate uncertainties from input data to output quantities. In this context, the proposed investigation is novel and will contribute to the development of many associated research areas in engineering, biomechanics, and materials science. Specific applications include soft biological materials (e.g., plants, articular cartilages, arterial walls, brain tissue) and engineering structures (e.g., soft actuators, 3D printing composites) at large strains.

In recent years, there has been a growing interest in stochastic modelling techniques for engineering and biomedical applications, where uncertainties in the material parameters calibrated to sparse and approximate observational data cannot be ignored. In the quest for estimating material uncertainties, stochastic finite elasticity introduces stochastic features into the finite elasticity theory in order to characterise the variability in the elastic responses of materials, which are rarely deterministic. Within this framework, stochastic hyperelastic materials are advanced phenomenological models described by a strain-energy function where the parameters are random variables characterised by probability density functions. These models rely on the notion of entropy (or uncertainty) and on the maximum entropy principle for a discrete probability distribution, and are able to propagate uncertainties from input data to output quantities. In this context, the proposed investigation is novel and will contribute to the development of many associated research areas in engineering, biomechanics, and materials science. Specific applications include soft biological materials (e.g., plants, articular cartilages, arterial walls, brain tissue) and engineering structures (e.g., soft actuators, 3D printing composites) at large strains.

### Planned Impact

A central challenge in predicting mechanical responses of many engineering and natural materials is the lack of quantitative characterisation of the uncertainties inherent in the experimental data and in the mathematical models derived from them. In particular, contact problems with large stresses and strains at adjoining surfaces can be found in many important applications, when creases are formed and self-contact takes place in soft materials, or in material structures where the attachment between cells are sufficiently weak so that cells separate and failure through the appearance of gaps between them occurs. Even though debonding through loss of contact is a mechanism for damage initiation and propagation in many natural and engineering materials, it has been insufficiently investigated. Moreover, for these materials, deterministic approaches, which are based on average data values only, can greatly underestimate or overestimate the critical load, and may misrepresent the location and extend of damage. In order to address these limitations, during this project, stochastic representations accounting for data dispersion will be developed that can significantly improve assessment and predictions. To achieve this, the proposed investigation will combine knowledge from nonlinear elasticity, mathematical analysis, numerical analysis, optimisation, applied probability and statistics. Presently, Continuum Mechanics and Statistics & Applied Probability are earmarked as research areas for growth, while Mathematical Analysis and Numerical Analysis continue to be maintained in terms of EPSRC/UKRI investment in Mathematical Sciences. In the UK, the study of how natural and manufactured materials behave when damaged underpins an ever increasing number of applications, and concerns several EPSRC Challenge Themes, including the multidisciplinary Manufacturing the Future and Healthcare Technologies, as well as the fundamental Mathematical, Engineering and Physical Sciences. Internationally, there is a growing interest in stochastic modelling techniques for engineering and biomedical applications, where the calibration of models using available data and the quantification of uncertainties in material parameters are of utmost importance. However, despite the fact that there are numerous challenges associated with the consideration and quantification of uncertainties in material responses, which arise out of incomplete and approximate information, currently, typical nonlinear elasticity applications lie outside the scope of main-stream stochastic modelling, and stochastic hyperelastic models are largely unexplored. Nevertheless, by building on existing deterministic solid mechanics approaches, stochastic elasticity has the potential to become the conventional approach of the future. In this context, the proposed investigation is novel and will contribute to ``The protection of the UK's long-term capability, and the expansion of multidisciplinary research'' as defined in EPSRC's 2016/17-2019/20 Delivery Plan. The long-term goal is to establish stochastic elasticity as an effective and powerful tool capable of dealing with large data sets for complex solid mechanics problems arising in modern applications. Specific applications include soft biological materials (e.g., plants, articular cartilages, arterial walls, brain tissue) and engineering structures (e.g., soft actuators, 3D printing composites) at large strains.

### Organisations

### Publications

Mihai LA
(2022)

*A mathematical model for the auxetic response of liquid crystal elastomers.*in Philosophical transactions. Series A, Mathematical, physical, and engineering sciences
Schweickert E
(2020)

*A note on non-homogeneous deformations with homogeneous Cauchy stress for a strictly rank-one convex energy in isotropic hyperelasticity*in International Journal of Non-Linear Mechanics
Mihai L
(2020)

*A plate theory for nematic liquid crystalline solids*in Journal of the Mechanics and Physics of Solids
Angela Mihai L
(2020)

*A pseudo-anelastic model for stress softening in liquid crystal elastomers.*in Proceedings. Mathematical, physical, and engineering sciences
Goriely A
(2022)

*A Rod Theory for Liquid Crystalline Elastomers*in Journal of Elasticity
Buze M
(2022)

*A Stochastic Framework for Atomistic Fracture*in SIAM Journal on Applied Mathematics
Mihai L
(2023)

*A theoretical liquid crystal elastomer model that mimics the elasticity of cat skin*in Mechanics of Soft Materials
Buze M
(2021)

*Atomistic modelling of near-crack-tip plasticity **in Nonlinearity
Mihai L
(2021)

*Instabilities in liquid crystal elastomers*in MRS Bulletin
Angela Mihai L
(2020)

*Likely cavitation and radial motion of stochastic elastic spheres*in NonlinearityDescription | The study of material properties has traditionally used deterministic approaches, based on ensemble averages, to quantify constitutive parameters. In practice, these parameters can meaningfully take on different values corresponding to possible outcomes of the experiments. From the modelling point of view, stochastic representations accounting for data dispersion are needed to improve assessment and predictions. We develop stochastic material models described by strain-energy densities where the parameters are characterised by probability distributions at a continuum level. To answer important questions, such as ``what is the influence of probabilistic parameters on predicted mechanical responses?'' and ``what are the possible equilibrium states and how does their stability depend on the material constitutive law?'', we focus on likely instabilities in hyperelastic bodies and in nematic liquid crystal elastomers. |

Exploitation Route | The current results are suitable to build on during the remaining time on the project. |

Sectors | Aerospace, Defence and Marine,Healthcare,Manufacturing, including Industrial Biotechology |