Effects of material anisotropy and inhomogeneity on cavitation for composite incompressible anisotropic nonlinearly elastic spheres

The effects of material anisotropy and inhomogeneity on void nucleation and growth in incompressible anisotropic nonlinearly elastic solids are examined. A bifurcation problem is considered for a composite sphere composed of two arbitrary homogeneous incompressible nonlinearly elastic materials which are transversely isotropic about the radial direction, and perfectly bonded across a spherical interface. Under a uniform radial tensile dead-load, a branch of radially symmetric configurations involving a traction-free internal cavity bifurcates from the undeformed configuration at sufficiently large loads. Several types of bifurcation are found to occur. Explicit conditions determining the type of bifurcation are established for the general transversely isotropic composite sphere. In particular, if each phase is described by an explicit material model which may be viewed as a generalization of the classic neo-Hookean model to anisotropic materials, phenomena which were not observed for the homogeneous anisotropic sphere nor for the composite neo-Hookean sphere may occur. The stress distribution as well as the possible role of cavitation in preventing interface debonding are also examined for the general composite sphere.     

Simple shear in nonlinear Cosserat elasticity: bifurcation and induced microstructure

We discuss the simple shear problem for a geometrically exact Cosserat model. In contrast to linear Cosserat elasticity, where the unique solution is available in closed form we exhibit a multitude of solutions to the nonlinear problem, even if the two fields of deformations φ and microrotations remain homogeneous. This motivates a search for new conditions on the microrotations which single out a unique, physically reasonable, response. The influence of material parameters, notably the Cosserat couple modulus μ _c and the internal length scale L _c on the response is also studied. For small Cosserat couple modulus μ _c  > 0 we observe a pitchfork bifurcation of the homogeneous response and for vanishing internal length L _c  = 0 and zero Cosserat couple modulus μ _c  = 0 the Cosserat model may show highly oscillating “microstructure” solutions which are energetically better than the homogeneous response. Thus, the large scale nonlinear Cosserat limit is not necessarily a classical limit.      

Localized Thickening of a Compressed Elastic Band

An infinite elastic band is compressed along its unbounded direction, giving rise to a continuous family of homogeneous configurations that is parameterized by the compression rate β < 1 (β = 1 when there is no compression). It is assumed that, for some critical value β ₀, the compression force as a function of β has a strict local extremum and that the linearized equation around the corresponding homogeneous configuration is strongly elliptic. Under these conditions, there are nearby localized deformations that are asymptotically homogeneous. When the compression force reaches a strict local maximum at β ₀, they describe localized thickening and they occur for values of β slightly smaller than β ₀. Since the material is supposed to be hyperelastic, homogeneous and isotropic, the localized deformations are not due to localized imperfections. The method follows the one developed by A. Mielke for an elastic band under traction: interpretation of the nonlinear elliptic system as an infinite dimensional dynamical system in which the unbounded direction plays the role of time, its reduction to a center manifold and the existence of a homoclinic solution to the reduced finite dimensional problem in [A. Mielke, Hamiltonian and Lagrangian fiows on center manifolds, Lecture Notes in Mathematics 1489. Springer, Berlin Heidelberg New York, 1991]. The main difference lies in the fact that Agmon's condition does not hold anymore and therefore the linearized problem cannot be analyzed as in Mielke's work.     

Equivariant bifurcation theory and symmetry breaking

A study is made of the failure of the Maximal Isotropy Subgroup Conjecture for the Weyl group seriesW(D) _k . As part of the investigation, a general genericity and stability theorem is proved for bifurcation diagrams in equivariant bifurcation theory. As well, a concept of determinacy for equivariant bifurcation theory is introduced and it is shown that, for all compact Lie groupsG and absolutely irreducibleG-representationsV, G-equivariant bifurcation problems onV are finitely determined.     

Homogeneous Equilibrium Configurations of a Hyperelastic Compressible Cube under Equitriaxial Dead-Load Tractions

Non-uniqueness, bifurcation and stability of homogeneous solutions to the equilibrium problem of a hyperelastic cube subject to equitriaxial dead-load tractions are investigated. Besides the basic and theoretical questions raised by the analysis, the study is motivated by the somewhat surprising feature of this nonlinear problem for which the symmetric load may give rise to asymmetric stable deformations. In reality, the equilibrium problem, formulated for general homogeneous compressible isotropic materials with polyconvex energy function, may exhibit primary and secondary bifurcations. A primary bifurcation occurs when there exist paths of equilibrium states that bifurcate from the primary path of three equal principal stretches. These bifurcation branches have two coinciding stretches and along them, through secondary bifurcations, other completely asymmetric bifurcation branches, which are characterized by all three stretches different, may risen. In this case, the cube transforms into an oblique parallelepiped. With increasing loads, they are also possible discontinuous paths of equilibria which evince prompt jumps in the deformation process. Of course, the set of asymmetric solutions admitted by the equilibrium problem depends on the specific form of the stored energy function adopted. In this paper, expressions governing the global development of asymmetric equilibrium branches are derived. In particular, conditions to have bifurcation points are individualized. For compressible neo-Hookean and Mooney-Rivlin materials a wide parametric analysis is carried out showing by means of graphs the most interesting branches. Finally, using the energy criterion, a detailed study is performed to assess the stability of the computed solutions.      

Asymptotic Resonance, Interaction of Modes and Subharmonic Bifurcation

We study the existence of small amplitude oscillations near elliptic equilibria of autonomous systems, which mix different normal modes. The reference problem is the Fermi-Pasta-Ulam β-model: a chain of nonlinear oscillators with nearest-neighborhood interaction. We develop a new bifurcation approach that locates secondary bifurcations from the unimodal primary branches. Two sufficient conditions for bifurcation are given: one involves only the arithmetic properties of the eigenvalues of the linearized system (asymptotic resonance), while the other takes into account the nonlinear character of the interaction between normal modes (nonlinear coupling). Both conditions are checked for the Fermi-Pasta-Ulam problem.     

Transition bifurcation branches in non-linear water waves

We are concerned with the numerical computation of progressive free surface gravity waves on a horizontal bed. They are regarded as families of bifurcation branches (λ,A)_Q of constant discharge Q. Numerically we determine two transition values Q₁ and Q₂ with corresponding transition bifurcation branches that classify waves into three disjoint branch sets B₁, B₂ and B₃. Their members are families of waves (λ,A)_Q satisfying the conditions 0<Q² ?Q, Q <Q² ?Q and Q <Q² <B/27, respectively. The bifurcation patterns are analysed in some detail from the computed bifurcation diagram, which shows that in B₁ bifurcation is to the left and the amplitude A increases as the wavelength λ decreases; in B₂ bifurcation is to the right and turning points are observed nearly at breaking point. In B₃ bifurcation is to the right and A increases monotonically with λ.     

Hopf bifurcation and heteroclinic orbit in a 3D autonomous chaotic system

In this paper we revisit a 3D autonomous chaotic system, which contains both the modified Lorenz system and the conjugate Chen system, presented in [Huang and Yang, Chaos Solitons Fractals 39:567–578, 2009]. First by citing two examples to show the errors and limitations for the local stability of the equilibrium point S ₊ obtained in this literature, we formulate a complete determining criterion for the local stability of S ₊ of this system. Although the local bifurcation problem of this system, mainly for Hopf bifurcation, etc., has been studied, the invoking of incorrect proposition leads to an incorrect result for Hopf bifurcation. We then renew the study of the Hopf bifurcation of this system by utilizing the Project Method. The global bifurcation problem, relatively speaking, should be more difficult than the local bifurcation problem for a given system. However, the global bifurcation problem of this system, to the best of our knowledge, has not been investigated yet in the literatures. So next we consider the global bifurcation problem for this system, mainly for the existence of homoclinic and heteroclinic orbits. Our results, one of which shows the existence of two heteroclinic orbits, not only correct and further supplement the ones obtained in the literature, but also give something new to theoretically help fully understand the occurrence of chaos.     

Mechanical effects of the self-organization of dislocations and related critical characteristics

The effects of the dislocation pattern formed due to the self-organization of the dislocations in crystals on the macroscopic hardening and dynamic internal friction (DIF) during deformation are studied. The classic dislocation models for the hardening and DIF corresponding to the homogeneous dislocation configuration are extended to the case for the non-homogeneous one. In addition, using the result of dislocation patterning deduced from the non-linear dislocation dynamics model for single slip, the correlation between the dislocation pattern and hardening as well as DIF is obtained. It is shown that in the case of the tension with a constant strain rate, the bifurcation point of dislocation patterning corresponds to the turning point in the stress versus strain and DIF versus strain curves. This result along with the critical characteristics of the macroscopic behavior near the bifurcation point is microscopically and macroscopically in agreement with the experimental findings on mono-crystalline pure aluminum at temperatures around 0.5T _m . The present study suggests that measuring the DIF would be a sensitive and useful mechanical means in order to study the critical phenomenon of materials during deformation. The project supported by the National Natural Science Foundation of China under the Grand 19702019 & 19891180-4 and by the Chinese Academy of Sciences under the Grand KJ951-1-201.     

A Collocation-Based Algorithm for Computing the Pull-In Range of a Class of Phase-Locked Loops

The pull-in range (ω_p) of a phase-locked loop (PLL) is defined as the maximum value of loop detuning ω_0s for which pull-in occurs from anywhere on the PLL's phase plane. That is, pull-in is guaranteed from anywhere on the phase plane if ω_0s < ω_p. Simple approximation is available for computing ω_p for the high gain PLL where saddle-node bifurcation occurs at ω_0s = ω_p. Unlike the high gain case, a simple approximation for ω_p is not available for the low gain case where bifurcation from a separatrix cycle occurs at ω_0s = ω_p. The vector field model for a class of second-order PLLs is shown to have rotational properties, which imply the existence of a separatrix cycle. The external stability of this separatrix cycle is an indicator of the type of bifurcation (saddle-node or separatrix cycle) which terminates the limit cycle associated with the PLL's stable false lock state and the PLL pulls-in (i.e. achieve phase lock). A formula is given for determining the separatrix cycle's stability, which indicates that these paratrix cycle is externally stable for small values of closed loop gain. A collocation-based algorithm is presented for computing the PLL's separatrix cycle and the value of pull-in range frequency ω_0s = ω_p at which a stable separatrix cycle exists.     

Buckling and postbuckling of concentric cylindrical tubes under external pressure

The problem considered involves a structure composed of two concentric and bonded tubes subjected to external and uniform pressure. Compression tests are conducted using structures formed by a thin-walled internal rubber tube and a thick-walled external foam tube. Experimental results are plotted under the form of a bifurcation diagram representing the inner cross-sectional area of the thin tube as a function of pressure. The buckling pressure P_b and the contact pressure P_co are determined from this non linear diagram. A numerical computation by the Finite Element Method (FEM) is used in order to calculate the Euler buckling pressure P_b and the results are compared with experimental data. It is shown that the buckling pressure and the associated buckling mode n, strongly depend upon the elastic and geometrical parameters of both the tubes. The experimental and numerical investigations are also extended to postbuckling behaviour. The contact between the opposite sides of the inner wall is occured with a buckling mode index n = 2, 3. This contact phenomenon is given rise to the discontinuity of a previous diagram and was characterized by the contact pressure P_co.     

Symmetry and secondary bifurcation of elastic structures

We consider non-linear bifurcation problems for elastic structures modeled by the operator equation F[w;α]=0 where F:X×R^k→Y,X,Y are Banach spaces and XY. We focus attention on problems whose bifurcation equations are of the form

f_i(α₁,α₂;λ,μ)=(a_iμ+b_iλ)α_i+p_iα_i³+q_iα_i∑_j=1,j≠i^kα_j+1²+α_ih_i(λ,μ;α₁,α₂,…α_k) i=1,2,…k

which emanates from bifurcation problems for which the linearization of F is Fredholm operators of index 0. Under the assumption of F being odd we prove an important theorem of existence of secondary bifurcation. Under this same assumption we prove a symmetry condition for the reduced equations and consequently we got an existence result for secondary bifurcation. We also include a stability analysis of the bifurcating solutions.     

Optimum lifting body shapes in hypersonic flow at high angles of attack

A variational procedure for the determination of lifting body configurations having a maximum lift-to-drag ratio K _max in hypersonic flight at high angles of attack , is proposed. It is based on an analytical solution to the problem for three-dimensional hypersonic flow over small aspect ratio wings using thin shock-layer theory. This reduces the variational problem of finding K _max, and the corresponding optimized wing shape, to the minimization of a linear functional subject to various constraints. The contributions of nonequilibrium thermochemical effects and laminar or turbulent viscous drag effects are also included in the problem formulation. The solution shows that optimized wings have an unbent forward part and a concave lower surface. Due to bifurcation in the optimization process, the planform may have either a sharp apex or a straight nose cut. Corresponding values of K _max() significantly exceed the limiting value K _N=cot for a flat wing. Real thermochemical effects and air viscosity are shown to cause a decrease in K _max and sometimes to influence the optimized wing geometry; however, the relative increment of K _max to K _N is still retained.     

Heat transfer enhancement in grooved channels due to flow bifurcations

The flow bifurcation scenario and heat transfer characteristics in grooved channels, are investigated by direct numerical simulations of the mass, momentum and energy equations, using the spectral element methods for increasing Reynolds numbers in the laminar and transitional regimes. The Eulerian flow characteristics show a transition scenario of two Hopf bifurcations when the flow evolves from a laminar to a time-dependent periodic and then to a quasi-periodic flow. The first Hopf bifurcation occurs to a critical Reynolds number Rec₁ that is significantly lower than the critical Reynolds number for a plane-channel flow. The periodic and quasi-periodic flows are characterized by fundamental frequencies ω₁ and m· ω₁+n·ω₂, respectively, with m and n integers. Friction factor and pumping power evaluations demonstrate that these parameters are much higher than the plane channel values. The time-average mean Nusselt number remains mostly constant in the laminar regime and continuously increases in the transitional regime. The rate of increase of this Nusselt number is higher for a quasi-periodic than for a periodic flow regime. This higher rate originates because better flow mixing develops in quasi-periodic flow regimes. The flow bifurcation scenario occurring in grooved channels is similar to the Ruelle-Takens-Newhouse transition scenario of Eulerian chaos, observed in symmetric and asymmetric wavy channels.     

Minimizing Sequences Selected via Singular Perturbations, and their Pattern Formation

We study the Cahn-Hilliard energy E _ɛ(u) over the unit square under the constraint of a constant mass m with (ɛ > 0) and without ɛ= 0) interfacial energy. Minimizers of E ₀(u) have no preferred pattern and we select patterns via sequences of conditionally critical points of E _ɛ(u) converging to minimizers as ɛ tends to zero. Those critical points are not minimizers if the singular limit has no minimal interface. We obtain them by a global bifurcation analysis of the Euler-Lagrange equations for E _ɛ(u) where the mass m is the bifurcation parameter. We make use of the symmetry of the unit square, and the elliptic maximum principle, in turn, implies that the location of maxima and minima is fixed for all solutions on global branches. This property is used to guarantee the existence of a singular limit and to verify the Weierstrass-Erdmann corner condition which proves its minimizing property. Accepted January 21, 2000?Published online November 24, 2000     

Coherent structures in countercurrent axisymmetric shear flows

The dynamical behaviors of coherent structures in countercurrent axisymmetric shear flows are experimentally studied.The forward velocity U1 and the velocity ratio R=(U1-U2)/(U1+U2),where U2 denotes the suction velocity,are consldered as the control parameters.Two kinds of vortex structures,i.e.,axisymmetric and helical structures,were discovered with respect to different reginmes in the R versus U1 diagram .In the case of U1 rangjing from 3 to 20m/s and R from 1 to 3,the axisymmetric structures plan an important role.Based on the dynamical behaviors of axisymmetric structures,a critical forward velocity U1cr=6.8m/s was defined,subsequently,the subcritical velocity regime:U1&gt;U1cr and the supercritical velocity regime:U1&lt;U1er,In the subcritical velocity regine,the flow system contains shear layer self-excited oscillations in a certain range of the velocity ratio with respect to any forward velocity.In the supercritical velocity regime,the effect of the velocity ratio could be explained by the relative movement and the spatial evolution of the axisymmetric structure undergoes the following stages:(1) Kelvin-Helmholtz instability leading to vortex rolling up,(2) first time vortex agglomeration.(3) jet colunn self-excited oscillation,(4) shear layer self-excited oscillation,(5)“ordered tearing“,(6) turbulence in the case of U1&lt;4m/s (the “ordered tearing“ does not exist when U1&gt;4m/s),correspondingly,the spatial evolution of the temporal asymptotic behavior of a dynamical system can be described as follows:(1) Hopt bifurcation,(5) chaos(“weak turbulence“)in the case of U1&lt;4m/s(superharmonic bifurcation does not exist when U1&gt;4m/s).The proposed new terms,superharmonic and reversed superbarmonic bifurcations,are characterized of the frequency doubling rather than the period doubling.A kind of unfamiliar vortices referred to as the helical structure was discovered experimentally when the forward velocity around 2m/s and the velocity range from 1.1 to 2.3,There are two base frequencies contained in the flow system and they could coexist as indicated by the Wigner-Ville-Distribution and the temporal asymptotic behavior of the dynamical system corresponding to the helical vortex could be described as 2-torus as indicted by the 3D reconstructed phase trajectory and correlation dimension.The scenario of the spatial evolution of helical structures could be described as follows:the jet column is separated into two parts at a certain spatial location and they entangle each other to form the helical vortex until the occurrence of those separated jet columns to reconnect further downstream with the result that the flow system evolves into turbulence in a catastrophic form.Correspondingly,the dynamical system evolves directly into 2-tiorus through the supercritical Hopf bifurcation followed by a transition from a quasi-periodic attractor to a strange attractor.In the case of U1=2m/s,the parametric evolution of the temporal asymptotic behavior of the dynamical system as the velocity ratio increases from 1 to 3 could be described as follows:(1)2-torus(Hopf bifurcation),(2) limit cycle(reversed Hopf bifurcation),(3) strange attractor (subbarmonic bifurcation).     

Wave Packets, Signaling and Resonances in a Homogeneous Waveguide

We solve the initial-boundary-value linear stability problem for small localised disturbances in a homogeneous elastic waveguide formally by applying a combined Laplace – Fourier transform. An asymptotic evaluation of the solution, expressed as an inverse Laplace – Fourier integral, is carried out by means of the mathematical formalism of absolute and convective instabilities. Wave packets, triggered by perturbations localised in space and finite in time, as well as responses to sources localised in space, with the time dependence satisfying e^−iωt + O(e^−ɛt ), for t → ∞, where Im ω₀ = 0 and ω > 0 , that is, the signaling problem, are treated. For this purpose, we analyse the dispersion relation of the problem analytically, and by solving numerically the eigenvalue stability problem. It is shown that due to double roots in a wavenumber k of the dispersion relation function D(k, ω), for real frequencies ω, that satisfy a collision criterion, wave packets with an algebraic temporal decay and signaling with an algebraic temporal growth, that is, temporal resonances, are present in a neutrally stable homogeneous waveguide. Moreover, for any admissible combination of the physical parameters, a homogeneous waveguide possesses a countable set of temporally resonant frequencies. Consequences of these results for modelling in seismology are discussed. This revised version was published online in August 2006 with corrections to the Cover Date.     

Cavitation for incompressible anisotropic nonlinearly elastic spheres

In this paper, the effect ofmaterial anisotropy on void nucleation and growth inincompressible nonlinearly elastic solids is examined. A bifurcation problem is considered for a solid sphere composed of an incompressible homogeneous nonlinearly elastic material which is transversely isotropic about the radial direction. Under a uniform radial tensile dead-load, a branch of radially symmetric configurations involving a traction-free internal cavity bifurcates from the undeformed configuration at sufficiently large loads. Closed form analytic solutions are obtained for a specific material model, which may be viewed as a generalization of the classic neo-Hookean model to anisotropic materials. In contrast to the situation for a neo-Hookean sphere, bifurcation here may occur locally either to the right (supercritical) or to the left (subcritical), depending on the degree of anisotropy. In the latter case, the cavity has finite radius on first appearance. Such a discontinuous change in stable equilibrium configurations is reminiscent of the snap-through buckling phenomenon of structural mechanics. Such dramatic cavitational instabilities were previously encountered by Antman and Negrón-Marrero [3] for anisotropiccompressible solids and by Horgan and Pence [17] forcomposite incompressible spheres.     

Vibro-impact dynamics near a strong resonance point

Two typical vibratory systems with impact are considered, one of which is a two-degree-of-freedom vibratory system impacting an unconstrained rigid body, the other impacting a rigid amplitude stop. Such models play an important role in the studies of dynamics of mechanical systems with repeated impacts. Two-parameter bifurcations of fixed points in the vibro-impact systems, associated with 1:4 strong resonance, are analyzed by using the center manifold and normal form method for maps. The single-impact periodic motion and Poincaré map of the vibro-impact systems are derived analytically. Stability and local bifurcations of a single-impact periodic motion are analyzed by using the Poincaré map. A center manifold theorem technique is applied to reduce the Poincaré map to a two-dimensional one, and the normal form map for 1:4 resonance is obtained. Local behavior of two vibro-impact systems, near the bifurcation points for 1:4 resonance, are studied. Near the bifurcation point for 1:4 strong resonance there exist a Neimark–Sacker bifurcation of period one single-impact motion and a tangent (fold) bifurcation of period 4 four-impact motion, etc. The results from simulation show some interesting features of dynamics of the vibro-impact systems: namely, the “heteroclinic” circle formed by coinciding stable and unstable separatrices of saddles, T _in, T _on and T _out type tangent (fold) bifurcations, quasi-periodic impact orbits associated with period four four-impact and period eight eight-impact motions, etc. Different routes of period 4 four-impact motion to chaos are obtained by numerical simulation, in which the vibro-impact systems exhibit very complicated quasi-periodic impact motions. The project supported by National Natural Science Foundation of China (50475109, 10572055), Natural Science Foundation of Gansu Province Government of China (3ZS061-A25-043(key item)). The English text was polished by Keren Wang.