The peculiar rheo-optical behavior of bisphenol A-polycarbonate and polymethylmethacrylate

The rheological and stress-optical behavior of the melts of several grades ob bisphenol-A-polycarbonate (PC) and polymethylmethacrylate (PMMA) is investigated. Pertinent flow birefringence measurements are carried out in a remodelled cone-plate apparatus [1]. The shear stress in the polymer melt is calculated from the dynamic moduli, which are determined separately. It is shown that the linear stress optical rule is obeyed. In this way, the stress-optical coefficient C of the melt can be determined. The low-Mw polycarbonates all behave as Maxwellian fluids. The main stress direction does not deviate significantly from 45°. In the temperature range from 160° to 260°C the stress-optical coefficients of the different grades lie between 3 and 4×10^–9 Pa^–1 and show a weak temperature dependence. The stress-optical coefficient of PMMA is about a factor of 100 lower and shows a peculiar temperature-dependence, changing its sign at 144°C. The results are discussed in terms of the anisotropy of the polarizability of the polymer chain.     

Dynamic viscoelastic behaviour of wheat flour doughs

Summary The general dynamic mechanical properties of wheat flour doughs have been investigated for a range of strains over a frequency spectrum of 0.032–32 Hz. An approximately linear stress-strain relationship was found to hold at the lower range of strains imposed. Departures from the linear response of the dough are explained in terms of a protein chain-starch granule interaction and the accompanying time dependence in terms of a stress-biased diffusion process. The influence of water and temperature on the dynamic moduli are shown, indicating that a frequency-temperature and a water absorption-frequency superposition principle is applicable. The transient stress relaxation modulus is calculated from the dynamic moduli to demonstrate the ability of dynamic methods to investigate the time scale inaccessible to transient techniques.     

Dynamic stress-strain characteristics of metals at elevated temperatures

A method for measuring the load, strain and strain rate inside a high-temperature environment is described. Results of tests on specimens of high-purity copper are reported. Results indicate that copper is strain-rate dependent within the temperature range 78°–1000° F up to strains of 0.6 percent and strain rates of 10³ sec^–1.Paper was presented at 1969 SESA Spring Meeting held in Philadelphia, Pa., on May 13–16.This work was sponsored by the Army Research Office (Durham) under Contract No. DA 31-124-ARO-D229.     

A non-linear uniaxial integral constitutive equation incorporating rate effects,creep and relaxation

A previously proposed first order non-linear differential equation for uniaxial viscoplasticity, which is non-linear in stress and strain but linear in stress and strain rates, is transformed into an equivalent integral equation. The proposed equation employs total strain only and is symmetric with respect to the origin and applies for tension and compression. The limiting behavior for large strains and large times for monotonic, creep and relaxation loading is investigated and appropriate limits are obtained. When the equation is specialized to an overstress model it is qualitatively shown to reproduce key features of viscoplastic behavior. These include: initial linear elastic or linear viscoelastic response: immediate elastic slope for a large instantaneous change in strain rate normal strain rate sensitivity and non-linear spacing of the stress-strain curves obtained at various strain rates; and primary and secondary creep and relaxation such that the creep (relaxation) curves do not cross. Isochronous creep curves are also considered. Other specializations yield wavy stress-strain curves and inverse strain rate sensitivity. For cyclic loading the model must be modified to account for history dependence in the sense of plasticity.     

The Relaxation of Two-well Energies with Possibly Unequal Moduli

The elastic energy of a multiphase solid is a function of its microstructure. Determining the infimum of the energy of such a solid and characterizing the associated optimal microstructures is an important problem that arises in the modeling of the shape memory effect, microstructure evolution, and optimal design. Mathematically, the problem is to determine the relaxation under fixed phase fraction of a multiwell energy. This paper addresses two such problems in the geometrically linear setting. First, in two dimensions, we compute the relaxation under fixed phase fraction for a two-well elastic energy with arbitrary elastic moduli and transformation strains, and provide a characterization of the optimal microstructures and the associated strain. Second, in three dimensions, we compute the relaxation under fixed phase fraction for a two-well elastic energy when either (1) both elastic moduli are isotropic, or (2) the elastic moduli are well ordered and the smaller elastic modulus is isotropic. In both cases we impose no restrictions on the transformation strains. We provide a characterization of the optimal microstructures and the associated strain. We also compute a lower bound that is optimal except possibly in one regime when either (1) both elastic moduli are cubic, or (2) the elastic moduli are well ordered and the smaller elastic modulus is cubic; for moduli with arbitrary symmetry we obtain a lower bound that is sometimes optimal. In all these cases we impose no restrictions on the transformation strains and whenever the bound is optimal we provide a characterization of the optimal microstructures and the associated strain. In both two and three dimensions the quasiconvex envelope of the energy can be obtained by minimizing over the phase fraction. We also characterize optimal microstructures under applied stress.     

Second strain gradient elasticity of nano-objects

Mindlin's second strain gradient continuum theory for isotropic linear elastic materials is used to model two different kinds of size-dependent surface effects observed in the mechanical behaviour of nano-objects. First, the existence of an initial higher order stress represented by Mindlin's cohesion parameter, b₀, makes it possible to account for the relaxation behaviour of traction-free surfaces. Second, the higher order elastic moduli, c_i, coupling the strain tensor and its second gradient are shown to significantly affect the apparent elastic properties of nano-beams and nano-films under uni-axial loading. These two effects are independent from each other and allow for separated identification of the corresponding material parameters. Analytical results are provided for the size-dependent apparent shear modulus of a nano-thin strip under shear. Finite element simulations are then performed to derive the dependence of the apparent Young modulus and Poisson ratio of nano-films with respect to their thickness, and to illustrate hole free surface relaxation in a periodic nano-porous material.     

Constitutive relationships for polymeric materials with power-law distributions of relaxation times

The linear relaxation modulus of polydisperse polymer melts and solutions can often be approximated by a power law,ct ^–m over some range of time,t. If, in addition, the nonlinear rheology is given by a separable integral equation, with a strain-dependent factor typical of those observed experimentally, then some commonly observed empirical rules and equations can be readily derived as approximations, namely the Cox-Merz relationship between complex viscosity and steady-state shear viscosity, Bersted's predictions of steady shear stress and first normal-stress difference from a truncated spectrum of linear relaxation times, and the observation of Koyama and coworkers that the ratio of the nonlinear to the linear time-dependent elongational viscosity is independent of strain rate, over a range of strain rates outside the linear regime.     

Experimental investigation of cyclic and time-dependent deformation of titanium alloy at elevated temperature

A novel cyclic deformation test program was undertaken to characterize macroscopic time dependent deformation of a titanium alloy for use in viscoplastic model development. All tests were conducted at a high homologous temperature, 650 °C, where there are large time dependent and loading rate dependent effects. Uninterrupted constant amplitude tests having zero mean stress or a tensile mean stress were conducted using three different control modes: strain amplitude and strain rate, stress amplitude and stress rate, and a hybrid stress amplitude and strain rate. Strain ratcheting occurred for all cyclic tests having a tensile mean stress and no plastic shakedown was observed. The shape of the strain ratcheting curve as a function of time is analogous to a creep curve having primary, steady state and tertiary regions, but the magnitude of the ratchet strains are higher than creep strains would be for a constant stress equal to the mean stress. Strain cycles interrupted with up to eight 2-h stress relaxation periods around the hysteresis loop, including hold times in each quadrant of the stress–strain diagram, were also conducted. Stress relaxation was path-dependent and in some cases the stress relaxed to zero. The cyclic behavior of these interrupted tests was similar even though each cycle was very complex. These results support constitutive model development by providing exploratory, characterization and validation data.     

High-temperature pressure-shear plate impact experiments using pure tungsten carbide impactors

The pressure-shear plate impact experiment has been modified to test materials at high temperatures (up to 700°C). Together with the high strain rates characteristic of this experiment (10⁶ s^–1), the high-temperature capability allows the shearing resistance of materials to be measured under conditions unattainable with other testing equipment. The compressive and shear responses of pure tungsten carbide at different temperatures are presented, as well as the results of one test on OFHC copper at a temperature of 691°C and a shear strain rate of 1.4×10⁶ s^–1.     

Noncontacting strain measurements during tensile tests

The application of an innovative noncontracting Doppler laser extensometer is presented. True axial strain has been measured during tensile tests conducted on stainless-steel metal sheets over a range of strain rates (from 10^–4 to 10² 1/s) and temperatures (from –40°C to 400°C). The laser radiation scattered at the surface of the specimen is recorded during the duration of the experiment. The signals are then used to determine the evolution of the axial strain, which is subsequently combined with the load signal to construct the stress-strain curve for the material. Excellent agreement has been obtained between the total elongation predicted by the laser measurements and the actual values measured from the specimens. This technique offers several advantages over traditional strain-measuring technologies.     

Micromechanics-based elastic damage modeling of particulate composites with weakened interfaces

A micromechanical framework is proposed to predict the effective elastic behavior and weakened interface evolution of particulate composites. The Eshelby’s tensor for an ellipsoidal inclusion with slightly weakened interface [Qu, J., 1993a. Eshelby tensor for an elastic inclusion with slightly weakened interfaces. Journal of Applied Mechanics 60 (4), 1048–1050; Qu, J., 1993b. The effect of slightly weakened interfaces on the overall elastic properties of composite materials. Mechanics of Materials 14, 269–281] is adopted to model spherical particles having imperfect interfaces in the composites and is incorporated into the micromechanical framework. Based on the Eshelby’s micromechanics, the effective elastic moduli of three-phase particulate composites are derived. A damage model is subsequently considered in accordance with the Weibull’s probabilistic function to characterize the varying probability of evolution of weakened interface between the inclusion and the matrix. The proposed micromechanical elastic damage model is applied to the uniaxial, biaxial and triaxial tensile loadings to predict the various stress–strain responses. Comparisons between the present predictions with other numerical and analytical predictions and available experimental data are conducted to assess the potential of the present framework.     

Non-linear Viscoelastic Model for Behaviour Characterization of Temporomandibular Joint Discs

Generally, the complex behaviour of the disc of the temporomandibular joint (TMJ) cannot be adequately represented using linear elastic or linear viscoelastic models. Since the disc is regularly subjected to large strain and stress levels, the study of its non-linear response under compression is of practical interest, especially for analysis of medical dysfunctions. With this aim, relaxation and creep tests were carried out using round specimens of diameters ranging between 4 and 6 mm cut off from the central, anterior, posterior, lateral and medial zones of porcine discs to investigate the regional mechanical properties differences. The experimental data results are fitted using Prony series, based on generalized Maxwell and Kelvin models, allowing the relaxation and creep moduli to be represented, respectively, as a function of the strain and stress. The results show that the non-linear material behaviour of this biological tissue is properly described by the proposed models, to be considered subsequently in numerical calculations.     

Rheological properties of skim milk gels at various temperatures; interrelation between the dynamic moduli and the relaxation modulus

The rheological properties of rennet-induced skim milk gels were determined by two methods, i.e., via stress relaxation and dynamic tests. The stress relaxation modulusG _c (t) was calculated from the dynamic moduliG andG by using a simple approximation formula and by means of a more complex procedure, via calculation of the relaxation spectrum. Either calculation method gave the same results forG _c (t). The magnitude of the relaxation modulus obtained from the stress relaxation experiments was 10% to 20% lower than that calculated from the dynamic tests.Rennet-induced skim milk gels did not show an equilibrium modulus. An increase in temperature in the range from 20° to 35 °C resulted in lower moduli at a given time scale and faster relaxation. Dynamic measurements were also performed on acid-induced skim milk gels at various temperatures andG _c (t) was calculated. The moduli of the acid-induced gels were higher than those of the rennet-induced gels and a kind of permanent network seemed to exist, also at higher temperatures. G storage shear modulus,N·m^–2; - G loss shear modulus,N·m^–2; - G _c calculated storage shear modulus,N·m^–2; - G _c calculated loss shear modulus,N·m^–2; - G _e equilibrium shear modulus,N·m^–2; - G _ec calculated equilibrium shear modulus,N·m^–2; - G(t) relaxation shear modulus,N·m^–2; - G _c (t) calculated relaxation shear modulus,N·m^–2; - G ^*(t) pseudo relaxation shear modulus,N·m^–2; - H relaxation spectrum,N·m^–2; - t time,s; - relaxation time,s; - angular frequency, rad·s^–1. Partly presented at the Conference on Rheology of Food, Pharmaceutical and Biological Materials, Warwick, UK, September 13–15, 1989 [33].     

Die Dehnviskosität des Polypropylens

Zusammenfassung Es wurden die Dehnviskositäten von Polypropylen-Schmelzen bei 180°C im Dehngeschwindigkeitsbereich 0,02 bis 1,0 s^–1 gemessen. Dabei wurden Proben von drei verschiedenen Herstellern und mit sehr verschiedenen Molmassen untersucht. Sämtliche Proben zeigten einen monotonen Anstieg der Dehnviskosität mit der Zeit. Die durch Extrapolation gewonnenen Gleichgewichts-Dehnviskositäten betrugen in allen Fällen das Dreifache der Null-Scherviskositäten. In keinem einzigen Fall konnte eine Dehnverfestigung der Schmelzen beobachtet werden. Nur bei unvollständig aufgeschmolzenen Proben (168°C Meßtemperatur) ergab sich der triviale Fall einer Dehnverfestigung als Folge der Dehnkristallisation. Kontrollmessungen an verzweigtem Polyethylen (bei 150°C) ergaben die erwartete Dehnverfestigung, während diese, ebenfalls erwartet, bei linearem Polyethylen ausblieb.

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The elongational viscosities of polypropylene melts were measured at 180°C with different elongation rates between 0.02 and 1.0 s^–1 respectively. Samples were studied from three different manufacturers and some having markedly different molecular masses. All samples showed a monotonic rise of the elongational viscosity with time. The extrapolated steady-state elongational viscosities were always three times the values of the corresponding zero-shear viscosities. In no case strain hardening could be observed. Only incompletely molten samples (temperature of measurement 168°C) showed a kind of strain hardening as a consequence of strain crystallization. Control measurements with branced polyethylene at 150°C showed the expected strain hardening while this, also expected, was not observed in the case of linear polyethylene.

Herrn Professor Dr. Rolf Sammet zum 65. Geburtstag gewidmet     

Propagation of a soliton along a fluid-filled pipe

In [1, 2] a mathematical model of the motion of a fluid in a pipe whose axis is a curve in space was discussed and certain simplifications of the problem were studied. The propagation of linear and nonlinear waves in the framework of the model was studied. In the present paper we consider a simple wave flow in a pipe with elastic walls suing one of the models introduced in [1], which, unlike [2], takes into account axial displacements of the pipe. The basic equations describing the propagation of waves in the pipe are obtained.Translated from Zhurnal Prikladnoi Mekhaniki i Technicheskoi Fiziki, No. 3, pp. 58–63, May–June, 1986.     

Bending of Composite Plates under Static and Dynamic Loading

The stationary bending of a two-dimensional elastic system of joined rectangular plates with different mechanical properties and boundary conditions is studied. A technique for solving the corresponding problem is proposed. It is based on approximate approaches in combination with a generalization of the force method. It is established that the static strain state of the system is essentially dependent on the difference between the elastic moduli of the constituent plates. It is also shown that with certain boundary conditions, the dependence of the first resonant frequency on the relative position of plates along the contact line is nonmonotonic__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 1, pp. 77–84, January 2005.     

A thermoviscoelastic model for amorphous shape memory polymers: Incorporating structural and stress relaxation

A thermoviscoelastic constitutive model is developed for amorphous shape memory polymers (SMP) based on the hypothesis that structural and stress relaxation are the primary molecular mechanisms of the shape memory effect and its time-dependence. This work represents a new and fundamentally different approach to modeling amorphous SMPs. A principal feature of the constitutive model is the incorporation of the nonlinear Adam–Gibbs model of structural relaxation and a modified Eyring model of viscous flow into a continuum finite–deformation thermoviscoelastic framework. Comparisons with experiments show that the model can reproduce the strain–temperature response, the temperature and strain-rate dependent stress–strain response, and important features of the temperature dependence of the shape memory response. Because the model includes structural relaxation, the shape memory response also exhibits a dependence on the cooling and heating rates.     

Elasticity and dynamics of particle gels in non-Newtonian melts

We investigate the relation between the structure and the viscoelastic behavior of a model polymer nanocomposite system based on a mixture of titanium dioxide (TiO₂) nanoparticles and polypropylene. Above a critical volume fraction, Φ _c, the elasticity of the hybrids dramatically increases, and the frequency dependence of the elastic and viscous moduli reflects the superposition of the independent responses of the suspending polymer melt and of an elastic particle network. In addition, the elasticity of the hybrids shows critical behavior around Φ _c. We interpret these observations by hypothesizing the formation of a transient network, which forms due to crowding of particle clusters. Consistent with this interpretation, we find a long-time, Φ-dependent, structural relaxation, which emphasizes the transient character of the structure formed by the particle clusters. For times below this characteristic relaxation time, the elasticity of the network is Φ-independent and reminiscent of glassy behavior, with the elastic modulus, G^′, scaling with frequency, ω, as G^′∼ω ^0.3. We expect that our analysis will be useful for understanding the behavior of other complex fluids where the elasticity of the components could be superimposed.