The Cauchy problem in one-dimensional nonlinear viscoelasticity

The initial value problem for a nonlinear hyperbolic Volterra equation which models the motion of an unbounded viscoelastic bar is studied. Under physically motivated assumptions, the existence of a unique, globally defined, classical solution is established provided the initial data are sufficiently smooth and small. Boundedness and asymptotic behavior are also discussed. This analysis is based on energy estimates in conjunction with properties of strongly positive definite kernels.     

On formation of singularities in one-dimensional nonlinear thermoelasticity

Dedicated to Bernard D. Coleman on the occasion of his 60^th birthday     

A nonlinear functional differential equation in Banach space with applications to materials with fading memory

Archive for Rational Mechanics and Analysis -     

Chain scission efficiency and reactive-ion etch resistance of alternating copolymers of styrene and olefins trisubstituted or tetrasubstituted with electron-withdrawing groups

Alternating copolymers of styrene (St) with electron-deficient olefins trisubstituted or tetrasubstituted with cyano and carboalkoxy groups have been subjected to ⁶⁰Co γ-radiolysis together with a series of copolymers of methyl methacrylate (MMA) and St. The chain scission susceptibility G_s—G_x determined by membrane osmometry drastically decreases as St is incorporated in poly(methyl methacrylate) (PMMA). Whereas the alternating St-MMA copolymer is slightly crosslinked upon irradiation, an alternating copolymer of St with diethyl 2-cyano-1,1-ethylenedicarboxylate maintains a fairly high degradation sensitivity (G_s—G_x = 1.2). The reactive-ion etch rates were determined for the series of polymers in CF₄/O₂ (92/8). The etch resistance is significantly increased by introduction of St units in PMMA, and the highly substituted alternating copolymer etches as slowly as the MMA(50)—St(50) copolymers. Thus the alternating copolymer of NCCH=C(CO₂Et)₂ with St behaves like PMMA when exposed to high-energy radiation but is comparable to PSt in plasma environments.     

The Lavrentiev Phenomenon in Nonlinear Elasticity

In 1985 J.M. Ball and V.J. Mizel raised the question of whether there exist nonlinearly elastic materials possessing a physically natural stored energy density, i.e., one which is independent of an observer's coordinate frame (objective) and is invariant under the group of orthogonal linear transformations of space (isotropic), as well as physically reasonable boundary value problems for such materials such that the infimum of the total stored energy for those continuous deformations of the material meeting the boundary condition (admissible deformations) which belong to a Sobolev space W ^{1 p} 2 for some p ₂>1 is strictly greater than its infimum for those admissible continuous deformations belonging to some Sobolev space W ¹ p ₁, p ₁<p ₂, despite the density of W ^{1 p} 2 in W ¹ p ₁. The question was motivated by M. Lavrentiev's demonstration in 1926 of the presence of such a gap for a 1-dimensional variational boundary value problem on a bounded interval whose smooth integrand satisfied the conditions of Tonelli's existence theorem (as well as the development of improved versions in the 1980's). The present article describes a positive response to the question raised in 1985. Namely, we provide examples of nonlinearly elastic materials in 2-dimensions and physically reasonable boundary value problems for these materials in which a positive gap exists between the infimum of the total stored energy over admissible continuous deformations belonging to a Sobolev space W ^{1 p} 2 and its infimum over admissible continuous deformations belonging to a Sobolev space W ^{1 p} 1, with p ₁<p ₂. The physical and computational significance of such results is also discussed.     

Lavrentiev''s phenomenon for totally unconstrained variational problems in one dimension

We give examples of one-dimensional variational problems with free ends that exhibit Lavrentiev's phenomenon, i.e. the infimum of the functional over one class X of admissible functions is strictly greater than the infimum over another class Y of admissible functions - even though X is dense in Y.     

On perturbations of differentiable semigroups

LetX be a Banach space and letA be the infinitesimal generator of a differentiable semigroup {T(t) |t ≥ 0}, i.e. aC ₀-semigroup such thatt ↦T(t)x is differentiable on (0, ∞) for everyx εX. LetB be a bounded linear operator onX and let {S(t) |t ≥ 0} be the semigroup generated byA +B. Renardy recently gave an example which shows that {S(t) |t ≥ 0} need not be differentiable. In this paper we give a condition on the growth of ‖T′(t)‖ ast ↓ 0 which is sufficient to ensure that {S(t) |t ≥ 0} is differentiable. Moreover, we use Renardy’s example to study the optimality of our growth condition. Our results can be summarized roughly as follows:

The Lavrentiev Gap Phenomenon in Nonlinear Elasticity

The main objective of this paper is to present examples of the Lavrentiev phenomenon within the framework of two-dimensional nonlinear elasticity. Loosely speaking, this phenomenon is associated with the sensitivity of the infimum in a variational problem to the regularity required of the competing mappings. We provide a physically natural stored energy density and reasonable, though nontraditional, boundary conditions such that the energy functional exhibits the Lavrentiev phenomenon with admissible classes that are subsets of the continuous deformations. The stored-energy density W that we produce is smooth, materially homogeneous, frame-indifferent, isotropic and polyconvex. Furthermore, the corresponding minimization problem is such that existence of a continuous minimizer follows from known results. The basis for our examples is a convex integrand W ₀ for which the Euler-Lagrange equations have a very special form. We show that the functional associated with this W ₀ exhibits the Lavrentiev phenomenon for certain problems; by making a perturbation to W ₀ , we create the stored-energy density W described in the previous paragraph. With other perturbations to the integrand W ₀ and modifications of the boundary conditions, we are able to produce additional examples of the Lavrentiev phenomenon. Finally, we note that the integrand we use is just one of a family of integrands that can be used to produce examples of the phenomenon. (Accepted October 18, 2002) Published online March 12, 2003 Communicated by J. M. Ball