Local Bifurcation Analysis of a Second Gradient Model for Deformations of a Rectangular Slab

In this paper we carry out a derivation of the equilibrium equations of nonlinear elasticity with an added second-gradient term proportional to a small parameter . These equations are given by a fourth order semilinear system of pdes. We discuss different types of possible boundary conditions for these equations. We then specialize the equations to a rectangular slab and study the linearized problem about a homogenous deformation. We show that these equations admit solutions representable as Fourier series in one of the independent variables. Furthermore, we obtain the characteristic equation for the eigenvalues (possible bifurcation points) for the linear problem and derive asymptotic representations for this equation for small . We used these expressions to show that in the limit as the characteristic equation for converges uniformly (in certain regions of the parameter space) to the corresponding characteristic equation for . When the base material () is that of a Blatz–Ko type, we get conditions for the existence of eigenvalues of the linear problem with and small. Our numerical results in this case indicate that the number of bifurcation points is finite when and that this number monotonically increases as . For the problem with we get conditions for the existence of local branches of non-trivial solutions.      

Generalized Stefan models accounting for a discontinuous temperature field

We construct a class of generalized Stefan models able to account for a discontinuous temperature field across a nonmaterial interface. The resulting theory introduces a constitutive scalar interfacial field, denoted by and called the equivalent temperature of the interface. A classical procedure, based on the interfacial dissipation inequality, relates the interfacial energy release to the interfacial mass flux and restricts the equivalent temperature of the interface. We show that previously proposed theories are obtained as particular cases when or or, more generally, when for We study in a particular constitutive framework the solidification of an under-cooled liquid and we are able to give a sufficient condition for the existence of travelling wave solutions.Received: 29 January 2003, Accepted: 18 October 2003, Published online: 3 February 2004PACS: 64.70.Dv, 68.35.Md, 68.35.Rh     

Regularity of Flow of Anisotropic Fluid

We study the steady flow of an anisotropic generalised Newtonian fluid under Dirichlet boundary conditions in a bounded domain . The fluid is characterised by a nonlinear dependence of the stress tensor on the symmetric gradient of the velocity vector field. We prove the existence of a C ^1,α-solution of this problem under certain assumptions on the growth of the elliptic term. The result is global: we prove the regularity up to the boundary of the domain Ω. This research was supported by the grant GACR 201/03/0934; partially also by the grants MSM 0021620839 and GAUK 262/2002/B-MAT/MFF.     

Strong Solutions for Generalized Newtonian Fluids

We consider the motion of a generalized Newtonian fluid, where the extra stress tensor is induced by a potential with p-structure (p = 2 corresponds to the Newtonian case). We focus on the three dimensional case with periodic boundary conditions and extend the existence result for strong solutions for small times from \tfrac{5}{3}$$ " align="middle" border="0"> (see [16]) to \tfrac{7}{5}.$$ " align="middle" border="0"> Moreover, for we improve the regularity of the velocity field and show that for all 0.$$ " align="middle" border="0"> Within this class of regularity, we prove uniqueness for all \tfrac{7}{5}.$$ " align="middle" border="0"> We generalize these results to the case when p is space and time dependent and to the system governing the flow of electrorheological fluids as long as      

On Dual Configurational Forces

The dual conservation laws of elasticity are systematically re-examined by using both Noether's variational approach and Coleman–Noll–Gurtin's thermodynamics approach. These dual conservation laws can be interpreted as the dual configurational force, and therefore they provide the dual energy–momentum tensor. Some previously unknown and yet interesting results in elasticity theory have been discovered. As an example, we note the following duality condition between the configuration force (energy–momentum tensor) and the dual configuration force (dual energy–momentum tensor) ,

A geometrically exact Cosserat shell-model including size effects,avoiding degeneracy in the thin shell limit. Part I: Formal dimensional reduction for elastic plates and existence of minimizers for positive Cosserat couple modulus

This contribution is concerned with a consistent formal dimensional reduction of a previously introduced finite-strain three-dimensional Cosserat micropolar elasticity model to the two-dimensional situation of thin plates and shells. Contrary to the direct modelling of a shell as a Cosserat surface with additional directors, we obtain the shell model from the Cosserat bulk model which already includes a triad of rigid directors. The reduction is achieved by assumed kinematics, quadratic through the thickness. The three-dimensional transverse boundary conditions can be evaluated analytically in terms of the assumed kinematics and determines exactly two appearing coefficients in the chosen ansatz. Further simplifications with subsequent analytical integration through the thickness determine the reduced model in a variational setting. The resulting membrane energy turns out to be a quadratic, elliptic, first order, non degenerate energy in contrast to classical approaches. The bending contribution is augmented by a curvature term representing an additional stiffness of the Cosserat model and the corresponding system of balance equations remains of second order. The lateral boundary conditions for simple support are non-standard. The model includes size-effects, transverse shear resistance, drilling degrees of freedom and accounts implicitly for thickness extension and asymmetric shift of the midsurface. The formal thin shell membrane limit without classical h ³-bending term is non-degenerate due to the additional Cosserat curvature stiffness and control of drill rotations. In our formulation, the drill-rotations are strictly related to the size-effects of the bulk model and not introduced artificially for numerical convenience. Upon linearization with zero Cosserat couple modulus we recover the well known infinitesimal-displacement Reissner-Mindlin model without size-effects and without drill-rotations. It is shown that the dimensionally reduced Cosserat formulation is well-posed for positive Cosserat couple modulus by means of the direct methods of variations along the same line of argument which showed the well-posedness of the three-dimensional Cosserat bulk model [72].Received: 16 April 2004, Accepted: 3 May 2004, Published online: 17 September 2004     

Vector potential–vorticity relationship for the Stokes flows: application to the Stokes eigenmodes in 2D/3D closed domain

The unsteady dynamics of the Stokes flows, where , is shown to verify the vector potential–vorticity ( ) correlation , where the field is the pressure-gradient vector potential defined by . This correlation is analyzed for the Stokes eigenmodes, , subjected to no-slip boundary conditions on any two-dimensional (2D) closed contour or three-dimensional (3D) surface. It is established that an asymptotic linear relationship appears, verified in the core part of the domain, between the vector potential and vorticity, , where is a constant offset field, possibly zero.     

Energetic formulation of multiplicative elasto-plasticity using dissipation distances

We introduce a new energetic formulation for the inelastic rate-independent behavior of standard generalized materials. This formulation is solely based on the classical elastic energy-storage potential and a dissipation potential , and it replaces the classical variational inequalities which describe the flow rules for the inelastic variables like the plastic deformation and the hardening parameters. The energetic formulation has the major advantage that it is defined for a larger class of processes since it does not involve any derivatives of the strains or the internal variables, thus allowing for an analysis of processes involving sharp interfaces, localization or microstructure. Two new quantities are derived from and . First, this is the global dissipation distance on the manifold of internal states. Second, the reduced stored-energy density contains the comprised information of the elastic and plastic material properties via minimization of over the new internal variable. Several stability concept are derived and used to analyze failure mechanism. Finally, a natural incremental method is proposed which reduces to a minimization problem and can be solved efficiently using .Received: 5 December 2002, Accepted: 10 February 2003, Published online: 27 June 2003PACS: 66.20.F2, 62.40.+i, 80.40.cmA. Mielke: Research partially supported by DFG within the SFB 404 Multifield Problems     

Fast Decays of Strong Global Solutions of the Navier–Stokes Equations

In the paper we study the asymptotic dynamics of strong global solutions of the Navier Stokes equations. We are concerned with the question whether or not a strong global solution w can pass through arbitrarily large fast decays. Avoiding results on higher regularity of w used in other papers we prove as the main result that for the case of homogeneous Navier–Stokes equations the answer is negative: If [0, 1/4) and δ₀ > 0, then the quotient remains bounded for all t ≥ 0 and δ∈[0, δ₀]. This result is not valid for the non-homogeneous case. We present an example of a strong global solution w of the non-homogeneous Navier–Stokes equations, where the exterior force f decreases very quickly to zero for while w passes infinitely often through stages of arbitrarily large fast decays. Nevertheless, we show that for the non-homogeneous case arbitrarily large fast decays (measured in the norm cannot occur at the time t in which the norm is greater than a given positive number.      

Developable Surfaces as Generators of the “Isobaric Solutions” to the Euler Equations

The simplest solutions to the Euler equations (1.1) for which the pressure vanishes identically are those representing the motion of lines parallel to a fixed direction moving in the same direction (each line with an independent, given, constant velocity). Are there many other solutions to this problem? If yes, is there a simple characterization of all the initial data (volume occupied by the fluid at time t = 0 and initial velocity that gives rise to the general solutions? In this paper we show that the answer to both questions is positive. We prove, in particular, that there is a natural correspondence between solutions in R² of this problem and (Cartesian pieces of) developable surfaces in R³. See Theorem 3.     

Boundary control of the generalized Korteweg–de Vries–Burgers equation

This paper considers the boundary control problem of the generalized Korteweg–de Vries–Burgers (GKdVB) equation on the interval [0, 1]. We derive a control law of the form and α is a positive integer, and prove that it guarantees L ²-global exponential stability, H ¹-global asymptotic stability, and H ¹-semiglobal exponential stability. Numerical results supporting the analytical ones for both the controlled and uncontrolled equations are presented using a finite element method.     

A study of stationary crack-tip deformation fields in thin sheets by computer vision

The in-plane deformation fields near a stationary crack tip for thin, single edge-notched (SEN) specimens, made from Plexiglas, 3003 aluminum alloy and 304 stainless steel, have been successfully obtained by using computer vision. Results from the study indicate that (a) in-plane deformations ranging from elastic to fully plastic can be obtained accurately by the method, (b) for U, and , the size of the HRR dominant zone is much smaller than forV and , respectively. Since these results are in agreement with recent analytical work, suggesting that higher order terms will be needed to accurately predict trends in the data, it is clear that the region where the first term in the asymptotic solution is dominant is dependent on the component of the deformation field being studied, (c) the HRR solution can be used to quantity only in regions where theplastic strains strongly dominate the elastic strain components (i.e., when ); forV, the HRR zone appears to extend somewhat beyond this region, (d) the displacement componentU does not have the HRR singularity anywhere within the measurement region for either 3003 aluminum or 304 SS. However, the displacement componentV agrees with the HRR slope up to the plastic-zone boundary in 3003 aluminum ( ) and over most of the region where measurements were obtained ( ) in 304 SS and (e) the effects of end conditions must be included in any finite-element model of typical SEN specimen geometries to accurately calculate theJ integral and the crack-tip fields.Paper was presented at the 1992 SEM Spring Conference on Experimental Mechanics held in Las Vegas, NV on June 8–11.     

On the Rank 1 Convexity of Stored Energy Functions of Physically Linear Stress-Strain Relations

The rank 1 convexity of stored energy functions corresponding to isotropic and physically linear elastic constitutive relations formulated in terms of generalized stress and strain measures [Hill, R.: J. Mech. Phys. Solids 16, 229–242 (1968)] is analyzed. This class of elastic materials contains as special cases the stress-strain relationships based on Seth strain measures [Seth, B.: Generalized strain measure with application to physical problems. In: Reiner, M., Abir, D. (eds.) Second-order Effects in Elasticity, Plasticity, and Fluid Dynamics, pp. 162–172. Pergamon, Oxford, New York (1964)] such as the St.Venant–Kirchhoff law or the Hencky law. The stored energy function of such materials has the form

On the Strong Ellipticity of the Anisotropic Linearly Elastic Materials

In this paper we derive necessary and sufficient conditions for strong ellipticity in several classes of anisotropic linearly elastic materials. Our results cover all classes in the rhombic system (nine elasticities), four classes of the tetragonal system (six elasticities) and all classes in the cubic system (three elasticities). As a special case we recover necessary and sufficient conditions for strong ellipticity in transversely isotropic materials. The central result shows that for the rhombic system strong ellipticity restricts some appropriate combinations of elasticities to take values inside a domain whose boundary is the third order algebraic surface defined by x ²+y ²+z ²−2xyz−1=0 situated in the cube , , . For more symmetric situations, the general analysis restricts combinations of elasticities to range inside either a plane domain (for four classes in the tetragonal system) or in an one-dimensional interval (for the hexagonal systems, transverse isotropy and cubic system). The proof involves only the basic statement of the strong ellipticity condition.      

The structure and evolution of a two-dimensional H2/O2/Ar cellular detonation

This paper reports on two-dimensional numerical simulation of cellular detonation wave in a / / mixture with low initial pressure using a detailed chemical reaction model and high order WENO scheme. Before the final equilibrium structure is produced, a fairly regular but still non-equilibrium mode is observed during the early stage of structure formation process. The numerically tracked detonation cells show that the cell size always adapts to the channel height such that the cell ratio is fairly independent of the grid sizes and initial and boundary conditions. During the structural evolution in a detonation cell, even as the simulated detonation wave characteristics suggest the presence of an ordinary detonation, the evolving instantaneous detonation state indicates a mainly underdriven state. As a considerable region of the gas mixture in a cell is observed to be ignited by the incident wave and transverse wave, it is further suggested that these two said waves play an essential role in the detonation propagation.Received: 16 September 2003, Accepted: 14 June 2004, Published online: 20 August 2004[/PUBLISHED]PACS: 47.40.-x, 82.40.Fp, 82.33.Vx, 83.85.PtX.Y. Hu: Correspondence to Current address: Institut für Strömungsmechanik, Technische Universität Dresden, 01062 Dresden, Deutschland     

Random Kick-Forced 3D Navier–Stokes Equations in a Thin Domain

We consider the Navier–Stokes equations in the thin 3D domain , where is a two-dimensional torus. The equation is perturbed by a non-degenerate random kick force. We establish that, firstly, when ε ≪ 1, the equation has a unique stationary measure and, secondly, after averaging in the thin direction this measure converges (as ε → 0) to a unique stationary measure for the Navier–Stokes equation on . Thus, the 2D Navier–Stokes equations on surfaces describe asymptotic in time, and limiting in ε, statistical properties of 3D solutions in thin 3D domains.     

Burnett equations for the ellipsoidal statistical BGK model

In order to discuss the agreement of the ellipsoidal statistical BGK (ES-BGK) model with the Boltzmann equation, Burnett equations are computed by means of the second-order Chapman-Enskog expansion of the ES-BGK model. It is found that the Burnett equations for the ES-BGK model with the correct Prandtl number are identical to the Burnett equations for the Boltzmann equation for Maxwell molecules (fifth-order power potentials). However, for other types of particle interaction, the Boltzmann Burnett equations cannot be reproduced from the ES-BGK model.Furthermore, the linear stability of the ES-BGK Burnett equations is discussed. It is shown that the ES-BGK Burnett equations are linearly stable for Prandtl numbers of and for , while they are linearly unstable for and .Received: 29 April 2003, Accepted: 20 June 2003PACS: 510.10.-y, 47.45.-n Correspondence to: Y. Zheng     

Reduced and Generalized Stokes Resolvent Equations in Asymptotically Flat Layers, Part II: H ∞-Calculus

We study the generalized Stokes equations in asymptotically flat layers, which can be considered as compact perturbations of an infinite (flat) layer Besides standard non-slip boundary conditions, we consider a mixture of slip and non-slip boundary conditions on the upper and lower boundary, respectively. In this second part, we use pseudodifferential operator techniques to construct a parametrix to the reduced Stokes equations, which solves the system in L^q-Sobolev spaces, 1 < q < , modulo terms which get arbitrary small for large resolvent parameters . This parametrix can be analyzed to prove the existence of a bounded H-calculus of the (reduced) Stokes operator.     

Asymptotic analysis of linearly elastic shells. II. Justification of flexural shell equations

We consider as in Part I a family of linearly elastic shells of thickness 2?, all having the same middle surfaceS=?(?)?R ³, whereω?R ² is a bounded and connected open set with a Lipschitz-continuous boundary, and?∈l ³ (?;R ³). The shells are clamped on a portion of their lateral face, whose middle line is?(γ ₀), whereγ ₀ is any portion of?ω withlength γ ₀>0. We make an essential geometrical assumption on the middle surfaceS and on the setγ ₀, which states that the space of inextensional displacements $$\begin{gathered} V_F (\omega ) = \{ \eta = (\eta _i ) \in H^1 (\omega ) \times H^1 (\omega ) \times H^2 (\omega ); \hfill \\ \eta _i = \partial _v \eta _3 = 0 on \gamma _0 ,\gamma _{\alpha \beta } (\eta ) = 0 in \omega \} , \hfill \\ \end{gathered}$$ where $\gamma _{\alpha \beta }$ (η) are the components of the linearized change is metric tensor ofS, contains non-zero functions. This assumption is satisfied in particular ifS is a portion of cylinder and?(γ ₀) is contained in a generatrix ofS. We show that, if the applied body force density isO(? ²) with respect to?, the fieldu(?)=(u _i (?)), whereu _i (?) denote the three covariant components of the displacement of the points of the shell given by the equations of three-dimensional elasticity, once “scaled” so as to be defined over the fixed domain Ω=ω×]?1, 1[, converges as?→0 inH ¹(Ω) to a limitu, which is independent of the transverse variable. Furthermore, the averageζ=1/2ts _?1 ¹ u dx ₃, which belongs to the spaceV _F (ω), satisfies the (scaled) two-dimensional equations of a “flexural shell”, viz., $$\frac{1}{3}\mathop \smallint \limits_\omega a^{\alpha \beta \sigma \tau } \rho _{\sigma \tau } (\zeta )\rho _{\alpha \beta } (\eta )\sqrt {a } dy = \mathop \smallint \limits_\omega \left\{ {\mathop \smallint \limits_{ - 1}^1 f^i dx_3 } \right\} \eta _i \sqrt {a } dy$$ for allη=(η _i ) ∈V _F (ω), where $a^{\alpha \beta \sigma \tau }$ are the components of the two-dimensional elasticity tensor of the surfaceS, $$\begin{gathered} \rho _{\alpha \beta } (\eta ) = \partial _{\alpha \beta } \eta _3 - \Gamma _{\alpha \beta }^\sigma \partial _\sigma \eta _3 + b_\beta ^\sigma \left( {\partial _\alpha \eta _\sigma - \Gamma _{\alpha \sigma }^\tau \eta _\tau } \right) \hfill \\ + b_\alpha ^\sigma \left( {\partial _\beta \eta _\sigma - \Gamma _{\beta \sigma }^\tau \eta _\tau } \right) + b_\alpha ^\sigma {\text{|}}_\beta \eta _\sigma - c_{\alpha \beta } \eta _3 \hfill \\ \end{gathered} $$ are the components of the linearized change of curvature tensor ofS, $\Gamma _{\alpha \beta }^\sigma$ are the Christoffel symbols ofS, $b_\alpha ^\beta$ are the mixed components of the curvature tensor ofS, andf ⁱ are the scaled components of the applied body force. Under the above assumptions, the two-dimensional equations of a “flexural shell” are therefore justified.     

Asymptotic analysis of linearly elastic shells. III. Justification of Koiter's shell equations

We consider as in Parts I and II a family of linearly elastic shells of thickness 2?, all having the same middle surfaceS=?(?)?R ³, whereω?R ² is a bounded and connected open set with a Lipschitz-continuous boundary, and? ∈ ?³ (?;R ³). The shells are clamped on a portion of their lateral face, whose middle line is?(γ ₀), whereγ ₀ is a portion of?ω withlength γ ₀>0. For all?>0, let $\zeta _i^\varepsilon$ denote the covariant components of the displacement $u_i^\varepsilon g^{i,\varepsilon }$ of the points of the shell, obtained by solving the three-dimensional problem; let $\zeta _i^\varepsilon$ denote the covariant components of the displacement $\zeta _i^\varepsilon$ a ⁱ of the points of the middle surfaceS, obtained by solving the two-dimensional model ofW.T. Koiter, which consists in finding $$\zeta ^\varepsilon = \left( {\zeta _i^\varepsilon } \right) \in V_K (\omega ) = \left\{ {\eta = (\eta _\iota ) \in {\rm H}^1 (\omega ) \times H^1 (\omega ) \times H^2 (\omega ); \eta _i = \partial _v \eta _3 = 0 on \gamma _0 } \right\}$$ such that $$\begin{gathered} \varepsilon \mathop \smallint \limits_\omega a^{\alpha \beta \sigma \tau } \gamma _{\sigma \tau } (\zeta ^\varepsilon )\gamma _{\alpha \beta } (\eta )\sqrt a dy + \frac{{\varepsilon ^3 }}{3} \mathop \smallint \limits_\omega a^{\alpha \beta \sigma \tau } \rho _{\sigma \tau } (\zeta ^\varepsilon )\rho _{\alpha \beta } (\eta )\sqrt a dy \hfill \\ = \mathop \smallint \limits_\omega p^{i,\varepsilon } \eta _i \sqrt a dy for all \eta = (\eta _i ) \in V_K (\omega ), \hfill \\ \end{gathered}$$ where $a^{\alpha \beta \sigma \tau }$ are the components of the two-dimensional elasticity tensor ofS, $\gamma _{\alpha \beta }$ (η) and $\rho _{\alpha \beta }$ (η) are the components of the linearized change of metric and change of curvature tensors ofS, and $p^{i,\varepsilon }$ are the components of the resultant of the applied forces. Under the same assumptions as in Part I, we show that the fields $\frac{1}{{2_\varepsilon }}\smallint _{ - \varepsilon }^\varepsilon u_i^\varepsilon g^{i,\varepsilon } dx_3^\varepsilon$ and $\zeta _i^\varepsilon$ a ⁱ , both defined on the surfaceS, have the same principal part as? → 0, inH ¹ (ω) for the tangential components, and inL ²(ω) for the normal component; under the same assumptions as in Part II, we show that the same fields again have the same principal part as? → 0, inH ¹ (ω) for all their components. For “membrane” and “flexural” shells, the two-dimensional model ofW.T. Koiter is therefore justified.