Elastic constants restrictions

The restrictions on engineering constants can also be used in the solution of practical engineering analysis problems. For example, consider a differential equation that has several solutions depending on the relative values of the coefficients in the differential equation. Those coefficients in a physical problem of deformation of a body involve the elastic constants. The restrictions on elastic constants can then be used to determine which solution to the differential equation is applicable. 

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... 

Because there is a large literature, we restrict ourselves to an interesting example, useful for understanding the axial support of thin mirrors. Consider a thin circular plate of radius a and thickness h, with elastic constant E and Poissons ratio v. Let this plate be axially loaded by gravity and assume we will support this plate against this load by N supports. This is shown in Fig. 2. 

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. 

The engineering properties of interest are the elastic constants in the principal material coordinates. If we restrict ourselves to transversely isotropic materials, the elastic properties needed are Ei, Ei, v, and G23, i.e. the axial modulus, the transverse modulus, the major Poisson s ratio, the in-plane shear modulus and the transverse shear modulus, respectively. All the elastic properties can be obtained from these five elastic constants. Since experimental evaluation of these parameters is costly and time-consuming, it becomes important to have analytical models to compute these parameters based on the elastic constants of the individual constituents of the composite. The goal of micromechanics here is to find the elastic constants of the composite as functions of the elastic constants of its constituents, as... 

Tadoroko et al. provide a new method for the calculation of the elasticity of an isolated helical chain and the distribution of strain energy to the internal co-ordinates under conditions that the rotational angle per monomer is constant. For the calculation of three-dimensional elastic constants, the space group symmetry of the unit cell may be used to reduce the memory size required for polymers such as poly(vinyl alcohol) which have unsymmetrical repeat units. The constraints imposed upon tie molecules in semi-crystalline polyethylene are considered to restrict them to three conformations, the deforma-... 

For all of the cases of substrate curvature induced by film stress that were considered in Section 2.1, it was assumed that both the film and substrate materials were isotropic. This provided a basis for a relatively transparent discussion of curvature phenomena and it led to results which have proven to be broadly useful. However, there are situations for which some understanding of the influence of material anisotropy of the film material or the substrate material, or perhaps of both materials, is important to know. Therefore, in this section, representative results on the influence of material anisotropy on substrate curvature are included. Results are established for two particular curvature formulas. In the first case, the film is presumed from the outset to be very thin compared to the substrate. Furthermore, the substrate is assumed to be isotropic and the film is considered to be generally anisotropic. In the second case discussed, no restriction is placed on the thickness of the film relative to the substrate, but both materials are assumed to be anisotropic. However, to obtain expressions for curvature which are not too complex to be interpretable, attention is limited to cases for which both the film and substrate materials are orthotropic, that their axes of orthotropy aligned with each other and that one axis of or-thotropy of each material is normal to the film-substrate interface. There is no connection between the values of orthotropic elastic constants of the two materials. Consideration of these two cases illustrates the most useful approaches for anisotropic materials generalizations for cases of greater complexity are evident. 

We could use this as a starting point for finite elasticity. It would be desirable to reduce the number of elastic constants a, b, and so on, by considerations such as material symmetry. Rather than developing a general theory of finite elasticity, however, we will introduce appropriate restrictions at an early stage, as appropriate for a representation of the behaviour of rubbers. The principal restrictions are driven by the simplifications that ... 

Studies of mechanical anisotropy in polymers have for the most part been restricted to drawn fibres and uniaxially drawn films, both of which show isotropy in a plane perpendicular to the direction of drawing. The number of independent elastic constants is reduced to five [3, p. 138]. Choosing the 3 direction as the axis of symmetry, the compliance matrix sy reduces to... 

Before giving analytical expressions for the director deformations in Freedericksz cells, we will summarize the magnetic and electrical methods. The advantage of electro-optical measurements is that the cell thickness does not enter the equations and is therefore ruled out as an error source. Furthermore, the electric field can always be considered strictly perpendicular to the sample plane. On the other hand, in the electric method conductivity effects can influence the measurements and exact knowledge of and is required to extract the second elastic constant from the birefringence or capacitance characteristics. Moreover, the electric measurement is restricted... 

I. Viscoelastic Solutions in Terms of Elastic Solutions. The fundamental result is the Classical Correspondence Principle. It is based on the observation that the time Fourier transform (FT) of the governing equations of Linear Viscoelasticity may be obtained by replacing elastic constants by corresponding complex moduli in the FT of the elastic field equations. It follows that, whenever those regions over which different types of boundary conditions are specified do not vary with time, viscoelastic solutions may be generated in terms of elastic solutions that satisfy the same boundary conditions. In practical terms this method is largely restricted to the non-inertial case, since then a wide variety of elastic solutions are available and transform inversion is possible. 

The right-hand side of (3.43) must always be non-negative for solutions 0 because f(0) is positive this results in restrictions upon possible solutions depending on the relative magnitudes of the Frank elastic constants. Notice also that (3.43) is undefined at and cannot be positive at 0rn = 0. 

To fulfil the requirement (6.14) the energy w must be non-negative, as in nematic theory. This leads to restrictions on the values of the elastic constants. Notice that w given by equation (6.26) is actually a quadratic form in the five basic deformation operators. As an example, we follow Carlsson et al [32] and imagine that a deformation of the smectic can be made where only the distortions related to layer bending occur. If we set Xi = (b V x c) and X2 = (c V x b) then the energy terms related to the Aij terms can be written as the quadratic form... 

The preceding restrictions on engineering constants for orthotropic materials are used to examine experimental data to see if they are physically consistent within the framework of the mathematical elasticity model. For boron-epoxy composite materials, Dickerson and DiMartino [2-3] measured Poisson s ratios as high as 1.97 for the negative of the strain in the 2-direction over the strain in the 1-direction due to loading in the 1-direction (v 2)- The reported values of the Young s moduli for the two directions are E = 11.86 x 10 psi (81.77 GPa) and E2 = 1.33x10 psi (9.17 GPa). Thus,... 

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