Problems in determining the elastic strain energy function for rubber

Lightly crosslinked natural rubber can be stretched by 600% or more, and recovers almost completely. It is often regarded as a model highly elastic material and characterized by a strain energy function to describe its stress-strain behavior under various types of deformation. A number of such functions have been proposed; some of them appear in current finite element programs. They are usually validated by comparison with measured stress-strain relations by Treloar [7] [L.R.G. Treloar, Stress-strain data for vulcanized rubber under various types of deformation, Trans. Faraday Soc. 40 (1944) 59-70] and Jones and Treloar [15] [D.F. Jones, L.R.G. Treloar, The properties of rubber in pure homogeneous strain, J. Phys. D Appl. Phys. 8 (1975) 1285-1304]. But Treloar pointed out that the relations at high strains became markedly irreversible, and he did not assign a strain energy function for strains greater than about 300%. Rivlin's universal relation between torsional stiffness and tensile stress [14] [R.S. Rivlin, Large elastic deformations of isotropic materials. Part V1: further results in the theory of torsion, shear and flexure, Philos. Trans. R. Soc. A 243 (1949) 251-288] is applied here to show that a typical elastic solid cannot be described by any strain energy function at strains greater than about 300%. Elastic strain energy functions for higher strains, or for other rubbery materials, are thus of doubtful value unless evidence for reversibility of stress-strain relations is adduced or the applicability of a strain energy function is demonstrated.     

Simulation of heat and chemical reactions on Reiner Rivlin fluid model for blood flow through a tapered artery with a stenosis

In the present article, we have analyzed the effects of heat and mass transfer on Reiner Rivlin fluid model for blood flow through a tapered artery with a stenosis. The constitutive equations for a Reiner Rivlin fluid have been modelled in cylindrical coordinates. A perturbation series in dimensionless Reiner Rivlin fluid parameter (λ ₁ ≪ 1) have been used to obtain explicit forms for the velocity, temperature, concentration, resistance impedance, wall shear stress and shearing stress at the stenosis throat. The graphical results of different type of tapered arteries i.e. converging tapering, diverging tapering, non-tapered artery have been examined for different parameters of interest.     

Strain energy function of an uncross-linked butadiene rubber estimated from planar extension

The derivatives of the strain energy function u with respect to the invariants of the strain tensor (I₁ and I₂) are estimated for uncross-linked butadiene rubber by using the BKZ constitutive equation. The derivatives at small deformations show anomalous behavior; namely, an upturn for ∂u/∂I₁ and a downturn for ∂u/∂I₂ take place, as is the case of cross-linked rubbers. At large deformations, u is well described by u = A₁(I₁ −3) + A₂(I₂ −3) with numerical constants A₁ and A₂. This behavior is also quite similar to that for cross-linked rubbers. The non-zero positive constant A₂ for the melt suggests that the non-zero value is due to neither the inhomogeneity in network structure nor high extension of constituent polymer chains.     

Convection in a Lid-Driven Heat-Generating Porous Cavity with Alternative Thermal Boundary Conditions

Mixed convection flow in a two-sided lid-driven cavity filled with heat-generating porous medium is numerically investigated. The top and bottom walls are moving in opposite directions at different temperatures, while the side vertical walls are considered adiabatic. The governing equations are solved using the finite-volume method with the SIMPLE algorithm. The numerical procedure adopted in this study yields a consistent performance over a wide range of parameters that were 10⁻⁴ ≤ Da ≤ 10⁻¹ and 0 ≤ Ra _I ≤ 10⁴. The effects of the parameters involved on the heat transfer characteristics are studied in detail. It is found that the variation of the average Nusselt number is non-linear for increasing values of the Darcy number with uniform or non-uniform heating condition.     

Measurements of heat transfer coefficients between turbulent liquid jets and a radially finned heated disk placed at the bottom of an open vertical circular cavity

 Experiments have been performed to assess the impact of an extended surface on the heat transfer enhancement for axisymmetric, turbulent liquid jet impingement on a heated round disk. The disk, with an array of integral radial fins mounted on its surface, is placed at the bottom of an open vertical circular cavity. Hydrodynamic and heat transfer data were obtained for a dielectric fluorocarbon liquid FC-77. For a fixed circular heater of diameter D=22.23 mm, several geometric parameters were tested: the nozzle diameter (4.42≤d≤9.27 mm), the confining wall diameter of the vertical cavity (22.23≤D _c≤30.16 mm), and the nozzle-to-heater spacing (0.5≤S/d≤5.0). The FC-77 flow rates varied from =0.2 to 11.0 l/min producing Reynolds numbers in the wide interval 700≤Re _d ≤44,000. For d=4.42 mm, the heat transfer response to the separation distance S/d was small but increased gradually with increasing nozzle diameter up to d=9.27 mm. The thermal resistance R _th increased with the confining wall diameter D _c and also with the nozzle diameter d. A minimum value of the thermal resistance of R _th,min=0.4 cm² K/W was attained for a combination of d=4.42 mm, D _c=22.23 mm, S/d=1, and =7.5 l/min. Based on a simplified heat transfer model, reasonable agreement was obtained between measured values of the thermal resistance and the R _th-predictions. The total fin effectiveness ɛ_f was shown to increase with increasing nozzle diameter, but was invariant with the flow rate (or the jet exit velocity). More than a three-fold heat transfer enhancement was realized through the addition of the array of integral radial fins on the heated round disk. Received on 30 August 2000 / Published online: 29 November 2001     

Secondary flow patterns and mixing in laminar pulsating flow through a curved pipe

Mixing by secondary flow is studied by particle image velocimetry (PIV) in a developing laminar pulsating flow through a circular curved pipe. The pipe curvature ratio is η = r ₀/r _c  = 0.09, and the curvature angle is 90°. Different secondary flow patterns are formed during an oscillation period due to competition among the centrifugal, inertial, and viscous forces. These different secondary-flow structures lead to different transverse-mixing schemes in the flow. Here, transverse mixing enhancement is investigated by imposing different pulsating conditions (Dean number, velocity ratio, and frequency parameter); favorable pulsating conditions for mixing are introduced. To obviate light-refraction effects during PIV measurements, a T-shaped structure is installed downstream of the curved pipe. Experiments are carried out for the Reynolds numbers range 420 ≤ Re_st ≤ 1,000 (Dean numbers 126.6 ≤ Dn ≤ 301.5) corresponding to non-oscillating flow, velocity component ratios 1 ≤ (β = U _max,osc/U _m,st) ≤ 4 (the ratio of velocity amplitude of oscillations to the mean velocity without oscillations), and frequency parameters 8.37 < (α = r ₀(ω/ν)^0.5) < 24.5, where α² is the ratio of viscous diffusion time over the pipe radius to the characteristic oscillation time. The variations in cross-sectional average values of absolute axial vorticity (|ζ|) and transverse strain rate (|ε|) are analyzed in order to quantify mixing. The effects of each parameter (Re_st, β, and α) on transverse mixing are discussed by comparing the dimensionless vorticities (|ζ _P |/|ζ _S |) and dimensionless transverse strain rates (|ε _P |/|ε _S |) during a complete oscillation period.     

Numerical investigation of natural convection of Al<Subscript>2</Subscript>O<Subscript>3</Subscript> nanofluid in vertical annuli

This paper presents the numerical study of internal free convection of Al₂O₃ water nanofluid in vertical annuli. Vertical walls are maintained at constant temperatures and horizontal walls are adiabatic. Results are validated by experimental data. Effect of nanofluids on natural convection is investigated as a function of geometrical and physical parameters and particle fractions for aspect ratio of 1 ≤ H/L ≤ 5, Grashof number of 10³ ≤ Gr ≤ 10⁵ and concentration of 0 ≤ ϕ ≤ 0.06. More than 330 different numerical cases are investigated to develop a new correlation for the Nusselt number. This correlation is presented as a function of Nusselt number of base fluid and particle fraction which is a linear decreasing function of particle fraction. The developed correlation for annuli is also valid for the natural convection of Al₂O₃ water nanofluid in a square cavity. Furthermore, the effect of the viscosity and conductivity models on the Nusselt number of nanofluids in cylindrical cavities are discussed.     

Further study of rubber-like elasticity： elastic potentials matching biaxial data＊

By virtue of the rational interpolation procedure and logarithmic strain, a direct approach is proposed to obtain elastic potentials that exactly match uniaxial data and shear data for elastomers. This approach reduces the determination of multiaxial elastic potentials to that of two one-dimensional potentials, thus bypassing usual cumbersome procedures of identifying a number of unknown parameters. Predictions of the suggested potential are derived for a general biaxial stretch test and compared with the classical data given by Rivlin and Saunders(Rivlin, R. S. and Saunders, D. W. Large elastic deformation of isotropic materials. VII: experiments on the deformation of rubber. Phill. Trans. Royal Soc. London A, 243, 251–288(1951)). Good agreement is achieved with these extensive data.     

A Molecular-Statistical Basis for the Gent Constitutive Model of Rubber Elasticity

Molecular constitutive models for rubber based on non-Gaussian statistics generally involve the inverse Langevin function. Such models are widely used since they successfully capture the typical strain-hardening at large strains. Limiting chain extensibility constitutive models have also been developed on using phenomenological continuum mechanics approaches. One such model, the Gent model for incompressible isotropic hyperelastic materials, is particularly simple. The strain-energy density in the Gent model depends only on the first invariant I ₁ of the Cauchy–Green strain tensor, is a simple logarithmic function of I ₁ and involves just two material parameters, the shear modulus μ and a parameter J _m which measures a limiting value for I ₁−3 reflecting limiting chain extensibility. In this note, we show that the Gent phenomenological model is a very accurate approximation to a molecular based stretch averaged full-network model involving the inverse Langevin function. It is shown that the Gent model is closely related to that obtained by using a Padè approximant for this function. The constants μ and J _m in the Gent model are given in terms of microscopic properties. Since the Gent model is remarkably simple, and since analytic closed-form solutions to several benchmark boundary-value problems have been obtained recently on using this model, it is thus an attractive alternative to the comparatively complicated molecular models for incompressible rubber involving the inverse Langevin function. This revised version was published online in June 2006 with corrections to the Cover Date.     

Maximum density effects on convective instability of horizontal plane poiseuille flows in the thermal entrance region

A linear stability analysis is used to study the conditions marking the onset of secondary flow in the form of longitudinal vortices for plane Poiseuille flow of water in the thermal entrance region of a horizontal parallel-plate channel by a numerical method. The water temperature range under consideration is 0∼30°C and the maximum density effect at 4°C is of primary interest. The basic flow solution for temperature includes axial heat conduction effect and the entrance temperature is taken to be uniform at far upstream location jackie=−∞ to allow for the upstream heat penetration through thermal entrance jackie=0. Numerical results for critical Rayleigh number are obtained for Peclet numbers 1, 10, 50 and thermal condition parameters (λ ₁, λ ₂) in the range of −2.0≤λ ₁≤−0.5 and −1.0≤λ ₂≤1.4. The analysis is motivated by a desire to determine the free convection effect on freezing or thawing in channel flow of water.     

Three-dimensional viscoelastic simulation of coextrusion process: comparison with experimental data

 Three-dimensional numerical simulation of viscoelastic coextrusion process has been performed and numerical results were compared with the experimental data of Karagiannis et al. (1990). By varying the magnitude of the second normal stress difference and its ratio of Fluid I and Fluid II, we were able to control the interface profile and the degree of encapsulation along the downstream direction. By increasing the parameter α (α_Fluid I=α_Fluid II) from 0.1 to 0.4 in the Giesekus model and increasing the α ratio (α_Fluid Iα_Fluid II) between Fluid I and Fluid II from 2.0 to 4.0 in the permissible range of realistic polymeric systems, the interface profile and the degree of encapsulation along the downstream direction were fitted with the experimental results. There was little difference between the numerical results and the experimental data in the interface profile and the degree of encapsulation along the downstream direction when the α ratio was set to 3.0 (0.3:0.1). Fluid I with larger magnitude of the second normal stress difference protrudes into Fluid II with smaller magnitude of the second normal stress difference around the symmetric plane, while Fluid II wraps around Fluid I near the side walls. As the ξ ₁ ratio (ξ _{1 ,Fluid I}ξ _{1 ,Fluid II}) increases from 1.0 to 3.0 for the two-mode Phan-Thien and Tanner model, it was found that the curvature of the interface profile increased, and the difference between the numerical results and the experimental data in the interface profile and the degree of encapsulation along the downstream direction was almost negligible when the ξ ₁ ratio was set to 3.0 (0.54:0.18). Although the parameters of viscoelastic models were fitted by using the shear viscosity data only, quantitative agreements between the numerical results and the experimental coextrusion data were quite satisfactory. Received: 24 April 2001 Accepted: 5 June 2001     

Effects of solid and slotted baffles on the convection characteristics of backward-facing step flow in a channel

This study investigates the enhancement of the laminar forced convection characteristics of backward-facing step flow in a two-dimensional channel through the installation of solid and slotted baffles onto the channel wall. The effects of the height of baffle H _b, inclination of baffle installation ϕ_b, height of slot in baffle H _t, inclination of slot in baffle ϕ_t, and distance between the backward-facing step and baffle D on the flow structure, temperature distribution and Nusselt number variation for the system at various Re are numerically explored. Results show that a slotted baffle can enhance the average Nusselt number for the heating section of channel plate by the maximum 190% when Pr=0.7, H _s=0.5, L=5, H _b ≤ 0.3, W _b ≤ 0.2, 0.1 ≤ D ≤ 0.5, 0° ≤ ϕ_b ≤ 45°, H _t ≤ 0.1, 0° ≤ ϕ_t ≤ 45° and 50 ≤ Re ≤ 400. As for the solid baffle, the enhancement may be up by 230%. The solid baffle might cause the re-separation of main stream, and consequently result in poor local heat transfer coefficient in the end region of heating section. This disadvantage can be obviously improved as the baffle is slotted. Besides the penalty of increase in pressure drop due to the baffle installation is much higher for the situation with solid baffle.     

Melt shear rheology of carbon nanofiber/polystyrene composites

The rheological behavior and morphology of carbon nanofiber/polystyrene (CNF/PS) composites in their melt phase have been characterized both through experimental measurements and modeling. Composites prepared in the two different processes of solvent casting and melt blending are contrasted; melt-blended and solvent-cast composites were each prepared with CNF loadings of 2, 5, and 10 wt%. A morphological study revealed that the melt blending process results in composites with shorter CNFs than in the solvent-cast composites, due to damage caused by the higher stresses the CNFs encounter in melt blending, and that both processes retain the diameter of the as-received CNFs. The addition of carbon nanofiber to the polystyrene through either melt blending or solvent casting increases the linear viscoelastic moduli, G′ and G″, and steady-state viscosity, η, in the melt phase monotonically with CNF concentration, more so in solvent cast composites with their longer CNFs. The melt phase of solvent-cast composites with higher CNF concentrations exhibit a plateau of the elastic modulus, G′, at low frequencies, an apparent yield stress, and large first normal stress difference, N ₁, at low strain rates, which can be attributed to contact-based network nanostructure formed by the long CNFs. A nanostructurally-based model for CNF/PS composites in their melt phase is presented which considers the composite system as rigid rods in a viscoelastic fluid matrix. Except for two coupling parameters, all material constants in the model for the composite systems are deduced from morphological and shear flow measurements of its separate nanofiber and polymer melt constituents of the composite. These two coupling parameters are polymer–fiber interaction parameter, σ, and interfiber interaction parameter, C _I. Through comparison with our experimental measurements of the composite systems, we deduce that σ is effectively 1 (corresponding to no polymer–fiber interaction) for all CNF/PS nanocomposites studied. The dependence of CNF orientation on strain rate which we observe in our experiments is captured in the model by considering the interfiber interaction parameter, C _I, as a function of strain rate. Applied to shear flows, the model predicts the melt-phase, steady-state viscosities, and normal stress differences of the CNF/PS composites as functions of shear rate, polymer matrix properties, fiber length, and mass concentration consistent with our experimental measurements.     

Degree of branching of polypropylene measured from Fourier-transform rheology

In this study, linear and branched polypropylenes (PP) were compared under medium strain amplitude oscillatory shear (usually strain amplitude range from 10 to 100%) with Fourier-transform rheology (FT rheology). On a log–log diagram, the third relative intensity (I ₃/I ₁), which is a parameter to represent nonlinearity, shows a linear relationship with the strain amplitude in the range of medium strain amplitude. The slope of I ₃/I ₁ of linear PP with various molecular weight and molecular weight distribution was 2 as most constitutive equations predict, while that of branched PP was 1.64, which is lower than that of linear PP. When the linear and branch PP were blended, the slope of I ₃/I ₁ was proportional to the composition of the branch PP. Therefore, it is suggested that the degree of branching can be defined in terms of the slope of I ₃/I ₁ under medium amplitude oscillatory shear.     

Plane-strain state of an elastoplastic body with a crack under mixed-mode loading

A mixed-mode (I + II) crack model with a plastic strip on its continuation under plane strain is proposed. The stress components within the strip are determined from the yield conditions, stress limitation, and relationship between the normal stress components defined via the principal stress state. The crack parameters are analyzed for the Mises yield condition. In the quasibrittle case, the governing system of equations includes stress intensity factors K _I, K _II, and T-stresses     

Rheological and mechanical properties of silica colloids: from Newtonian liquid to brittle behaviour

Rheological and mechanical properties of aqueous mono-disperse silica suspensions (Ludox? HS40) are investigated as a function of particle volume fraction (ϕ _p ranging from 0.22 to 0.51) and water content, using shear rate tests, oscillatory methods, indentation and an ultrasonic technique. As the samples are progressively dried, four regimes are identified; they are related to the increasing particle content and the existence and behaviour of the electrical double layer (EDL) around each particle. For 0.22 ≤ ϕ _p ≤ 0.30), the suspensions are stable due to the strong electrostatic repulsion between particles and show Newtonian behaviour (I). As water is removed, the solution pH decreases and the ionic strength increases. The EDL thickness therefore slowly decreases, and screening of the electrostatic repulsion increases. For 0.31 ≤ ϕ _p ≤ 0.35, the suspensions become turbid and exhibit viscoelastic (VE) shear thinning behaviour (II), as they progressively flocculate. For 0.35 ≤ ϕ _p ≤ 0.47, the suspensions turn transparent again and paste-like, with VE shear thinning behaviour and high elastic modulus (III). At higher particle concentration, the suspensions undergo a glass transition and behave as an elastic brittle solid (IV, ϕ _p = 0.51).     

A method for measuring mode I crack tip constraint under static and dynamic loading conditions

A novel experimental technique for measuring crack tipT-stress, and hence in-plane crack tip constraint, in elastic materials has been developed. The method exploits optimal positioning of stacked strain gage rosette near a mode I crack tip such that the influence of dominant singular strains is negated in order to determineT-stress accurately. The method is demonstrated for quasi-static and low-velocity impact loading conditions and two values of crack length to plate width ratios (a/W). By coupling this new method with the Dally-Sanford single strain gage method for measuring the mode I stress intensity factorK _I , the crack tip biaxiality parameter is also measured experimentally. Complementary small strain, static and dynamic finite element simulations are carried out under plane stress conditions. Time histories ofK _I andT-stress are computed by regression analysis of the displacement and stress fields, respectively. The experimental results are in good agreement with those obtained from numerical simulations. Preliminary data for critical values ofK _I and β for dynamic experiments involving epoxy specimens are reported. Dynamic crack initiation toughness shows an increasing trend as β becomes more negative at higher impact velocities.     

A proof of the Benjamin-Feir instability

The existence and linear stability problem for the Stokes periodic wavetrain on fluids of finite depth is formulated in terms of the spatial and temporal Hamiltonian structure of the water-wave problem. A proof, within the Hamiltonian framework, of instability of the Stokes periodic wavetrain is presented. A Hamiltonian center-manifold analysis reduces the linear stability problem to an ordinary differential eigenvalue problem on ℝ⁴. A projection of the reduced stability problem onto the tangent space of the 2-manifold of periodic Stokes waves is used to prove the existence of a dispersion relation Λ(λ,σ, I ₁, I ₂)=0 where λ ε ℂ is the stability exponent for the Stokes wave with amplitude I ₁ and mass flux I ₂ and σ is the “sideband’ or spatial exponent. A rigorous analysis of the dispersion relation proves the result, first discovered in the 1960's, that the Stokes gravity wavetrain of sufficiently small amplitude is unstable for F ε (0,F₀) where F ₀ ≈ 0.8 and F is the Froude number.     

Two-Parameter Model of a Mode I Crack in an Elastoplastic Body under Plane-Strain conditions

The Dugdale crack model is generalized to the case of plane strain. The governing equations are set up to determine the stresses in the plastic zone. Numerical results from specific problems are analyzed and compared with those for plane stress state and other cases. A relationship between the crack model and K _I-T theory is established in the case of small-scale yielding at the crack tip __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 6, pp. 44–55, June 2005.