Abstract: The transient mixed problem for an elastic semi-strip is investigated in this article. The semi-strip is loaded by a transient load at the center of its short edge. The longitudinal sides of the semi-strip are fixed. The initial problem is reduced to a one-dimensional problem with the help of Laplace and Fourier transforms. The analytical solution of the one-dimensional boundary problem is constructed with the help of matrix differential calculation apparatus. The original functions are found with the help of mutual inversion of Laplace–Fourier transform. Here the Laplace originals were derived by the new approach based on expansion of Laplace transform in a multiple series. The wave field of the semi-strip was investigated depending on time and geometric parameters. PubDate: 2021-03-14

Abstract: In many micro- and macro-scale systems, collective dynamics occurs from the coupling of small spatially segregated, but dynamically active, units through a bulk diffusion field. This bulk diffusion field, which is both produced and sensed by the active units, can trigger and then synchronize oscillatory dynamics associated with the individual units. In this context, we analyze diffusion-induced synchrony for a class of cell-bulk ODE–PDE system in \({\mathbb {R}}^2\) that has two spatially segregated dynamically active circular cells of small radius. By using strong localized perturbation theory in the limit of small cell radius, we calculate the steady-state solution and formulate the linear stability problem. For Sel’kov intracellular reaction kinetics, we analyze how the effect of bulk diffusion can trigger, via a Hopf bifurcation, either in-phase or anti-phase intracellular oscillations that would otherwise not occur for cells that are uncoupled from the bulk medium. Phase diagrams in parameter space where these oscillations occur are presented, and the theoretical results from the linear stability theory are validated from full numerical simulations of the ODE–PDE system. In addition, the two-cell case is extended to study the onset of synchronous oscillatory instabilities associated with an infinite hexagonal arrangement of small identical cells in \({\mathbb {R}}^2\) with Sel’kov intracellular kinetics. This analysis for the hexagonal cell pattern relies on determining a new, computationally efficient, explicit formula for the regular part of a certain periodic reduced-wave Green’s function. PubDate: 2021-03-14

Abstract: We address a boundary-value problem involving a Poisson–Boltzmann equation that models the electrostatic potential of a channel formed by parallel plates with an electrolyte solution confined between the plates. We show the existence and uniqueness of solution to the problem, with special (particular) solutions as bounds, namely, a Debye–Hückel type solution as lower bound and a Gouy–Chapman type solution as upper bound. Our results are based on the maximum principle for elliptic equations and are useful for characterizing the behavior of the solutions. Also, we introduce a numerical scheme based on the Chebyshev pseudo-spectral method to calculate approximate solutions. This method is applied in conjunction with a multidomain procedure that attempts to capture the dramatic exponential increase/decay of the solution near the plates. PubDate: 2021-03-14

Abstract: The nonlinear along-fibre shear stress–strain relationship for unidirectionally fibre-reinforced composites has been investigated in this paper aiming at its applications in general 3D stress conditions in a consistent manner. So far, such relationship has only been addressed in plane stress conditions. In this paper, it has been shown that its straightforward generalisation to 3D stress states lacks objectivity, which is a basic requirement for all theoretical studies of physical problems. A new formulation is proposed based on the stress invariants and the complementary strain energy guided by the rational theoretical framework of nonlinear elasticity. A consistent and objective stress–strain relationship has been obtained and verified through an example of application to a torsion problem. PubDate: 2021-03-14

Abstract: We consider the elastic stress near a hole with corners in an infinite plate under biaxial stress. The elasticity problem is formulated using complex Goursat functions, resulting in a set of singular integro-differential equations on the boundary. The resulting boundary integral equations are solved numerically using a Chebyshev collocation method which is augmented by a fractional power term, derived by asymptotic analysis of the corner region, to resolve stress singularities at corners of the hole. We apply our numerical method to the test case of the hole formed by two partially overlapping circles, which can include either a corner pointing into the solid or a corner pointing out of the solid. Our numerical results recover the exact stress on the boundary to within relative error \(10^{-3}\) for modest computational effort. PubDate: 2021-03-13

Abstract: A stagnant free-surface flow is an instantaneous flow field of pure acceleration with zero velocity and a deformed surface. There exists a potential-flow acceleration field. With zero velocity and the acceleration field given, there is a limiting free-surface position which possesses one peak at its point of highest elevation. By complex analysis, it can be shown that the surface peak has a right angle. We elaborate on an elementary model of two-dimensional stagnant free-surface flow with a peak. Our model may serve to describe a situation of maximal single-wave run-up with a given energy at a uniformly sloping beach. The highest possible run-up of an incoming solitary wave corresponds to zero kinetic energy. It encompasses an idealized situation where the kinetic wave energy is converted into potential energy in a water mass piling up along the slope to become stagnant at one single moment. Multipoles with singularities outside the fluid domain may give rise to a smooth and gradual deceleration needed for a non-breaking run-up process. A pair of dipoles with an orientation perpendicular to a given slope represents the stagnant acceleration fields with the highest surface peak spatially concentrated along the slope. Thereby, a one-parameter family of surface shapes is constituted, only dependent on the slope angle. The initial flow field, the initial free surface, the initial isobars and the geometric parameters are all calculated for different slope angles. PubDate: 2021-03-13

Abstract: In his landmark paper, Keller (J Appl Phys 34:991–993, 1963) obtained an approximation for the effective conductivity of a composite medium made out of a densely packed square array of perfectly conducting circular cylinders embedded in a conducting medium. We here examine Keller’s problem for the case of square cylinders, considering both a full-pattern and a checkerboard configurations. The dense limit is handled using matched asymptotic expansions, where the conduction problem is separately analyzed in the the “local” narrow gap between adjacent cylinders and the “global” region outside the gap. The conduction problem in both regions is solved using conformal-mapping techniques. PubDate: 2021-03-10

Abstract: Three weakly nonlinear but fully dispersive Whitham–Boussinesq systems for uneven bathymetry are studied. The derivation and discretization of one system is presented. The numerical solutions of all three are compared with wave gauge measurements from a series of laboratory experiments conducted by Dingemans (Comparison of computations with Boussinesq-like models and laboratory measurements. Delft Hydraulics memo H168412, 1994). The results show that although the models are mathematically similar, their accuracy varies dramatically. PubDate: 2021-03-09

Abstract: Problems of water wave propagation over an infinite step in the presence of a thin vertical barrier of four different geometrical configurations are investigated in this paper. For each configuration of the barrier, the problem is reduced to solving an integral equation or a coupled integral equation of first kind involving horizontal component of velocity below or above the barrier and above the step. The integral equations are solved employing Galerkin approximation in terms of simple polynomials multiplied by appropriate weight functions whose forms are dictated by the edge conditions at the corner of the step and at the submerged end(s) of the barrier. The reflection and transmission coefficients are then computed and depicted graphically against the wave number. PubDate: 2021-03-09

Abstract: Residual stresses in an unloaded configuration of an elastic material have a significant influence on the response of the material from that configuration, but the effect of residual stress on the stability of the material, whether loaded or unloaded, has only been addressed to a limited extent. In this paper we consider the level of residual stress that can be supported in a thick-walled circular cylindrical tube of non-linearly elastic material without loss of stability when subjected to fixed axial stretch and either internal or external pressure. In particular, we consider the tube to have radial and circumferential residual stresses, with a simple form of elastic constitutive law that accommodates the residual stress, and incremental deformations restricted to the cross section of the tube. Results are described for a tube subject to a level of (internal or external) pressure characterized by the internal azimuthal stretch. Subject to restrictions imposed by the strong ellipticity condition, the emergence of bifurcated solutions is detailed for their dependence on the level of residual stress and mode number. PubDate: 2021-03-04

Abstract: The paper presents a study of the propagation and interaction of weakly nonlinear plane waves in isotropic and transversely isotropic media. It begins with a definition of stored energy functions of considered hyperelastic models. The equation of elastodynamics as well as the first-order quasilinear hyperbolic system for plane waves are provided. The eigensystem for this system is determined to study three-wave interaction coefficients. The main part of the paper concerns a discussion of these coefficients. Applying the weakly nonlinear asymptotics method, it is shown that in the case of transverse isotropy the inviscid Burgers’ equation describes an evolution of a single quasi-shear wave. The result contradicts the case of isotropy, where the equation with quadratic nonlinearity cannot describe any shear wave propagation. The paper ends with an example of numerical solutions for the obtained evolution equation. PubDate: 2021-03-04

Abstract: A straight elastic fibre is usually perceived as a one-dimensional structural component, and its similarity with a cylindrical rod makes its concept analogous, if not equivalent with the concept of an elastic spring. This analogy enables this communication to match the one-dimensional response of a relevant viscoelastic fibre with that of a viscoelastic spring and, hence, to describe its one-dimensional behaviour in the light of a new, generalised viscoelastic spring model. The model shares simultaneously properties of an elastic spring and an inelastic damper (dashpot) and this communication is interested on its applicability at small strain only. However, the form of its constitutive equation, which is based on the combined action of an internal energy function and a viscous flow potential, is non-linear as well as differential and, also, implicit in the stress. The model enables a posteriori determination of (i) the manner that the elastic and the inelastic parts of the fibre strain are assembled and form the observed total deformation, (ii) the part of stress that creates recoverable work and the part of stress wasted in energy dissipation, and (iii) the amount of work stored in the material as well as the amount of energy dissipation during the fibre deformation. A detailed analysis is presented for the case that small-strain, steady viscoelastic deformation takes place in a spatially homogeneous manner. This includes a complete relevant solution of the problem of interest and is accompanied by an adequate set of corresponding qualitative numerical results. PubDate: 2021-03-03

Abstract: This work presents new semi-analytical solutions for the combined fully developed electro-osmotic pressure-driven flow in microchannels of viscoelastic fluids, described by the generalised Phan-Thien–Tanner model (gPTT) recently proposed by Ferrás et al. (Journal of Non-Newtonian Fluid Mechanics, 269:88–99, 2019). This generalised version of the PTT model presents a new function for the trace of the stress tensor—the Mittag–Leffler function—where one or two new fitting constants are considered in order to obtain additional fitting flexibility. The semi-analytical solution is obtained under sufficiently weak electric potential that allows the Debye–Hückel approximation for the electrokinetic fields and for thin electric double layers. Based on the solution, the effects of the various relevant dimensionless numbers are assessed and discussed, such as the influence of \(\varepsilon Wi^2\) , of the parameters \(\alpha \) and \(\beta \) of the gPTT model, and also of \({\bar{\kappa }}\) , the dimensionless Debye–Hückel parameter. We conclude that the new model characteristics enhance the effects of both \(\varepsilon Wi^2\) and \({\bar{\kappa }}\) on the velocity distribution across the microchannels. The effects of a high zeta potential and of the finite size of ions are also studied numerically. PubDate: 2021-03-03

Abstract: An implicit constitutive relation is proposed to study transversely isotropic bodies. The relation is obtained assuming the existence of a Gibbs potential that depends on the second Piola–Kirchhoff stress tensor, from which the Green Saint-Venant strain tensor is obtained as the derivative with respect to the stress. The responses of unconstrained as well as inextensible bodies are studied, and some boundary value problems are analysed. An inextensible body, where the constraint of inextensibility appears only in tension is also considered. PubDate: 2021-02-24

Abstract: Resin infusion is a pressure-gradient-driven composite manufacturing process in which the liquid resin is driven to flow through and fill in the void space of a porous composite preform prior to the heat treatment for resin solidification. It usually is a great challenge to design both the infusion system and the infusion process meeting the manufacturing requirements, especially for large-scale components of aircraft and wind turbine blades. Aiming at addressing the key concerns about flow fronts and air bubble entrapment, the present study proposes a modelling framework of the multiphase flow of resin and air in a dual scale porous medium, i.e. a composite preform. A finite strain formulation is discussed for the fluid–solid interaction during an infusion process. The present study bridges the gap between the microscopic observation and the macroscopic modelling by using the averaging method and first principle method, which sheds new light on the high-fidelity finite element modelling. PubDate: 2021-02-24

Abstract: We describe a multiphysics model of the collagen structure of the cornea undergoing a progressive localized reduction of the stiffness, preluding to the development of ectasia and keratoconus. The architecture of the stromal collagen is assumed to follow the simplified two-family model proposed in Pandolfi et al. (A microstructural model of cross-link interaction between collagen fibrils in the human cornea. Philos Trans R Soc A 377:20180079, 2019), where the mechanical stiffness of the structure is supplied by transversal bonds within the fibrils of the same family (inter-crosslink bonds) and across the fibrils of the two families (intra-crosslink bonds). In Pandolfi et al. (A microstructural model of cross-link interaction between collagen fibrils in the human cornea. Philos Trans R Soc A 377:20180079, 2019), it was shown that the loss of the spherical shape due to the protrusion of a cone can be ascribed to the mechanical weakening of the intra-crosslink bonds in the central region of the collagen structure. In the present study, the reduction of bond stiffness is coupled to an evolutive pathologic phenomenon, modeled as a reaction–diffusion process of a normalized scalar field. We assume that the scalar field is a concentration-like measure of the degeneration of the chemical bonds stabilizing the structural collagen. We follow the evolution of the mechanical response of the system in terms of shape change, according to the propagation of the degeneration field, and identify the critical loss of mechanical stability resulting in the typical bulging of keratoconus corneas. PubDate: 2021-02-24

Abstract: A multilayered rectangular flexoelectric structure is considered. This composite has a unit cell of N-constituents which belong to the cubic crystal symmetry. Using the two-scale asymptotic homogenization method, explicit expressions of the local problems are derived. Simple closed-form formulas of the effective stiffness, piezoelectric, dielectric, and flexoelectric tensors are given, based on the solutions of local problems of stratified multilayered composites with perfect contact at the interface. These formulas provide information about the symmetry of the homogenized structure. As a numerical example, a bilaminate composite where the layers are perpendicular to the \(x_3\) axis is studied, and the constituents are Barium Titanate and Gallium Arsenide. The composite after the homogenization process exhibits flexoelectric properties with tetragonal \(\bar{4}2\) m crystal symmetry. The effective properties are computed for several constituents of volume fractions. PubDate: 2021-02-24

Abstract: In this work, an eco-epidemic predator–prey model with media-induced response function for the interaction of humans with adulterated food is developed and studied. The human population is divided into two main compartments, namely, susceptible and infected. This system has three equilibria; trivial, disease-free and endemic. The trivial equilibrium is forever an unstable saddle position, while the disease-free state is locally asymptotically stable under a threshold of delay parameter \(\tau \) as well as \({\mathcal {R}}_0<1\) . The sufficient conditions for the local stability of the endemic equilibrium point are further explored when \(\min \{{\mathcal {R}}_0,{\mathcal {R}}_0^*\}>1\) . The conditions for the occurrence of the stability switching are also determined by taking infection delay time as a critical parameter, which concludes that the delay can produce instability and small amplitude oscillations of population masses via Hopf bifurcations. Further, we study the stability and direction of the Hopf bifurcations using the center manifold argument. Furthermore, some numerical simulations are conducted to validate our analytical findings and discuss their biological inferences. Finally, the normalized forward sensitivity index is used to perform the sensitivity analysis of \({\mathcal {R}}_0\) and \({\mathcal {R}}_0^*\) . PubDate: 2021-02-20

Abstract: In offshore engineering design, nonlinear wave models are often used to propagate stochastic waves from an input boundary to the location of an offshore structure. Each wave realization is typically characterized by a high-dimensional input time-series, and a reliable determination of the extreme events is associated with substantial computational effort. As the sea depth decreases, extreme events become more difficult to evaluate. We here construct a low-dimensional characterization of the candidate input time series to circumvent the search for extreme wave events in a high-dimensional input probability space. Each wave input is represented by a unique low-dimensional set of parameters for which standard surrogate approximations, such as Gaussian processes, can estimate the short-term exceedance probability efficiently and accurately. We demonstrate the advantages of the new approach with a simple shallow-water wave model based on the Korteweg–de Vries equation for which we can provide an accurate reference solution based on the simple Monte Carlo method. We furthermore apply the method to a fully nonlinear wave model for wave propagation over a sloping seabed. The results demonstrate that the Gaussian process can learn accurately the tail of the heavy-tailed distribution of the maximum wave crest elevation based on only \(1.7\%\) of the required Monte Carlo evaluations. PubDate: 2021-02-15

Abstract: An efficient time-adaptive multigrid algorithm is used to solve a range of normal and oblique droplet impacts on dry surfaces and liquid films using the Depth-Averaged Form (DAF) method of the governing unsteady Navier–Stokes equations. The dynamics of a moving three-phase contact line on dry surfaces is predicted by a precursor film model. The method is validated against a variety of experimental results for droplet impacts, looking at factors such as crown height and diameter, spreading diameter and splashing for a range of Weber, Reynolds and Froude numbers along with liquid film thicknesses and impact angles. It is found that, while being a computationally inexpensive methodology, the DAF method produces accurate predictions of the crown and spreading diameters as well as conditions for splash, however, underpredicts the crown height as the vertical inertia is not included in the model. PubDate: 2021-02-15