## Theoretical Solid Mechanics

We carry out research in elastic waves and vibrations in waveguides; the mathematical theory of nonlinear elasticity; structural vibrations induced by fluid flow; the mathematics of thin elastic layers; ground vibrations produced by high speed trains; and waves in multi-layered elastic media. For more information on PhD opportunities, please see here.

Stability of structures has always been an important consideration in engineering designs since loss of stability will in general compromise the load-bearing capacity of structures. Loss of stability is usually manifested through buckling (e.g. a plastic ruler may buckle when compressed; a submarine may buckle/collapse if it dives deep enough). Mathematically, buckling corresponds to *bifurcation* from one solution to another solution.

There are two types of buckling patterns: periodic or localized. If you squeeze your skin, you will see periodic patterns. One strand of our recent research is concerned with situations where the buckling pattern is localized. Localized bulging of inflated tubular party balloons is the simplest example (see picture above). A major difference between the two types of bifurcation problems is that the critical load for periodic buckling can be determined by a *linear* analysis, but for localized buckling linear analysis gives very little information and the necessary *nonlinear* analysis is much more challenging.

Other examples of localized buckling include necking of stretched metal bars, cavitation (hole formation) in 3D stretching of solid balls, knot formation in twisted hose pipes, collapse of pipelines in deep water, phase transition in shape memory alloys. We are hoping that understanding localized bulging in an inflated rubber tube will provide insight into the entire class of localized buckling problems.

We are the **first** to recognize that localized bulging of an inflated rubber tube is a bifurcation problem [1]. This was achieved through the use of a membrane theory under which the governing equations are ordinary differential equations and so we only had to deal with a *finite* dimensional (spatial) dynamical system. A significant advance was made in [2] where we considered a rubber tube of *arbitrary *thickness that are governed by nonlinear partial differential equations. We are able to show that the bifurcation condition for localized bulging takes a very simple form, namely that the Jacobian determinant of the internal pressure and axial force, as functions of two deformation measures, is equal to zero. Guided by this new result, a new set of experiments was carried out to confirm the theoretical predictions [3]. In particular, it is shown that localized bulging is indeed extremely sensitive to material or geometrical imperfections. More recently, a weakly nonlinear analysis was shown to be possible even for a rubber tube of arbitrary thickness [4] (the difficulty here is that the governing nonlinear PDEs all have variable coefficients). Excellent agreement with finite element simulations was demonstrated.

[1] Post-bifurcation analysis of a thin-walled hyperelastic tube under inflation, IJNM (2008)

[2] Localized bulging in an inflated cylindrical tube of arbitrary thickness – the effect of bending stiffness, JMPS (2016)

[3] An experimental study of localized bulging in inflated cylindrical tubes guided by newly emerged analytical results, JMPS (2019)

[4] Weakly nonlinear analysis of localized bulging of an inflated hyperelastic tube of arbitrary wall thickness, JMPS (2020)

**Collabarators:**Profs Shibin Wang, Linan Li, Yang Liu, Tianjin University, China

We have developed new mathematical methods for analysing elastic wave propagation and vibration in engineering structures [1,2]. In both papers, the results were obtained by mathematical deduction from the governing equations, rather than from kinematic hypotheses or modelling assumptions, and the papers employed a high level of mathematical analysis throughout.

The first determined very accurate approximations to the shape of the elastic field in a layer which can be expressed in terms of a small number of elementary functions, but retain accuracy up to very high frequency. The second gave highly accurate formulae for the frequency and wavelength of the waves expected in certain types of elastic duct.

[1] Chapman CJ, Sorokin SV, 2016 A class of reduced-order models in the theory of waves and stability. Proc. R. Soc. A 472: 20150703

[2] Chapman CJ, Sorokin SV, 2017 The deferred limit method for long waves in a curved waveguide. Proc. R. Soc. A 473:20160900

**Collaborator: **

Sergey Sorokin, Aalborg University, Denmark

Understanding the behaviour of materials with small inclusions or defects is of paramount importance in engineering. This is particularly the case when such materials form components of engineering assemblies frequently used by society, such as bridges and buildings. We develop powerful methods to create approximate formulas enabling us to understand the response of materials with large collections of small defects. These methods can account for the shape and size of the small defects as well as how these inclusions interact. They provide efficient alternatives to numerical procedures commonly used by engineers in addressing the above problems.

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We have developed these approximations for problems involving bodies with clusters of small impurities. Our approximations are capable of treating scenarios involving sparse collections of small defects through to dense clusters of impurities.

[1] Green's Kernels and Meso-Scale Approximations in Perforated Domains, Electrostatics (Springer Book)

[2] Mesoscale Models and Approximate Solutions for Solids Containing Clouds of Voids, Elastostatics (MMS 2016)

[3] Asymptotic Analysis of Solutions to Transmission Problems in Solids with Many Inclusions, Heat conduction (SIAM JAM 2017)

[4] Eigenvalue Problem in a Solid with Many Inclusions: Asymptotic Analysis, Vibration (MMS 2017)

**Collaborators:**Prof. Vladimir Maz’ya, Leningrad University, Russia & Linköping University, Sweden

Prof. Alexander Movchan, University of Liverpool, UK

Buckling is viewed as something undesirable in traditional engineering (e.g. buckling of a column in a bridge). However in recent decades it has come to be realized that buckling patterns can be harnessed to serve useful purposes. For instance, lotus leaves are water repellent due to the delicate micro-structures on their surfaces; such micro-structures can be mimicked by buckling patterns. The simplest setup to produce controllable stress-induced patterns is a film/substrate bilayer (our skin is a natural example!). Thus, understanding pattern formation on a film/substrate bilayer has applications at the micrometer and submicrometer scales ranging from cell patterning, optical grating, to creation of surfaces with desired wetting and adhesion properties. When the film is much harder than the substrate, the initial periodic wrinkling would be followed by period-doubling secondary bifurcations as compression continues. We developed an asymptotic method for predicting the critical compression for the secondary bifurcation [1]. This was built on our earlier paper [2] where it was shown, **for the first time**, that there exists a critical stiffness ratio such that on one side (stiff film) of this ratio the bifurcation is supercritical and on the other side the bifurcation is subcritical. Generally speaking, when the bifurcation is super-critical, we observe periodic patterns, whereas when the bifurcation is sub-critical, we observe localized patterns such as creases [3]. The review article [4] results from collaborations with our colleagues in the Institute of Science and Technology in Medicine who are using dynamic wrinkling patterns to guide cell growth/differentiations.

[1] An Asymptotic Analysis of the Period-Doubling Secondary Bifurcation in a Film/Substrate Bilayer, SIAM JAM (2015)

[2] On the imperfection sensitivity of a coated elastic half-space, PRS (1999)

[3] Buckling of a coated elastic half-space when the coating and substrate have similar material properties, PRS (2015)

[4] Biomedical Applications of Wrinkling Polymers. Recent Progress in Materials. (2020)

**Collaborators: **

Prof. Zongxi Cai, Tianjin University, China

Prof. Pasquale Ciarletta, Politecnico di Milano, Italy

Prof. Ying Yang, School of Pharmacy and Bioengineering, Keele University

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We study how materials undergo fracture. This is a process that embeds several phenomena occurring at multiple scales of the material. Conventional models of failure in continuous solids are not able to capture all information concerning these multi-scale processes. To help address this, we model the failure of lattice materials that represent the microstructure of a continuous solid. Some questions we ask are:

- Can one predict the dynamic failure phenomenon induced by waves propagating in this structure?
- Is the crack propagation steady, following a straight path, or is its behaviour more complicated?
- What speed does the crack possess?
- What other dynamic processes appear as a result of this failure?

These questions are addressed in the following series of monograph and papers.

[1] Maz’ya, V., Movchan, A., Nieves, M., (2013):*Green’s Kernels and Meso-scale Approximations in Perforated Domains*, Lecture Notes in Math. 2077, Springer.

[2] Maz’ya, V.G., Movchan, A.B., Nieves, M.J (2014):

*Mesoscale approximations for solutions of the Dirichlet problem in a perforated elastic body*, J. Math. Sci. (N.Y.) 202, no. 2, 215-244.

[3] Maz’ya, V.G., Movchan, A.B., Nieves, M.J (2016):

*Mesoscale models and approximate solutions for solids containing clouds of voids*, Multiscale Model. Simul. 14, no.1, 138-172.

[4] Nieves, M.J. (2017):

*Asymptotic analysis of solutions to transmission problems in solids with many inclusions*, SIAM J. Appl. Math. 77 (4), 1417-1443.

[5] Maz’ya, V.G., Movchan, A.B., Nieves, M.J. (2017):

*Eigenvalue problem in a solid with many inclusions: asymptotic analysis*, Multiscale Model. Simul., 15, no. 2, 1003–1047.

[6] Maz’ya, V., Movchan, A.B., Nieves, M.J. (2020):

*On meso-scale approximations for vibrations of membranes with lower-dimensional clusters of inertial inclusions*

*Algebra i Analiz*, no. 3, 219-237.

**Collaborators:**Dr Nikolai Gorbushin, École Supérieure de Physique et de Chimie Industrielles, France

Prof. Ian Jones, Liverpool John Moores University, UK

Prof. Gennady Mishuris, Aberysywth University, UK

Prof. Leonid Slepyan, Tel Aviv University, Israel

Prof. Andrea Piccolroaz, University of Trento, Italy

*Soft* solids such as hydro-gels, creams, foams, rubbers, pressure-sensitive adhesives are being increasingly used in various hi-tech applications (e.g. tissue engineering, flexible electronics and soft robotics). A major difference between such soft solids and traditional *hard* engineering materials is that surface tension may no longer be negligible for soft solids.

Due to surface tension, a beading instability takes place in a long enough fluid column that results in the breakup of the column (known as the Plateau-Rayleigh instability). Similarly, a soft elastic cylinder under axial stretching can develop an instability if the surface tension is large enough. Understanding the mechanisms leading to such a shape transformation is of major importance since, for instance, axonal beading is triggered by mechanical trauma and is also believed to precede axonal atrophy in Alzheimer's disease, Parkinson's disease, and other neurodegenerative conditions. We have carried out a systematic nonlinear analysis of this phenomenon and shown that whereas fluid beading follows a supercritical bifurcation solid beading or necking is a subcritical localized instability akin to phase transition. Our weakly and fully nonlinear analysis shows precisely how beading initiates and evolves into a “two-phase” state under three different loading scenarios.

Necking, beading, and bulging in soft elastic cylinders (2020)

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**Collaborators: **

Prof. Alain Goriely, Oxford University

Dr. Lishuai Jin, Harvard University

Dielectric elastomers have received much attention in recent decades due to their use in high performance actuators for a variety of applications. An important technical problem involving such materials is the loss of stability when a dielectric membrane coated with soft electrodes is subjected to in-plane stretching as well as an electric field. Our recent work [1] corrects two widespread misconceptions that have often appeared in the literature. The first one is that when the voltage reaches a maximum in uniform loading, corresponding to marginal violation of the Hessian stability criterion under dead mechanical loading, the membrane will thin down uniformly (commonly referred to as pull-in instability), leading sometimes to dielectric failure. The second one is that since the Hessian stability criterion refers to stability with respect to spatially uniform perturbations, it only has limited applicability and so other stability criteria might be more relevant. We show that when the Hessian stability criterion is marginally violated, i.e. when the determinant of the Hessian matrix vanishes, localized necking will take place and will, in the absence of any constraint, rapidly evolve into a two-phase deformation. We have also developed a reduced model to describe the leading-order bending stiffness effect in dielectric membranes [2].

[1] Localized necking of a dielectric membrane, EML (2018)[2] A reduced model for electrodes-coated dielectric plates, IJNM (2018)

**Collaborators:**Prof. Luis Dorfmann, Tufts University, USA

Prof. Yuxin Xie, Tianjin University, China

#### Academic Staff

#### Projects

- Waves in layered and curved media (
**Professsor C. J. Chapman**) - Localised buckling and pattern formation in soft materials (
**Professor Yibin Fu**) - Asymptotic analysis of thin elastic structures (
**Professor J. D. Kaplunov**) - Dynamic failure in structured materials (
**Dr Mike Nieves**) - Explicit asymptotic models for meta-surfaces (
**Dr D. Prikazchikov**) - Vibrations of high-contrast materials (
**Dr L. Prikazchikov****a**)

#### PhD students

##### Current

- Dominic Emery (Supervisor: Yibin Fu)
- Ali Mohammed Mubaraki (Supervisor: Danila Prikazchikov)
- Roza Sabirova (Supervisors: Julius Kaplunov and Danila Prikazchikov)
- Andreas Chorozoglou (Supervisor: Julius Kaplunov)

##### Recent

- Mohammed Alkinidri (Supervisor: Julius Kaplunov)
- Ali Althobaiti (Supervisor: Yibin Fu)
- Saad Althobaiti (Supervisor: Danila Prikazchikov)
- Ahmed Alzaidi (Supervisor: Julius Kaplunov)
- Roman Chebakov (Supervisors: Julius Kaplunov and Graham Rogerson)
- Marta Garau (Supervisor: Mike Nieves)
- Geethamala Francis (Supervisor: Yibin Fu)
- Maha Helmi (Supervisor: Ludmila Prikazchikova)
- Olga Sergushova (Supervisor: Danila Prikazchikov)
- Anna Shestakova (Supervisor: Julius Kaplunov)
- Peter Wootton (Supervisor: Julius Kaplunov)

See Publications