Unique Poisson’s ratio

The problem will be solved for the case where the viscoelastic half-plane is characterized by a discrete spectrum model (Sect. 1.6). The more general continuous spectrum model is discussed by Golden (1977). The proportionality assumption (Sect. 1.9) will be adopted for the material so that a unique Poisson s ratio exists. Therefore, from (1.6.25, 28, 29), (3.5.20) and (3.5.22), we have... 

Problem 3.9.1 Show that if a constant load is applied at = 0 to a viscoelastic material with unique Poisson s ratio, the creep function of which tends to a finite value, then the contact region expands monotonically to some finite interval. Recall (3.2.12). 

We now explore these relationships for the case of the standard linear solid. A unique Poisson s ratio v will be assumed. We have, from (3.1.15), (3.2.12) and (1.6.2P),... 

Strictly, (4.6.1) should be an inequality stating that the left-hand side is greater than or equal to the right, in which case conditions (4.6.16- 18) become inequalities. These conditions have the same form as the Griffith criterion for crack extension for an elastic body with which is an instantaneous inverse modulus, replacing the elastic inverse modulus. If a unique Poisson s ratio exists, then... 

Problem 5.4.1 If the material has a unique Poisson s ratio and the shear creep function of the material has the step function form... 

Typical materials for the electrolyte are YSZ, samaria doped ceria (SDC), and LaGaC>3. The intrinsic property of the thermal expansion behavior of an electrolyte depends only on the material species. However, the other mechanical properties (Young s modulus, Poisson s ratio, and strength) depend on the morphology through the manufacturing processes. Accordingly, the reported mechanical properties are not unique. The reported thermal expansion coefficient (TEC) and other mechanical properties for the electrolyte materials are listed in Table 10.1. 

For an isotropic body, there are only two stress components that are independent of the other. This means that while different loadings and strains can be imposed, the material constants relating stress and strain are not all unique. There are six common elastic constants (Table 6.1), including Poisson s ratio, defined as the ratio of the lateral strain accompanying a longitudinal strain, V = -eiilsn- Since only two are unique for an isotropic body, each elastic constant can be expressed as a function of any other pair these expressions are tabulated in Table 6.1 in terms of Poisson s ratio (Mott and Roland, 2009). 

From (4.8.9), we see that (t) is positive, since k(t) is a positive function (this is clear at least when Poisson s ratio is unique, since then k(t) is related to the derivative of the creep function). Similarly, remembering (4.8.7), we have that N (t) is negative so the right-hand side of (4.8.12) is negative and decreasing. Therefore, no solution can occur for positive x. It will occur first at x = -c, since this is the point on the negative axis at which the function on the left is maximum. Therefore, the condition is... 

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