Maxwell model stress-strain relation

The Maxwell model consists of a spring and dashpot in series as shown in Figure 5.11(a). The equations for the stress-strain relations are... 

The Maxwell Model. In the above development, discussion moves from elastic behavior to viscoelastic descriptions of material behavior. In a simple sense, viscoelasticity is the behavior exhibited by a material that has both viscous and elastic elements in its response to a deformation or load. In early days, this was often represented by elastic or viscous mechanical elements combined in different ways (9-12). The simplest models are two element models that contain a viscous element (dashpot) and an elastic element (spring). The dashpot is assumed to follow a Newtonian fluid constitutive law in which the stress is related directly to the strain rate by the following expression ... 

To relate the viscoelastic behavior of plastics with an S-S curve the popular Maxwell model is used, this mechanical model is shown in Fig. 3.8. This model is useful for the representation of stress relaxation and creep with Newtonian flow analysis that can be related to plastic s non-Newtonian flow behavior. It consists of a spring [simulating modulus of elasticity (E)] in series with a dashpot of coefficient of viscosity (ij)- It is an isostress model (with stress the strain (e) being the sum of the individual strains in the spring and dashpot. 

The Maxwell model allows the approximation of elastic and creep strain under static stress as a function of time by assigning elastic compliance leading to instant elastic strain to the spring part of the model. The flow due to creep is represented by the linear slope related to the single dashpot element with linear viscous flow properties. 

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. 

These equations are often used in terms of complex variables such as the complex dynamic modulus, E = E + E", where E is called the storage modulus and is related to the amount of energy stored by the viscoelastic sample. E" is termed the loss modulus, which is a measure of the energy dissipated because of the internal friction of the polymer chains, commonly as heat due to the sinusoidal stress or strain applied to the material. The ratio between E lE" is called tan 5 and is a measure of the damping of the material. The Maxwell mechanical model provides a useful representation of the expected behavior of a polymer however, because of the large distribution of molecular weights in the polymer chains, it is necessary to combine several Maxwell elements in parallel to obtain a representation that better approximates the true polymer viscoelastic behavior. Thus, the combination of Maxwell elements in parallel at a fixed strain will produce a time-dependent stress that is the sum of all the elements ... 

Another way to introduce fractional derivatives is through rheological models of fractional order. In particular, the fractional Maxwell element corresponds to a spring in series with a fractional damper. The one-dimensional linear stress, <7, versus strain, e, relation of a spring in parallel with the fractional Maxwell element can expressed in terms of fractional derivatives [171], e.g.,... 

The models described in the preceding section are useful in developing mathematical relations between stress and strain in viscoelastic polymers and in giving insight to their response to creep, relaxation and other types of loading. Consider again the Maxwell fluid from Fig. 3.21,... 

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