Geometric continuum mechanics

Geometric Continuum Mechanics ( GCM) is a new formulation of Continuum Mechanics ( CM) based on the requirement of Geometric Naturality ( GN). According to GN, in introducing basic notions, governing principles and constitutive relations, the sole geometric entities of space-time to be involved are the metric field and the motion along the trajectory. The additional requirement that the theory should be applicable to bodies of any dimensionality, leads to the formulation of the Geometric Paradigm ( GP) stating that push-pull transformations are the natural comparison tools for material fields. This basic rule implies that rates of material tensors are Lie-derivatives and not derivatives by parallel transport. The impact of the GP on the present state of affairs in CM is decisive in resolving questions still debated in literature and in clarifying theoretical and computational issues. As a consequence, the notion of Material Frame Indifference ( MFI) is corrected to the new Constitutive Frame Invariance ( CFI) and reasons are adduced for the rejection of chain decompositions of finite elasto-plastic strains. Geometrically consistent notions of Rate Elasticity ( RE) and Rate Elasto-Visco-Plasticity ( REVP) are formulated and consistent relevant computational methods are designed.     

Global Solution to the Incompressible Oldroyd-B Model in the Critical L p Framework: the Case of the Non-Small Coupling Parameter

This paper is dedicated to the global well-posedness issue of the incompressible Oldroyd-B model in the whole space \({\mathbb{R}^d}\) with \({d \geqq 2}\) . It is shown that this set of equations admits a unique global solution in a certain critical L ^p -type Besov space provided that the initial data, but not necessarily the coupling parameter, is small enough. As a consequence, even through the coupling effect between the equations of velocity u and the symmetric tensor of constrains τ is not small, one may construct the unique global solution to the Oldroyd-B model for a class of large highly oscillating initial velocity. The proof relies on the estimates of the linearized systems of (u, τ) and \({(u, \mathbb{P}{\rm div}\tau)}\) which may be of interest for future works. This result extends the work by Chemin and Masmoudi (SIAM J Math Anal 33:84–112, 2001) to the non-small coupling parameter case.     

Periodic solutions to some problems of n-body type

We prove the existence of at least one T-periodic solution to a dynamical system of the type $$ - m_i \ddot u_i = \sum\limits_{j = 1,j \ne i}^n {\triangledown V_{ij} (u_i - u_j ,{\text{ }}t)}$$ (1) where the potentials V _ij are T-periodic in t and singular at the origin, u _i ε R ^k i=1, ..., n, and k≧3. We also provide estimates on the H ¹ norm of this solution. The proofs are based on a variant of the Ljusternik-Schnirelman method. The results here generalize to the n-body problem some results obtained by Bahri & Rabinowitz on the 3-body problem in [6].     

Global Existence for a System of Non-Linear and Non-Local Transport Equations Describing the Dynamics of Dislocation Densities

In this paper, we study the global in time existence problem for the Groma-Balogh model describing the dynamics of dislocation densities. This model is a two-dimensional model where the dislocation densities satisfy a system of transport equations such that the velocity vector field is the shear stress in the material, solving the equations of elasticity. This shear stress can be expressed as some Riesz transform of the dislocation densities. The main tool in the proof of this result is the existence of an entropy for this system.     

On Cesàro-Volterra Method in Orthotropic Saint-Venant Beam

The linearly elastic and orthotropic Saint-Venant beam model, with a spatially constant Poisson tensor and fiberwise homogeneous elastic moduli, is investigated by a coordinate-free approach. A careful reasoning reveals that the elastic strain, fulfilling the whole set of differential conditions of integrability and a differential condition imposed by equilibrium, is defined on the whole ambient space in which the beam is immersed. At this stage the shape of the beam cross-section is inessential and Cesàro-Volterra formula provides the general integral of the differential conditions of kinematic compatibility. The cross-section geometrical shape comes into play only when differential and boundary equilibrium conditions are imposed to evaluate the warping displacement field. The treatment of an orthotropic Saint-Venant beam is applied to investigate about the locations of the shear and twist centres. It is shown that the position of the shear centre can be expressed in terms of the sole cross-section twist warping. The advantage with respect to treatments in the literature is that the solution of a single Neumann-like problem is required.     

Comparison of dynamical cores for NWP models: comparison of COSMO and Dune

We present a range of numerical tests comparing the dynamical cores of the operationally used numerical weather prediction (NWP) model COSMO and the university code Dune, focusing on their efficiency and accuracy for solving benchmark test cases for NWP. The dynamical core of COSMO is based on a finite difference method whereas the Dune core is based on a Discontinuous Galerkin method. Both dynamical cores are briefly introduced stating possible advantages and pitfalls of the different approaches. Their efficiency and effectiveness is investigated, based on three numerical test cases, which require solving the compressible viscous and non-viscous Euler equations. The test cases include the density current (Straka et al. in Int J Numer Methods Fluids 17:1–22, 1993), the inertia gravity (Skamarock and Klemp in Mon Weather Rev 122:2623–2630, 1994), and the linear hydrostatic mountain waves of (Bonaventura in J Comput Phys 158:186–213, 2000).     

The Two-Dimensional Euler Equations on Singular Domains

We establish the existence of global weak solutions of the two-dimensional incompressible Euler equations for a large class of non-smooth open sets. Loosely, these open sets are the complements (in a simply connected domain) of a finite number of obstacles with positive Sobolev capacity. Existence of weak solutions with L ^p vorticity is deduced from a property of domain continuity for the Euler equations that relates to the so-called γ-convergence of open sets. Our results complete those obtained for convex domains in Taylor (Progress in Nonlinear Differential Equations and their Applications, Vol. 42, 2000), or for domains with asymptotically small holes (Iftimie et al. in Commun Partial Differ Equ 28(1–2), 349–379, 2003; Lopes Filho in SIAM J Math Anal 39(2), 422–436, 2007).     

Busemann Functions for the N-Body Problem

Following ideas in Maderna and Venturelli (Arch Ration Mech Anal 194:283–313, 2009), we prove that the Busemann function of the parabolic homotetic motion for a minimal central coniguration of the N-body problem is a viscosity solution of the Hamilton–Jacobi equation and that its calibrating curves are asymptotic to the homotetic motion.     

A boundary value problem associated with the second painlevé transcendent and the Korteweg-de Vries equation

The differential equation considered is \(y'' - xy = y|y|^\alpha \) . For general positive α this equation arises in plasma physics, in work of De Boer & Ludford. For α=2, it yields similarity solutions to the well-known Korteweg-de Vries equation. Solutions are sought which satisfy the boundary conditions (1) y(∞)=0 (2) (1) $$y{\text{(}}\infty {\text{)}} = {\text{0}}$$ (2) $$y{\text{(}}x{\text{) \~( - }}\tfrac{{\text{1}}}{{\text{2}}}x{\text{)}}^{{{\text{1}} \mathord{\left/ {\vphantom {{\text{1}} \alpha }} \right. \kern-\nulldelimiterspace} \alpha }} {\text{ as }}x \to - \infty $$ It is shown that there is a unique such solution, and that it is, in a certain sense, the boundary between solutions which exist on the whole real line and solutions which, while tending to zero at plus infinity, blow up at a finite x. More precisely, any solution satisfying (1) is asymptotic at plus infinity to some multiple kA i(x) of Airy's function. We show that there is a unique k^*(α) such that when k=k^*(α) the condition (2) is also satisfied. If 0 ^*, the solution exists for all x and tends to zero as x→-∞, while if k>k ^* then the solution blows up at a finite x. For the special case α=2 the differential equation is classical, having been studied by Painlevé around the turn of the century. In this case, using an integral equation derived by inverse scattering techniques by Ablowitz & Segur, we are able to show that k^*=1, confirming previous numerical estimates.     

Macroscopic Response of¶Nematic Elastomers via Relaxation of¶a Class of SO(3)-Invariant Energies

We obtain an explicit formula for the relaxation of the free-energy density for nematic elastomers proposed by Bladon, Terentjev&;Warner (Phys. Rev. E 47 (1993), 3838–3840). The proof is based on a characterization of the level sets of the relaxed energy. In particular, the construction uses only laminates within laminates and it identifies those deformations that correspond to simple laminates.     

Phase behavior and effects of microstructure on viscoelastic properties of a series of polylactides and polylactide/poly(ε-caprolactone) copolymers

Dynamic viscoelastic measurements were combined with differential scanning calorimetry (DSC) and atomic force microscopy (AFM) analysis to investigate the rheology, phase structure, and morphology of poly(l-lactide) (PLLA), poly(ε-caprolactone) (PCL), poly(d,l-lactide) (PDLLA) with molar composition l-LA/d-LA?=?53:47, and poly(l-lactide-co-ε-caprolactone) (PLAcoCL) with molar composition l-LA/CL?=?67:33. After melt conformation, both copolymers PDLLA and PLAcoCL were found to be amorphous whereas PLLA and PCL presented partial crystallinity. The copolymers and PCL were considered as thermorheologically simple according to the rheological methods employed. Therefore, data from different temperatures could be overlapped by a simple horizontal shift (a _T) on elastic modulus, G′, and loss modulus, G′, versus frequency graph, generating the corresponding master curves. Moreover, these master curves showed a dependency of G″≈ω and G′≈ω ² at low frequencies, which is a characteristic of homogeneous melts. For the first time, fundamental viscoelastic parameters, such as entanglement modulus G _N ⁰ and reptation time τ _d, of a PLAcoCL copolymer were obtained and correlated to chain microstructure. PLLA, by contrast, was unexpectedly revealed as a thermorheologically complex liquid according to the failure observed in the superposition of the phase angle (δ) versus the complex modulus (G*); this result suggests that the narrow window for rheological measurements, chosen to be close to the melting point centered at 180 °C thus avoiding thermal degradation, was not sufficient to assure an homogeneous behavior of PLLA melts. The understanding of the melt rheology related to the chain microstructural aspects will help in the understanding of the complex phase structures present in medical devices.     

A connection formula for the second Painlevé transcendent

We consider the second Painlevé transcendent $$\frac{{d^2 y}}{{dx^2 }} = xy + 2y^3 .$$ It is known that if y(x) ～ k Ai (x) as x → + ∞, where ?1<k<1 and Ai (x) denotes Airy's function, then $$y(x) \sim d|x|^{ - \tfrac{1}{4}} sin\{ \tfrac{2}{3}|x|^{\tfrac{3}{2}} - \tfrac{3}{4}d^2 1n|x| - c\} ,$$ where the constants d, c depend on k. This paper shows that $$d^2 = \pi ^{ - 1} 1n(1 - k^2 )$$ , which confirms a conjecture by Ablowitz & Segur.     

A Quantitative Modulus of Continuity for the Two-Phase Stefan Problem

We derive the quantitative modulus of continuity $$\omega(r)=\left[ p+\ln \left( \frac{r_0}{r}\right)\right]^{-\alpha (n, p)},$$ which we conjecture to be optimal for solutions of the p-degenerate two-phase Stefan problem. Even in the classical case p = 2, this represents a twofold improvement with respect to the early 1980’s state-of-the-art results by Caffarelli– Evans (Arch Rational Mech Anal 81(3):199–220, 1983) and DiBenedetto (Ann Mat Pura Appl 103(4):131–176, 1982), in the sense that we discard one logarithm iteration and obtain an explicit value for the exponent α(n, p).     

Sharp Trace Hardy–Sobolev-Maz’ya Inequalities and the Fractional Laplacian

In this work we establish trace Hardy and trace Hardy–Sobolev–Maz’ya inequalities with best Hardy constants for domains satisfying suitable geometric assumptions such as mean convexity or convexity. We then use them to produce fractional Hardy–Sobolev–Maz’ya inequalities with best Hardy constants for various fractional Laplacians. In the case where the domain is the half space, our results cover the full range of the exponent ${s \in}$ (0, 1) of the fractional Laplacians. In particular, we give a complete answer in the L ² setting to an open problem raised by Frank and Seiringer (“Sharp fractional Hardy inequalities in half-spaces,” in Around the research of Vladimir Maz’ya. International Mathematical Series, 2010).     

Berechnung von Transportgrößen mit Hilfe der Statistischen Physik: Ein Überblick

Transport coefficientsL(k, ω) are calculated by two methods, a) Using a kinetic equation for distribution functions (Boltzmann, Landau) and b) the linear response to a periodic disturbance in space and time (Kubo). The regions of validity of both methods are discussed, some results are compared with experiment and some general properties of the Kubo formula forL (k, ω) are given.     

Singular Limiting Induced from Continuum Solutions and the Problem of Dynamic Cavitation

In the works of Pericak-Spector and Spector (Arch Rational Mech Anal. 101:293–317, 1988, Proc. Royal Soc. Edinburgh Sect A 127:837–857, 1997) a class of self-similar solutions are constructed for the equations of radial isotropic elastodynamics that describe cavitating solutions. Cavitating solutions decrease the total mechanical energy and provide a striking example of non-uniqueness of entropy weak solutions (for polyconvex energies) due to point-singularities at the cavity. To resolve this paradox, we introduce the concept of singular limiting induced from continuum solution (or slic-solution), according to which a discontinuous motion is a slic-solution if its averages form a family of smooth approximate solutions to the problem. It turns out that there is an energetic cost for creating the cavity, which is captured by the notion of slic-solution but neglected by the usual entropic weak solutions. Once this cost is accounted for, the total mechanical energy of the cavitating solution is in fact larger than that of the homogeneously deformed state. We also apply the notion of slic-solutions to a one-dimensional example describing the onset of fracture, and to gas dynamics in Langrangean coordinates with Riemann data inducing vacuum in the wave fan.