On Anisotropic Polynomial Relations for the Elasticity Tensor

In this paper, we explore new conditions for an elasticity tensor to belong to a given symmetry class. Our goal is to propose an alternative approach to the identification problem of the symmetry class, based on polynomial invariants and covariants of the elasticity tensor C, rather than on spectral properties of the Kelvin representation. We compute a set of algebraic relations which describe precisely the orthotropic ( $[\mathbb {D}_{2}]$ ), trigonal ( $[\mathbb {D}_{3}]$ ), tetragonal ( $[\mathbb {D}_{4}]$ ), transverse isotropic ([SO(2)]) and cubic ( $[\mathbb {O}]$ ) symmetry classes in $\mathbb {H}^{4}$ , the highest-order irreducible component in the decomposition of $\mathbb {E}\mathrm {la}$ . We provide a bifurcation diagram which describes how one “travels” in $\mathbb {H}^{4}$ from a given isotropy class to another. Finally, we study the link between these polynomial invariants and those obtained as the coefficients of the characteristic or the Betten polynomials. We show, in particular, that the Betten invariants do not separate the orbits of the elasticity tensors.     

C 1 Solutions for Semi-Implicit Systems of Differential Equations

The present note is a continuation of the author??s effort to study the existence of continuously differentiable solutions to the semi-implicit system of differential equations (1) $$f(x^{\prime}(t)) = g(t, x(t))$$ (2) $$\quad x(0) = x_0,$$ where

${\quad\Omega_g \subseteq \mathbb{R} \times\mathbb{R}^n}$ is an open set containing (0, x ₀) and ${g:\Omega_g \rightarrow\mathbb{R}^n}$ is a continuous function,

${\quad\Omega_f \subseteq \mathbb{R}^n}$ is an open set and ${f:\Omega_f\rightarrow\mathbb{R}^n}$ is a continuous function.

The transformation of (1)?C(2) into a solvable explicit system of differential equations is trivial if f is locally injective around an element ${\gamma\in \Omega_f\cap f^{-1}(g(0,x_0))}$ . In this paper, we study (1)?C(2) when such a translation is not possible because of the inherent multivalued nature of f ^?1.     

A Strategy to Locate Fixed Points and Global Perturbations of ODE’s: Mixing Topology with Metric Conditions

In this paper we discuss a topological treatment for the planar system 0.1 $$\begin{aligned} z'=f(t,z)+g(t,z) \end{aligned}$$ where $f:\mathbb {R}\times \mathbb {R}^{2}\longrightarrow \mathbb {R}^{2}$ and $g:\mathbb {R}\times \mathbb {R}^{2}\longrightarrow \mathbb {R}^{2}$ are $T$ -periodic in time and $g(t,z)$ is bounded. Namely, we study the effect of $g(t,z)$ in two different frameworks: isochronous centers and time periodic systems having subharmonics. The main tool employed in the proofs consists of a topological strategy to locate fixed points in the class of orientation preserving embedding under the condition of some recurrence properties. Generally speaking, our topological result can be considered as an extension of the main result in Brown (Pac J Math 143:37–41, 1990) (concerning two cycles) to any recurrent point.     

Dynamical Parallelepipeds in Minimal Systems

For a topological dynamical system $(X,T)$ ( X , T ) and $d\in \mathbb N $ d ∈ N , the associated dynamical parallelepiped $\mathbf{Q}^{[d]}$ Q [ d ] was defined by Host–Kra–Maass. For a minimal distal system it was shown by them that the relation $\sim _{d-1}$ ～ d ? 1 defined on $\mathbf{Q}^{[d-1]}$ Q [ d ? 1 ] is an equivalence relation; the closing parallelepiped property holds, and for each $x\in X$ x ∈ X the collection of points in $\mathbf{Q}^{[d]}$ Q [ d ] with first coordinate $x$ x is a minimal subset under the face transformations. We give examples showing that the results do not extend to general minimal systems.     

On the Stokes and Oseen Problems with Singular Data

We consider the steady Stokes and Oseen problems in bounded and exterior domains of ${\mathbb{R}^n}$ of class C ^k-1,1 (n = 2, 3; k ≥ 2). We prove existence and uniqueness of a very weak solution for boundary data a in ${W^{2-k-1/q,q} (\partial\Omega)}$ . If ${\Omega}$ is of class ${C^\infty}$ , we can assume a to be a distribution on ${\partial\Omega}$ .     

Hadamard Variational Formula for the Green’s Function of the Boundary Value Problem on the Stokes Equations

For every ${\varepsilon > 0}$ , we consider the Green’s matrix ${G_{\varepsilon}(x, y)}$ of the Stokes equations describing the motion of incompressible fluids in a bounded domain ${\Omega_{\varepsilon} \subset \mathbb{R}^d}$ , which is a family of perturbation of domains from ${\Omega\equiv \Omega_0}$ with the smooth boundary ${\partial\Omega}$ . Assuming the volume preserving property, that is, ${\mbox{vol.}\Omega_{\varepsilon} = \mbox{vol.}\Omega}$ for all ${\varepsilon > 0}$ , we give an explicit representation formula for ${\delta G(x, y) \equiv \lim_{\varepsilon\to +0}\varepsilon^{-1}(G_{\varepsilon}(x, y) - G_0(x, y))}$ in terms of the boundary integral on ${\partial \Omega}$ of ${G_0(x, y)}$ . Our result may be regarded as a classical Hadamard variational formula for the Green’s functions of the elliptic boundary value problems.     

Properties of Monodromic Points on Center Manifolds in {\mathbb {R}}^3 via Lie Symmetries

In this work we use Lie symmetries to investigate monodromic points on center manifolds of a singularity of an analytic vector field ${\mathcal {X}}$ in ${\mathbb {R}}^3$ . We investigate how whether the singularity is a focus or a center, is analytically normalizable or not, and is linearizable or not is reflected in the centralizer and normalizer of ${\mathcal {X}}$ .     

Spatiotemporal complexity of a three-species ratio-dependent food chain model

In this paper, we investigate the complex dynamics of a ratio-dependent spatially extended food chain model. Through a detailed analytical study of the reaction–diffusion model, we obtain some conditions for global stability. On the basis of bifurcation analysis, we present the evolutionary process of pattern formation near the coexistence equilibrium point $(N^*,P^*,Z^*)$ via numerical simulation. And the sequence cold spots $\rightarrow $ stripe–spots mixtures $\rightarrow $ stripes $\rightarrow $ hot stripe–spots mixtures $\rightarrow $ hot spots $\rightarrow $ chaotic wave patterns controlled by parameters $a_1$ or $c_1$ in the model are presented. These results indicate that the reaction–diffusion model is an appropriate tool for investigating fundamental mechanism of complex spatiotemporal dynamics.     

A Note: An Invariance Theorem for Mappings II

Conditions on a domain D in ${\mathbb{R}^n}$ are given so that if f is a continuous mapping of ${\overline{D}}$ into ${\mathbb{R}^n}$ , is an open mapping on the interior of D and maps the boundary of D into the closure of D then f maps the entire set into its closure, i.e. ${\overline{D}}$ is invariant. This is an improvement over a previous result where f was required to be injective (one-to-one) since a locally injective map on the interior of D is an open map.     

Mixed Convection Boundary-Layer Flow Over a Vertical Surface Embedded in a Porous Material Subject to a Convective Boundary Condition

The mixed convection boundary-layer flow on one face of a semi-infinite vertical surface embedded in a fluid-saturated porous medium is considered when the other face is taken to be in contact with a hot or cooled fluid maintaining that surface at a constant temperature $T_\mathrm{{f}}$ . The governing system of partial differential equations is transformed into a system of ordinary differential equations through an appropriate similarity transformation. These equations are solved numerically in terms of a dimensionless mixed convection parameter $\epsilon $ and a surface heat transfer parameter $\gamma $ . The results indicate that dual solutions exist for opposing flow, $\epsilon <0$ , with the dependence of the critical values $\epsilon _\mathrm{{c}}$ on $\gamma $ being determined, whereas for the assisting flow $\epsilon >0$ , the solution is unique. Limiting asymptotic forms for both $\gamma $ small and large and $\epsilon $ large are also discussed.     

Global Compensated Compactness Theorem for General Differential Operators of First Order

Let A ₁(x, D) and A ₂(x, D) be differential operators of the first order acting on l-vector functions ${u= (u_1, \ldots, u_l)}$ in a bounded domain ${\Omega \subset \mathbb{R}^{n}}$ with the smooth boundary ${\partial\Omega}$ . We assume that the H ¹-norm ${\|u\|_{H^{1}(\Omega)}}$ is equivalent to ${\sum_{i=1}^2\|A_iu\|_{L^2(\Omega)} + \|B_1u\|_{H^{\frac{1}{2}}(\partial\Omega)}}$ and ${\sum_{i=1}^2\|A_iu\|_{L^2(\Omega)} + \|B_2u\|_{H^{\frac{1}{2}}(\partial\Omega)}}$ , where B _i  = B _i (x, ν) is the trace operator onto ${\partial\Omega}$ associated with A _i (x, D) for i = 1, 2 which is determined by the Stokes integral formula (ν: unit outer normal to ${\partial\Omega}$ ). Furthermore, we impose on A ₁ and A ₂ a cancellation property such as ${A_1A_2^{\prime}=0}$ and ${A_2A_1^{\prime}=0}$ , where ${A^{\prime}_i}$ is the formal adjoint differential operator of A _i (i = 1, 2). Suppose that ${\{u_m\}_{m=1}^{\infty}}$ and ${\{v_m\}_{m=1}^{\infty}}$ converge to u and v weakly in ${L^2(\Omega)}$ , respectively. Assume also that ${\{A_{1}u_m\}_{m=1}^{\infty}}$ and ${\{A_{2}v_{m}\}_{m=1}^{\infty}}$ are bounded in ${L^{2}(\Omega)}$ . If either ${\{B_{1}u_m\}_{m=1}^{\infty}}$ or ${\{B_{2}v_m\}_{m=1}^{\infty}}$ is bounded in ${H^{\frac{1}{2}}(\partial\Omega)}$ , then it holds that ${\int_{\Omega}u_m\cdot v_m \,{\rm d}x \to \int_{\Omega}u\cdot v \,{\rm d}x}$ . We also discuss a corresponding result on compact Riemannian manifolds with boundary.     

Non-uniqueness of solutions in asymptotically self-similar shock reflections

The present study addresses the self-similar problem of unsteady shock reflection on an inclined wedge. The start-up conditions are studied by modifying the wedge corner and allowing for a finite radius of curvature. It is found that the type of shock reflection observed far from the corner, namely regular or Mach reflection, depends intimately on the start-up condition, as the flow “remembers” how it was started. Substantial differences were found. For example, the type of shock reflection for an incident shock Mach number $M=6.6$ and an isentropic exponent $\gamma =1.2$ changes from regular to Mach reflection between $44^\circ $ and $45^\circ $ when a straight wedge tip is used, while the transition for an initially curved wedge occurs between $57^\circ $ and $58^\circ $ .     

Characteristics of Wave Propagation in The Saturated Thermoelastic Porous Medium

Took into consideration the coupling effect of thermo, hydraulics and mechanics, a set of thermo–hydro-mechanical coupled wave equations for fluid–saturated soil are developed. In these wave equations, the $P_{3}$ -wave in solid phase and $P_{4}$ -wave in fluid phase are coupled into $T$ -wave in fluid–saturated soil by the assumption that the temperature of the solid phase is equal to the temperature of liquid phase at the same position. The dispersion equations for the thermo-elastic wave, which can be degraded to the equations for elastic wave in fluid–saturated soil, are derived from the above equations by introducing four potential functions. Then, these equations are solved numerically. The characteristics of wave phase velocity, attenuation and the effect of thermal expansion, initial temperature and porosity, etc., on phase velocities of $P_{1}$ -, $P_{2}$ -, and $T$ -wave are discussed. As a reference, the characteristics of the propagation of elastic waves in fluid–saturated soil are also studied. The computation results show that (1) the phase velocity of $P_{1}$ -wave obtained by the theory of thermoporoelascity (THM) is faster than that by the theory of poroelasticity (HM); (2) the attenuation of $P_{1}$ -wave obtained by either the theory of THM or HM are consistent; (3) the dissemination characteristics of $P_{2}$ -wave are almost consistent; (4) the phase velocity of $T$ -wave is the slowest among the three compressional waves; and (5) The attenuation versus frequency characteristic of $T$ -wave is similar to that of $P_{2}$ -wave.     

Regularity and Asymptotic Behavior of Nonlinear Stefan Problems

We study the following nonlinear Stefan problem $$\left\{\begin{aligned}\!\!&u_t\,-\,d\Delta u = g(u) & &\quad{\rm for}\,x\,\in\,\Omega(t), t > 0, \\ & u = 0 \, {\rm and} u_t = \mu|\nabla_{x} u|^{2} &&\quad {\rm for}\,x\,\in\,\Gamma(t), t > 0, \\ &u(0, x) = u_{0}(x) &&\quad {\rm for}\,x\,\in\,\Omega_0,\end{aligned} \right.$$ where ${\Omega(t) \subset \mathbb{R}^{n}}$ ( ${n \geqq 2}$ ) is bounded by the free boundary ${\Gamma(t)}$ , with ${\Omega(0) = \Omega_0}$ , μ and d are given positive constants. The initial function u ₀ is positive in ${\Omega_0}$ and vanishes on ${\partial \Omega_0}$ . The class of nonlinear functions g(u) includes the standard monostable, bistable and combustion type nonlinearities. We show that the free boundary ${\Gamma(t)}$ is smooth outside the closed convex hull of ${\Omega_0}$ , and as ${t \to \infty}$ , either ${\Omega(t)}$ expands to the entire ${\mathbb{R}^n}$ , or it stays bounded. Moreover, in the former case, ${\Gamma(t)}$ converges to the unit sphere when normalized, and in the latter case, ${u \to 0}$ uniformly. When ${g(u) = au - bu^2}$ , we further prove that in the case ${\Omega(t)}$ expands to ${{\mathbb R}^n}$ , ${u \to a/b}$ as ${t \to \infty}$ , and the spreading speed of the free boundary converges to a positive constant; moreover, there exists ${\mu^* \geqq 0}$ such that ${\Omega(t)}$ expands to ${{\mathbb{R}}^n}$ exactly when ${\mu > \mu^*}$ .     

On the Symmetry of Energy-Minimising Deformations in Nonlinear Elasticity II: Compressible Materials

Consider a homogeneous, isotropic, hyperelastic body occupying the region ${A = \{{\bf x}\in\mathbb{R}^{n}\, : \,a <\,|{\bf x} |\,< b \}}$ in its reference state and subject to radially symmetric displacement, or mixed displacement/traction, boundary conditions. In Part I (Sivaloganathan and Spector in Arch Ration Mech Anal, 2009, in press) the authors restricted their attention to incompressible materials. For a large-class of polyconvex constitutive relations that grow sufficiently rapidly at infinity it was shown that to each nonradial isochoric deformation of A there corresponds a radial isochoric deformation that has strictly less elastic energy than the given deformation. In this paper that analysis is further developed and extended to the compressible case. The key ingredient is a new radial-symmetrisation procedure that is appropriate for problems where the symmetrised mapping must be one-to-one in order to prevent interpenetration of matter. For the pure displacement boundary-value problem, the radial symmetrisation of an orientation-preserving diffeomorphism u : A → A* between spherical shells A and A* is the deformation $${\bf u}^{\rm rad}({\bf x})=\frac{r(R)}{R}{\bf x}, \quad R=|{\bf x}|,\qquad\qquad\qquad\qquad(0.1)$$ that maps each sphere ${S_R\subset\,A}$ , of radius R > 0, centred at the origin into another such sphere ${S_r={\bf u}^{\rm rad}(S_R)\subset\,A^*}$ that encloses the same volume as u(S _R ). Since the volumes enclosed by the surfaces u(S _R ) and u ^rad (S _R ) are equal, the classical isoperimetric inequality implies that ${{{\rm Area}( {\bf u}^{\rm rad} (S_R))\leqq {\rm Area}({\bf u} (S_R))}}$ . The equality of the enclosed volumes together with this reduction in surface area is then shown to give rise to a reduction in total energy for many of the constitutive relations used in nonlinear elasticity. These results are also extended to classes of Sobolev deformations and applied to prove that the radially symmetric solutions to these boundary-value problems are local or global energy minimisers in various classes of (possibly nonsymmetric) deformations of a thick spherical shell.     

Experimental Characterization of Porosity Structure and Transport Property Changes in Limestone Undergoing Different Dissolution Regimes

Limestone dissolution by $\hbox {CO}_2$ -rich brine induces critical changes of the pore network geometrical parameters such as the pore size distribution, the connectivity, and the tortuosity which govern the macroscopic transport properties (permeability and dispersivity) that are required to parameterize the models, simulating the injection and the fate of $\hbox {CO}_2$ . A set of four reactive core-flood experiments reproducing underground conditions ( $T = 100\,^{\circ }\hbox {C}$ and $P = 12$ MPa) has been conducted for different $\hbox {CO}_2$ partial pressures $(0.034 < P_{\mathrm{CO}_2}< 3.4\; \hbox {MPa})$ in order to study the different dissolution regimes. X-ray microtomographic images have been used to characterize the changes in the structural properties from pore scale to Darcy scale, while time-resolved pressure loss and chemical fluxes enabled the determination of the sample-scale change in porosity and permeability. The results show the growth of localized dissolution features associated with high permeability increase for the highest $P_{\mathrm{CO}_2}$ , whereas dissolution tends to be more homogeneously distributed for lower values of $P_{\mathrm{CO}_2}$ . For the latter, the higher the $P_{\mathrm{CO}_2}$ , the more the dissolution patterns display ramified structures and permeability increase. For the lowest value of $P_{\mathrm{CO}_2}$ , the preferential dissolution of the calcite cement associated with the low dissolution kinetics triggers the transport that may locally accumulate and form a microporous material that alters permeability and produces an anti-correlated porosity–permeability relationship. The combined analysis of the pore network geometry and the macroscopic measurements shows that $P_{\mathrm{CO}_2}$ regulates the tortuosity change during dissolution. Conversely, the increase of the exponent value of the observed power law permeability–porosity trend while $P_{\mathrm{CO}_2}$ increases, which appears to be strongly linked to the increase of the effective hydraulic diameter, depends on the initial rock structure.     

On the Elastic Energy Density of Constrained Q-Tensor Models for Biaxial Nematics

Within the Landau–de Gennes theory, the order parameter describing a biaxial nematic liquid crystal assigns a symmetric traceless 3 × 3 matrix Q with three distinct eigenvalues to every point of the region Ω occupied by the system. In the constrained case of matrices Q with constant eigenvalues, the order parameter space is diffeomorphic to the eightfold quotient ${\mathbb{S}^3/\mathcal{H}}$ of the 3-sphere ${\mathbb{S}^3}$ , where ${\mathcal{H}}$ is the quaternion group, and a configuration of a biaxial nematic liquid crystal is described by a map from Ω to ${\mathbb{S}^3/\mathcal{H}}$ . We express the (simplest form of the) Landau–de Gennes elastic free-energy density as a density defined on maps ${q: \Omega \to \mathbb{S}^3}$ , whose functional dependence is restricted by the requirements that (1) it is well defined on the class of configuration maps from Ω to ${\mathbb{S}^3/\mathcal{H}}$ (residual symmetry) and (2) it is independent of arbitrary superposed rigid rotations (frame indifference). As an application of this representation, we then discuss some properties of the corresponding energy functional, including coercivity, lower semicontinuity and strong density of smooth maps. Other invariance properties are also considered. In the discussion, we take advantage of the identification of ${\mathbb{S}^3}$ with the Lie group of unit quaternions ${Sp(1) \cong SU(2)}$ and of the relations between quaternions and rotations in ${\mathbb{R}^3}$ and ${\mathbb{R}^4}$ .     

Γ-Convergence Analysis of Systems of Edge Dislocations: the Self Energy Regime

This paper deals with the elastic energy induced by systems of straight edge dislocations in the framework of linearized plane elasticity. The dislocations are introduced as point topological defects of the displacement-gradient fields. Following the core radius approach, we introduce a parameter ${\varepsilon > 0}$ representing the lattice spacing of the crystal, we remove a disc of radius ${\varepsilon}$ around each dislocation and compute the elastic energy stored outside the union of such discs, namely outside the core region. Then, we analyze the asymptotic behaviour of the elastic energy as ${\varepsilon \rightarrow 0}$ , in terms of Γ-convergence. We focus on the self energy regime of order ${\log\frac{1}{\varepsilon}}$ ; we show that configurations with logarithmic diverging energy converge, up to a subsequence, to a finite number of multiple dislocations and we compute the corresponding Γ-limit.