Viscoelasticity, non-linear

When the magnitude of response of a material is proportional to the strength of the imposed field, a linear relationship exists. Typical examples in the field of rheology are Hooke s law  

Gum rubbers in general exhibit linear behaviour at small deformations. For studying mixing of rubber, we must treat large deformations. In this case, modulus is not only a function of time, but also of strain the relationship is called non-linear. An example with elongational measurements is already stated in Equation 6.15 

When the material behaviour is linearly viscoelastic for a sample (Equation 6.18), E t) plotted against t forms a curve, which is unique for a given sample usually a log E t) versus log t plot is used. When the material behaviour is non-linear, as in Equation (6.15), it requires both the t-axis and -axis to present the sample characteristics as a curved surface. 

Where two different samples are concerned, the two curved surfaces must be compared and the process is rather cumbersome. In addition, many measurements are required to construct the characteristic surface. 

There have been various attempts to simplify the situation they are, in general, attempts to linearise the non-linear behaviour. The linearisation is possible in some cases and not 

the center deflection of the beam is a product of the maximum stress at the outer elements of the beam, and at the center a geometrical term (L / 6h)) divided by Young s modulus E(i), which is now time-dependent because of the viscoelastic relaxations in the beam, and decreases with time under stress as additional inelastic strains build up. However, the stresses in the viscoelastic beam continue to remain unaltered since they depend only on the applied forces and moments, which remain constant. 

We must note in passing that, if the beam of interest had been constrained at any point by a local reaction support of a different stiffness with a different time dependence, then the problem would not have been of a statically determinate character and the simple procedure would not apply. The solution of such problems would require energy methods as discussed, e.g., by Timoshenko (1930). 

Polymers of all types, glassy or semi-crystalline, have a much more protracted transition from small-strain elastic behavior to fully developed plastic flow than do metals, which can stretch the transition over a quite large strain of the order of 0.05. This is a consequence of the much lower level of crystallinity in polymers than in metals and because the thermally assisted unit inelastic transformation events, occurring primarily in the amorphous component, are in the form of isolated sessile shear transformations in relatively equi-axed small-volume 

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This type of response is referred to as non-linear viscoelastic but as it is not amenable to simple analysis it is often reduced to the form... 

Because of the assumption that linear relations exist between shear stress and shear rate (equation 3.4) and between distortion and stress (equation 3.128), both of these models, namely the Maxwell and Voigt models, and all other such models involving combinations of springs and dashpots, are restricted to small strains and small strain rates. Accordingly, the equations describing these models are known as line viscoelastic equations. Several theoretical and semi-theoretical approaches are available to account for non-linear viscoelastic effects, and reference should be made to specialist works 14-16 for further details. 

J.L. Leblanc, Investigating the non-linear viscoelastic behavior of filled mbber compounds through Fourier transform rheometry. Rubber Chem. TechnoL, 78, 54—75, 2005. 

C. Friedrich, K. Mattes, and D. Schulze, Non-linear Viscoelastic Properties of Polymer Melts as Analyzed by LAOS-FT Experiments, lUPAC Macro 2004, Paris, France, July 4—9, 2004, Paper 6.1.3. 

J.L. Leblanc, Fourier Transform rheometry A new tool to investigate intrinsically non-linear viscoelastic materials, Ann. Trans. Nordic Rheol. Soc., 13, 3-21, 2005. 

Non-linear viscoelastic flow phenomena are one of the most characteristic features of polymeric liquids. A matter of very emphasised interest is the first normal stress difference. It is a well-accepted fact that the first normal stress difference Nj is similar to G, a measure of the amount of energy which can be stored reversibly in a viscoelastic fluid, whereas t12 is considered as the portion that is dissipated as viscous flow [49-51]. For concentrated solutions Lodge s theory [52] of an elastic network also predicts normal stresses, which should be associated with the entanglement density. 

The term y(t,t ) is the shear strain at time t relative to the strain at time t. The use of a memory function has been adopted in polymer modelling. For example this approach is used by Doi and Edwards11 to describe linear responses of solution polymers which they extended to non-linear viscoelastic responses in both shear and extension. 

The most surprising result is that such simple non-linear relaxation behaviour can give rise to such complex behaviour of the stress with time. In Figure 6.3(b) there is a peak termed a stress overshoot . This illustrates that materials following very simple rules can show very complex behaviour. The sample modelled here, it could be argued, can show both thixotropic and anti-thixotropic behaviour. One of the most frequently made non-linear viscoelastic measurements is the thixotropic loop. This involves increasing the shear rate linearly with time to a given... 

Figure 6.3 Plot of a simple non-linear viscoelastic response for (a) the stress relaxation as a function of the applied strain, (b) stress as a function of time at a shear strain y = 1 and (c) viscosity as a function of shear stress. (r (0) = 33Pas, rj(co) = 3 Pas, a = 1, P = 0.1, m = 0.35 and t = Is). Continued overleaf...

Contents Chain Configuration in Amorphous Polymer Systems. Material Properties of Viscoelastic Liquids. Molecular Models in Polymer Rheology. Experimental Results on Linear Viscoelastic Behavior. Molecular Entan-lement Theories of Linear iscoelastic Behavior. Entanglement in Cross-linked Systems. Non-linear Viscoelastic-Properties. 

Note 5 Creep is sometimes described in terms of non-linear viscoelastic behaviour, leading, for example, to evaluation of recoverable shear and steady-state recoverable shear compliance. The definitions of such terms are outside the scope of this document. 

The mechanical response of polypropylene foam was studied over a wide range of strain rates and the linear and non-linear viscoelastic behaviour was analysed. The material was tested in creep and dynamic mechanical experiments and a correlation between strain rate effects and viscoelastic properties of the foam was obtained using viscoelasticity theory and separating strain and time effects. A scheme for the prediction of the stress-strain curve at any strain rate was developed in which a strain rate-dependent scaling factor was introduced. An energy absorption diagram was constructed. 14 refs. 

The effect of gas compression on the uniaxial compression stress-strain curve of closed-cell polymer foams was analysed. The elastic contribution of cell faces to the compressive stress-strain curve is predicted quantitatively, and the effect on the initial Young s modulus is said to be large. The polymer contribution was analysed using a tetrakaidecahedral cell model. It is demonstrated that the cell faces contribute linearly to the Young s modulus, but compressive yielding involves non-linear viscoelastic deformation. 3 refs. 

Only a few non-linear viscoelastic properties have been studied with polymers of well-characterized structure. The most prominent of these is the shear-rate dependence of viscosity. Considerable data have now been accumulated for several polymers, extending over a wide range of molecular weights and concen-... 

The dynamics of block copolymers melts are as intriguing as their thermodynamics leading to complex linear viscoelastic behaviour and anisotropic diffusion processes. The non-linear viscoelastic behaviour is even richer, and the study of the effect of external fields (shear, electric. ..) on the alignment and orientation of ordered structures in block copolymer melts is still in its infancy. Furthermore, these fields can influence the thermodynamics of block copolymer melts, as recent work has shown that phase transition lines shift depending on the applied shear. The theoretical understanding of dynamic processes in block copolymer melts is much less advanced than that for thermodynamics, and promises to be a particularly active area of research in the coming years. 

The exact solutions of the linear elasticity theory only apply for small strains, and under idealised loading conditions, so that they should at best only be treated as approximations to the real behaviour of materials under test conditions. In order to describe a material fully we need to know all the elastic constants and, in the case of linear viscoelastic materials, relaxed and unrelaxed values of each, a distribution of relaxation times and an activation energy. While for non-linear viscoelastic materials we cannot obtain a full description of the mechanical properties. 

Our reason for stressing the concept of representative volume element is that it seems to provide a valuable dividing boundary between continuum theories and molecular or microscopic theories. For scales larger than the RVE we can use continuum mechanics (classical and large strain elasticity, linear and non-linear viscoelasticity) and derive from experiment useful and reproducible properties of the material as a whole and of the RVE in particular. Below the scale of the RVE we must consider the micromechanics if we can - which may still be analysable by continuum theories but which eventually must be studied by the consideration of the forces and displacements of polymer chains and their interactions. 

Figure 2.34 Schematic of Newtonian, elastic, linear, and non-linear viscoelastic regimes as a function of deformation and Deborah number during deformation of polymeric materials.

Viscoelasticity has already been introduced in Chapter 1, based on linear viscoelasticity. However, in polymer processing large deformations are imposed on the material, requiring the use of non-linear viscoelastic models. There are two types of general non-linear viscoelastic flow models the differential type and the integral type. 

The question whether ammonia treated wood shows linear or non-linear viscoelastic behavior has not been answered so far. 

The function F(t — t ) is related, as with the temporary network model of Green and Tobolsky (48) discussed earlier, to the survival probability of a tube segment for a time interval (f — t ) of the strain history (58,59). Finally, this Doi-Edwards model (Eq. 3.4-5) is for monodispersed polymers, and is capable of moderate predictive success in the non linear viscoelastic range. However, it is not capable of predicting strain hardening in elongational flows (Figs. 3.6 and 3.7). 

D. Aciemo, F. P. La Mantia, G. Marrucci, and G. Titomanlio, A Non-linear Viscoelastic Model with Structure-dependent Relaxation Times I. Basic Formulation, J. Non-Newt. Fluid Meek, 1, 125-146 (1976). 

S. Shrivastava and J. Tang, Large Deformation Finite Element Analysis of Non-linear Viscoelastic Membranes With Reference to Thermoforming, J. Strain Anal. 28, 31 (1993). 

The quantities r and r] in equation (8.34) depend on the invariants of the tensor rik in accordance with equation (8.32). We ought to note that the behaviour of a non-linear viscoelastic liquid in a non-steady state would be different, if a dependence of the material parameters r and r] on the tensor velocity gradients or on the stress tensor is assumed. This is a point which is sometimes ignored. In any case, if r and r) are constant, equation (8.34) belongs to the class of equations introduced and investigated by Oldroyd (1950). 

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