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”.      

A comparison of extensional viscosity measurements from various RME rheometers

The transient uniaxial extensional viscosity η _e of linear low density polyethylene (LLDPE) has been measured using the commercial Rheometric Scientific RME and the Münstedt Tensile Rheometer in an effort to compare the performance of available extensional rheometers. The RME indicated a significant strain hardening of the LLDPE, especially at a strain rate of 1 s⁻¹. In contrast, the Münstedt rheometer showed the LLDPE to be only slightly strain hardening. This artificial strain hardening effect in the RME resulted from the strain rate applied to the sample, determined from the sample deformation, being up to 20% less than the set strain rate. These results initiated a round-robin experiment in which the same LLDPE was tested on several RMEs in various locations around the world. All but one of the RMEs indicated a deviation between set and applied strain rates of at least 10%, especially at strain rates above 0.1 s⁻¹. The strain rate deviation was found to depend strongly on the value of the basis length L ₀ , and may result from the upper pair of belts not properly gripping the sample during extension. Thus visual inspection of the sample deformation is necessary to determine the applied strain rate. The most accurate measurements of η _e with respect to the strain rate deviation were obtained when the correct L ₀ value and belt arrangement were used. A list of recommendations for running an RME test is provided. Future work focusing on the fluid mechanics during the test may identify fully the cause of the strain rate deviation, but from a practical point of view the problem can be corrected for in the determination of η _e . Received: 27 September 2000/Accepted: 5 February 2001     

Elastic Fields for the Ellipsoidal Cavity Problem

Lur’e (Three-dimensional Problem of the Theory of Elasticity. Interscience, New York, 1964, §6.9) presented an approach to solve the problem of an ellipsoidal cavity in a linear, elastic and isotropic medium loaded by uniform principal stresses at infinity. In this paper we show that the approach by Lur’e may have no solution. Derivation mistakes are first pointed out in his (6.9.22), (6.9.23), (6.9.30) and (6.9.31). With the correct expressions, we then prove the coefficient matrix in his (6.9.32) to be singular. Therefore constants A,A ₄,A ₅ may have no solution. The problem lies in the harmonic functions chosen by Lur’e for the Papkovich-Neuber solution. From the solutions obtained by the Eshelby equivalent inclusion method, the present paper derives new Papkovich-Neuber harmonic functions for the ellipsoidal cavity problem.     

Pointwise and Other Decay Estimates for an Isotropic Elastic Strip

This paper is concerned with an homogeneous isotropic linear elastic strip, in plane strain. It is supposed that its lateral boundaries are displacement-free and that the deformation is generated by actions on the ends. A cross-sectional measure of deformation, complementing that of a previous paper, is defined and shown to satisfy a generalised convexity condition in the axial variable x ₁, for materials with negative Poisson's ratio σ. An enhanced measure is subsequently defined, and, in the case of a semi-infinite strip, is shown to yield pointwise exponential decay estimates for both the axial and the transverse displacement components for materials with positive Poisson's ratio. This revised version was published online in July 2006 with corrections to the Cover Date.     

Quasi-static cylindrical cavity expansion in an elastoplastic compressible Mises solid

The self-similar elastoplastic field induced by quasi-static expansion of a pressurized cylindrical cavity is investigated for Mises solids under the assumption of plane-strain. Material behavior is modeled by the elastoplastic J₂ flow theory with the standard hypoelastic version. The theory accounts for elastic-compressibility and allows for arbitrary strain-hardening (or softening) in the plastic range. A formulation of the exact governing equations is presented and analyzed in detail for the remote elastic field and for asymptotic plastic behavior near the cavity wall, along with numerical investigations for the entire deformation zone. An analytical solution was obtained under the axially-hydrostatic assumption (axial stress coincides with hydrostatic stress) within an error of about 2% or less as compared to the exact, numerically evaluated, value of cavitation pressure. Two ad-hoc compressibility approximations for cavitation pressure are suggested. These relations, which give very accurate results, appear to provide tight lower and upper bounds on the exact value of cavitation pressure within an error of less than 0.5%.     

Use of physical methods and electron microscopy in analyzing structural changes in Ni3Fe alloy

Results of studies of structural and mechanical properties ofNi ₃ Fe alloy are presented. The densities and types of dislocations in the alloy, the parameters of dislocation interaction, and the energy expended in deformation ofNi ₃ Fe specimens in various initial states are estimated. The strain hardening behavior of orderedNi ₃ Fe alloy is shown to be influenced by a number of external factors, such as deformation or γ-irradiation. Tomsk State Architectural University, Tomsk 634003. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 3, pp. 211–215, May–June, 1999.     

The Effects of Compressibility on Inhomogeneous Deformations for a Class of Almost Incompressible Isotropic Nonlinearly Elastic Materials

Experimental data for simple tension suggest that there is a power–law kinematic relationship between the stretches for large classes of slightly compressible (or almost incompressible) non-linearly elastic materials that are homogeneous and isotropic. Here we confine attention to a particular constitutive model for such materials that is of generalized Varga type. The corresponding incompressible model has been shown to be particularly tractable analytically. We examine the response of the slightly compressible material to some nonhomogeneous deformations and compare the results with those for the corresponding incompressible model. Thus the effects of slight compressibility for some basic nonhomogeneous deformations are explicitly assessed. The results are fundamental to the analytical modeling of almost incompressible hyperelastic materials and are of importance in the context of finite element methods where slight compressibility is usually introduced to avoid element locking due to the incompressibility constraint. It is also shown that even for slightly compressible materials, the volume change can be significant in certain situations.      

An adaptation of finite linear viscoelasticity theory for rubber-like viscoelasticity by use of a generalized strain measure

In this paper a simplified three-dimensional constitutive equation for viscoelastic rubber-like solids is derived by employing a generalized strain measure and an asymptotic expansion similar to that used by Coleman and Noll (1961) in their derivation of finite linear viscoelasticity (FLV) theory. The first term of the expansion represents exactly the time and strain separability relaxation behavior exhibited by certain soft polymers in the rubbery state and in the transition zone between the glassy and rubbery states. The relaxation spectra of such polymers are said to be deformation independent. Retention of higher order terms of the asymptotic expansion is recommended for treating deformation dependent spectra.Certain assumptions for the solid theory are relaxed in order to obtain a constitutive equation for uncross-linked liquid materials which exhibit large elastic recovery properties.Apart from the strain energyW(I₁,I ₂), which alternatively characterizes the long-time elastic response of solids or the instantaneous elastic response of elastic liquids, only the linear viscoelastic relaxation modulus is required for the first-order theory. Both types of material functions can be obtained, in theory, from simple laboratory testing procedures. The constitutive equations for solids proposed by Chang, Bloch and Tschoegl (1976) and a special form of K-BKZ theory for elastic liquids are shown to be particular cases of the first-order theory.Previously published experimental data on a cross-linked styrene-butadiene rubber (SBR) and an uncross-linked polyisobutylene (PIB) rubber is used to corroborate the theory.     

Material Property Differentiation in Indentation Testing Using Secondary Sensors

Current in vivo and in situ testing procedures are dominated by indentation. The major challenge for this testing technique is in finding a unique solution to the “inverse problem” i.e., defining an appropriate constitutive framework and obtaining material properties consistent with the indentation force–displacement data. Much of the information related to the interplay between shear and bulk compliance in the deformation field beneath the indenter is lost when capturing this single output. We propose a material testing method that follows the well proven path of conventional indentation methods, but enriches the signal by acquiring displacement data not only for the actuated indenter, but also for a set of offset, passive secondary sensors. We use finite element (FE) simulations involving three cases of materials: (a) linear elastic, (b) hyperelastic and (c) time-dependent to demonstrate the benefit of these additional sensors. The results indicate that the addition of these secondary sensors can help to discern between materials with varying degrees of compressibility.     

The Remarkable Nature of Radially Symmetric Deformation of Spherically Uniform Linear Anisotropic Elastic Solids

Antman and Negron-Marrero [1] have shown the remarkable nature of a sphere of nonlinear elastic material subjected to a uniform pressure at the surface of the sphere. When the applied pressure exceeds a critical value the stress at the center r=0 of the sphere is infinite. Instead of nonlinear elastic material, we consider in this paper a spherically uniform linear anisotropic elastic material. It means that the stress-strain law referred to a spherical coordinate system is the same for any material point. We show that the same remarkable nature appears here. What distinguishes the present case from that considered in [1] is that the existence of the infinite stress at r=0 is independent of the magnitude of the applied traction σ0 at the surface of the sphere. It depends only on one nondimensional material parameter κ. For a certain range of κ a cavitation (if σ0>0) or a blackhole (if σ0<0) occurs at the center of the sphere. What is more remarkable is that, even though the deformation is radially symmetric, the material at any point need not be transversely isotropic with the radial direction being the axis of symmetry as assumed in [1]. We show that the material can be triclinic, i.e., it need not possess a plane of material symmetry. Triclinic materials that have as few as two independent elastic constants are presented. Also presented are conditions for the materials that are capable of a radially symmetric deformation to possess one or more symmetry planes. This revised version was published online in August 2006 with corrections to the Cover Date.     

Nano-Indentation Hardness and Strain Hardening of Silicon,Sodium Chloride and Tungsten Crystals

Background

A previous review of micro- and nano-indentation hardness tests and their analyses gave emphasis to obtaining measurements of continuous nano-indentation load, P, versus, depth, h, recordings that monitor the full elastic–plastic deformation behavior of a localized crystal volume [1].

Objective

Attention is given to determining the complete, indentation-based, elastic–plastic deformation properties of the local volume, including the initial crystal elastic deformation behavior and, especially to evaluation of post pop-in plastic strain hardening.

Method

Stress–strain calculations are presented for an initial Hertzian elastic loading and follow-on crystal micro- and nano-scale plastic deformation responses [2].

Results

Applied load, P, dependencies on contact diameters, d_i, of silicon crystals are compiled on the basis of elastic, plastic and cracking predictions, giving indication at the lowest P values of an indentation size effect (ISE) for the crystal hardness. Elastic–plastic stress–strain curves are presented for sodium chloride and tungsten crystals. The hardness and strain hardening calculations also demonstrate an influence of the ISE.

Conclusions

The exceptional plastic strain hardening behaviors scale dimensionally with corresponding dislocation interactions and sessile reactions within the very localized plastic indentation zones. There is usefulness in determining elastic modulus values from the initial loading record. Micro- and nano-scale dislocation interactions/reactions account for the high stress and strain hardening levels as well as the occurrence of an ISE.

     

Measurement of length-scale and solution of cantilever beam in couple stress elasto-plasticity

Owing to the absence of proper analytical solution of cantilever beams for couple stress/strain gradient elasto-plastic theory, experimental studies of the cantilever beam in the micro-scale are not suitable for the determination of material length-scale. Based on the couple stress elasto-plasticity, an analytical solution of thin cantilever beams is firstly presented, and the solution can be regarded as an extension of the elastic and rigid-plastic solutions of pure bending beam. A comparison with numerical results shows that the current analytical solution is reliable for the case of σ0 〈〈 H 〈〈 E, where σ0 is the initial yield strength, H is the hardening modulus and E is the elastic modulus. Fortunately, the above mentioned condition can be satisfied for many metal materials, and thus the solution can be used to determine the material length-scale of micro-structures in conjunction with the experiment of cantilever beams in the micro-scale.     

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.     

Lorentz group SOo(5, 1) for perfect elastoplasticity with large deformation and a consistency numerical scheme

How to effectively deal with non-linearity and accurately fulfill the consistency condition is essential for modeling and computing in plasticity. Utilizing the concepts of two phases and homogeneous coordinates, we obtain a linear representation of a constitutive model of perfect elastoplasticity with large deformation, and, furthermore, a linear irreducible representation, which contains a five-order spin tensor. The underlying vector space is found to be the projective realization of a composite space resulting from a surgery on Minkowski spacetime ⁵⁺¹. The irreducible representation in the vector space admits of the projective proper orthochronous Lorentz group PSO_o(5, 1) in the on (or elastoplastic) phase and the special Euclidean group SE(5) in the off (or elastic) phase. The input path dictates symmetry switching between the two groups. Based on such symmetry a numerical scheme is devised which preserves the consistency condition for every time step. The consistency scheme is shown to be stabler, more accurate, and more efficient than the current numerical schemes developed directly based upon the model itself, because the new scheme preserves the internal symmetry SO_o(5, 1) of the model in the on phase so as to locate the stress point automatically on the yield surface at each time step without iterations at all.     

Asymptotic crack-tip fields for dynamic crack growth in compressive power-law hardening materials

The effect of material compressibility on the stress and strain fields for a mode-I crack propagating steadily in a power-law hardening material is investigated under plane strain conditions. The plastic deformation of materials is characterized by the J₂ flow theory within the framework of isotropic hardening and infinitesimal displacement gradient. The asymptotic solutions developed by the present authors [Zhu, X.K., Hwang K.C., 2002. Dynamic crack-tip field for tensile cracks propagating in power-law hardening materials. International Journal of Fracture 115, 323–342] for incompressible hardening materials are extended in this work to the compressible hardening materials. The results show that all stresses, strains, and particle velocities in the asymptotic fields are fully continuous and bounded without elastic unloading near the dynamic crack tip. The stress field contains two free parameters σ_eq0 and s₃₃₀ that cannot be determined in the asymptotic analysis, and can be determined from the full-field solutions. For the given values of σ_eq0 and s₃₃₀, all field quantities around the crack tip are determined through numerical integration, and then the effects of the hardening exponent n, the Poisson ratio ν, and the crack growth speed M on the asymptotic fields are studied. The approximate behaviors of the proposed solutions are discussed in the limit of ν → 0.5 or n → ∞.     

Existence of Exact Solutions for the Eversion of Elastic Cylinders

The existence of eversion deformations of an elastic cylinder into another right circular cylinder is studied. It is found that such deformations are associated with strain-energy functions of separable form W(λ₁,λ₂,λ₃) =φ(λ₁)+φ(λ₂)+φ (λ₃), and thus can serve as a test for materials of this kind. Under certain constitutive assumptions, there always exists a cylindrical eversion deformation that satisfies exact pointwise traction free boundary conditions over the entire surface of the cylinder. For such solutions the cavity must remain open upon eversion. This revised version was published online in August 2006 with corrections to the Cover Date.     

Modelling of pressure–strain correlation in compressible turbulent flow

Previous studies carried out in the early 1990s conjectured that the main compressible effects could be associated with the dilatational effects of velocity fluctuation. Later, it was shown that the main compressibility effect came from the reduced pressure-strain term due to reduced pressure fluctuations. Although better understanding of the compressible turbulence is generally achieved with the increased DNS and experimental research effort, there are still some discrepancies among these recent findings. Analysis of the DNS and experimental data suggests that some of the discrepancies are apparent if the compressible effect is related to the turbulent Mach number, Mt. From the comparison of two classes of compressible flow, homogenous shear flow and inhomogeneous shear flow （mixing layer）, we found that the effect of compressibility on both classes of shear flow can be characterized in three categories corresponding to three regions of turbulent Mach numbers： the low-Mr, the moderate-Mr and high-Mr regions. In these three regions the effect of compressibility on the growth rate of the turbulent mixing layer thickness is rather different. A simple approach to the reduced pressure-strain effect may not necessarily reduce the mixing-layer growth rate, and may even cause an increase in the growth rate. The present work develops a new second-moment model for the compressible turbulence through the introduction of some blending functions of Mt to account for the compressibility effects on the flow. The model has been successfully applied to the compressible mixing layers.     

Influence of molecular parameters on the stress dependence of viscous and elastic properties of polypropylene melts in shear

The stress dependencies of the steady-state viscosity η and, particularly, that of the steady-state elastic compliance J _e of various linear isotactic polypropylenes (PP) and one long-chain branched PP are investigated using creep-recovery tests. The creep stresses applied range from 2 to 10,000 Pa. In order to discuss the stress-dependent viscosity η and elastic compliance J _e with respect to the influence of the weight average molar mass M _w and the polydispersity factor M _w/M _n the PP are characterized by SEC–MALLS. For the linear PP, linear steady-state elastic compliances J_e⁰J_{\rm e}^0 in the range of 10^− 5–10^− 3 Pa^− 1 are obtained depending on the molar mass distribution. J_e⁰J_{\rm e}^0 of the LCB-PP is distinctly higher and comes to lie at around 10^− 2 Pa^− 1. J_e⁰J_{\rm e}^0 is found to be independent of M _w but strongly dependent on polydispersity. η and J _e decrease with increasing stress. For the linear PP, J _e as a function of the stress τ is temperature independent. The higher M _w/M _n the stronger is the shear thinning of η and the more pronounced is the stress dependence of J _e. For the LCB-PP, the strongest stress dependence of η and J _e is observed. Furthermore, for all PP J _e reacts more sensitively to an increasing stress than η. A qualitative explanation for the stronger stress dependence of J _e compared to η is given by analyzing the contribution of long relaxation times to the viscosity and elasticity.     

A rheological approach to the stability of humic acid/clay colloidal suspensions

Steady-state, oscillatory, and transient rheological determinations were used to assess the stability of homoionic sodium montmorillonite (NaMt) suspensions at constant ionic strength (10^–2 mol/l NaCl) and different pH values, after adsorption of humic acid (HA) on the particles. The adsorption of the latter was first spectrophotometrically determined, at pH 3 and 9. While at pH 9 adsorption saturation was observed, at pH 3 the adsorption density continued to grow up to the maximum equilibrium HA concentration reached (∼200 mg/l). Considering the similarity between the structure of edge surfaces of NaMt particles and the surfaces of silica and alumina, the adsorption of HA was also investigated on the latter solids. The results suggest that at pH 3 humic acids adsorb preferentially on edge surfaces, mainly through electrostatic attraction with positively charged aluminol groups. This hypothesis is indirectly confirmed by zeta potential, ζ, values: while HA concentration has little effect on ζ for silica, the addition of HA yields the zeta potential of alumina increasingly negative for all pH values. Using shear stress vs shear rate plots, the yield stress of NaMt was determined as a function of particle concentration, C, for pH 3, 5, 7, and 9, with and without addition of 50 mg/l HA. The yield stress, σ_y, was fitted with a power law σ_y∝C ⁿ ; it was found that n values as high as 12 are characteristic of NaMt suspensions at pH 9 in the presence of HA. This indicates a strong stabilizing effect of humic acid. This stabilization was confirmed by oscillometric measurements, as the storage modulus G′ in the viscoelastic linear region also scales with C, displaying large n values at neutral and basic pHs in the presence of HA. The modulus (in the viscoelastic linear region, for a frequency ν=1 Hz) was found to increase with time, but G′ was lower at any time when HA was added, a consequence of the stabilization provided by HA. Similarly, creep-recovery experiments demonstrated that NaMt suspensions containing HA displayed a less elastic behavior, and a permanent deformation. Modeling the results as a Kelvin-Voigt model allowed one to establish a new scaling law of the reciprocal instantaneous deformation with C. As before, high values of n were found for suspensions at pH 9 in the presence of HA.     

Heat transfer in the flow of a viscoelastic fluid over a stretching surface

An analysis is made of heat transfer in the boundary layer of a viscoelastic fluid flowing over a stretching surface. The velocity of the surface varies linearly with the distance x from a fixed point and the surface is held at a uniform temperature T _w higher than the temperature T _∞ of the ambient fluid. An exact analytical solution for the temperature distribution is found by solving the energy equation after taking into account strain energy stored in the fluid (due to its elastic property) and viscous dissipation. It is shown that the temperature profiles are nonsimilar in marked contrast with the case when these profiles are found to be similar in the absence of viscous dissipation and strain energy. It is also found that temperature at a point increases due to the combined influence of these two effects in comparison with its corresponding value in the absence of these two effects. A novel result of this analysis is that for small values of x, heat flows from the surface to the fluid while for moderate and large values of x, heat flows from the fluid to the surface even when T _w >T _∞. Temperature distribution and the surface heat flux are determined for various values of the Prandtl number P, the elastic parameter K ₁ and the viscous dissipation parameter a. Numerical solutions are also obtained through a fourth-order accurate compact finite difference scheme. Received on 14 October 1997