Showing papers in "Archive for Rational Mechanics and Analysis in 1987"

Fine phase mixtures as minimizers of energy

Abstract: Solid-solid phase transformations often lead to certain characteristic microstructural features involving fine mixtures of the phases. In martensitic transformations one such feature is a plane interface which separates one homogeneous phase, austenite, from a very fine mixture of twins of the other phase, martensite. In quartz crystals held in a temperature gradient near the α-β transformation temperature, the α-phase breaks up into triangular domains called Dauphine twins which become finer and finer in the direction of increasing temperature. In this paper we explore a theoretical approach to these fine phase mixtures based on the minimization of free energy.

The gradient theory of phase transitions and the minimal interface criterion

Abstract: In this paper I prove some conjectures of GURTIN [15] concerning the Van der Waals-Cahn-Hilliard theory of phase transitions. Consider a fluid, under isothermal conditions and confined to a bounded container 12 Q R', whose Gibbs free energy, per unit volume, is a prescribed function Wo of the density distribution u. The classical problem (cf. GURTIN [16]) of determining the stable configurations of the fluid is to minimize the total energy of the fluid, E(u) = f Wo(u(x)) dx, t2

Injectivity and self-contact in nonlinear elasticity

Abstract: Let Ω be a bounded open connected subset of ℝ3 with a sufficiently smooth boundary. The additional condition ∫ det ▽ψ dx ≦ vol ψ(Ω) is imposed on the admissible deformations ψ: ¯Ω → ℝ of a hyperelastic body whose reference configuration is ¯Ω. We show that the associated minimization problem provides a mathematical model for matter to come into frictionless contact with itself but not interpenetrate. We also extend J. Ball's theorems on existence to this case by establishing the existence of a minimizer of the energy in the space W1,p(Ω;ℝ3), p > 3, that is injective almost everywhere.

Phase transitions in one-dimensional nonlinear viscoelasticity: admissibility and stability

Abstract: For the motion of a one-dimensional viscoelastic material of rate type with a non-monotonic stress-strain relation, a mixed initial boundary value problem is considered. A simple existence theory is outlined, based on a novel transformation of the equation into the form of a degenerate reaction-diffusion system. This leads to new results characterizing the regularity of weak solutions. It is shown that each solution tends strongly to a stationary state asymptotically in time. Stable stationary states are characterized. Stable states may contain coexistent phases, i.e. they may have discontinuous strain. They need not be minimizers of energy in the strong sense of the calculus of variations; “metastable” and “absolutely stable” phases may coexist in a stable state. Furthermore, such states do arise as long-time limits of smooth solutions. Beyond the above, “hysteresis” and “creep” phenomena are exhibited in a model of a loaded viscoelastic bar. Also, a viscosity criterion is proposed for the admissibility of propagating waves in the associated purely elastic model. This criterion is then applied to describe the formation of some propagating phase boundaries in a loaded elastic bar.

Hamiltonian structures and stability for rigid bodies with flexible attachments

Abstract: The dynamics of a rigid body with flexible attachments is studied. A general framework for problems of this type is established in the context of Poisson manifolds and reduction. A simple model for a rigid body with an attached linear extensible shear beam is worked out for illustration. Second, the Energy-Casimir method for proving nonlinear stability is recalled and specific stability criteria for our model example are worked out. The Poisson structure and stability results take into account vibrations of the string, rotations of the rigid body, their coupling at the point of attachment, and centrifugal and Coriolis forces.

A regularity theorem for minimizers of quasiconvex integrals

Abstract: We prove C1,α partial regularity for minimizers of functionals with quasiconvex integrand f(x, u, Du) depending on vector-valued functions u. The integrand is required to be twice continuously differentiable in Du, and no assumption on the growth of the derivatives of f is made: a polynomial growth is required only on f itself.

On the Uniqueness of Flow of a Navier-Stokes Fluid Due to a Stretching Boundary

Abstract: When a sheet of polymer is extruded continuously from a die, it entrains the ambient fluid and a boundary layer develops. Such a boundary layer is markedly different from that in the Blasius flow past a flat plate in that the boundary layer grows in the direction of the motion of the sheet, starting at the die. Sakiadis [1]–[3] was the first to study such a boundary layer flow due to a rigid flat continuous surface moving in its own plane. Later, Erickson, Fan & Fox [4] studied the boundary layer due to the motion of a porous flat plate when the transverse velocity at the surface is non-zero.

Convex solutions of certain elliptic equations have constant rank hessians

Abstract: In this note we first consider solutions u of (1.1) Au = f ( u , Au) > 0 in a region ~2 of Euclidean n space (R"). Here Vu and Au denote the gradient and Laplaeian of u. We assume that f has H61der continuous second partial derivatives on some open set containing the range of the function x ~ (u(x), Vu(x)), x E ~ . We also assume that f is strictly positive, with 0.2) 2 ( L y (., Vu) f ( . , 7u)f , , ( . , Vu) > 0, that is, 1If(., 7u) is convex in u. Let H denote the Hessian matrix of u. Our main result is

Stability and Folds

Abstract: It is known that when one branch of a simple fold in a bifurcation diagram represents (linearly) stable solutions, the other branch represents unstable solutions. The theory developed here can predict instability of some branches close to folds, without knowledge of stability of the adjacent branch, provided that the underlying problem has a variational structure. First, one particular bifurcation diagram is identified as playing a special role, the relevant diagram being specified by the choice of functional plotted as ordinate. The results are then stated in terms of the shape of the solution branch in this distinguished bifurcation diagram. In many problems arising in elasticity the preferred bifurcation diagram is the loaddisplacement graph. The theory is particularly useful in applications where a solution branch has a succession of folds. The theory is illustrated with applications to simple models of thermal selfignition and of a chemical reactor, both of which systems are of Emden-Fowler type. An analysis concerning an elastic rod is also presented.

On the positivity of the second variation in finite elasticity

Abstract: where f is a vector-valued function on a bounded open domain O (1%" that satisfies f = d (d given) on a portion of the boundary 9 . Here (Vf) e = OfffSxj, i , j = 1, 2, . . . , n. I f f is a sufficiently smooth solution of the corresponding Euler-Lagrange equations, then a further necessary condition for it to be a weak relative minimizer (a local minimizer in the Cx(~) topology) is that the second variation of I at f be nonnegative2'3; i.e., ~IAu) := f Vu. G[Vu] _> o I2

Central configurations of the N-body problem via equivariant Morse theory

Abstract: In this paper we use the equivariant Morse theory to give an estimate of the minimal number of central configurations in the N-body problem in ℝ3. In the case of equal masses we prove that the planar central configurations are saddle points for the potential energy. From this we deduce the presence of non-planar central configurations, for every N ≧ 4. The principal difficulty in applying Morse theory is that the potential function is defined on a manifold on which the group O(3) does not act freely. To avoid this problem the equivariant cohomology functor is applied in order to obtain the Morse inequalities.

On the Behavior at Infinity of Solutions of Elliptic Systems with a Finite Energy Integral

Abstract: Asymptotic behavior in a neighborhood of infinity is studied in this paper for solutions of an elliptic system of order 2m with constant complex coefficients. It is supposed that the weighted Dirichlet integral is bounded. Our considerations include solutions with finite energy for the system of linear elasticity (see Theorem 3). A class of solutions periodic in some independent variables is also studied in this paper (the E. Sanchez-Palencia problem).

Eventual C∞-Regularity and Concavity for Flows in One-Dimensional Porous Media

Abstract: We study the regularity and the asymptotic behavior of the solutions of the initial value problem for the porous medium equation $${u_t} = {eft( {{u^m}} ight)_{xx}}inQ = R imes eft( {0,nfty } ight),$$ $$ueft( {x,0} ight) = {u_0}eft( x ight)forx n R,$$

Existence of positive solutions for semilinear elliptic equations in general domains

Abstract: The main purpose of this paper is to prove some new existence theorems for positive solutions to the Dirichlet problem $$ elta u(x) + f(u(x)) = 0,x n mega ,$$ (11) $$ u(x) = 0,x n artial mega $$ (12)

Singular solutions for some semilinear elliptic equations

Abstract: Let B R = {x ∈ ℝ N ; |x| < R} with N ≥ 2. Consider a function u which satisfies $$[CD egin{array}{*{20}{c}}{u n {C^2}({B_R}ackslash 0 ),u eqslant 0 on {B_R}ackslash 0 } { -elta u + {u^p} = 0 on {B_R}ackslash 0 } nd{array}$$ (1)

Well-posedness of the fundamental boundary value problems for constrained anisotropic elastic materials

Abstract: We consider the equations of linear homogeneous anisotropic elasticity admitting the possibility that the material is internally constrained, and formulate a simple necessary and sufficient condition for the fundamental boundary value problems to be well-posed. For materials fulfilling the condition, we establish continuous dependence of the displacement and stress on the elastic moduli and ellipticity of the elasticity system. As an application we determine the orthotropic materials for which the fundamental problems are well-posed in terms of their Young's moduli, shear moduli, and Poisson ratios. Finally, we derive a reformulation of the elasticity system that is valid for both constrained and unconstrained materials and involves only one scalar unknown in addition to the displacements. For a two-dimensional constrained material a further reduction to a single scalar equation is outlined.