Linear viscoelastic solids

Here we describe the strain history with the Finger strain tensor C 1(t t ) as proposed by Lodge [55] in his rubber-like liquid theory. This equation was found to describe the stress in deforming polymer melts as long as the strains are small (second strain invariant below about 3 [56] ). The permanent contribution GcC 1 (r t0) has to be added for a linear viscoelastic solid only. C 1(t t0) is the strain between the stress free state t0 and the instantaneous state t. Other strain measures or a combination of strain tensors, as discussed in detail by Larson [57], might also be appropriate and will be considered in future studies. A combination of Finger C 1(t t ) and Cauchy C(t /. ) strain tensors is known to express the finite second normal stress difference in shear, for instance. 

This material is a linear viscoelastic solid and is described by the multiple Maxwell model with an additional term, the spring elasticity... 

Note 3 Comparison with the general definition of a linear viscoelastic behaviour shows that the polynomial P(D) and Q D) are of order one, go = afpi, a = PiiP and po = adai. Hence, the standard linear viscoelastic solid is a solid (ago>0) having instantaneous elasticity (/"(D) and 0(D) are of the same order). 

Note 4 The standard linear viscoelastic solid may be represented by ... 

Note 5 The standard linear viscoelastic solid can be used to represent both creep and stress relaxation in materials in terms of single retardation and relaxation times, respectively. 

Note 3 The relaxation time of a standard linear viscoelastic solid is r= /p( = P a. ... 

Lockett FJ, "Non-linear Viscoelastic Solids", Acad Press, Inc, London, 1972. 

The models have been developed mainly for semi-crystalline polymers, which in general show the largest mechanical anisotropy, but some of the discussion is equally relevant to oriented non-crystalline polymers. Although an oriented polymer is strictly a non-linear viscoelastic solid (see Chapters 10 and 11) the present discussion is restricted to theoretical models which represent linear elastic or linear viscoelastic behaviour. 

Since by definition the viscoelastic material in shear exhibits both the behavior governed by equation 6 and than represented by equation 8, the constitutive relation for the linear viscoelastic solid can be written as... 

W. K. Waldron and A. S. Wineman, Shear and Normal Stress Effects in Finite Circular Shear of a Compressible Non-linear Viscoelastic Solid Int. J. Nonlinear Mech. 31,345-369(1996). 

Hooke s law describes the behaviour of a linear elastic solid and Newton s law that of a linear viscous liquid. A simple constitutive relation for the behaviour of a linear viscoelastic solid is obtained by combining these two laws ... 

Figure 4.2 (a) Deformation of an elastic solid (b) deformation of a linear viscoelastic solid... 

When an instantaneous strain is applied to an ideal elastic solid a frnite and constant stress will be recorded. For a linear viscoelastic solid subjected to a nominally instantaneous strain the initial stress will be proportional to the applied strain and will decrease with time (Figure 4.4), at a rate characterized by the relaxation time r. This behaviour is called stress relaxation. For amorphous linear polymers at high temperatures the stress may eventually decay to zero. In the following discussion we shall ignore transient behaviour. 

The creep deformation of a linear viscoelastic solid under constant stress can be represented by the model shown, in which a spring is in series with a Kelvin unit. 

For linear viscoelastic solids the torsional modulus is complex, and may be written as G = G + 1G2. When the damping is small, it is justified to replace Gi for G in Equation (5.1), hence... 

Lockett, F. J., Non-Linear Viscoelastic Solids, Academic Press, London, 1972. 

Gottenberg W, Christensen R (1972) Prediction of the transient response of a linear viscoelastic solid. J Appl Mech 6 448-450... 

Rajagopal K, Wineman A (2008) A quasi-correspondence principle for Quasi-Linear viscoelastic solids. Mech Time-Depend Mater 12 1-14... 

Bradshaw, R., Brinson, L.C., A Sign Control Method for Pitting and Interconverting Material Functions for Linearly Viscoelastic Solids Mech. of Time-Dependent Materials, Vol. 1, No. 1 / March, 1997, p. 85-108. 

The effect of applying a similar loading programme to a linear viscoelastic solid has several similarities (Figure 5.2(b)). In the most general case, the total strain e is the sum of three separate parts e, and ez. e and are often termed the immediate elastic deformation and the delayed elastic deformation respectively, ez is the Newtonian flow, that is that part of the deformation, which is identical with the deformation of a viscous liquid obeying Newton s law of viscosity. 

Figure 5.6 The creep behaviour of a linear viscoelastic solid.

Figure 11.4 (a) Loading programme, (b) deformation and (c) direct comparison of creep ec(t) and recovery edt) for a non-linear viscoelastic solid. 

Knauss, W.G., Dietmann, H. (1970) Crack propagation under variable load histories in linearly viscoelastic solids. Int. J. Eng. Sci. 8, 643-656... 

McCartney, L.N. (1978) Crack propagation in linear viscoelastic solids some new results. Int. J. Fract. 14, 547-554... 

Rivlin, R. S., "Non-Linear Viscoelastic Solids," SIAM Review,... 

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