Behavior of Piezoelectric Beams under Axially Non-uniform Distributed Loading

The paper presents a generic solution methodology for a quasi-static homogeneous monoclinic piezoelectric beam under axially distributed electric and mechanical surface loads and body forces expressed as polynomials of degree K≥ 0 of the axis variable. (In the absence of any electrical loading, this problem is known as the Almansi–Michell problem). The stress and the electrical displacement components are presented as a set of polynomials of degree ≤K+2 of the axis variable (“solution hypothesis”) containing 4K unknown tip loading constants and 3K stress functions of two variables. The cases K=0,1 stand for uniform or linear distributed loads in the axis direction. The analysis is initiated by the Kth level and continues down to lower levels. The main result of this work generalizes the “elastic” solution given recently by O. Rand and the first author (2005). Examples of solutions for axially uniform distributed loads (K=0), and equilibrium in which the stress and the electrical displacement do not depend on the axis variable, are presented. The applications to constant body loads and a hydrostatic pressure are considered.      

Small and large strain rheology of wheat gluten

 The rheological properties of wheat gluten were studied under both small and large deformation and compared with those of the parent flours. The limiting strain of linear viscoelastic behaviour of gluten doughs, 3 × 10⁻², was an order of magnitude larger than that of the flour doughs, 10⁻³. The role of starch in the lower limiting strain of flour doughs was indicated by the exponential decrease in the limiting strain of gluten-starch mixtures with greater quantities of starch. Large strain measurements showed gluten doughs possessed greater shear and elongational viscosities than flour doughs and these differences were greatest at lower shear and elongation rates (0.01 and 0.1 s⁻¹). The larger viscosities of flour and gluten doughs at the low strain rates help to stabilise and prevent the collapse of gas bubbles during bread fermentation and baking. Increasing starch levels in gluten-starch mixtures, at either constant or optimal water levels, lowered the elongational viscosity. Dynamic measurements were, however, more sensitive to the level of water added to the gluten-starch mixtures. The storage modulus decreased with increasing starch levels when constant water levels were used to prepare the mixtures, but when optimal water levels were used the storage modulus increased. Gluten and starch are major contributors to the large and small strain rheological properties of flour doughs; however, gluten-starch mixtures were unable to duplicate exactly the rheological properties of flour doughs, indicating that other flour components such as pentosans, lipids and water soluble proteins also influence dough rheology. Received: 20 March 2001 Accepted: 11 July 2001     

Experimental and conceptual problems in the rheological characterization of wheat flour doughs

Wheat flour dough is an industrially important material and a better understanding of its rheological behavior could have long ranging impact on the agricultural and the food processing industries. However, rheological characterization of dough is proving to be difficult due to a range of testing issues and anomalies in flow behavior. In a cone-and-plate rheometer wheat flour doughs “roll-out” of the gap before steady state viscosities can be established, as discussed by Bloksma and Nieman (1975). However, the mirror image of the transient viscosity-time plot obtained using a cone-and-plate viscometer has been used to obtain an estimate of steady shear viscosity behavior (Gleissle, 1975). To check this transient methodology for doughs, a second method, in addition to cone-and-plate transient flow, for determination of the shear viscosity, was needed. For this, capillary extrusion was chosen. Both a piston-driven and pressure driven capillary rheometer were employed. End corrections were determined to provide information on both the shear viscosity and, following Binding (1988), the extensional viscosity of the doughs. There are few data available on end corrections for doughs, though published data by Kieffer indicate that the corrections are unexpectedly very high. In this present work it was found that the end correction experiments were very difficult and imprecise in part due to the time-dependent nature of the doughs and difficulties in preparing replicate batches required to compare dies of differing L/R values. Further it was unexpectedly found that the samples, though prepared by normal mixing procedures to the “optimum” level, were so heterogeneous that large fluctuations in the pressure at constant output rate (in the piston-driven rheometer) and in output rate at constant pressure (in the pressure-driven instrument) were observed. These fluctuations could be eliminated by overmixing of the doughs, but overmixed doughs are of little practical interest. Although the problems encountered in this work were significant, it was encouraging that even these preliminary studies indicate that rheological measurements are effective in differentiating between spring and winter wheats. Defining a constitutive model for dough rheology still remains a major challenge, as results from one type of testing do not corroborate the findings from a different type of testing. Received: 19 May 1998 Accepted: 27 July 1998     

The investigation on dynamic fracture behavour of materials under compressive-shear combined stress waves

In this paper, an improved plate impact experimental technique is presented for studying dynamic fracture mechanism of materials, under the conditions that the impacting loading is provided by a single pulse and the loading time is in the sub-microsecond range. The impacting tests are carried out on the pressure-shear gas gun. The loading rate achieved is dK/dt∼10⁸ MPa m^1/2s⁻¹. With the elimination of influence of the specimen boundary, the plane strain state of a semi-infinite crack in an infinite elastic plate is used to simulate the deformation fields of crack tip. The single pulses are obtained by using the “momentum trap” technique. Therefore, the one-time actions of the single pulse are achieved by eradicating the stress waves reflected from the specimen boundary or diffracted from the crack surfaces. In the current study, some important phenomena have been observed. The special loading of the single pulse can bring about material damage around crack tip, and affect the material behavior, such as kinking and branching of the crack propagation. Failure mode transitions from mode I to mode II crack are observed under asymmetrical impact conditions. The mechanisms of the dynamic crack propagation are consistent with the damage failure model. The project supported by the National Natural Science Foundation of China (No. 19672066 and 18981180-4) and the Key Project of Chinese Academy of Sciences (No. KJ951-1-20)     

Dynamic flight stability of hovering insects

The equations of motion of an insect with flapping wings are derived and then simplified to that of a flying body using the “rigid body” assumption. On the basis of the simplified equations of motion, the longitudinal dynamic flight stability of four insects (hoverfly, cranefly, dronefly and hawkmoth) in hovering flight is studied (the mass of the insects ranging from 11 to 1,648 mg and wingbeat frequency from 26 to 157 Hz). The method of computational fluid dynamics is used to compute the aerodynamic derivatives and the techniques of eigenvalue and eigenvector analysis are used to solve the equations of motion. The validity of the “rigid body” assumption is tested and how differences in size and wing kinematics influence the applicability of the “rigid body” assumption is investigated. The primary findings are: (1) For insects considered in the present study and those with relatively high wingbeat frequency (hoverfly, drone fly and bumblebee), the “rigid body” assumption is reasonable, and for those with relatively low wingbeat frequency (cranefly and howkmoth), the applicability of the “rigid body” assumption is questionable. (2) The same three natural modes of motion as those reported recently for a bumblebee are identified, i.e., one unstable oscillatory mode, one stable fast subsidence mode and one stable slow subsidence mode. (3) Approximate analytical expressions of the eigenvalues, which give physical insight into the genesis of the natural modes of motion, are derived. The expressions identify the speed derivative M _u (pitching moment produced by unit horizontal speed) as the primary source of the unstable oscillatory mode and the stable fast subsidence mode and Z _w (vertical force produced by unit vertical speed) as the primary source of the stable slow subsidence mode. The project supported by the National Natural Science Foundation of China (10232010 and 10472008).     

Eshelby Tensor of a Polygonal Inclusion and its Special Properties

Nozaki and Taya (ASME J. Appl. Mech. 64 (1997) 495–502) analyzed the elastic field in a convex polygonal inclusion in an infinite body. By numerical analysis, they found that, when the shape of the inclusion is a regular polygon, “the strain at the center of inclusion” and “the strain energy per unit volume of inclusion” have strange and remarkable properties: these values are the same as those of a circular inclusion and are invariant for inclusion's orientation if the shape of the inclusion is not a square. In this paper, we first derive a simple, exact expression of the Eshelby tensor for an arbitrary polygonal inclusion. Using the expression, we then show a mathematical explanation why these special properties appear. This revised version was published online in July 2006 with corrections to the Cover Date.     

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.     

Discontinuous Solutions of the Navier-Stokes Equations for Multidimensional Flows of Heat-Conducting Fluids

We prove the global existence of weak solutions of the Navier-Stokes equations for compressible, heat-conducting fluids in two and three space dimensions when the initial density is close to a constant in L ²∩L ^∞, the initial temperature is close to a constant in L ², and the initial velocity is small in H ^s ∩L ⁴, where s=0 when n=2 and when n=3. (The L ^p norms must be weighted slightly when n=2.) In particular, the initial data may be discontinuous across a hypersurface of ⁿ . A great deal of qualitative information about the solution is obtained. For example, we show that the velocity, vorticity, and temperature are relatively smooth in positive time, as is the “effective viscous flux”F, which is the divergence of the velocity minus a certain multiple of the pressure. We find that F plays a central role in the entire analysis, particularly in closing the required energy estimates and in understanding rates of regularization near the initial layer. Moreover, F is precisely the quantity through which the hyperbolicity of the corresponding equations for inviscid fluids shows itself, an effect which is crucial for obtaining time-independent pointwise bounds for the density. (Accepted June 13, 1996)     

Numerical study of transient laminar natural convection heat transfer over a sphere subjected to a constant heat flux

Transient laminar natural convection over a sphere which is subjected to a constant heat flux has been studied numerically for high Grashof numbers (10⁵ ≤ Gr ≤ 10⁹) and a wide range of Prandtl numbers (Pr = 0.02, 0.7, 7, and 100). A plume with a mushroom-shaped cap forms above the sphere and drifts upward continuously with time. The size and the level of temperature of the transient cap and plume stem decrease with increasing Gr and Pr. Flow separation and an associated vortex may appear in the wake of the sphere depending on the magnitude of Gr and Pr. A recirculation vortex which appears and grows until “steady state” is attained was found only for the very high Grashof numbers (10⁵ ≤ Gr ≤ 10⁹) and the lowest Prandtl number considered (Pr = 0.02). The appearance and subsequent disappearance of a vortex was observed for Gr = 10⁹ and Pr = 0.7. Over the lower hemisphere, the thickness of both the hydrodynamic (δ_H) and the thermal (δ_T) boundary layers remain nearly constant and the sphere surface is nearly isothermal. The surface temperature presents a local maximum in the wake of the sphere whenever a vortex is established in the wake of the sphere. The surface pressure recovery in the wake of the sphere increases with decreasing Pr and with increasing Gr. For very small Pr, unlike forced convection, the ratio δ_T/δ_H remains close to unity. The results are in good agreement with experimental data and in excellent agreement with numerical results available in the literature. A correlation has also been presented for the overall Nusselt number as a function of Gr and Pr.     

Extensional-flow-induced crystallization of isotactic polypropylene

A filament stretching extensional rheometer with a custom-built oven was used to investigate the effect of uniaxial flow on the crystallization of polypropylene. Prior to stretching, samples were heated to a temperature well above the melt temperature to erase their thermal and mechanical histories and the Janeschitz-Kriegl protocol was applied. The samples were stretched at extension rates in the range of 0.01 s^-1 ￡ [(e)\dot] ￡ 0.75 s^-10.01\,\mbox{s}^{-1}\le \dot{{\varepsilon }}\le 0.75\,{\rm s}^{-1} to a final strain of ε = 3.0. After stretching, the samples were allowed to crystallize isothermally. Differential scanning calorimetry was applied to the crystallized samples to measure the degree of crystallinity. The results showed that a minimum extension rate is required for an increase in percent crystallization to occur and that there is an extension rate for which percent crystallization is maximized. No increase in crystallization was observed for extension rates below a critical extension rate corresponding to a Weissenberg number of approximately Wi = 1. Below this Weissenberg number, the flow is not strong enough to align the contour path of the polymer chains within the melt and as a result there is no change in the final percent crystallization from the quiescent state. Beyond this critical extension rate, the percent crystallization was observed to increase to a maximum, which was 18% greater than the quiescent case, before decaying again at higher extension rates. The increase in crystallinity is likely due to flow-induced orientation and alignment of contour path of the polymer chains in the flow direction. Polarized light microscopy verified an increase in number of spherulites and a decrease in spherulite size with increasing extension rate. In addition, small angle X-ray scattering showed a 7% decrease in inter-lamellar spacing at the transition to flow-induced crystallization. Although an increase in strain resulted in a slight increase in percent crystallization, no significant trends were observed. Crystallization kinetics were examined as a function of extension rate by observing the time required for molten samples to crystallize under uniaxial flow. The crystallization time was defined as the time at which a sudden increase in the transient force measurement was observed. The crystallization time was found to decrease as one over the extension rate, even for extension rates where no increase in percent crystallization was observed. As a result, the onset of extensional-flow-induced crystallization was found to occur at a constant value of strain equal to ε _c  = 5.8.     

Rheo-optical investigations of lyotropic mesophases of polymeric surfactants

The shear orientation of hexagonal and lamellar liquid crystalline phases of polymeric surfactants was investigated by rheo-optical techniques (flow birefringence (Δn), small-angle light scattering) as well as by nuclear magnetic resonance and optical microscopy. The evolution of birefringence in the hexagonal phase is discussed for simple and oscillatory shear, and an alignment of rodlike micelles along the flow direction was found. A shear induced formation of vesicles (“onions”) is observed with the lamellar phase. They displayed a characteristic four-lobe pattern in depolarized light scattering. Above a critical shear stress vesicles were degraded and perpendicularly aligned lamellae (i.e. with their normal along the vorticity direction) were obtained. A comparison of experiments performed at constant stress and constant rate revealed that the vesicle to planar lamellae transition occurred above a critical shear stress. The behavior of the polysoap lyotropic mesophases under shear, i.e. the strain dependent alignment in the hexagonal phase, the shear induced formation of vesicles, and a transition to planar lamellae in the lamellar phase, is very similar to the behavior of lyotropic mesophases formed by low molar mass surfactants or amphiphilic block copolymers. The geometrical constraints that are introduced when amphiphilic side groups are fixed to a polymer backbone do not significantly alter the response of the mesophase to a shear deformation. Received: 4 May 1999 /Accepted: 19 July 1999     

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.     

Pointwise Estimates and Stability for Dispersive–Diffusive Shock Waves

We study the stability and pointwise behavior of perturbed viscous shock waves for a general scalar conservation law with constant diffusion and dispersion. Along with the usual Lax shocks, such equations are known to admit undercompressive shocks. We unify the treatment of these two cases by introducing a new wave-tracking method based on “instantaneous projection”, giving improved estimates even in the Lax case. Another important feature connected with the introduction of dispersion is the treatment of a non-sectorial operator. An immediate consequence of our pointwise estimates is a simple spectral criterion for stability in all L ^p norms, p≥ 1 for the Lax case and p > 1 for the undercompressive case. Our approach extends immediately to the case of certain scalar equations of higher order, and would also appear suitable for extension to systems. Accepted May 29, 2000?Published online November 16, 2000     

Free volume dependence of polymer viscosity

The Simha–Somcynsky (S–S) equation of state (eos) was used to compute the free volume parameter, h, from the pressure–volume–temperature (PVT) dependencies of eight molten polymers. The predicted by eos variation of h with T and P was confirmed by the positron annihilation lifetime spectroscopy; good agreement was found for h(P = constant) = h(T) as well as for h(T = constant) = h(P). Capillary shear viscosity (η) data of the same polymers (measured at three temperatures and six pressures up to 700 bars), were plotted as logη vs 1/h, the latter computed for T and P at which η was measured. In previous works, such a plot for solvents and silicone oils resulted in a “master curve” for the liquid, in a wide range of T and P. However, for molten polymers, no superposition of data onto a “master curve” could be found. The superposition could be obtained allowing the characteristic pressure reducing parameter, P*, to vary. The necessity for using a “rheological” characteristic pressure reducing parameter, P*_R = κP*, with κ = 1 to 2.1 indicates that the free volume parameter extracted from the thermodynamic equilibrium data may not fully describe the dynamic behavior. After eliminating possibility of other sources for the deviation, the most likely culprit seems to be the presence of structures in polymer melts at temperatures above the glass transition, T _g. For example, it was observed that for amorphous polymers at T ≅ 1.52T _g the factor κ = 1, and the deviation vanish.     

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”.      

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.     

Shear-induced long-range alignment of BCC-ordered block copolymers

Effect of large shear on an asymmetric block copolymer with nanospherical domains has been studied using rheology and small angle X-ray scattering. The material investigated was a triblock copolymer poly[styrene-b-(ethylene-co-butylene)-b-styrene] swollen in a midblock-selective solvent. When cooled below the order–disorder transition temperature (T _ODT), the system forms a locally ordered structure of grains with body-centered cubic (BCC) lattice. Isothermal shearing, either at constant rate or with large amplitude oscillatory shear (LAOS) at low frequencies and strain amplitude greater than or equal to 2.0, leads to the destruction of the BCC lattice (isothermal “shear melting”). Upon cessation of the shear, the BCC structure recovers with kinetics similar to the one after thermal quench from above T _ODT. Under certain experimental conditions, LAOS leads to alignment of the BCC lattice. The lattice orientation depends primarily on shearing frequency. At low frequencies, there exists an upper and lower bound on strain amplitude where monodomain textures can be obtained. Upon alignment, the modulus drops by about 30% of that of the polycrystalline structure. Measurement of rheological properties offers an indirect method for distinguishing between polycrystalline structure (grains) and monodomain texture.     

An Exact Derivation from Three-Dimensions of the Linear Equations for the Bending of an Elastically Monoclinic Plate and Associated Three-Dimensional Solutions

The exact linear three-dimensional equations for a elastically monoclinic (13 constant) plate of constant thickness are reduced without approximation to a single 4th order differential equation for a thickness-weighted normal displacement plus two auxiliary equations for weighted thickness integrals of a stress function and the normal strain. The 4th order equation is of the same form as in classical (Kirchhoff) theory except the unknown is not the midsurface normal displacement. Assuming a solution of these plate equations, we construct so-called modified Saint-Venant solutions—“modified” because they involve non-zero body and surface loads. That is, solutions of the exact three-dimensional elasticity equations that exhibit no boundary layers and that are subject to a special set of body and surface loads that leave the analogous plate loads arbitrary.     

A Novel “Subset Splitting” Procedure for Digital Image Correlation on Discontinuous Displacement Fields

Digital Image Correlation (DIC) is an easy to use yet powerful approach to measure displacement and strain fields. While the method is robust and accurate for a variety of applications, standard DIC returns large error and poor correlation quality near displacement discontinuities such as cracks or shear bands. This occurs because the subsets used for correlation can only capture continuous deformations from the reference to the deformed image. As a result the regions around discontinuities are typically removed from the area of interest, before or after analysis. Here, a novel approach is proposed which enables the subset to split in two sections when a discontinuity is detected. This method enables the measurement of “displacement jumps”, and also of displacements and strains right by the discontinuity (for example a crack profile or residual strains in the wake). The method is validated on digitally created images based on mode I and mode II asymptotic displacement fields, for both sub-pixel and super-pixel crack opening displacements. Finally, an actual fracture experiment on a high density polyethylene (HDPE) specimen demonstrates the robustness of the method on actual images. Compared to other methods capable of handling discontinuities, this novel “subset-splitting” procedure offers the advantage of being a direct extension of the now popular standard DIC, and can therefore be implemented as an “upgrade” to that method.