Stress–strain behaviour of poly(ethylene terephthalate) (PET) during large plastic deformation by plane strain compression: the relation between stress–strain curve and thermal history, temperature and strain rate

This study presents an experimental investigation of the large plastic deformation of poly(ethylene terephthalate) (PET) submitted to plane strain compression. PET samples, obtained by injection moulding, annealed and non-annealed, were deformed using a specific compression device developed for this purpose. The obtained stress–strain curves at different temperatures and strain rates are useful for engineering applications and show a significant temperature dependence and a minor dependence on the strain rate. A softening temperature as a minimum temperature necessary to initiate deformation when a minimum, almost zero, stress is applied is introduced. This temperature, at the zero stress and strain limit, we denominate “Stress–Strain independent softening Temperature (T _SOF)”. The T _SOF values, 104 and 113°C for non-annealed and annealed PET, respectively, have been obtained using three different strain rates, indicating that the property is sensitive to the thermal history of the material.     

Extended Polar Decompositions for Plane Strain

In a finite deformation at a particle of a continuous body, a triad of infinitesimal material line elements is said to be “unsheared” when the angles between the three pairs of line elements of the triad suffer no change. In a previous paper, it has been shown that there is an infinity of unsheared oblique triads. With each oblique unsheared triad may be associated an “extended polar decomposition” F = QG = HQ of the deformation gradient F, in which Q is a rotation tensor, and G, H are not symmetric. Both G and H have the same real eigenvalues which are the stretches of the elements of the triad. In this paper, a detailed analysis of extended polar decompositions is presented in the case when the finite deformation is that of plane strain. Then, we may deal with a 2 × 2 deformation gradient F′ = Q′G′ = H′Q′ instead of the full 3 × 3 tensor F. In this case, the extended polar decompositions are associated with “unsheared pairs,” i.e., pairs of infinitesimal material line elements in the plane of strain which suffer no change in angle in the deformation. If one arm of an unsheared pair is chosen in the plane of strain, then, in general, its companion in the plane is determined. It follows that all possible extended polar decompositions may then be described in terms of a single parameter, the angle that the chosen arm makes with a coordinate axis in the plane. Explicit expressions for G′ and H′ are obtained, and various special cases are discussed. In particular, we note that the expressions for G′ and H′ remain valid even when the chosen arm is along a “limiting direction,” that is the direction of a line element which has no companion element in the plane forming an unsheared pair with it. The results are illustrated by considering the cases of simple shear and of pure shear.Dedicated to Professor Piero Villaggio as a symbol of our friendship and esteem.     

On the Implementation of the Eigenvalue Method for Limit Cycle Determination in Nonlinear Systems

In many practical systems, limit cycles can be predicted with suitable precision by frequency domain methods using describing functions. Within such an approach, limit cycles can be predicted using the “eigenvalue method” [Somieski, G., Nonlinear Dynamics 26(1), 2001, 3–22]. This contribution presents a novel and advantageous implementation of this method, using singular value instead of eigenvalue calculations, and enhancing computational efficiency by avoiding a so called “frequency iteration”. An erratum to this article is available at .     

Two-group kinetic theory of vehicular traffic

Summary We study a two-group model of vebicular traffic oli a highway which generalizes the Prigogine-Herman model. According to the two-group theory, drivers are divided into “slow” and “fast”. Correspondingly, we introduce a slow and a fast desired distribution representing the program that drivers of each of the two groups wish to follow. We first prove the existence and the uniqueness of a solution belonging to a suitable Banach space by using the theory of semigroups. Then, we indicate some sufficient conditions for the stability of an x and t independent solution.

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Sommario Si generalizza il modello di Prigogine-Herman del traffico automobilistico su un'autostrada, introducendo due gruppi di guidatori: i “lenti” ed i “veloci”. Ognuno dei due gruppi riene caratterizzato da un'opportuna funzione “desiderio”, che traduce i programmi che i guidatori intendono seguire nel corso del loro viaggio. Facendo poi uso della teoria dei semigruppi, si prova l'esistenza e l'unicità di una soluzione appartenente ad un opportuno spazio di Banach e si danno condizioni sufficienti per la stabilità.

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Work performed under the auspices of the National Research Council (C.N.R., Gruppo Nazionale per la Fisica Matematica).As the choice of the appropriate model for these problems is still a controversial matter, the Editorial Board welcomes any comment or discussion from the readers.     

THE PROBLEMS OF THE NONLINEAR UNSYMMETRICAL BENDING FOR CYLINDRICALLY ORTHOTROPIC CIRCULAR PLATE（II）

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Decomposition of the Deformations of a Thin Shell. Asymptotic Behavior of the Green-St Venant’s Strain Tensor

We investigate the behavior of the deformations of a thin shell, whose thickness δ tends to zero, through a decomposition technique of these deformations. The terms of the decomposition of a deformation v are estimated in terms of the L ²-norm of the distance from ∇ v to SO(3). This permits in particular to derive accurate nonlinear Korn’s inequalities for shells (or plates). Then we use this decomposition technique and estimates to give the asymptotic behavior of the Green-St Venant’s strain tensor when the “strain energy” is of order less than δ ^3/2.     

On the Time Decay of the Solutions of the Navier–Stokes System

We investigate the relationship between the time decay of the solutions u of the Navier–Stokes system on a bounded open subset of and the time decay of the right-hand sides f. In suitable function spaces, we prove that u always inherits at least part of the decay of f, up to exponential, and that the decay properties of u depend only upon the amount and type (e.g., exponential, or power-like) of decay of f. This is done by first making clear what is meant by “type” and “amount” of decay and by next elaborating upon recent abstract results pointing to the fact that, in linear and nonlinear PDEs, the decay of the solutions is often intimately related to the Fredholmness of the differential operator. This work was done while the second author was visiting the Bernoulli Center, EPFL, Switzerland, whose support is gratefully acknowledged.     

Exploration of the rotational power consumption of a rigid flapping wing

The development of Micro Air Vehicles with flapping wings is inspired from the observation and study of natural flyers such as insects and birds. This article explores the rotational power consumption of a flapping wing using a mechanical flapper at Re ≃ 4,500. This mechanical flapper is simplified to a 2D translation and a rotation in a water tank. Moreover, the wing kinematics are reduced to a linear translation and a rotation for the purpose of our study. We introduce the notion of non-ideal flapper and associated non-ideal rotational power. Such non-ideal devices are defined as consuming power for adding and removing mechanical power to and from the flow, respectively. First, we use a traditional symmetrical wing kinematic which is a simplified kinematic inspired from natural flyers. The lift coefficient of this flapping is about C _L ≃ 1.5. This symmetrical wing kinematic is chosen as a reference. Further, wing kinematics with asymmetric rotations are then compared with this one. These new kinematics are built using a differential velocity defined according to the translational kinematics, a time lag and a distance, r _kp. The analogy of this distance is discussed as a key point to follow along the chord. First, the wing kinematics are varied keeping a similar shape for the profiles of the angular velocity. It is shown that when compared to the reference wing kinematic, a 10% reduction in the rotational power is obtained whilst the lift is reduced by 9%. Second, we release the limitation to a similar shape for the profiles of the angular velocity leading to a novel shape for the angular velocity profile named here as “double bump” profile. With these new wing kinematics, we show that a 60% reduction in the non-ideal rotational power can be achieved whilst the lift coefficient is only reduced by 1.7%. Such “double bump kinematics” could then be of interest to increase the endurance of Micro Air Vehicles.     

Influence of elastic material compressibility on parameters of an expanding spherical stress wave

We investigated the influence of elastic material compressibility on parameters of an expanding spherical stress wave. The material compressibility is represented by Poisson’s ratio, ν, in this paper. The stress wave is generated by a pressure produced inside a spherical cavity surrounded by the isotropic elastic material. The analytical closed form formulae determining the dynamic state of the mechanical parameters (displacement, particle velocity, strains, stresses, and material density) in the material have been derived. These formulae were obtained for surge pressure p(t) = p ₀ = const inside the cavity. From analysis of these formulae, it is shown that the Poisson’s ratio substantially influences the course of material parameters in space and time. All parameters intensively decrease in space together with an increase of the Lagrangian coordinate, r. On the contrary, these parameters oscillate versus time around their static values. These oscillations decay in the course of time. We can mark out two ranges of parameter ν values in which vibrations of the parameters are “damped” at a different rate. Thus, Poisson’s ratio in the range below about 0.4 causes intense decay of parameter oscillations. On the other hand in the range 0.4 < ν < 0.5, i.e. in quasi-incompressible materials, the “damping” of parameter vibrations is very low. In the limiting case when ν = 0.5, i.e. in the incompressible material, “damping” vanishes, and the parameters harmonically oscillate around their static values. The abnormal behaviour of the material occurs in the range 0.4 < ν < 0.5. In this case, an insignificant increase of Poisson’s ratio causes a considerable increase of the parameter vibration amplitude and decrease of vibration “damping”.      

On Shearing, Stretching and Spin

An analysis is presented of stretching, shearing and spin of material line elements in a continuous medium. It is shown how to determine all pairs of material line elements at a point x, at time t, which instantaneously are not subject to shearing. For a given pair not subject to shearing, a formula is presented for the determination of a third material line element such that all three form a triad not subject to shearing, instantaneously. It is seen that there is an infinity of such triads not subject to shearing. A new decomposition of the velocity gradient L is introduced. In place of the classical decomposition of Cauchy and Stokes, L=d+w, where d is the stretching tensor and w is the spin tensor, the new decomposition is L=?+ℳ, where ?, called the ldquo;modified” stretching tensor, is not symmetric, and ℳ, called the “modified” spin tensor, is skew-symmetric – the tensor ? being chosen so that it has three linearly independent real right (and left) eigenvectors. The physical interpretation of this decomposition is that the material line elements along the three linearly independent right eigenvectors of ? instantaneously form a triad not subject to shearing. They spin as a rigid body with angular velocity μ (say) associated with ℳ. Also, for each decomposition L=?+ℳ, there is a decomposition L=? ^T +, where is also skew-symmetric. The triad of material line elements along the right eigenvectors of ? ^T (the set reciprocal to the right eigenvectors of ?) is also instantaneously not subject to shearing and rotates with angular velocity (say) associated with . It is seen that the vorticity vector ω is the mean of the two angular velocities μ and , ω =(μ+)/2. For irrotational motion, ω =0, so that μ=-; any triad of material line elements suffering no shearing rotates with angular velocity equal and opposite to that of the reciprocal triad of material line elements. It is proved that provided d is not spherical, there is an infinity of choices for ? and ℳ in the decomposition L=?+ℳ. Two special types of decompositions are introduced. The first type is called “CCS-decomposition” (where CCS is an abbreviation for Central Circular Section). It is associated with the infinite family of triads (not subject to shearing) with a common edge along the normal to one plane of central circular section of an ellipsoid ? associated with the stretching tensor, and the two other edges arbitrary in the other plane of central circular section of ?. There are two such CCS-decompositions. The second type is called “triangular decomposition”, because, in a rectangular cartesian coordinate system, ? has three off-diagonal zero elements. There are six such decompositions. Received 14 November 2000 and accepted 2 August 2001     

Visual observations of flow structure and melting front morphology in horizontal ice plate melting from above into a mixture

Visual observations reveal a complicated flow in the liquid melt and a melting front configuration resulting from horizontal ice plate melting from above into a 20 wt% calcium chloride aqueous solution. The initial temperature of the ice plate and the mixture are both −5°C. Small scale “mountain and valley” structures (∼1 mm) appear on the flat melting front just after melting begins, which have been called “sharkskin”. Innumerable upward and downward flows appear near the sharkskin and are controlled by its “mountain and valley” structure. These typical flows will considerably promote the melting of the ice plate to be 30% larger as compared to the numerically predicted results assuming a flat melting front (i.e., without the sharkskin), and also by three times larger compared with the results for melting from below.     

The “Arch of equilibration” of Charles Hutton (1772)

This paper is a historical re-visitation of the work of C. Hutton, a mathematician of the XVIII century, who wrote an innovative essay on the shape of arches, whose intrados he identifies with the curve of pressure, that he names “arch of equilibration”. His solution was mostly ignored by his contemporaries and after the middle of the XIX century forgotten. But Hutton’s problem is important, also because it offers the first example of an inverse problem in structural mechanics.     

Lubricated squeezing flow of thin slabs of wheat flour dough: comparison of results at constant plate speed and constant extension rates

Lubricated squeezing flow experiments on wheat flour dough have, until now, mostly been performed in constant plate speed mode (CPS), i.e. at a permanently increasing extension rate. We have compared the results obtained under the CPS and constant extension rate (CER) modes using one of the very few commercial rheometers that allow operation in the CER mode. In both cases, and at any constant biaxial strain, a power law could be fitted to the stress versus extension rate data, the “consistency index” (K) increasing continuously with the strain and the “flow behaviour index” (n) being constant only up to a low strain value (≈0.25) and then decreasing. When compared to the CER mode, the CPS mode produced higher K and n values. For wheat flour doughs, an increase in K with extension may be associated to a strain-hardening phenomenon but the roles of viscoelasticity and lubricant thinning are discussed.     

Smoothing Estimates for Boltzmann Equation with Full-Range Interactions: Spatially Homogeneous Case

In this work, we are concerned with the regularities of the solutions to the Boltzmann equation with physical collision kernels for the full range of intermolecular repulsive potentials, r ^−(p−1) with p > 2. We give new and constructive upper and lower bounds for the collision operator in terms of standard weighted fractional Sobolev norms. As an application, we get the global entropy dissipation estimate which is a little stronger than that described by Alexandre et al. (Arch Rational Mech Anal 152(4):327–355, 2000). As another application, we prove the smoothing effects for the strong solutions constructed by Desvillettes and Mouhot (Arch Rational Mech Anal 193(2):227–253, 2009) of the spatially homogeneous Boltzmann equation with “true” hard potential and “true” moderately soft potential.     

Vorticity and Regularity for Viscous Incompressible Flows under the Dirichlet Boundary Condition. Results and Related Open Problems

In reference [7] it is proved that the solution of the evolution Navier–Stokes equations in the whole of R ³ must be smooth if the direction of the vorticity is Lipschitz continuous with respect to the space variables. In reference [5] the authors improve the above result by showing that Lipschitz continuity may be replaced by 1/2-H?lder continuity. A central point in the proofs is to estimate the integral of the term (ω · ∇)u · ω, where u is the velocity and ω = ∇ × u is the vorticity. In reference [4] we extend the main estimates on the above integral term to solutions under the slip boundary condition in the half-space R ₊³. This allows an immediate extension to this problem of the 1/2-H?lder sufficient condition. The aim of these notes is to show that under the non-slip boundary condition the above integral term may be estimated as well in a similar, even simpler, way. Nevertheless, without further hypotheses, we are not able now to extend to the non slip (or adherence) boundary condition the 1/2-H?lder sufficient condition. This is not due to the “nonlinear" term (ω · ∇)u · ω but to a boundary integral which is due to the combination of viscosity and adherence to the boundary. On the other hand, by appealing to the properties of Green functions, we are able to consider here a regular, arbitrary open set Ω.      

An analogue rotated vector field of polynomial system

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Near wake properties of axisymmetric bluff body flows

Experiments have been made to measure some of the near wake properties of axisymmetric bluff body flows with fixed points of separation, including the detention or residence time of fluid borne scalar entities, base pressure coefficient, wake bubble length parameter and shape parameter. Measurements were made in smooth and turbulent air flow for Reynolds number in the range 2×10³<Re<4×10⁴. The results for a given bluff body were found to be uniquely controlled by a free-stream turbulence parameter. The data for all the shapes of bluff body in the class under consideration were found to collapse into unique inter-relationships by the introduction of the face pressure coefficient as a quantitative measure of “bluffness”. This paper was originally presented at the 14th International Congress of Theoretical and Applied Mechanics, Delft (August–September, 1976).     

A New Approach to the Existence of Weak Solutions of the Steady Navier–Stokes System with Inhomogeneous Boundary Data in Domains with Noncompact Boundaries

We prove the existence of a weak solution to the steady Navier–Stokes problem in a three dimensional domain Ω, whose boundary ∂Ω consists of M unbounded components Γ¹, . . . , Γ ^M and N − M bounded components Γ ^M+1, . . . , Γ ^N . We use the inhomogeneous Dirichlet boundary condition on ∂Ω. The prescribed velocity profile α on ∂Ω is assumed to have an L ³-extension to Ω with the gradient in L ²(Ω)^3×3. We assume that the fluxes of α through the bounded components Γ ^M+1, . . . , Γ ^N of ∂Ω are “sufficiently small”, but we impose no restriction on the size of fluxes through the unbounded components Γ¹, . . . , Γ ^M .     

Dual leading-edge vortex structure for flow over a simplified butterfly model

The dye visualization experiments show that a dual leading-edge vortex (LEV) structure exists on the suction side of a simplified butterfly model of Papilio ulysses at α = 8°−12°. Furthermore, the results of particle image velocimetry (PIV) measurement indicate that the axial velocity of the primary (outer) vortex core reaches the lower extreme value while a transition from a “wake-like” to a “jet-like” axial velocity profile occurs. The work reveals for the first time the existence of dual LEV structure on the butterfly-like forward-sweep wing configuration.     

Measure-theoretic foundations of statistical mechanics

The basic formulas of classical equilibrium statistical mechanics are derived from well-known theorems in measure theory and ergodic theory. The method used is a generalization of the methods of Khinchin and Grad and deals with several, in fact a “complete set”, of “invariants” or “integrals of the motion”. Most of the results are simple corollaries of Birkhoff's ergodic theorem, and since time-averages are used, the whole approach is characterized by an absence of statistical “ensembles” and probability notions. In the course of the development a “generalized temperature” is introduced, and a generalization of the second law of thermodynamics is derived. Formulas for the “microcanonical”, “canonical”, and “grand canonical” distributions appear as special cases of the general theory.