Elastic constants definition

Here E is Young modulus. Comparison with Equation (3.95) clearly shows that the parameter k, usually called spring stiffness, is inversely proportional to its length. Sometimes k is also called the elastic constant but it may easily cause confusion because of its dependence on length. By definition, Hooke s law is valid when there is a linear relationship between the stress and the strain. Equation (3.97). For instance, if /q = 0.1 m then an extension (/ — /q) cannot usually exceed 1 mm. After this introduction let us write down the condition when all elements of the system mass-spring are at the rest (equilibrium) ... 

Note A flexo-electric domain occurs when Ae< Anel/K where e is the flexo-electric coefficient and K is the elastic constant, assuming Ki = Kt, = K and e = -e = e (see Definitions 5.3 and 5.16). 

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). 

To construct the elastic Green function for an isotropic linear elastic solid, we make a special choice of the body force field, namely, f(r) = fo5(r), where fo is a constant vector. In this case, we will denote the displacement field as Gik r) with the interpretation that this is the component of displacement in the case in which there is a unit force in the k direction, fo = e. To be more precise about the problem of interest, we now seek the solution to this problem in an infinite body in which it is presumed that the elastic constants are uniform. In light of these definitions and comments, the equilibrium equation for the Green function may be written as... 

The full definition of the adiabatic elastic constants is given by... 

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. 

A definition of these angles is given in Fig, 1, The deformation profile is dependent on the dielectric constants C and Ej., the elastic constants for splay, twist and bend Kn, K22> 33 the total twist (90-23q), the tilt angle at the surface of the substrates ao, and the applied voltage. The optical response depends in addition on the refractive indices ne and viq and the ratio of wavelength to cell thickness. For display applications a finite tilt at the surfaces is required to avoid areas of opposite tilt. Therefore the deformation profiles are calculated for various combinations of K33/K11, Ae/ej and using Berreman s program. All calculations are performed for 10 im cells and a pretilt ao=l°. 

Other definitions of elastic constants are sometimes used (2). The Lame constants k and p are related to K and G in the following manner ... 

Consider a material that is symmetric about two planes the X1-X2 and JC2-JC3 planes as shown in Fig. 11.6. It must be expected that the elastic constants in both coordinate systems (unprimed and primed) are identical, i.e. Cy = C y. The definition of the coordinate systems leads to... 

The relationships between elastic constants which must be satisfied for an isotropic material impose restrictions on the possible range of values for the Poisson s ratio of -1 < v <. In a similar manner, there are restrictions in orthotropic and transversely isotropic materials. These constraints are based on considerations of the first law of thermodynamics [15]. Moreover, these constraints imply that both the stiffness and compliance matrices must be positive-definite, i.e. each major diagonal term of both matrices must be greater than 0. 

For crystals with a triclinic structure, which possess only a center of symmetry and exhibit the most general elastic anisotropy, a complete characterization requires all 21 independent constants. The existence of a higher degree of crystal symmetry can further reduce the number of independent elastic constants needed for proper description. In such cases, some elastic constants may vanish and members of some subsets of constants may be related to each other in some definite way, depending on the crystal symmetry. Finally, it should be noted that, while the total number of material constants required to characterize a material of a certain class is independent of the coordinate axes used to represent components of stress or strain, the values of particular components of Qj do depend on the material reference axes chosen for their representation. 

The main approximation of the FEAt coupling method is therefore given by the truncation of the Taylor series expansion [1]. Within the framework of linear local elasticity theory, it is enough to retain (and match) terms only up to the second order. However, if nonlinear elastic effects need to be included, third-order elastic constants must also be matched. Moreover, because the first-order elastic constants in the continuum are zero by definition, such a matching condition requires the interatomic potential to yield zero stress in a perfect lattice. 

Hence, the stated above results demonstrate nonzero contribution of noncrystalline regions in yield stress even for such semicrystalline polymers, which have devitrificated amorphous phase in testing conditions. At definite conditions noncrystalline regions contribution can be prevailed. Polymers yield stress and elastic constants proportionality is not a general rule and is fulfilled only at definite conditions. 

We will now consider in more detail some of the alignment or director field patterns around different defect structures in chiral nematics. Using the simple one elastic constant approximation (i.e., k as for the nematic case above) and the definition of the chiral director (i.e., n=(cos0, sin0, 0), 6=kz, and 0=0 see Eq. (1)) in the free energy density expression, (Eq. 2) gives... 

J-g -K ) for alumina and 55.8 J-mof -K ( = 0.453 J-g -K" ) for zirconia. These values are in satisfactory agreement with literature values [Munro 1997, NIST 2002, Salmang Scholze 1982]. With this input information at hand. Equations (40) and (41) (the latter in connection with approximate values for the shear and bulk moduli, cf Table 8 below for the definite values) can now be used to obtain estimates for the differences that have to be expected at room temperature between adiabatic elastic constants (measured via dynamic techniques) and isothermal elastic constants (measured via static techniques). For alumina and zirconia... 

There are various kinds of elastic constants, some of which are illustrated in Fig. 20. They are correlated with one another by their definitions, as shown in Table 4. When two independent kinds of constants are given, we can derive other elastic constants. 

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