Spectral Stability of Ideal-Gas Shock Layers

Extending recent results in the isentropic case, we use a combination of asymptotic ODE estimates and numerical Evans-function computations to examine the spectral stability of shock-wave solutions of the compressible Navier–Stokes equations with ideal gas equation of state. Our main results are that, in appropriately rescaled coordinates, the Evans function associated with the linearized operator about the wave (i) converges in the large-amplitude limit to the Evans function for a limiting shock profile of the same equations, for which internal energy vanishes at one end state; and (ii) has no unstable (positive real part) zeros outside a uniform ball |λ| ≦ Λ. Thus, the rescaled eigenvalue ODE for the set of all shock waves, augmented with the (nonphysical) limiting case, form a compact family of boundary-value problems that can be conveniently investigated numerically. An extensive numerical Evans-function study yields one-dimensional spectral stability, independent of amplitude, for gas constant γ in [1.2, 3] and ratio ν/μ of heat conduction to viscosity coefficient within [0.2, 5] (γ ≈ 1.4, ν/μ ≈ 1.47 for air). Other values may be treated similarly but were not considered. The method of analysis extends also to the multi-dimensional case, a direction that we shall pursue in a future work.     

Photomechanics of light-activated polymers

Light-activated polymers are an exciting class of modern materials that respond mechanically when irradiated by light at particular wavelengths. While details of the mechanisms that connect the optical excitation to mechanical behavior are complex and differ from material to material, there is sufficient commonality among them to permit the development of a generalized modeling framework to describe the photomechanics. The features shared by light-activated polymers involve light interacting with the material, which triggers photochemical reactions that alter the structure of the crosslinked polymer network. Many such structural alterations result in an evolution of the polymer network, and subsequent macroscopic deformation. When this process is appropriately executed it can enable a photomechanical shape-memory effect. In this paper, we develop a three-dimensional finite-deformation modeling framework to describe the photomechanical response of light-activated polymer systems. This framework integrates four coupled phenomena that contribute to macroscopic photomechanical behavior: photophysics, photochemistry, chemomechanical coupling, and mechanical deformation. The chemomechanical coupling consists of chemically induced structural alterations of the crosslinked network that result in subsequent deformation. We describe this behavior through a decomposition of the crosslinked network into two components consisting of an original network and a photochemically altered network; both evolve during photomechanical deformation. The modeling framework presented in this paper is sufficiently general that it is applicable to light-activated polymer systems that operate with various mechanisms in each of the four areas. Using this modeling approach, we develop constitutive models for two recently developed light-activated polymer systems [Lendlein, A., Hongyan, J., Junger, O., Langer, R., 2005. Light-induced shape-memory polymers. Nature 434 (7035) 879; Scott, T.F., Schneider, A.D., Cook, W.D., Bowman, C.N., 2005. Photoinduced plasticity in crosslinked polymers. Science 308 (5728) 1615]. For the material developed by Scott and his co-workers we validate our model by measuring and numerically simulating photo-induced stress relaxation and bending deformation and obtain good agreement between measurements and predictions. Finally, we use the model to study the effects of photomechanical parameters (applied strain magnitude, irradiation time and intensity, and photoabsorber concentration) and the behavior of the network evolution rule on the material response.     

Determination of contact parameters for discrete element method simulations of granular systems

Both linear-spring-dashpot （LSD） and non-linear Hertzian-spring-dashpot （HSD） contact models are commonly used for the calculation of contact forces in Discrete Element Method （DEM） simulations of granular systems. Despite the popularity of these models, determination of suitable values for the contact parameters of the simulated particles such as stiffness, damping coefficient, coefficient of restitution, and simulation time step, is not altogether obvious. In this work the relationships between these contact parameters for a model system where a particle impacts on a flat base are examined. Recommendations are made concerning the determination of these contact parameters for use in DEM simulations.     

Interfacially active particles in droplet/matrix blends of model immiscible homopolymers: Particles can increase or decrease drop size

Particles have been shown to adsorb at the interface between immiscible homopolymer melts and to affect the morphology of blends of those homopolymers. We examined the effect of such interfacially active particles on the morphology of droplet/matrix blends of model immiscible homopolymers. Experiments were conducted on blends of polydimethylsiloxane and 1,4-polyisoprene blended in either a 20:80 or 80:20 weight ratio. The effects of three different particle types, fluoropolymer particles, iron particles, and iron oxyhydroxide particles, all at a loading of 0.5 vol.%, were examined by rheology and by direct flow visualization. Particles were found to significantly affect the strain recovery behavior of polymer blends, increasing or decreasing the ultimate recovery, slowing down or accelerating the recovery kinetics, and changing the dependence of these parameters on the applied stress prior to cessation of shear. These rheological observations were found to correlate reasonably well with particle-induced changes in drop size. The particles can both increase as well as decrease the drop size, depending on the particle type, as well as on which phase is continuous. The cases in which particles cause a decrease in drop size are analogous to the particle stabilization of “Pickering emulsions” well-known from the literature on oil/water systems. We hypothesize that cases in which particles increase drop size are analogous to the “bridging–dewetting” mechanism known in the aqueous foam literature.     

Nonlinear Stability of Time-Periodic Viscous Shocks

In order to understand the nonlinear stability of many types of time-periodic travelling waves on unbounded domains, one must overcome two main difficulties: the presence of embedded neutral eigenvalues and the time-dependence of the associated linear operator. This problem is studied in the context of time-periodic Lax shocks in systems of viscous conservation laws. Using spatial dynamics and a decomposition into separate Floquet eigenmodes, it is shown that the linear evolution for the time-dependent operator can be represented using a contour integral similar to that of the standard time-independent case. By decomposing the resulting Green’s distribution, the leading order behavior associated with the embedded eigenvalues is extracted. Sharp pointwise bounds are then obtained, which are used to prove that time-periodic Lax shocks are linearly and nonlinearly stable under the necessary conditions of spectral stability and minimal multiplicity of the translational eigenvalues. The latter conditions hold, for example, for small-oscillation time-periodic waves that emerge through a supercritical Hopf bifurcation from a family of time-independent Lax shocks of possibly large amplitude.     

Boundary Layer Stability¶in Real Vanishing Viscosity Limit

In the previous paper [20], an Evans function machinery for the study of boundary layer stability was developed. There, the analysis was restricted to strongly parabolic perturbations, that is to an approximation of the form u _t +(F(u)) _x =ν(B(u)u _x ) _x $ (ν≪1) with an “elliptic” matrix B. However, real models, like the Navier–Stokes approximation of the Euler equations for a gas flow, involve incompletely parabolic perturbations: B is not invertible in general. We first adapt the Evans function to this realistic framework, assuming that the boundary is not characteristic, neither for the hyperbolic first order system u _t +(F(u)) _x = 0, nor for the perturbed system. We then apply it to the various kinds of boundary layers for a gas flow. We exhibit some examples of unstable boundary layers for a perfect gas, when the viscosity dominates heat conductivity. Received: 27 November 2000/ Accepted: 16 March 2001     

Forbidden minors to graphs with small feedback sets

Finite obstruction set characterizations for lower ideals in the minor order are guaranteed to exist by the graph minor theorem. In this paper we characterize several families of graphs with small feedback sets, namely k₁-F V S , k₂-F E S and (k₁,k₂)-F V /E S , for small integer parameters k₁ and k₂. Our constructive methods can compute obstruction sets for any minor-closed family of graphs, provided the pathwidth (or treewidth) of the largest obstruction is known.     

Continuous cellulose fiber-reinforced cellulose ester composites III. Commercial matrix and fiber options

Thermoplastic composites were prepared using two continuous regenerated cellulose fiber types, rayon and lyocell, and with several different commercially-available thermoplastic cellulose esters as matrix. Matrix options included cellulose acetate propionate (CAP), and several cellulose acetate butyrates (CAB) with different butyryl content, having different molecular weights and different methods of plasticization (adipates and very low molecular weight cellulose ester fractions). Choice of cellulose ester type was generally found to have little or no effect on mechanical properties. A significant effect, however, was revealed for fiber type. The lyocell-based composites thereby were reflective of the greater stiffness of a fiber produced from anisotropic solution state. Their modulus consistently exceeded 20GPa whereas the rayon fiber-based composites had moduli between 6 and 8GPa. The latter, however, possessed failure strains that were 3 to 4 times greater than their stiffer counterparts.     

N‐Acetyl‐l‐phenyl­alanine

The structure of the title compound, C₁₁H₁₃NO₃, is characterized by a two‐dimensional infinite network of intermolecular N—H?O and O—H?O hydrogen bonds.     

Microstructure in a Cubic to Orthorhombic Transition

Microstructures for a cubic to orthorhombic transition are constructed using a geometrically nonlinear, thermoelastic theory of martensitic transformations. Such microstructures are of interest because they provide low energy paths along which a specimen can transform. The particular microstructures considered are the twinned martensite, austenite–martensite, wedge, triangle, and diamond. More specifically, all possible twins are found along with the corresponding twinning elements and magnitude of the twin shear. Further, two kinds of austenite–martensite microstructures are studied: those with a single variant of martensite and those with twinned martensite. The regions in the space of transformation stretches in which each of these microstructures exist are determined, and the shape strains and habit plane normals are found as well. In addition, special microstructures, the wedge, triangle, and diamond, are constructed with both the austenite-single variant and austenite-twinned martensite microstructures. These special microstructures are of interest because they provide a mechanism through which the transformation may proceed more easily, and they are possible only in alloys with particular transformation stretches. Numerically computed level curves in the space of the stretches are presented on which the special microstructures are possible. These results may be useful in providing guidelines for alloy design.