Elastic constants isotropic 80 83

Here Fq is tire free energy of the isotropic phase. As usual, tire z direction is nonnal to tire layers. Thus, two elastic constants, B (compression) and (splay), are necessary to describe tire elasticity of a smectic phase [20,19, 86]. 

Mechanical Properties. The hexagonal symmetry of a graphite crystal causes the elastic properties to be transversely isotropic ia the layer plane only five independent constants are necessary to define the complete set. The self-consistent set of elastic constants given ia Table 2 has been measured ia air at room temperature for highly ordered pyrolytic graphite (20). With the exception of these values are expected to be representative of... 

In solids of cubic symmetry or in isotropic, homogeneous polycrystalline solids, the lateral component of stress is related to the longitudinal component of stress through appropriate elastic constants. A representation of these uniaxial strain, hydrostatic (isotropic) and shear stress states is depicted in Fig. 2.4. Such relationships are thought to apply to many solids, but exceptions are certainly possible as in the case of vitreous silica [88C02]. 

Fig. 2.4. Within the elastic range it is possible to relate uniaxial strain data obtained under shock loading to isotropic (hydrostatic) loading and shear stress. Such relationships can only be calculated if elastic constants are not changed with the finite amplitude stresses.

In the perfectly elastic, perfectly plastic models, the high pressure compressibility can be approximated from static high pressure experiments or from high-order elastic constant measurements. Based on an estimate of strength, the stress-volume relation under uniaxial strain conditions appropriate for shock compression can be constructed. Inversely, and more typically, strength corrections can be applied to shock data to remove the shear strength component. The stress-volume relation is composed of the isotropic (hydrostatic) stress to which a component of shear stress appropriate to the... 

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... 

An isotropic and non-magnetie amorphous alloy has only two independent second order elastic constants. The other elastic moduli are related through the equations [30] ... 

It can be shown that for the cross-terms 221 = 2i2, 2si = 2b. and so on, so that of the initial 36 values, there are only 21 independent elastic constants necessary to completely define an anisotropic volume without any geometrical symmetry (not to be confused with matrix symmetry). The number of independent elastic constants decreases with increasing geometrical symmetry. For example, orthorhombic symmetry has 9 elastic constants, tetragonal 6, hexagonal 5, and cubic only 3. If the body is isotropic, the number of independent moduli can decrease even fmther, to a limiting... 

Upper and lower bounds on the elastic constants of transversely isotropic unidirectional composites involve only the elastic constants of the two phases and the fiber volume fraction, Vf. The following symbols and conventions are used in expressions for mechanical properties Superscript plus and minus signs denote upper and lower bounds, and subscripts / and m indicate fiber and matrix properties, as previously. Upper and lower bounds on the composite axial tensile modulus, Ea, are given by the following expressions ... 

For isotropic materials there are only two independent constants, which may be taken as Cn and c44 (the relationship between the various isotropic elastic constants is given in Table 6.1 at the end of this section). The isotropic stiffness tensor may be obtained by substituting c12 = 0n — 2c44 in the cubic stiffness matrix. 

In many cases considerable simplification is possible, because of the constraints imposed on the number of independent elastic constants. For cubic symmetry, for which the elastic stiffness tensor has only three independent constants as given in (6.29), the elements of T, are given in Table 11.1(b), and for hexagonal symmetry the elements are given in Table 11.1(c). If c12 = Cn - 2c44 were to be substituted in Table 11.1(b) the isotropic elements would... 

The fundamental quantities in elasticity are second-order tensors, or dyadicx the deformation is represented by the strain thudte. and the internal forces are represented by Ihe stress dyadic. The physical constitution of the defurmuble body determines ihe relation between the strain dyadic and the stress dyadic, which relation is. in the infinitesimal theory, assumed lo be linear and homogeneous. While for anisotropic bodies this relation may involve as much as 21 independent constants, in the euse of isotropic bodies, the number of elastic constants is reduced lo two. 

We may then write the arrays for the elastic constants for various symmetries, the two most useful being hexagonal (also transverse isotropy as in fibre symmetry) and isotropic. Hexagonal gives ... 

For anisotropic materials torsion is discussed in the books by Love, Lekhnitskii175 and Hearmon185. The torque M now depends not upon one elastic constant only, as in the isotropic case, but upon two. This makes the determination of shear modulus by a torsion test a difficult task and requires careful experimentation. Early work on this for polymers was done by Raumann195, by Ladizesky and Ward205 and by Arridge and Folkes165. 

In various experiments different elastic constants are being determined with a torsion pendulum, for instance, the shear modulus, G, is measured, with creep or vibrations in elongation or in bending the Young s modulus, (tensile modulus), E. For an isotropic material the relation between E and G is as follows ... 

Elastic constants for elastically isotropic polycrystalline materials. A measure of the manner in which a polycrystalline body responds to small external forces in the elastic regime. 

The big difference between normal isotropic liquids and nematic liquids is the effect of anisotropy on the viscous and elastic properties of the material. Liquid crystals of low molecular weight can be Newtonian anisotropic fluids, whereas liquid crystalline polymers can be rate and strain dependent anisotropic non-Newtonian fluids. The anisotropy gives rise to 5 viscosities and 3 elastic constants. In addition, the effective flow properties are determined by the flow dependent and history dependent texture. This all makes the rheology of LCPs extremely complicated. 

Amorphous solids and polycrystalline substances composed of crystals of arbitrary symmetry arranged with a perfectly disordered or random orientation are elastically isotropic macroscopically (taken as a whole). They may be described by nine elastic constants, which may be reduced to two independent (effective) elastic constants. 

With elastically anisotropic materials the elastic behavior varies with the crystallographic axes. The elastic properties of these materials are completely characterized only by the specification of several elastic constants. For example, it can be seen from Table 10.3 that for a cubic monocrystal, the highest symmetry class, there are three independent elastic-stiffness constants, namely, Cn, C12, and C44. By contrast, polycrystalline aggregates, with random or perfectly disordered crystallite orientation and amorphous solids, are elastically isotropic, as a whole, and only two independent elastic-stiffness coefficients, C44 and C12, need be specified to fully describe their elastic response. In other words, the fourth-order elastic modulus tensor for an isotropic body has only two independent constants. These are often referred to as the Lame constants, /r and A, named after French mathematician Gabriel Lame (1795-1870) ... 

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