Elastic compliance constants

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... 

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... 

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)... 

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... 

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. 

Here Ctju are the stilfness constants and Sijki are the compliance constants. They form two symmetric fourth-rank tensors with 81 elements inverse one to another. For the triclinic symmetry only 21 elements are independent because the strain and stress tensors are symmetric. Consequently the indices i,j and k, I can be permuted and also can be permuted one pair with another. For a crystal symmetry higher than triclinic the number of independent elastic constants is less than 21. 

Thus one would expect from a (6x6) matrix of the elastic stiffness coefficients (c,y) or compliance coefficients (sy) that there are 36 elastic constants. By the application of thermodynamic equilibrium criteria, cy (or Sjj) matrix can be shown to be symmetrical cy =cji and sy=Sji). Therefore there can be only 21 independent elastic constants for a completely anisotropic solid. These are known as first order elastic constants. For a crystalline material, periodicity brings in elements of symmetry. Therefore symmetry operation on a given crystal must be consistent with the representation of the elastic quantities. Thus for example in a cubic crystal the existence of 3C4 and 4C3 axes makes several of the elastic constants equal to each other or zero (zero when under symmetry operation cy becomes -cy,). As a result, cubic crystal has only three independent elastic constants (cu== C22=C33, C44= css= and Ci2=ci3= C2i=C23=C3i=C32). Cubic Symmetry is the highest that can be attained in a crystalline solid but a glass is even more symmetrical in the sense that it is completely isotropic. Therefore the independent elastic constants reduce further to only two, because C44=( c - C i)l2. 

The next step is to calculate the constant of proportionality between the stress and the strain, the elastic compliance matrix. This is the inverse of the elastic constant matrix (the second derivative of energy with respect to strain), which is determined by again expanding the lattice energy to second order ... 

These corrections to the elastic constants were calculated using the above equation and compared with those determined numerically. The numerical approach was to apply small stresses (Sj Voigt notation) of + 0.2 GPa and —0.2 GPa with the elastic compliances related to the resulting strains, e, to e6, by... 

The elastic compliance tensor was then inverted to evaluate the elastic constants. 

The compliance tensor for background rock matrix is a general expression however, in the current work, it is defined by elastic constants. For an assumed transversely anisotropic material, the tensor is defined by five elastic constants (Ej, E2, Vi, V2, and Gt -Young s modulus in the horizontal plane. Young s modulus in the vertical plane, Poisson s ratio in the horizontal plane, Poisson s ratio in the vertical plane, and shear modulus in the vertical plane of the background rock mass, respectively). The compliance tensor for fractures is defined by ... 

If this process is repeated, one finds only three values of Poisson s ratio are needed, not six. For fiber-reinforced materials, the number of elastic constants may be further reduced if other symmetries appear. For example, in some materials short fibers are randomly oriented in a plane and this gives transverse isotropy. That is, there is an elastically isotropic plane but the stiffness and compliance constants will be different normal to this plane (five elastic constants are needed). 

Many ceramics are used in a random polycrystalline form and thus, it is useful to be able to predict the elastic constants from those of the single crystals. The approaches outlined in the last two sections are used for this procedure by considering the random polycrystal as an infinite number of phases with all possible orientations. For example, Voigt and Reuss used a technique based on averaging the stiffness or compliance constants and obtained upper and lower bounds. The Voigt upper bounds for the bulk (B) and shear (/i) moduli of the composite can be written as... 

The general version of Hooke s Law involves either the elastic compliance or elastic constants. This array of elastic constants is... 

Studies of mechanical anisotropy in polymers have been made on specimens of two distinct types. Uniaxially drawn filaments or films have fibre symmetry, with isotropy in the plane perpendicular to the draw direction. Films drawn at constant width or films drawn uniaxially and subsequently rolled and annealed under closely controlled conditions, show orthorhombic symmetry. For fibre symmetry (also called transverse isotropy) the number of independent elastic constants reduces to five and the compliance matrix is... 

For orthorhombic symmetry there are nine independent elastic constants and the compliance matrix is... 

The calculation of the elastic constants of the partially oriented polymer can be made in two ways. One can assume either uniform strain throughout the aggregate, which involves a summation of stiffness constants, or uniform stress, which implies a summation of compliance constants. In the former case the tractions across the boundaries of the unit do not satisfy the stress equilibrium conditions in the latter case there is discontinuity... 

For an isotropic aggregate, the stiffness averaging procedure had been proposed by Voigt, ° and the compliance averaging procedure by Reuss, many years previously. Each had been used to compare the elastic constants of single crystals with those of an isotropic aggregate of single crystals (see for example Ref. 12). 

The equations which predict the elastic constants of a partially oriented polymer involve orientation functions to define the orientation of the aggregate units. For example, the average extensional compliance S33 for a transversely isotropic aggregate of transversely isotropic structural units is given by S 33 = Su sin 0+S33Cos" 0-l-(2Si3-l-544) sin 0cos 9. 

The significance of these experiments is that the Poisson s ratio values, which yield Si 2 and Si 3, were obtained directly, with an accuracy of about 5%, on the same small area of material over which extensional compliances were measured. With all compliances measured accurately over a range of strains it was possible to test under what circumstances the deformation proceeded at constant volume. By comparison the pioneering work of Raumann had been so limited in its accuracy that the data were not inconsistent with the deformation occurring at constant volume—a situation in which the behaviour may be described in terms of only 3, rather than 5, elastic constants. In the earlier measurements of Ward and his colleagues V13 was measured directly, but with an accuracy no better than 10%, while V12 was determined indirectly, applying an expression containing other compliances obtained with different test samples. 

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... 

Since, in our consideration, both the stress and the strain tensors are symmetrical about the diagonal, there are only six independent stress and strain elements, making many of the elastic compliances equal to each other. This makes it possible to simplify the generalized Hooke s law so that it involves at most 36 elastic compliances or constants, but requires the introduction of a shorthand notation both for stress and for strain for a unique representation that is referred to as the Voigt notation that we state as follows, for stresses and strains ... 

These two alternatives give the formal connection of the elastic constants and elastic compliances. They can be abbreviated as matrix products as... 

---