Constants, elastic compliance elasticity

For isotropic elastic solids there are only two independent elastic constants, or compliances. While Young s modulus E and the shear modulus // are the most widely used, we shall choose as the two physically independent pair of moduli the shear modulus /i and the bulk modulus K, where the first gauges the shear response and the second the bulk or volumetric response. However, in stating the linear elastic response in the equations below we still choose the more compact pair of E and //. Thus, for the six strain elements we have... 

Table 7.17 Material coefficients of PVDF polymer thin film (Nalwa 1995). Elastic constants and compliances, piezoelectric, pyroelectric and thermal expansion coefficients are listed.

The atomic arrays of intermetallic compounds may affect the orientation dependence of the elastic constants. Table 4 shows the ratio of the interatomic distance in the [ijk] direction to that in the [001] direction, and the associated stiffness-constant and compliance-constant ratios for various intermetallics (Nakamura, 1991a). Here, andS[j, t] represent c,... 

In Section 2.2, the stress-strain relations (generalized Hooke s law) for anisotropic and orthotropic as well as isotropic materials are discussed. These relations have two commonly accepted manners of expression compliances and stiffnesses as coefficients (elastic constants) of the stress-strain relations. The most attractive form of the stress-strain relations for orthotropic materials involves the engineering constants described in Section 2.3. The engineering constants are particularly helpful in describing composite material behavior because they are defined by the use of very obvious and simple physical measurements. Restrictions in the form of bounds are derived for the elastic constants in Section 2.4. These restrictions are useful in understanding the unusual behavior of composite materials relative to conventional isotropic materials. Attention is focused in Section 2.5 on stress-strain relations for an orthotropic material under plane stress conditions, the most common use of a composite lamina. These stress-strain relations are transformed in Section 2.6 to coordinate systems that are not aligned with the principal material... 

The biaxially oriented PET sheets have been extensively studied with regard to their mechanical anisotropy and all nine independent elastic constants have been determined by a variety of experimental techniques 38,39). The complete set of compliances for a one-way drawn sheet of draw ratio 5 1 is shown in Table 7. It is interesting to note that these compliances clearly reflect the two major structural features, the high chain axis orientation and the preferential orientation of the benzene rings... 

Even in cases where the rigid polymer forms the continuous phase, the elastic modulus is less than that of the pure matrix material. Thus two-phase systems have a greater creep compliance than does the pure rigid phase. Many of these materials craze badly near their yield points. When crazing occurs, the creep rate becomes much greater, and stress relaxes rapidly if the deformation is held constant. 

Table 8 The basic elastic constants g and ec, the highest filament values of the modulus ( ) and the strength (q,), together with the average values of the creep compliance (/(f)) at 20 °C (ratio of creep rate and load stress) and the interchain bond for a variety of organic polymer fibres... 

The electric properties of polymers are also related to their mechanical behavior. The dielectric constant and dielectric loss factor are analogous to the elastic compliance and mechanical loss factor. Electric resistivity is analogous to viscosity. Polar polymers, such as ionomers, possess permanent dipole moments. These polar materials are capable of storing... 

Equations (6) and (7) express these relationships. are the elastic compliance constants OC are the linear thermal expansion coefficients 4 and d jj,are the direct and converse piezoelectric strain coefficients, respectively Pk are the pyroelectric coefficients and X are the dielectric susceptibility constants. The superscript a on Pk, Pk, and %ki indicates that these quantities are defined under the conditions of constant stress. If is taken to be the independent variable, then O and are the dependent quantities ... 

Using the elastic compliances saP which are related to the elastic constants by (see e.g. Barron 1998)... 

The coefficients Cn are called elasticity constants and the coefficients Su elastic compliance constants (Azaroff, 1960). Generally, they are described jointly as elasticity constants and constitute a set of strictly defined, in the physical sense, quantities relating to crystal structure. Their experimental determination is impossible in principle, since Cu = (doildefei, where / i, and hence it would be necessary to keep all e constant, except et. It is easier to satisfy the necessary conditions for determining Young s modulus E, when all but one normal stresses are constant, since... 

Generally, the elastic properties of crystals should be described by 36 elasticity constants Cit but usually a proportion of them are equal to zero or are interrelated. It follows that in crystals, the tensors (2.6) and (2.7) are symmetric tensors, owing to which the number of elastic compliance coefficients is reduced, e.g., in the triclinic configuration, from 36 to 21 (Table 2.1). With increasing symmetry, the number of independent co-... 

The elastic constants derived by Van Fo Fy and Savin are as follows. (The symmetry axis is 3, c is the concentration of the circular reinforcing phase in a hexagonal array. The compliance constants Sy are quoted)... 

The set of constitutive parameters contains the (drained) elastic volumetric compliance C and two poroelastic constants the Biot stress coefficient b, and the unconstrained storage coefficient Sa = d(/dp a which can be expressed as So- = bB 1C ([13]), where B is the Skempton pore pressure coefficient. The other three parameters, a, f3, and 7 quantify the physico-chemical interactions. Both a and (3 are constrained to vary from 0 when there is no chemical interaction to 1 when the salt ions are trapped in the pore network (this limiting case is referred to as the perfect ion exclusion membrane model ). The coefficient 7 can simply be approximated by 7 x0/n, where n is the porosity of the shale. 

An excellent reference describing appropriate ways of measuring the piezoelectric coefficients of bulk materials is the IEEE Standard for Piezoelectricity [1], In brief, the method entails choosing a sample with a geometry such that the desired resonance mode can be excited, and there is little overlap between modes. Then, the sample is electrically excited with an alternating field, and the impedance (or admittance, etc.) is measured as a function of frequency. Extrema in the electrical responses are observed near the resonance and antiresonance frequencies. As an example, consider the length extensional mode of a vibrator. Here the elastic compliance under constant field can be measured from... 

Where sE, sD are the elastic compliances and the superscripts denote the parameters held constant similarly sx denotes the permittivity measured at constant stress (the usual condition) to measure the permittivity at constant strain ex is more difficult to accomplish. 

The elastic compliance s, (and stiffness c) has very different values depending upon whether the electric field E within the material is maintained at zero (the component is short-circuited ), or whether the electric displacement D remains constant (the component is open-circuited ). The short circuit value is specified by the superscript E and the open circuit condition by the superscript D. Similarly the permittivity can be measured with the specimen free to deform, that is at constant stress sx, or in the clamped state,... 

By means of these elastic constants and the orientation distribution of the symmetry axes with respect to the fibre axis, the Compliance (S = 1/E = reciprocal modulus) of the fibre could be calculated. 

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