A Standard Linear Solid

Assuming that an acceptable solution exists, say for T = T2, condition (4.8.16), determining the interval of crack closure, has the form 

As the value of R(T) begins to ascend, the crack will begin to reopen smoothly, but this will not be complete before 6 = 2n is reached. Beyond this point, the picture becomes increasingly complex and will not be explored. 

Finally, it is of interest to distinguish the class of solids for which (4.8.23) cannot be satisfied. This is case (ii), described above. The minimum value of R(T) occurs at 

Problem 4.8.1 Observe that the problem considered in this section provides an example of a receding contact problem as described in Sect. 2.9, and use it to illustrate the results given there. 

In this chapter, stress distributions and displacements are calculated for viscoelastic bodies loaded in plane strain and containing cracks. Conditions are derived for the extension of those cracks. 

---

For uniaxial tension performed at constant strain rate, how would the stiffness of a standard linear solid change with decreasing strain rate ... 

It is straightforward to calculate the response of a standard linear solid in a constant strain-rate test, and this is given by... 

This model is a convenient starting point in that it represents the simplest material which contains no special, degenerate features in a sense that will become clear when we discuss limiting cases of it - namely the Maxwell and Voigt materials. A standard linear solid has a relaxation function of the form... 

Problem 1,6.2 Show that for a model with one decay time, or a standard linear solid, relations (1.6.1 p) may be put in the form... 

VI. Alternative Expansion and Contraction. The general problem is discussed in Sect. 3.10 and the steady-state limit in Sect. 3.11. The detailed solution for a standard linear solid and a sinusoidal load is contained in (3.11.45-60). Numerical solutions may be obtained without difficulty. 

We now give more explicit expressions for these quantities in the case of a standard linear solid. Equation (3.11.12) gives that, for odd /,... 

The problem of a stationary crack (i.e. a crack that remains constant in length) subject to an alternating time-dependent normal applied load has been studied for a standard linear solid in Sect. 4.4. It is found that the crack is always either entirely open or entirely closed. Furthermore, in contrast with elastic theory, the crack remains open for some time after the load becomes compressive, and opens up again before the load becomes tensile. Also, the stress... 

These are in fact the same as for the non-inertial problem. The results (7.3.11 -13) for the stress ahead of the crack and the stress intensity factor have been given by Willis (1%7), for a standard linear solid, and Walton (1982) for a general material, using quite different methods to the one outlined here. 

III. Tearing Mode Crack Problem. This problem is solved, utilizing constraints imposed by Causality. The form of the tearing stress off the crack face on the line of fracture is given by (7.3.11) for the case where the velocity is in the range of speeds of sound of the medium [see (7.3.4)] and by (7.3.13a) for the subsonic case. The quantity is defined by (7.3.1 p) for a standard linear solid. The stress intensity factor for the two cases is given by (7.3.12) and (7.3.13b), respectively. 

It is apparent therefore that the Superposition Principle is a convenient method of analysing complex stress systems. However, it should not be forgotten that the principle is based on the assumption of linear viscoelasticity which is quite inapplicable at the higher stress levels and the accuracy of the predictions will reflect the accuracy with which the equation for modulus (equation (2.33)) fits the experimental creep data for the material. In Examples (2.13) and (2.14) a simple equation for modulus was selected in order to illustrate the method of solution. More accurate predictions could have been made if the modulus equation for the combined Maxwell/Kelvin model or the Standard Linear Solid had been used. 

Here the term ik is the retardation time. It is given by the product of the compliance of the spring and the viscosity of the dashpot. If we examine this function we see that as t -> 0 the compliance tends to zero and hence the elastic modulus tends to infinity. Whilst it is philosophically possible to simulate a material with an infinite elastic modulus, for most situations it is not a realistic model. We must conclude that we need an additional term in a single Kelvin model to represent a typical material. We can achieve this by connecting an additional spring in series to our model with a compliance Jg. This is known from the polymer literature as the standard linear solid and Jg is the glassy compliance ... 

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

The Standard Linear Solid Model combines the Maxwell Model and a like Hook spring in parallel. A viscous material is modeled as a spring and a dashpot in series with each other, both of which other, both of which are in parallel with a lone spring. For this model, the governing constitutive relation is ... 

Therefore under a constant stress, the modeled material will instantaneously deform to some strain, which is the elastic portion of the strain, and after that it will continue to deform and asynptotically approach a steady-state strain. This last portion is the viscous part of the strain. Although the Standard Linear Solid Model is more accurate than the Maxwell and Kelvin-Voigt models in predicting material responses, mathematically it returns inaccurate results for strain under specific loading conditions and is rather difficult to calculate. 

Since neither model adequately describes the behavior of real viscoelastic materials, a combination of the classic elements is often made to gain closer representation. The most common configuration is called the standard linear solid4 configuration, and it is illustrated in Figure 6.6. A more accurate representation of actual behavior can be obtained by a composite of multiple elements of the standard linear solid configuration into a multi-element model (Figure 6.7) with an array of coefficients for each element. 

The standard linear solid (SLS) is a more complicated model than the two previously considered. It combines series and parallel elements, as shown in fig. 7.9, and can describe both stress-relaxation and creep. For stress-relaxation the spring a remains at the original strain and only E), nd rj are involved in the relaxation. Hence r = r]/E, but the stress relaxes to eE, not to zero. For creep it can be shown that t = (l/ a + 1 Unlike the Voigt model, the SLS exhibits an immediate response, e = a/ E + E, because the two springs in parallel can extend immediately. Thus the SLS is a much better model than either of the simpler models. 

Fig. 9. Number of attractors correctly discovered from the same random set by a HEDA with a standard linear learning rule (square markers and solid error bars) and a HEDA trained with the log-Hebbian rule. In both cases, the target function contained n(2 Inn) attractors and the graph shows how many of them were found.

Figure 5.14 Common viscoelastic models a) Voigt/Kelvin model b) Zener model/standard linear solid.

The simplest way to obtain the behavior discussed in connection with Fig. 5.9 is to place a second spring in series with a Voigt model. This is shown in Fig. 5.14(b) and is known as the Zener model or standard linear solid. The constitutive equation is found by simply adding the strains from the spring and Voigt model (Eqs. (5.41) and (5.48))... 

Figure 5.15 Variation of modulus of standard linear solid as a function of time under a constant stress.

If the standard linear solid (SLS) is unloaded from a constant stress, the spring (modulus ,) closes immediately and the elastic strain is removed. The anelastic strain then decays to zero as the second spring closes the dashpot, i.e., there is complete recovery. Under the action of a constant strain, the SLS model will also show stress relaxation but, in this case, the time constant, Tf =rf /(E +E2). In applying a constant stress to the SLS model, the strain can be considered to lag behind the stress, both on loading and unloading. This lag concept is also very important in considering the effect of a dynamic stress or strain. 

Figure 5.16 Stress-strain behavior for standard linear solid subjected to a sinusoidal stress. The system is elastic with a high modulus at very high frequencies and a lower modulus at low frequencies. At intermediate frequencies, hysteresis develops and the loss passes through a maximum.

Some materials demonstrate anelastic behavior and this is often modeled by a spring in series with a parallel spring and dashpot unit (standard linear solid, SLS). 

The Zener model (or standard linear solid). The model may be represented as a spring in series with a Kelvin model, as in (a), or as a spring in parallel with a Maxwell model, as in (b). The significant properties inherent in the Zener model include (i) two time constants, one for constant stress and one for constant strain r (ii) an instantaneous strain at t>0 when subject to a step-function stress and (iii) full recovery following removal of the stress. For the Kelvin and Maxwell models, see Problem 4.8. 

Voigt element n. This is a Voight model which is a component, together with other Voight or Maxwell components, of a more complex viscoelastic model system, such as the standard linear solid. 

---