We derive a global higher regularity result for weak solutions of the linear relaxed micromorphic model on smooth domains. The governing equations consist of a linear elliptic system of partial differential equations that is coupled with a system of Maxwell-type. The result is obtained by combining a Helmholtz decomposition argument with regularity results for linear elliptic systems and the classical embedding of $H(\operatorname {div};\Omega )\cap H_0(\operatorname {curl};\Omega )$ into $H^1(\Omega )$.

Mathematical models of polyelectrolyte gels are often simplified by assuming the gel is electrically neutral. The rationale behind this assumption is that the thickness of the electric double layer (EDL) at the free surface of the gel is small compared to the size of the gel. Hence, the thin-EDL limit is taken, in which the thickness of the EDL is set to zero. Despite the widespread use of the thin-EDL limit, the solutions in the EDL are rarely computed and shown to match to the solutions for the electrically neutral bulk. The aims of this paper are to study the structure of the EDL and establish the validity of the thin-EDL limit. The model for the gel accounts for phase separation, which gives rise to diffuse interfaces with a thickness described by the Kuhn length. We show that the solutions in the EDL can only be asymptotically matched to the solutions for an electrically neutral bulk, in general, when the Debye length is much smaller than the Kuhn length. If the Debye length is similar to or larger than the Kuhn length, then phase separation can be initiated in the EDL. This phase separation spreads into the bulk of the gel and gives rise to electrically charged layers with different degrees of swelling. Thus, the thin-EDL limit and the assumption of electroneutrality only generally apply when the Debye length is much smaller than the Kuhn length.

Thin spray-on liners (TSLs) have been found to be effective for structurally supporting the walls of mining tunnels and thus reducing the occurrence of rock bursts, an effect primarily due to the penetration of cracks by the liner. Surface tension effects are thus important. However, TSLs are also used to simply stabilize rock surfaces, for example, to prevent rock fall, and in this context crack penetration is desirable but not necessary, and the tensile and shearing strength and adhesive properties of the liner determine its effectiveness. We examine the effectiveness of nonpenetrating TSLs in a global lined tunnel and in a local rock support context. In the tunnel context, we examine the effect of the liner on the stress distribution in a tunnel subjected to a geological or mining event. We show that the liner has little effect on stresses in the surrounding rock and that tensile stresses in the rock surface are transmitted across the liner, so that failure is likely to be due to liner rupture or detachment from the surface. In the local rock support context, loose rock movements are shown to be better achieved using a liner with small Young’s modulus, but high rupture strength.

In this paper, we consider the time decay of the solutions to some problems arising in strain gradient thermoelasticity. We restrict to the two-dimensional case, and we assume that two dissipative mechanisms are introduced, the temperature and the mass dissipation. First, we show that this problem is well-posed proving that the operator defining it generates a contractive semigroup of linear operators. Then, assuming that the function involving the coupling terms is elliptic, the exponential decay of the solutions is concluded as well as the analyticity of the solutions. Finally, we describe how to obtain the exponential stability in the case of hyperbolic dissipation.

When an explosive burns, gaseous products are formed as a result. The interaction of the burning solid and gas is not well understood. More specifically, the process of the gaseous product heating the explosive is yet to be explored in detail. The present work sets out to fill some of that gap using mathematical modelling: this aims to track the temperature profile in the explosive. The work begins by modelling single-step reactions using a simple Arrhenius model. The model is then extended to include three-step reaction. An alternative asymptotic approach is also employed. There is close agreement between results from the full reaction-diffusion problem and the asymptotic problem.

Neodymium magnets were independently discovered in 1984 by General Motors and Sumitomo. Today, they are the strongest type of permanent magnets commercially available. They are the most widely used industrial magnets with many applications, including in hard disk drives, cordless tools and magnetic fasteners. We use a vector potential approach, rather than the more usual magnetic potential approach, to derive the three-dimensional (3D) magnetic field for a neodymium magnet, assuming an idealized block geometry and uniform magnetization. For each field or observation point, the 3D solution involves 24 nondimensional quantities, arising from the eight vertex positions of the magnet and the three components of the magnetic field. The only unknown in the model is the value of magnetization, with all other model quantities defined in terms of field position and magnet location. The longitudinal magnetic field component in the direction of magnetization is bounded everywhere, but discontinuous across the magnet faces parallel to the magnetization direction. The transverse magnetic fields are logarithmically unbounded on approaching a vertex of the magnet.

Tissue engineering aims to grow artificial tissues in vitro to replace those in the body that have been damaged through age, trauma or disease. A recent approach to engineer artificial cartilage involves seeding cells within a scaffold consisting of an interconnected 3D-printed lattice of polymer fibres combined with a cast or printed hydrogel, and subjecting the construct (cell-seeded scaffold) to an applied load in a bioreactor. A key question is to understand how the applied load is distributed throughout the construct. To address this, we employ homogenisation theory to derive equations governing the effective macroscale material properties of a periodic, elastic–poroelastic composite. We treat the fibres as a linear elastic material and the hydrogel as a poroelastic material, and exploit the disparate length scales (small inter-fibre spacing compared with construct dimensions) to derive macroscale equations governing the response of the composite to an applied load. This homogenised description reflects the orthotropic nature of the composite. To validate the model, solutions from finite element simulations of the macroscale, homogenised equations are compared to experimental data describing the unconfined compression of the fibre-reinforced hydrogels. The model is used to derive the bulk mechanical properties of a cylindrical construct of the composite material for a range of fibre spacings and to determine the local mechanical environment experienced by cells embedded within the construct.

This article presents a novel monolithic numerical method for computing flow-induced stresses for problems involving arbitrarily-shaped stationary boundaries. A unified momentum equation for a continuum consisting of both fluids and solids is derived in terms of velocity by hybridizing the momentum equations of incompressible fluids and linear elastic solids. Discontinuities at the interface are smeared over a finite thickness around the interface using the signed distance function, and the resulting momentum equation implicitly takes care of the interfacial conditions without using a body-fitted grid. A finite volume approach is employed to discretize the obtained governing equations on a Cartesian grid. For validation purposes, this method has been applied to three examples, lid-driven cavity flow in a square cavity, lid-driven cavity flow in a circular cavity, and flow over a cylinder, where velocity and stress fields are simultaneously obtained for both fluids and structures. The simulation results agree well with the results found in the literature and the results obtained by COMSOL Multiphysics®.

In this paper, a method is proposed for extracting fracture parameters in anisotropic thermoelasticity cracking via interaction integral method within the framework of extended finite element method (XFEM). The proposed method is applied to linear thermoelastic crack problems. The numerical results of the stress intensity factors (SIFs) are presented and compared with those reported in related references. The good agreement of the results obtained by the developed method with those obtained by other numerical solutions proves the applicability of the proposed approach and confirms its capability of efficiently extracting thermoelasticity fracture parameters in anisotropic materials.

Similarity solution is investigated for the synchronous grouting of shield tunnel under the vertical non-axisymmetric displacement boundary condition in the paper. The synchronous grouting process of shield tunnel was simplified as the cylindrical expansion problem, which was based on the mechanism between the slurry and stratum of the synchronous grouting. The stress harmonic function on the horizontal and vertical ground surfaces is improved. Based on the virtual image technique, stress function solutions and Boussinesq's solution, elastic solution under the vertical non-axisymmetric displacement boundary condition on the vertical surface was proposed for synchronous grouting problems of shield tunnel. In addition, the maximum grouting pressure was also obtained to control the vertical displacement of horizontal ground surface. The validity of the proposed approach was proved by the numerical method. It can be known from the parameter analysis that larger vertical displacement of the horizontal ground surface was induced by smaller tunnel depth, smaller tunnel excavation radius, shorter limb distance, larger expansion pressure and smaller elastic modulus of soils.

This paper presents a relatively simple numerical method to investigate the flow and heat transfer of laminar power-law fluids over a semi-infinite plate in the presence of viscous dissipation and anisotropy radiation. On one hand, unlike most classical works, the effects of power-law viscosity on velocity and temperature fields are taken into account when both the dynamic viscosity and the thermal diffusivity vary as a power-law function. On the other hand, boundary layer equations are derived by Taylor expansion, and a mixed analytical/numerical method (a pseudosimilarity method) is proposed to effectively solve the boundary layer equations. This method has been justified by comparing its results with those of the original governing equations obtained by a finite element method. These results agree very well especially when the Reynolds number is large. We also observe that the robustness and accuracy of the algorithm are better when thermal boundary layer is thinner than velocity boundary layer.

We would like to present a method to compute the incompatibility operator in any system of curvilinear coordinates (components). The procedure is independent of the metric in the sense that the expression can be obtained by means of the basis vectors only, which are first defined as normal or tangential to the domain boundary, and then extended to the whole domain. It is an intrinsic method, to some extent, since the chosen curvilinear system depends solely on the geometry of the domain boundary. As an application, the in-extenso expression of incompatibility in a spherical system is given.

Common silicate glasses are among the most brittle of the materials. However, on warming beyond the glass transition temperature Tg glass transforms into one of the most plastic known materials. Bulk metallic glasses exhibit similar phenomenology, indicating that it rests on the disordered structure instead on the nature of the chemical bonds. The micromechanics of a solid with bulk amorphous structure is examined in order to determine the most basic conditions the system must satisfy to be able of plastic flow. The equations for the macroscopic flow, consistent with the constrictions imposed at the atomic scale, prove that a randomly structured bulk material must be either a brittle solid or a liquid, but not a ductile solid. The theory permits to identify a single parameter determining the difference between the brittle solid and the liquid. However, the system is able of perfect ductility if the plastic flow proceeds in two dimensional plane layers that concentrate the strain. Insight is gained on the nature of the glass transition, and the phase occurring between glass transition and melting.

Dislocations are line defects in crystalline materials. The Peierls-Nabarro models are hybrid models that incorporate atomic structure of dislocation core into continuum framework. In this paper, we present a numerical method for a generalized Peierls-Nabarro model for curved dislocations, based on the fast multipole method and the iterative grid redistribution. The fast multipole method enables the calculation of the long-range elastic interaction within operations that scale linearly with the total number of grid points. The iterative grid redistribution places more mesh nodes in the regions around the dislocations than in the rest of the domain, thus increases the accuracy and efficiency. This numerical scheme improves the available numerical methods in the literature in which the long-range elastic interactions are calculated directly from summations in the physical domains; and is more flexible to handle problems with general boundary conditions compared with the previous FFT based method which applies only under periodic boundary conditions. Numerical examples using this method on the core structures of dislocations in Al and Cu and in epitaxial thin films are presented.

The current geothermal and volcanic activity in the North Island of New Zealand is explained as a consequence of Pacific and Australian plate interactions over the last 20 million years. The primary hypothesis is that the Kermadec subduction zone has for the last 20 million years or more been retreating in a south-easterly direction at about five centimetres per year. It is surmised that this motion and interaction with another subduction zone almost at right angles to it under the North Island resulted in plate tearing due to the incompatibility of the plate geometry where these subduction zones interacted. The nature and consequences of this plate tearing are partially revealed in published maps of the plate currently under the North Island. If the subducted parts of this plate, as shown in Eiby’s maps, [G. A. Eiby, “The New Zealand sub-crustal rift”, New Zeal. J. Geol. Geophy.7 (1964) 109–133] are straightened, then the plate edge lies on a curve giving a rough picture of their position before being torn and subducted by the Kermadec trench motion. This map of the tear suggests the shape of the edge of a missing plate segment torn from the plate, and implies a rotation of the upper North Island, clockwise approximately 20 degrees, about a point just south of the Thames estuary. A consequence of this plate tearing is that the solid retreating crustal wave generating magma pressure beneath the crest of the solid wave has the potential to inject significant basaltic magma into the crust through the tears. These intrusive magma fluxes have the ability to generate geothermal fields and rhyolitic lavas from crustal melts. This could explain the geothermal activity along the Coromandel peninsula five to seven million years ago, the ignimbrite outcrops about Lake Taupo and the current geothermal and volcanic activity stretching from Taupo to Rotorua.

A three-layer compartmental model of the geological structure in the Taupo Volcanic Zone of New Zealand is developed, based on the assumptions of isostasy (constant geostatic pressure at 25 km depth) and a constant rate of volcanism. The upper layer consists of volcanic infill to a depth of about 2500 m, then a middle layer of greywacke-like material to a depth of about 15 km, and a lower layer of andesitic-like material to a depth of 25 km. Our model assumptions predict that the area of each layer increases at a constant rate; that there is a constant ratio between the rate of energy production from volcanic activity and geothermal convection; and that there is the possibility of an abrupt change from rhyolitic to basaltic volcanism, if the middle layer becomes sufficiently thin. Two models are considered: a rifting and a spreading model. Both models predict the lower layer has an andesitic-like density. The spreading model has difficulty matching heat output with observed extension rates. The rifting model predicts the observed extension rates, but requires very deep circulation of groundwater to be consistent with observed chemical and isotopic properties of geothermal fluids.