Elastic stiffness modulus, complex

The second characterisation (b) is more often used in practice. In linear elastic multi-layer calculations, for instance, the absolute value of the complex modulus, lE" , which is equivalent to the stiffness modulus, , is used as input. 

The SC solutions appear to run into difficulties when there is a large elastic mismatch between the constituent phases, for example, at high concentrations of a rigid phase in a compliant matrix or of a porous phase in a stiff matrix. The latter situation will be discussed in Section 3.6. One approach to this problem is known as the Generalized Self-Consistent Approach, the concept behind which is illustrated in Fig. 3.14. Instead of a single inclusion in an effective medium, a composite sphere is introduced into the medium. As in the composite sphere assemblage discussed in the last section, the relative size of the spheres reflects the volume fraction, i.e., V = a bf as before. Interestingly, this approach leads to the HS bounds for the bulk modulus. The solution for the shear modulus is complex but can be written in a closed form. 

Modulus the ratio of stress to strain, which is a measure of stiffness of a polymer. A high modulus polymer is stiff and has very low elongation. Some systems, especially solutions of high molecular weight or cross-linked polymers, show complex behavior under stress. They posses an elastic modulus, G, and a loss modulus, G", representing the recoverable and irrecoverable strain, respectively. 

This offers the advantage of being a single real quantity describing the stiffness of the material, but information about the relative viscous and elastic character of the material is lost. Use of the complex modulus magnitude IG I is particularly prevalent in data characterizing measurements on fluids in shear. 

For both elastic and viscoelastic materials, the response of the contact ((X a) in Figure 2) is simply related to the contact radius a(t). Starting from equation 1, assuming the cantilever to be perfectly elastic with torsional stiffness Ke, and describing the contact response by a complex shear modulus G =, ... 

Permanent deformation The rutting resistance of the binder is represented by the stiffness of the binder at high temperatures that one would expect in use. This is represented by G /sin(5), where G is the complex shear modulus and 5 is the phase angle determined by the dynamic shear rheometry, DSR, measured at 10 rad s (1.59 Hz). The complex modulus can be considered as the total resistance of the binder to deformation under repeated shear, and consists of elastic modulus, G and loss modulus, G" (recoverable and non-recoverable components). The relative amounts of recoverable and non-recoverable deformation are indicated by the phase angle, 5. The asphalt binder will not recover or rebound from deformation if d = 90°. 

In almost all of these equations, there are two important input variables. One of these is the elastic modulus of the material, E, as described in Chapter 3. This is then combined with the stiffness of the structure, which is determined by its moment of inertia, or I. E and I can be measured (or calculated) in any given direction, or in any mode of motion. In bending applications, the applicable modulus of elasticity is the flexural modulus, and the moment of inertia will depend on the shape of the structure. While the equations to determine it can be complex, I is almost always based on the cube of the thickness (Figure 5.4). 

Additionally, the solution path could utilize the correspondence principle discussed previously, where the elastic solution is obtained and the real modulus or stiffness is replaced with the complex modulus or stiffness. However, let us explore the classic method of directly solving the equation of motion. For the first case, consider sinusoidal vibration of the base. 

---