Experiments on the postbuckling behavior of circular cylindrical shells under compression

Detailed experimental studies are performed on the postbuckling behavior of circular cylindrical shells under compression, by using polyester test cylinders with the geometric parameterZ ranging from 20 to 1000. In each case, variations of the equilibrium load, circumferential wave number and maximum inward and outward deflections, with applied edge shortenings, are clarified. Contour lines for typical postbuckling configurations are also shown. It is found that, as the cylinder is compressed beyond the primary buckling, secondary bucklings take place successively with diminishing wave numbers, and that postbuckling equilibrium loads become significantly lower than those at buckling asZ increases. Further, for short shells withZ≦100, the buckled waveforms are always symmetric with one-tier diamond buckles, while for longer shells, asymmetric postbuckling patterns with two tiers of buckles dominate.     

Non-piezoactivity in piezoacoustics: basic properties and topological features

The existence conditions of zero electric fields E and zero electric displacements D are studied for bulk acoustic waves in piezoelectric crystals. General equations are derived for lines of zero electric fields, E(m)=0, and for specific points m ₀ of vanishing electric displacements, D(m ₀)=0, on the unit sphere of propagation directions m ²=1. The obtained equations are solved for a series of examples of particular crystal symmetry. It is shown that the vectors D _α (m) being generally orthogonal to the wave normal m are characterized by definite orientational singularities in the vicinity of m ₀ and can be described by the Poincaré indices n=0, ±1 or ±2. The algebraic expressions for the indices n are found both for unrestricted anisotropy and for a series of particular cases.     

On the use of interaction tensors to describe and predict rod interactions in rod suspensions

Recently, Férec et al. (2009a) proposed a model for nondilute rod-like suspensions, where particle interactions are taking into account via a micromechanical approach. The derived governing equation used the well-known second- and fourth-order orientation tensors (a ₂ and a ₄ ) and novel second- and fourth-order interaction tensors (b ₂ and b ₄ ). To completely close the model, it is necessary to express a ₄ , b ₂ , and b ₄ in terms of a ₂ . This paper gives the general framework to elaborate these new relations. Firstly, approximations for b ₂ are developed based on linear combinations of a ₂ and a ₄ . Moreover, a new closure approximation is also derived for b ₄ , based on the orthotropic fitted closure approach. Unknown parameters are determined by a least-square fitting technique with assumed exact solutions constructed from the probability distribution function (PDF). As numerical solutions for the PDF are difficult to obtain given the nonlinearity of the problem, a combination of steady state solutions is used to generate PDF designed to cover uniformly the entire domain of possible orientations. All these proposed approximations are tested against the particle-based simulations in a variety of flow fields. Improvements of the different approximations are observed, and the couple iORW-CO4P3 gives efficient results.     

Finite Amplitude Transverse Waves in Special Incompressible Viscoelastic Solids

We consider the propagation of finite amplitude plane transverse waves in a class of homogeneous isotropic incompressible viscoelastic solids. It is assumed that the Cauchy stress may be written as the sum of an elastic part and a dissipative viscoelastic part. The elastic part is of the form of the stress corresponding to a Mooney–Rivlin material, whereas the dissipative part is a linear combination of A ₁, A ₁ ² and A ₂, where A ₁, A ₂ are the first and second Rivlin–Ericksen tensors. The body is first subject to a homogeneous static deformation. It is seen that two finite amplitude transverse plane waves may propagate in every direction in the deformed body. It is also seen that a finite amplitude circularly polarized wave may propagate along either n ⁺ or n ^?, where n ⁺, n ^? are the normals to the planes of the central circular section of the ellipsoid x?B ^?1 x=1. Here B is the left Cauchy–Green strain tensor corresponding to the finite static homogeneous deformation.     

Solution of three-dimensional problems for thick elastic shells by the method of reference surfaces

A new method for solving the elasticity problem for thick and thin shells is proposed. The method is based on the concept of reference surfaces inside the shell. According to this method, N reference surfaces are introduced in the body of the shell so that they are parallel to the midsurface and located at the Chebyshev polynomial nodes, which permits taking the displacement vectors u ¹, u ², …, u ^N of these surfaces for the desired functions. This choice of the desired functions allows one to represent the resolving equations of the proposed theory of higher-order shells in a sufficiently concise form and obtain deformation relations which permit describing the shell displacements as motions of a rigid body.     

Compatibility Equations of Nonlinear Elasticity for Non-Simply-Connected Bodies

Compatibility equations of elasticity are almost 150 years old. Interestingly, they do not seem to have been rigorously studied, to date, for non-simply-connected bodies. In this paper we derive necessary and sufficient compatibility equations of nonlinear elasticity for arbitrary non-simply-connected bodies when the ambient space is Euclidean. For a non-simply-connected body, a measure of strain may not be compatible, even if the standard compatibility equations (“bulk” compatibility equations) are satisfied. It turns out that there may be topological obstructions to compatibility; this paper aims to understand them for both deformation gradient F and the right Cauchy-Green strain C = F ^T F. We show that the necessary and sufficient conditions for compatibility of deformation gradient F are the vanishing of its exterior derivative and all its periods, that is, its integral over generators of the first homology group of the material manifold. We will show that not every non-null-homotopic path requires supplementary compatibility equations for F and linearized strain e. We then find both necessary and sufficient compatibility conditions for the right Cauchy-Green strain tensor C for arbitrary non-simply-connected bodies when the material and ambient space manifolds have the same dimensions. We discuss the well-known necessary compatibility equations in the linearized setting and the Cesàro-Volterra path integral. We then obtain the sufficient conditions of compatibility for the linearized strain when the body is not simply-connected. To summarize, the question of compatibility reduces to two issues: i) an integrability condition, which is d(F dX) = 0 for the deformation gradient and a curvature vanishing condition for C, and ii) a topological condition. For F dx this is a homological condition because the equation one is trying to solve takes the form dφ = F dX. For C, however, parallel transport is involved, which means that one needs to solve an equation of the form dR/ ds = RK, where R takes values in the orthogonal group. This is, therefore, a question about an orthogonal representation of the fundamental group, which, as the orthogonal group is not commutative, cannot, in general, be reduced to a homological question.     

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.     

Nonsymmetric buckle patterns in progressive plastic buckling

This paper presents the results of a series of experiments on the progressive plastic buckling of cylindrical shells under axial compressive load. It shows that, for shell bodies with anR/t less than 100, the normal axisymmetric ring buckling will develop into nonsymmetric patterns. We demonstrate that there exists also a class of shells within this thickness-radius range for which nonsymmetric plastic buckling always occurs without the prior formation of a ring. It appears from the limited number of tests made that, for a particularR, R/t, material and rate of loading, there is a critical value ofL, above which there is a high probability of the buckle pattern developing in a nonsymmetric fashion. It seems probable, too, that there are bands ofR/t for a particularL/R, R, material and rate of loading for which the buckle number will be constant. The experiments tend to indicate that the postbuckling efficiency of the shell decreases with increasing buckle number. The nonsymmetric patterns demonstrated appear to be inextensional deformations. They are very similar in character to the Yoshimura pattern which occurs as the limiting case for thin shells in axial compression and, under impact loading. Load-displacement histories are presented for some of the various modes of failure demonstrated.     

A thermodynamic approach to isotropic-nematic phase transitions in liquid crystals

We propose a dynamical model for (non-isothermal) phase transitions in liquid crystals. Macroscopic motions of the liquid crystal (LC) are neglected, while the coupling with the electromagnetic field is considered. The LC is described in terms of the classical order tensor Q, which is split as Q=s N, where N is a normalized tensor; two independent evolution laws are given for s and N. The model includes an evolutive equation for the temperature field obtained from an appropriate form of the energy balance, in which the internal powers associated to the equations for s and N are accounted for. The thermodynamic restrictions in the constitutive relations which ensure the Clausius–Duhem inequality have been pointed out.     

In the Chapman-Enskog Expansion¶the Burnett Coefficients Satisfy¶the Universal Relation ω3+ω4+θ3= 0

The Chapman–Enskog expansion when applied to a gas of spherical molecules yields formal expressions for the stress deviator P and energy-flux vector q, P=μP ⁽¹⁾+μ² P ⁽²⁾+…, q=μq ⁽¹⁾+μ² q ⁽²⁾+…. The Burnett terms P ⁽²⁾, q ⁽²⁾ depend on 11 coefficients ω _i , 1≦i≦6, θ&; _i , 1≦i≦ 5. This paper shows that ω₃+ω₄+θ₃= 0.     

Minimum tube diameters for steady propagation of gaseous detonations

Recent experimental results on detonation limits are reported in this paper. A parametric study was carried out to determine the minimum tube diameters for steady detonation propagation in five different hydrocarbon fuel–oxygen combustible mixtures and in five polycarbonate test tube diameters ranging from 50.8 mm down to a small scale of 1.5 mm. The wave propagation in the tube was monitored by optical fibers. By decreasing the initial pressure, hence the sensitivity of the mixture, the onset of limits is indicated by an abrupt drop in the steady detonation velocity after a short distance of travel. From the measured wave velocities inside the test tube, the critical pressure corresponding to the limit and the minimum tube diameters for the propagation of the detonation can be obtained. The present experimental results are in good agreement with previous studies and show that the measured minimum tube diameters can be reasonably estimated on the basis of the \(\lambda \) /3 rule over a wide range of conditions, where \(\lambda \) is the detonation cell size. These new data shall be useful for safety assessment in process industries and in developing and validating models for detonation limits.     

Regularity for Energy-Minimizing Area-Preserving Deformations

In this paper we establish the square integrability of the nonnegative hydrostatic pressure p, that emerges in the minimization problem $$\inf_{\mathcal{K}}\int_{\varOmega}|\nabla \textbf {v}|^2, \quad\varOmega\subset \mathbb {R}^2 $$ as the Lagrange multiplier corresponding to the incompressibility constraint det?v=1 a.e. in Ω. Our method employs the Euler-Lagrange equation for the mollified Cauchy stress C satisfied in the image domain Ω ^?=u(Ω). This allows to construct a convex function ψ, defined in the image domain, such that the measure of the normal mapping of ψ controls the L ² norm of the pressure. As a by-product we conclude that $\textbf {u}\in C^{\frac{1}{2}}_{\textrm {loc}}(\varOmega)$ if the dual pressure (introduced in Karakhanyan, Manuscr. Math. 138:463, 2012) is nonnegative.     

A Refinement of the Local Serrin-Type Regularity Criterion for a Suitable Weak Solution to the Navier–Stokes Equations

We formulate a new criterion for regularity of a suitable weak solution v to the Navier–Stokes equations at the space-time point (x ₀, t ₀). The criterion imposes a Serrin-type integrability condition on v only in a backward neighbourhood of (x ₀, t ₀), intersected with the exterior of a certain space-time paraboloid with vertex at point (x ₀, t ₀). We make no special assumptions on the solution in the interior of the paraboloid.     

Plane-Strain Problems for a Class of Gradient Elasticity Models��A Stress Function Approach

The plane strain problem is analyzed in detail for a class of isotropic, compressible, linearly elastic materials with a strain energy density function that depends on both the strain tensor ?? and its spatial gradient ???. The appropriate Airy stress-functions and double-stress-functions are identified and the corresponding boundary value problem is formulated. The problem of an annulus loaded by an internal and an external pressure is solved.     

On the effect of non-smooth Coulomb damping on flutter-type self-excitation in a non-gyroscopic circulatory 2-DoF-system

This article deals with self-excited vibrations, attractivity of stationary solutions, and the corresponding bifurcation behavior of two-dimensional differential inclusions of the type $\mathbf{M}\mathbf{q}'' + \mathbf{D}\mathbf{q}' + (\mathbf{K} + \bar{\mu}\mathbf{N})\mathbf{q} \in-\mathbf{R}\operatorname{Sign}(\mathbf{q}')$ . For the smooth case R=0, the equilibrium may become unstable due to non-conservative positional forces stemming from the circulatory matrix N. This type of instability is usually referred to as flutter instability and the loss of stability is related to a Hopf bifurcation of the steady state, which occurs for a critical parameter $\bar{\mu}= \bar{\mu}_{\mathrm{crit}}$ . For R≠0, the steady state is a set of equilibria, which turns out to be attractive for all values of the bifurcation parameter $\bar{\mu}$ . Depending on $\bar{\mu}$ , the basin of attraction of the equilibrium set can be infinite or finite. The transition from an infinite to a finite basin of attraction occurs at the stability threshold $\bar{\mu}_{\mathrm{crit}}$ of the underlying smooth problem. For the finite basin of attraction, its size is proportional to the Coulomb friction and inverse-proportional to $(\bar{\mu}- \bar{\mu}_{\mathrm{crit}})$ . By adding Coulomb damping the notion of steady state stability for the smooth problem is replaced by the question whether the basin of attraction of the steady state is infinite or finite. Simultaneously, the local Hopf-bifurcation is replaced by a global bifurcation. This implies that in the presence of Coulomb damping the occurrence of self-excited vibrations can only be investigated with regard to the perturbation level.     

Existence of Time-Periodic Solutions for a Magnetoelastic System in Bounded Domains

We investigate the existence of periodic solutions for a semilinear (nonlinearly coupled) magnetoelastic system in bounded, simply connected, three-dimensional domains with boundaries of class C ². The mathematical model includes a nonlinear mechanical dissipation like ρ(u′)=|u′| ^p u′ and a periodic forcing function of period T. We prove the existence of T-periodic weak solutions when p∈[3,4] (p=0 being a simpler case). In the corresponding two-dimensional case, the existence result holds under the assumption that p≥2.     

Parallel transport and defects on nematic shells

Nematic shells are thin films of nematic liquid crystal deposited on the boundary of colloidal particles, where liquid crystal molecules may freely glide, while remaining tangent to the surface substrate. The surface nematic order is described here by an appropriate tensor field Q, which vanishes wherever a defect occurs in the molecular order. We show how the classical concept of parallel transport on a manifold introduced by Levi-Civita can be adapted to this setting to define the topological charge m of a defect. We arrive at a simple formula to compute m from a generic representation of Q. In a number of separate appendices, we revisit in a unified language several, apparently disparate applications of Levi-Civita’s parallel transport.