Elastic constant matrix

The Halpin-Tsai equations represent a semiempirical approach to the problem of the significant separation between the upper and lower bounds of elastic properties observed when the fiber and matrix elastic constants differ significantly. The equations employ the rule-of-mixture approximations for axial elastic modulus and Poisson s ratio [Equations. (5.119) and (5.120), respectively]. The expressions for the transverse elastic modulus, Et, and the axial and transverse shear moduli, Ga and Gf, are assumed to be of the general form... 

The strains on the lattice are equal to the stress divided by the elastic constant matrix ... 

Sometimes the failure occurs by propagation of a crack that starts at the top and travels downward until the interface is completely debonded. In this case, the fracture mechanics analysis using the energy balance approach has been applied [92] in which P, relates to specimen dimensions, elastic constants of fiber and matrix, initial crack length, and interfacial work of fracture (W,). 

Specific results are calculated for SiC fiber-glass matrix composites with the elastic constants given in Table 4.1. A constant embedded fiber length L = 2.0 mm, and constant radii a = 0.2 mm and B = 2.0 mm are considered with varying matrix radius b. The stress distributions along the axial direction shown in Fig. 4.31 are predicted based on micromechanics analysis, which are essentially similar to those obtained by FE analysis for the two extremes of fiber volume fraction, V[, shown in Fig. 4.32. The corresponding FAS distribution calculated based on Eqs. (4.90) and (4.120), and IFSS at the fiber-matrix interface of Eqs. (4.93) and (4.132) are plotted along the axial direction in Fig. 4.32. 

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. 

The matrix C can be inverted to give the corresponding matrix S, with the relationships to the conventional elastic constants E, G, v as shown. 

The sets of equations are solved by the assumption of periodic waves and, by expansion in powers of the wave number, a relation is found for the limiting case of long waves so that the elements of the dynamical matrix elastic constants of the continuum. It is also possible to derive the Raman frequencies from the lattice dynamics analysis but this does not seem to have been done for polymer crystals, though they have been derived for example, for NaCl and for diamond. 

Determine the form of the matrix C of second-order elastic constants for crystals with the point group 0. 

Because nanocomposites are made from different phases with different thermal expansion coefficients and elastic moduli, they inevitably develop residual thermal stress during cooling after sintering. Assuming the dispersion phase is spherical particulate in the matrix material, residual stresses can be developed due to differences in the thermal expansion and elastic constants between the matrix and the particles [23] ... 

Background At elevated temperatures the rapid application of a sustained creep load to a fiber-reinforced ceramic typically produces an instantaneous elastic strain, followed by time-dependent creep deformation. Because the elastic constants, creep rates and stress-relaxation behavior of the fibers and matrix typically differ, a time-dependent redistribution in stress between the fibers and matrix will occur during creep. Even in the absence of an applied load, stress redistribution can occur if differences in the thermal expansion coefficients of the fibers and matrix generate residual stresses when a component is heated. For temperatures sufficient to cause the creep deformation of either constituent, this mismatch in creep resistance causes a progres-... 

As with spontaneous strains, the elastic constants of a crystal have symmetry properties. Symmetry-adapted combinations of the elastic constants are obtained by diagonalising the elastic constant matrix for a given crystal class, and the eigenvalues are then associated with different irreducible representations of the point group. Each is then also associated with a particular symmetry-adapted strain. Manipulations of this type only need to be done once for all possible changes in crystal class and the results are available in tabulated form in the literature (e g. Table 6 of Carpenter and Salje 1998). In practice, the most important process is the derivation of the Landau free energy expansion for a... 

There have been many efforts for combining the atomistic and continuum levels, as mentioned in Sect. 1. Recently, Santos et al. [11] proposed an atomistic-continuum model. In this model, the three-dimensional system is composed of a matrix, described as a continuum and an inclusion, embedded in the continuum, where the inclusion is described by an atomistic model. The model is validated for homogeneous materials (an fee argon crystal and an amorphous polymer). Yang et al. [96] have applied the atomistic-continuum model to the plastic deformation of Bisphenol-A polycarbonate where an inclusion deforms plastically in an elastic medium under uniaxial extension and pure shear. Here the atomistic-continuum model is validated for a heterogeneous material and elastic constant of semi crystalline poly( trimethylene terephthalate) (PTT) is predicted. 

The whole simulations are performed according to the following procedure. First, elastic constants of PTT in amorphous phase Camor are calculated using the atomistic modeling, which will be used as input values for the matrix CP in the atomistic-continuum model. Second, elastic constants of PTT in crystalline phase CCTSt are also evaluated in the same manner as those of amorphous PTT. Third, the atomistic-continuum model is validated for heterogeneous material by comparing the calculated elastic constant for the system of infinite lamellas with its exact solution. Finally, elastic constants of semicrystalline PTT with dif-... 

The Hashin-Strikman formulae were obtained by using a variational method to determine the upper Kc. ic and the lower Kn, p bounds of the effective elastic properties for an inhomogeneous medium [131]. The upper bound Kc, ic corresponds to a composite structure in which spherical inclusions with elastic constants K2, p2 are placed in a matrix of elastic constants K, Pj in the following, it is assumed that K > K2, pj > p2. The lower bound Kn, p is obtained when the components are permuted—that is, when the matrix is described by K2, p2 and the spherical inclusions are described by K, pj. 

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 ... 

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