Shear modulus definition

Let us examine the instability oi strained thin films. In Fig. 3, thin films of30 ML are coherently bonded to the hard substrates. The film phase has a misfit strain, e = 0.01, relative to the substrate phase, and the periodic length is equal to 200 a. The three interface energies are identical to each other = yiv = y = Y Both phases are elastically isotropic, but the shear modulus of the substrate is twice that of the film (p = 2p). On the left-hand side, an infinite-torque condition is imposed to the substrate-vapor and film-substrate interfaces, whereas torque terms are equal to zero on the right. In the absence of the coherency strain, these films are stable as their thickness is well over 16 ML. With a coherency strain, surface undulations induced by thermal fluctuations become growing waves. By the time of 2M, six waves are definitely seen to have established, and these numbers are in agreement with the continuum linear elasticity prediction [16]. 

The problem of definition of modulus applies to all tests. However there is a second problem which applies to those tests where the state of stress (or strain) is not uniform across the material cross-section during the test (i.e. to all beam tests and all torsion tests - except those for thin walled cylinders). In the derivation of the equations to determine moduli it is assumed that the relation between stress and strain is the same everywhere, this is no longer true for a non-linear material. In the beam test one half of the beam is in tension and one half in compression with maximum strains on the surfaces, so that there will be different relations between stress and strain depending on the distance from the neutral plane. For the torsion experiments the strain is zero at the centre of the specimen and increases toward the outside, thus there will be different torque-shear modulus relations for each thin cylindrical shell. Unless the precise variation of all the elastic constants with strain is known it will not be possible to obtain reliable values from beam tests or torsion tests (except for thin walled cylinders). 

The case for increased organization in hydrophobic films caused by nonhydrogen-bonding solvents is not as well documented. It is difficult for us to see how benzene can increase the shear modulus of paraffin wax or how tetrahydrofuran can do the same for poly (vinyl chloride). It is possible that we are dealing with a van der Waals bonding in swollen polymers in these cases but much more evidence would be needed before a definite theory can be advanced. 

It is interesting to note the similarity between the definitions of the shear viscosity (Equation 13.1) and the shear modulus (Equation 13.2), which leads to a relationship (Equation 13.3) between r and G in terms of the ratio of the shear strain (y) divided by the shear rate (y). 

Most polycrystalline solids are considered to be isotropic, where, by definition, the material properties are independent of direction. Such materials have only two independent variables (that is elastic constants) in matrix (7.3), as opposed to the 21 elastic constants in the general anisotropic case. The two elastic constants are the Young modulus E and the Poisson ratio v. The alternative elastic constants bulk modulus B and shear modulus /< can also be used. For isotropic materials, n and B can be found from E and t by a set of equations, and on the contrary. 

In the first set no reliable values were obtained for the shear modulus. Also the tests on polyurethane were rather unsuccessful. In the second set more reliable results were obtained. However, since the number of specimens in each test series was limited, one should consider the results in Table 5 only as indicative and to save time they should not be used as definitive material data. 

The development of the physical chemistry of rubber was greatly aided by the clear definition of an "ideal" state for this material. An ideal rubber is an amorphous, isotropic solid. The liquidlike structure of rubber was discovered very soon after the technique of X-ray scattering was developed. An isotropic material is characterized by physical properties that do not depend on the orientation of the sample. The deformation of an isotropic solid can be characterized by only two unique moduli the modulus of compression, K, and the shear modulus, G. A solid is characterized by equilibrium dimensions that are functions of temperature, pressure, and the externally imposed constraints. It is convenient to define a shape vector, L, whose components are the length, width, and height of a rectangular parallelepiped. For a system with no external constraints, the shape vector can be expressed as ... 

Eiastic Materiai Functions. The material functions that are used here for an isotropic, elastic material are the shear modulus G, the extensional modulus E, the bulk modulus K, and the Poisson s ratio v. However, any two of these provides the full set of information needed to describe such a material, as they are not all independent. For simple deformations, the definitions of these moduli are... 

Adhesive systems can be divided into rigid, flexible structural and elastic bonding groups. A fourth group comprises sealants. These adhesive and sealant systems have a different, reversible elastic deformation and shear modulus (Fig. 6). Reversible elastic deformation is explained in more detail in the Glossary/ Definition of Terms at the end of this chapter. 

Elastic modulus is a quantitative measure of the stiffness or rigidity of a material. For example, for homogeneous isotropic substances in tension, the strain (e) is related to the applied stress (o) by the equation E = o/e, where E is defined as the elastic modulus. A similar definition of shear modulus (g) applies when the strain is shear. 

Because the material properties are direction-dependent in a cubic crystal, they have to be stated together with the corresponding direction. According to the definition, the load direction has to be stated for Young s modulus Ei. Because the shear stress Tij and shear strain -y j have two indices, two indices are needed for the shear modulus Gij. Poisson s ratio relates strains in two directions. Here the second index j denotes the direction of the strain that causes the transversal contraction in the direction marked by the first index i eu = If the coordinate system is aligned with the axes... 

According to Eq. (11) the cell ensemble should be characterised by the universal mean relaxation mode time , i.e. by the mean relaxation time = Tkin (Eq. 9). Hence, the distortions of coc produced during cell growth should bring about a defined and typical frequency dispersion . To describe this we define the complex density n(a>c) (in analogy to the definition of a complex shear modulus) by... 

There is a single dimensionless group, XVjL, which is known as the Weissenberg number, denoted by various authors as We or Wi. (We is more common, but it can lead to confusion with the Weber number, so Wi will be used here.) The shear rate in any viscometric flow is equal to a constant multiplied by V/L, so it readily follows that the ratio of the first normal stress difference to the shear stress is equal to twice that constant multiphed by Wi. Hence, Wi can be interpreted as the relative magnitude of elastic (normal) stresses to shear stresses in a viscometric flow. The ratio of the shear stress to the shear modulus, G, is sometimes known as the recoverable shear and is denoted Sr. Sr differs from Wi for a Maxwell fluid only by the constant that multiplies F jL to form the shear rate for a given flow. In fact, many authors define Wi as the product of the relaxation time and the shear rate, in which case Wi = Sr. It is important to keep the various definitions of Wi in mind when comparing results from different authors. 

In the theories for dilute solutions of flexible molecules based on the bead-spring model, the contribution of the solute to the storage shear modulus, loss modulus, or relaxation modulus is given by a series of terms the magnitude of each of which is proportional to nkT, i.e., to cRTjM, as in equation 18 of Chapter 9 alternatively, the definition of [C ]y as the zero-concentration limit of G M/cRT (equations 1 and 6 of Chapter 9) implies that all contributions are proportional to nkT. Each contribution is associated with a relaxation time which is proportional to [ri Ti)sM/RT-, the proportionality constant (= for r i) depends on which theory applies (Rouse, Zimm, etc.) but is independent of temperature, as is evident, for example, in equation 27 of Chapter 9. Thus the temperature dependence of viscoelastic properties enters in four variables [r ], t/j, T explicitly, and c (which decreases slightly with increasing temperature because of thermal expansion). 

Returning to the static moduH, the subject of this section. Fig. 12.1 serves as a reminder of the definitions of the engineering quantities bulk modulus (K), shear modulus (G), Young s modulus E), and Poisson s ratio (v), which in an isotopi-cally elastic soHd (fine, randomly textured grains) are simply related to each other according to ... 

E ( denotes the axial elastic modulus of the fiber, where as V)2f is the longitudinal Poisson s ratio of the fiber, determined by measuring the radial contraction under an axial tensile load in the fiber axis direction and Gm denotes the matrix shear modulus. It should be noted that the negative sign in the expression for the shear stress is introduced to be consistent with the definition of an interfacial shear stress in classical theory of elasticity. The radial stress at the interface is given by ... 

When one molecnle grows to span the entire network, the system is at the gel point. The gel point is best determined from measurements of flow behaviour. It is the point at which the zero-shear viscosity of the system becomes infinite and the system develops a shear modulus. However, these definitions are not very helpfnl practically, becanse it is difficult to measure very large values of viscosity or a very small shear modulus. A more useful definition relies on measnrements of the dynamic shear modnli, the gel point being defined by the condition G = G". The gel point is also signalled by the onset of insolnbility of the three-dimensional network. 

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