Independent stiffness constants

The elastic behavior of a material is completely characterized by a set of independent stiffness constants C,-. For an orthorhombic material with symmetry axes along the 1, 2 and 3 directions of a Cartesian coordinate system, there are nine independent stiffness constants ... 

Similar measurements for wave propagating in the 1—2 plane and 1-3 plane give the two sets of stiffness constants, (C, and (Cjj, C33, Cj3, C ), respectively. Therefore, the three series of velocity measurements give the nine independent stiffness constants, C C... 

With all the independent stiffness constants known, it is possible to calculate the Young s modulus at an angle 9 relative to the draw axis. This is a valuable parameter which is very difficult to measure on a conventional tensile machine. As shown in the Young s modulus vs. 6 plots at 2 = 15 (Figure 14.20), the Young s modulus increases with increasing PLC content at all angles, but the reinforcement effect becomes weaker as 0 increases. 

It is seen that ultrasonic techniques have two major advantages high accuracy and the ability to determine the complete set of elastic constants for a sample of small size. As a result of these capabilities, all five independent stiffness constants have been obtained for extruded rods of PLC of high draw ratio (2 = 15) and small diameter (0.8 mm). Moreover, it has been possible to study the skin-core structure in injection molded PLC by monitoring the variation in stiffness with position. For blends of a PLC and a thermoplastic or glass fiber-reinforced PLC, successful correlation has been obtained between the modulus data and the orientation of the PLC fibrils or glass fibers. 

It follows that b AlbepSi = cy = b Albetbej = cp also, Sy = sjf, whence the number of independent stiffness constants and compliances is reduced from 36 to 21. We may then integrate Eq. (5.10.16) to obtain the work or isothermal free energy per unit volume needed to produce the strain ej, i.e. the strain... 

In zinc blende (cubic) crystals, there are only three independent stiffness constants Cii, C12, and C44. The above relations then take the form... 

Note that the transformed reduced stiffness matrix Qy has terms in all nine positions in contrast to the presence of zeros in the reduced stiffness matrix Qy. However, there are still only four independent material constants because the lamina is orthotropic. In the general case with body coordinates x and y, there is coupling between shear strain and normal stresses and between shear stress and normal strains, i.e., shear-extension coupling exists. Thus, in body coordinates, even an orthotropic lamina appears to be anisotropic. However, because such a lamina does have orthotropic characteristics in principal material coordinates, it is called a generally orthotropic lamina because it can be represented by the stress-strain relations in Equation (2.84). That is, a generally orthotropic lamina is an orthotropic lamina whose principai material axes are not aligned with the natural body axes. 

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... 

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) ... 

From Table 10.3, there are six independent elastic-stiffness constants Cn, C12, C13, C33, C44, and C66. Substitution of these relations into Eq. 10.21, and using the relations given in Eqs. 10.22-10.24, gives ... 

Table 12.5 The matrix C of the stiffness constants for all Lane classes. The matrix S of the compliance constants is identical, only C is replaced by S. The last column gives the number of the independent constants.

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. 

Figures 14.28 and 14.29 show the stiffness constants of 30wt% glass fiber-reinforced Vectra A as functions of the position along the width. The position dependence and the pattern of anisotropy are similar to those of Vectra A (Figures 14.10-14.12) but the magnitudes of the stiffnesses are higher owing to the reinforcing effects of the fibers. In the top layer the stiffnesses are independent of position, and C33 > c, and C33 > These results imply that not...

For single crystals with transverse dimensions large enough to permit a plane wave condition to be attained, the results are unambiguous and virtually free from theoretical assumptions. Five independent elastic constants (stiffnesses or compliances) are required to describe the linear elastic stress-strain relations for hexagonal materials. Only three independent constants are required for cubic (y-Ce, Eu, Yb) materials. Since there are no single crystal elastic constant data for the cubic rare earth metals, this discussion will concentrate on the relationships for hexagonal symmetry. 

This chapter began by describing briehy the elasticity of anisotropic materials, providing the fundamental relationships and the allowed simplihcations by the existence of material planes of symmetry. The current unidirectional composites are usually classihed as transversely isotropic materials. In this case, only hve independent elastic constants are necessary to fully characterize unidirectional composites. The micromechanics provides the analytical and numerical approaches to predict the elastic constants based on the elastic properties of the composite constituents. Several classical closed formulas are revisited and compared with experimental data. Finally, stiffness and compliance transformations are given in the context of unidirectional composites. Experimental data are used to assess theoretical predictions and illustrate the off-axis in-plane elastic properties. 

For an isotropic polymer there are two independent elastic constants, and the two alternative schemes predict a value for the isotropic shear compliance 544 and the isotropic shear stiffness C44 respectively. 

Show that the film appears to be orthorhombic (or orthotropic), with five independent elastic stiffness constants, in a global coordinate system with axes coinciding with the crystallographic directions [101], [010] and [101]. 

This frictionless assumption is often appropriate for very stiff materials where adhesive forces are relatively unimportant, but it is often not the case for softer materials such as elastomers, where adhesive forces play a very important role. In these cases, a full-friction boundary condition, where sliding of the two surfaces is not allowed, is often more appropriate, In many important cases (contact of a very thick, incompressible elastic layer, for example) there is little or no practical difference for these two boundary conditions. Nevertheless, in the discussion that follows, we are careful to indicate that boundary condition (frictionless or full-friction) that formally applies in each case. In all cases we assume that the contacting materials are isotropic and homogeneous, each being characterized by two independent elastic constants. 

For tetragonal monocrystals (6 independent elastic constants) the general form of the orthorhombic (orthotropic) stiffness matrix. Equation (15), together with conditions (17) applies. The same holds true for materials with higher than tetragonal symmetry. In these cases, however, the following additional conditions hold for the non-zero elements ... 

In the case of elastically-isotropic solids there are only two independent elastic constants, Cn and Cn because 2c44 = Cu - C12. Since glassy polymers and randomly oriented semi-crystalline polymers fall into this category it is worth considering how the stiffness constants can be related to quantities such as Young s modulus, E, Poisson s ratio v, shear modulus G and bulk modulus K which are measured directly. The shear modulus G relates the shear stress 04 to the angle of shear 74 through the equation... 

It is clear that since the four constants, G, K, E and v can be expressed in terms of three stiffness constants, only two of which are independent, then G, K, E and v must all be related to each other. It is a matter of simple algebraic manipulation to show that, for example... 

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