The prediction of time-dependent non-linear stresses in viscoelastic materials: III. Memory-function non-linearity

The new measure I of strain developed in Part I has been used in a single-integral constitutive equation containing a memory functionμ dependent on both history s and the second strain-rate invariant ii_D. Three methods of calculating this non-linear memory function from experimental data are presented, and their relative merits are discussed. When applied to polyisobutylene solution data all three methods yield similar results and show an initially positive memory function decreasing to slightly negative values before finally decaying to zero. At amounts of shear a less than 1.5,μ is equal to the linear value but for a ≈2, μ falls rapidly to much less than the linear value. These results are in agreement with earlier work on superposed oscillatory and continuous shear, which was interpreted in terms of a shear-rate dependent relaxation spectrum truncated at a time inversely proportional to the shear rate. The present results also shed light on the objection that is sometimes raised to memory functions that are dependent on the strain rate.     

Non-linear combined axial and torsional shear wave propagation in an incompressible hyperelastic solid

Finite amplitude combined axial and torsional shear wave propagation in an incompressible isotropic hyperelastic solid is considered. When the strain energy function of the solid is a non-linear function offI₁,− 3) and (I₂− 3), where I₁, and I₂are the first and second basic invariants of the left Cauchy-Green tensor, the two second order partial differential equations governing the propagation of the axial and torsional waves are non-linear and coupled. These two coupled equations are equivalent to a hyperbolic system of first order partial differential equations and a modification of the MacCormack finite difference scheme is used to obtain numerical solutions of this system. Numerical results, which show the effect of the coupling, are presented for boundary-initial value problems of propagation into initially unstressed and initially stressed regions at rest.     

Determination of the non-linear parameter (mobility factor) of the Giesekus constitutive model using LAOS procedure

The paper introduces a novel procedure to determine the non-linear parameter of the Giesekus model, in relation to the characterization of the non-linear oscillatory shear regime of viscoelastic polymer solutions based on polyacrylamide. Instead of using the shear-thinning viscosity as the representative non-linear effect, the third harmonic in the Fourier spectrum of the shear stress response signal is considered for computing the mobility factor. The fluid is subjected to large amplitude oscillatory shear (LAOS) and its response is recorded. Deviations of this signal from the sinusoidal form are specific to each material and gives both qualitative and quantitative measures of the non-linearity. By fitting the material response with the corresponding numerical solutions of the n-modes Giesekus constitutive relation, one can extract the values of the non-linear α_i-parameters that describe the fluid rheology. It is demonstrated that this procedure, which can be successfully applied to semi-concentrated polymer solutions, provides better results than the classical viscosity-fit method.     

TTS in LAOS: validation of time-temperature superposition under large amplitude oscillatory shear

The validation of time-temperature superposition of non-linear parameters obtained from large amplitude oscillatory shear is investigated for a model viscoelastic fluid. Oscillatory time sweeps were performed on a 11?wt.% solution of high molecular weight polyisobutylene in pristane as a function of temperature and frequency and for a broad range of strain amplitudes varying from the linear to the highly non-linear regime. Lissajous curves show that this reference material displays strong non-linear behaviour when the strain amplitude is exceeding a critical value. Elastic and viscous Chebyshev coefficients and alternative non-linear parameters were obtained based on the framework of Ewoldt et al. (J Rheol 52(6):1427?C1458, 2008) as a function of temperature, frequency and strain amplitude. For each strain amplitude, temperature shift factors a _T (T) were calculated for the first order elastic and viscous Chebyshev coefficients simultaneously, so that master curves at a certain reference temperature T _ref were obtained. It is shown that the expected independency of these shift factors on strain amplitude holds even in the non-linear regime. The shift factors a _T (T) can be used to also superpose the higher order elastic and viscous Chebyshev coefficients and the alternative moduli and viscosities onto master curves. It was shown that the Rutgers-Delaware rule also holds for a viscoelastic solution at large strain amplitudes.     

Predicting extrusion instabilities of commercial polyethylene from non-linear rheology measurements

Processing at the highest possible throughput rates is essential from an economical point of view. However, various flow instabilities and extrudate distortions like sharkskin, stick slip, and gross melt fracture (GMF) may limit the production rate of high-quality products. Predicting the process conditions leading to the occurrence of rheological instabilities is the key for improving product quality, process control, and optimization. Large-amplitude oscillatory shear (LAOS) and FT-rheology were used to quantify the non-linear rheological behavior and instabilities of a series of well-characterized commercial polyethylene (PE). From the latter, we derive the critical non-linearity parameter, F _0,c, which corresponds to the normalized intensity of the third harmonic at the critical strain amplitude, γ _0,C (defined by the appearance of the second harmonic), normalized by γ _0,C . The F _0,c is correlated with the high molecular mass fraction of the polymers and with the Deborah numbers. Linear rheological parameters and molecular structures were related to F _0,c. An experimental correlation between F _0,c of commercial PE melts and pressure fluctuations associated with flow instabilities (sharkskin) was established both for capillary rheometry and extrusion.     

On shear band formation: I. Constitutive relationship for a dual yield model

A phenomenological constitutive relation, for capturing the shear band formation in a rate-independent elastic-plastic material, is established. The model takes into account both the J₂-isotropic flow and a threshold shear stress-based flow. The elastic-plastic constitutive tensor is expressed explicity in terms of elastic constants, the deviatoric stress tensor, the direction of the principal shear velocity-strain, and other material constants. This model particularly facilitates the resolution of the formation of the shear band even under material hardening conditions and does not demand an a priori knowledge of the orientation of the shear band. This is incorporated in an FEM, and the plane strain tensile test of Anand and Spitzig [1980] is numerically simulated. The computed results compare favorably with the experimental data. The shear band emerges more naturally as a solution to the boundary value problem, unlike the situations in solutions based on classical bifurcation methods. Nevertheless, the usefulness of the local instability condition (Ortiz et al. [1987]) is also demonstrated.     

Transient shear rheology of carbon nanofiber/polystyrene melt composites

We characterize the transient shear rheology of polystyrene/carbon nanofiber composites. Our experimental measurements of the composites show increasing stress overshoot responses to transient shear as the carbon nanofiber concentration increases. We also find the steady state viscosity reached at long times during application of a constant shear rate increases with increasing carbon nanofiber concentration. Flow reversal experiments show the effects of nanofiber orientation and structural evolution on the composite's rheological response.We present a microstructurally based constitutive model where all but two parameters are determined by rheological characterization of the pure polymer and the shape and concentration of the nanoparticles. The Folgar-Tucker constant, C_I, is treated as a fitting parameter, while several definitions for the shape factors A, B, C and F are evaluated. We make note of the effects each parameter has on the model's predictions. We find that the constitutive model is in agreement with our experimentally measured transient shear rheology of the PS/CNF melt composites for the CNF concentrations and shear rates presented.     

A stability in the large investigation for a non-linear second order two-degree-of-freedom system with periodic excitation

An extension to an algorithm due to Simpson has been developed for the analysis of a non-linear second order two-degree-of-freedom system with external periodic excitation. The form of equations considered arises from the study of mechanical systems with a single concentrated weak non-linearity and the method assumes a solution made up of harmonic terms whose amplitudes vary slowly in time. The system considered is such that in the absence of external excitation, it possesses a stable equilibrium point and an unstable limit cycle arising from a sub-critical Hopf bifurcation. When forcing is applied, the stable equilibrium point may then be replaced by a stable periodic attractor, and the limit cycle by an unstable multi-periodic attractor. The method has been applied to the problem of locating these attractors, and if they exist, of finding the stable attractor's basin of attraction in terms of initial conditions. The method reduces the problem from a search in four-dimensional phase space to a search for a boundary in a plane defined by amplitudes a₁ and a₂ in the assumed form of the solution.The method was applied to three non-linear systems in which the non-linearity was due to either a linear spring with a small amount of cubic hardening or a linear spring with freeplay. Agreement was shown to be good in those cases where the non-linearity was weak. However, the method would not be expected to give such accurate results if the non-linear effect was more significant. This was illustrated for a case involving the freeplay non-linearity.     

Rivlin's Representation Formula is Ill-Conceived for the Determination of Response Functions via Biaxial Testing

The experimental determination of a strain energy function W for a rubber specimen must address departures from an elastic ideal in a rational fashion. Herein, such a rational experimental method is developed for biaxial stretching experiments and applied to rubber data in the literature. It is shown that Rivlin's representation formula is experimentally ill-conceived because experimental error is magnified to the extent that error obscures trends in the response function plots. Upon developing direct tensor expressions for the response function calculations, we show that Rivlin's representation formula (or any such constitutive law that has high covariance amongst the response terms) magnifies experimental error greatly. By “high covariance”, we mean the inner product amongst the response terms in the constitutive law is nearly equal to the maximum possible value - i.e., the product of their magnitudes. Moreover, we show that the second partials of W with respect to I ₁ and I ₂ should approach infinity as the strain decreases. Using an alternate set of invariants with minimal covariance (i.e., a null inner product amongst the response terms), a W for rubber can be determined forthwith.     

An investigation of the stability in the large for an autonomous second-order two-degree-of-freedom system

An extension to an algorithm due to Simpson has been developed for the analysis of a second-order two-degree-of-freedom autonomous system. The form of equations considered arises from the study of mechanical systems with a single concentrated non-linearity and the method assumes a solution made up of harmonic terms whose amplitudes vary slowly in time. For a system possessing a stable equilibrium point and an unstable limit cycle arising from a subcritical Hopf bifurcation, the method has been applied to the problem of predicting the basin of attraction of the equilibrium point. The method reduces the problem from a search in four-dimensional phase space to a search for a boundary in a plane defined by amplitudes a₁ and a₂ in the assumed form of the solution. The method was applied to four weakly non-linear systems in which the non-linearity was due to either a linear spring with a small amount of cubic hardening or a linear spring with freeplay. Agreement was shown to be good in the cases considered. However, it would be expected that the method would not give such accurate results if the non-linear effect was more significant. This was illustrated for the case of the cubic hardening non-linearity.     

Nonlinear vibration analysis of micro-plates based on strain gradient elasticity theory

In this study, non-linear free vibration of micro-plates based on strain gradient elasticity theory is investigated. A general form of Mindlin’s first-strain gradient elasticity theory is employed to obtain a general Kirchhoff micro-plate formulation. The von Karman strain tensor is used to capture the geometric non-linearity. The governing equations of motion and boundary conditions are obtained in a variational framework. The Homotopy analysis method is employed to obtain an accurate analytical expression for the non-linear natural frequency of vibration. For some specific values of the gradient-based material parameters, the general plate formulation can be reduced to those based on some special forms of strain gradient elasticity theory. Accordingly, three different micro-plate formulations are introduced, which are based on three special strain gradient elasticity theories. It is found that both geometric non-linearity and size effect increase the natural frequency of vibration. In a micro-plate having a thickness comparable with the material length scale parameter, the strain gradient effect on increasing the non-linear natural frequency is higher than that of the geometric non-linearity. By increasing the plate thickness, the strain gradient effect decreases or even diminishes. In this case, geometric non-linearity plays the main role on increasing the natural frequency of vibration. In addition, it is shown that for micro-plates with some specific thickness to length scale parameter ratios, both geometric non-linearity and size effect have significant role on increasing the frequency of non-linear vibration.     

A robust hyper-elastic beam model under bi-axial normal-shear loadings

In this paper, a simple and robust constitutive model is proposed to simulate mechanical behaviors of hyper-elastic materials under bi-axial normal-shear loadings in the finite strain regime. The Mooney–Rivlin strain energy function is adopted to develop a two-dimensional (2D) normal-shear constitutive model within the framework of continuum mechanics. A motion field is first proposed for combined normal and shear deformations. The deformation gradient of the proposed field is calculated and then substituted into right Cauchy–Green deformation tensor. Constitutive equations are then derived for normal and shear deformations. They are two explicit coupled equations with high-level polynomial non-linearity. In order to examine capabilities of the developed hyper-elastic model, uniaxial tensile responses and non-linear stability behaviors of moderately thick straight and curved beams undergoing normal axial and transverse shear deformations are simulated and compared with experiments. Fused deposition modeling technique as a 3D printing technology is implemented to fabricate hyper-elastic beam structures from soft poly-lactic acid filaments. The printed specimens are tested under tensile/compressive in-plane and compressive out-of-plane forces. A finite element formulation along with the Newton–Raphson and Riks techniques is also developed to trace non-linear equilibrium path of beam structures in large defamation regimes. It is shown that the model is capable of predicting non-linear equilibrium characteristics of hyper-elastic straight and curved beams. It is found that the modeling of shear deformation and finite strain is essential toward an accurate prediction of the non-linear equilibrium responses of moderately thick hyper-elastic beams. Due to simplicity and accuracy, the model can serve in the future studies dealing with the analysis of hyper-elastic structures in which two normal and shear stress components are dominant.     

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.     

Basic equations for the microscopic theory of plasticity of polycrystalline materials with given texture

An expression for the yield stress of anisotropic materials is applied to the anisotropic strength of hard rolled copper foils whose crystallographic texture is known. We assume that this crystallographic texture is the only cause of the anisotropic plastic behaviour of the material. The model used for the yield stress is also used to deduce:

1. Stress-strain relations for isotropic polycrystalline materials;
2. A formula for the fully plastic strain tensor, applied to anisotropic hard rolled copper foils.

For the anisotropic copper foils considered the calculated curves of the yield stress and of the strain tensor as a function of the angle x between rolling and tensile direction agree qualitatively with the measured values. However, the theory is not complete, since the yield stress and the plastic strain tensor are both a function of a parameter Q, the fraction of the number of available crystallographic slip planes on which the maximum shear stress has reached the critical value τ_a. We assume that for “fully” plastic deformation a certain critical fraction Q _e of the total number of slip planes has to be active. The fraction Q _e is called the critical active quantity. With the parameter Q _e we adjust the calculated curves to the measured ones. The dependence of Q _e on the properties of the material (e.g. the crystallographic texture) is discussed in Appendix I.     

On a formulation for anisotropic elastoplasticity at finite strains invariant with respect to the intermediate configuration

The paper is concerned with a formulation of anisotropic finite strain inelasticity based on the multiplicative decomposition of the deformation gradient F=F_eF_p. A major feature of the theory is its invariance with respect to rotations superimposed on the inelastic part of the deformation gradient. The paper motivates and shows how such an invariance can be achieved. At the heart of the formulation is the mixed-variant transformation of the structural tensor, defined as the tensor product of the privileged directions of the material as given in a reference configuration, under the action of F_p. Issues related to the plastic material spin are discussed in detail. It is shown that, in contrast to the isotropic case, any flow function formulated purely in terms of stress quantities, necessarily exhibits a non-vanishing plastic material spin. The possible construction of spin-free rates is discussed as well, where it is shown that the flow rule must then depend not only on the stress but on the strain as well.     

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.     

Shear viscosity of slightly-elastic concentrated suspensions at low and high shear rates

A quasi-static asymptotic analysis is employed to investigate the elastic effects of fluids on the shear viscosity of highly concentrated suspensions at low and high shear rates. First a brief discussion is presented on the difference between a quasi-static analysis and the periodic-dynamic approach. The critical point is based on the different order-of-contact time between particles. By considering the motions between a particle withN near contact point particles in a two-dimensional “cell” structure and incorporating the concept of shear-dependent maximum packing fraction reveals the structural evolution of the suspension under shear and a newly asymptotic framework is devised. In order to separate the influence of different elastic mechanisms, the second-order Rivlin-Ericksen fluid assumption for describing normal-stress coefficients at low shear rates and Harnoy's constitutive equation for accounting for the stress relaxation mechanism at high shear rates are employed. The derived formulation shows that the relative shear viscosity is characterized by a recoverable shear strain,S _R at low shear rates if the second normal-stress difference can be neglected, and Deborah number,De, at high shear rates. The predicted values of the viscosities increase withS _R , but decrease withDe. The role ofS _R in the matrix is more pronounced than that ofDe. These tendencies are significant when the maximum packing fraction is considered to be shear-dependent. The results are consistent with that of Frankel and Acrivos in the case of a Newtonian suspension, except for when the different divergent threshhold is given as [1 ? (Φ/Φ _m )^1/2] ? 1.     

A new two-point deformation tensor and its relation to the classical kinematical framework and the stress concept

Starting from the issue of what is the correct form for a Legendre transformation of the strain energy in terms of Eulerian and two-point tensor variables we introduce a new two-point deformation tensor, namely H=(F−F^−T)/2, as a possible deformation measure involving points in two distinct configurations. The Lie derivative of H is work conjugate to the first Piola–Kirchhoff stress tensor P. The deformation measure H leads to straightforward manipulations within a two-point setting such as the derivation of the virtual work equation and its linearization required for finite element implementation. The manipulations are analogous to those used for the Lagrangian and Eulerian frameworks. It is also shown that the Legendre transformation in terms of two-point tensors and spatial tensors require Lie derivatives. As an illustrative example we propose a simple Saint Venant–Kirchhoff type of a strain-energy function in terms of H. The constitutive model leads to physically meaningful results also for the large compressive strain domain, which is not the case for the classical Saint Venant–Kirchhoff material.     

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.