Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences最新文献

As the size of a layered structure scales down, the adhesive layer thickness correspondingly decreases from macro- to micro-scale. The influence of the material microstructure of the adhesive becomes more pronounced, and possible size effect phenomena can appear. This paper describes the mechanical behaviour of composites made of two solids, bonded together by a thin layer, in the framework of strain gradient and micropolar elasticity. The adhesive layer is assumed to have the same stiffness properties as the adherents. By means of the asymptotic methods, the contact laws are derived at order 0 and order 1. These conditions represent a formal generalization of the hard elastic interface conditions. A simple benchmark equilibrium problem (a three-layer composite micro-bar subjected to an axial load) is developed to numerically assess the asymptotic model. Size effects and non-local phenomena, owing to high strain concentrations at the edges, are highlighted. The example proves the efficiency of the proposed approach in designing micro-scale-layered devices.This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'.

A class of variational inequalities describing the equilibrium of elastic Timoshenko plates whose boundary is in contact with the side surface of an inclined obstacle is considered. At the plate boundary, mixed conditions of Dirichlet type and a non-penetration condition of inequality type are imposed on displacements in the mid-plane. The novelty consists of modelling oblique interaction with the inclined obstacle which takes into account shear deformation and rotation of transverse cross-sections in the plate. For proposed problems of equilibrium of the plate contacting the inclined obstacle, the unique solvability of the corresponding variational inequality is proved. Under the assumption that the variational solution is smooth enough, optimality conditions are obtained in the form of equilibrium equations and relations revealing the mechanical properties of integrated stresses, moments and generalized displacements on the contact part of the boundary. Accounting for complementarity type conditions owing to the contact of the plate with the inclined obstacle, a primal-dual variational formulation of the obstacle problem is derived. A semi-smooth Newton method based on a generalized gradient is constructed and performed as a primal-dual active-set algorithm. It is advantageous for efficient numerical solution of the problem, provided by a super-linear estimate for the corresponding iterates in function spaces. This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'.

The paper investigates a problem concerning the equilibrium of a solid body containing a thin rigid inclusion and a crack. It is assumed that the body is hyperelastic, therefore, it is described within the framework of finite strain theory. One of the peculiarities of this problem is a global injectivity constraint, which prevents the body, the crack faces and the inclusion from both mutual and self penetration. First, the paper deals with the differential formulation of the problem. Next, we consider energy minimization, showing that the latter provides the weak formulation of the former. Finally, the existence of the weak solution is demonstrated through the use of the variational technique.This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'.

In this article, we study general properties of distributed shape derivatives admitting a volumetric tensor representation of order two. We obtain a general result providing a range of expressions for the shape derivative, with the distributed shape derivative at one end of the range and the standard Hadamard formula at the other end. We further apply this result to a cost functional depending on the solution of a fourth-order elliptic equation, and obtain the distributed shape derivative in the case of open sets, and the Hadamard formula for sets of class [Formula: see text]. We also consider the case of polygons, for which a description of the weak singularities of the solution appearing in the neighbourhood of the vertices is required to obtain the Hadamard formula. This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'.

We consider the new boundary value problem for the generalized Boussinesq model of heat transfer under the inhomogeneous Dirichlet boundary condition for the velocity and under mixed boundary conditions for the temperature. It is assumed that the viscosity, thermal conductivity and buoyancy force in the model equations, as well as the heat exchange boundary coefficient, depend on the temperature. The mathematical apparatus for studying the inhomogeneous boundary value problem under study based on the variational method is being developed. Using this apparatus, we prove the main theorem on the global existence of a weak solution of the mentioned boundary value problem and establish sufficient conditions for the problem data ensuring the local uniqueness of the weak solution that has the additional property of smoothness with respect to temperature. This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'.

This study examines a class of time-dependent constitutive equations used to describe viscoelastic materials under creep in solid mechanics. In nonlinear elasticity, the strain response to the applied stress is expressed via an implicit graph allowing multi-valued functions. For coercive and maximal monotone graphs, the existence of a solution to the quasi-static viscoelastic problem is proven by applying the Browder-Minty fixed point theorem. Moreover, for quasi-linear viscoelastic problems, the solution is constructed as a semi-analytic formula. The inverse viscoelastic problem is represented by identification of a design variable from non-smooth measurements. A non-empty set of optimal variables is obtained based on the compactness argument by applying Tikhonov regularization in the space of bounded measures and deformations. Furthermore, an illustrative example is given for the inverse problem of isotropic kernel identification. This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'.

The intermittent storage of hydrogen in subsurface porous media such as depleted gas fields could be pivotal to a successful energy transition. Numerical simulations investigate the intermittent storage of hydrogen in a porous, depleted subsurface reservoir. Various parametric studies are performed to assess the effect of mechanical properties of the reservoir (i.e. Young's modulus, Poisson's ratio, Biot coefficient and permeability) on the induced fault slip of a single through-going fault that transverses the entire reservoir. Simulations are run using a three-dimensional, finite element, fully coupled hydromechanical code with explicit representations of layers and faults. The effect of the domain mesh refinement and fault mesh refinement on the fault slip versus operation time solution is investigated. The fault is observed to slip in two distinct events, one during the second injection period and one in the third injection period. The fault is not observed to slip during the storage or withdrawal periods. It is found that in order to minimize seismic risk, a reservoir rock with high Young's modulus (>40 GPa), high Poisson's ratio (>0.30) and high Biot coefficient (>0.65) would be preferable for hydrogen storage. Reservoir rocks of low Young's modulus (10-30 GPa), intermediate Poisson's ratio (0.00-0.30) and low-to-intermediate Biot coefficient (0.25-0.65), at high injection rates, were found to have higher potential of inducing large seismic events.This article is part of the theme issue 'Induced seismicity in coupled subsurface systems'.

Injection-induced seismicity and aseismic slip often involve the reactivation of long-dormant faults, which may have extremely low permeability prior to slip. In contrast, most previous models of fluid-driven aseismic slip have assumed linear pressure diffusion in a fault zone of constant permeability and porosity. Slip occurs within a frictional shear crack whose edge can either lag or lead pressure diffusion, depending on the dimensionless stress-injection parameter that quantifies the prestress and injection conditions. Here, we extend this foundational work by accounting for permeability enhancement and dilatancy, assumed to occur instantaneously upon the onset of slip. The fault zone ahead of the crack is assumed to be impermeable, so fluid flow and pressure diffusion are confined to the interior, slipped part of the crack. The confinement of flow increases the pressurization rate and reduction of fault strength, facilitating crack growth even for severely understressed faults. Suctions from dilatancy slow crack growth, preventing propagation beyond the hydraulic diffusion length. Our new two-dimensional and three-dimensional solutions can facilitate the interpretation of induced seismicity data sets. They are especially relevant for faults in initially low permeability formations, such as shale layers serving as caprock seals for geologic carbon storage, or for hydraulic stimulation of geothermal reservoirs.This article is part of the theme issue 'Induced seismicity in coupled subsurface systems'.

Hydraulic stimulation is a critical process for increasing the permeability of fractured geothermal reservoirs. This technique relies on coupled hydromechanical processes induced through pressurized fluid injection into the rock formation. The injection of fluids causes poromechanical stress changes that can lead to fracture slip and shear dilation, as well as tensile fracture opening and propagation, so-called mixed-mechanism stimulation. The effective permeability of the rock is particularly enhanced when new fractures connect with pre-existing fractures. While hydraulic stimulation can significantly improve the productivity of fractured geothermal reservoirs, the process is also related to induced seismicity. Hence, understanding the coupled physics is central, for both reservoir engineering and seismic risk mitigation. This article presents a modelling approach for simulating the deformation, propagation and coalescence of fractures in porous media under the influence of anisotropic stress and fluid injection. It uses a coupled hydromechanical model for poroelastic, fractured media. Fractures are governed by contact mechanics and a fracture propagation model. For numerical solutions, we employ a two-level approach, combining a finite volume method for poroelasticity with a finite element method for fracture propagation. The study investigates the impact of injection rate, matrix permeability and stress anisotropy on stimulation outcomes.This article is part of the theme issue 'Induced seismicity in coupled subsurface systems'.