Elastic-compliance-constant matrix

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

Since the elastic strain energy is a unique function of state, which is independent of how that state was reached, it is possible to demonstrate that the elastic-compliance and elastic-constant matrixes, as defined above, must be symmetrical. This follows directly from the observation, for example, that... 

The tensor of elastic compliancy S T) = C T), and die elastic constant matrix (ignoring the terms due to electron-rotation interaction) is equal to ... 

In this system of equations the piezoelectric charge constant d indicates the intensity of the piezo effect is the dielectric constant for constant T and is the elastic compliance for constant E eft is the transpose of matrix d. The mentioned parameters are tensors of the first to fourth order. A simplification is possible by using the symmetry properties of tensors. Usually, the Cartesian coordinate system in Fig. 6.12a is used, with axis 3 pointing in the direction of polarization of the piezo substance (see below) [5,6]. 

Here D is the vector of the dielectric displacement (size 3x1, unit C/m ), S is the strain (size 6x1, dimension 1), E is a vector of the electric field strength (size 3x1, unit V/m) and T is a vector of the mechanical tension (size 6x1, unit N/m ). As the piezoelectric constants depend on the direction in space they are described as tensors e- is the permittivity constant also called dielectric permittivity at constant mechanical tension T (size 3x3, unit F/m) and 5 , is the elastic compliance matrix (size 6x6, unit m /N). The piezoelectric charge coefficient df " (size 6x3, unit C/N) defines the dielectric displacement per mechanical tension at constant electrical field and (size 3x6, unit m/V) defines the strain per eiectric fieid at constant mechanical tension [84], The first equation describes the direct piezo effect (sensor equation) and the second the inverse piezo effect (actuator equation). 

Even in cases where the rigid polymer forms the continuous phase, the elastic modulus is less than that of the pure matrix material. Thus two-phase systems have a greater creep compliance than does the pure rigid phase. Many of these materials craze badly near their yield points. When crazing occurs, the creep rate becomes much greater, and stress relaxes rapidly if the deformation is held constant. 

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 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 ... 

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 elastic constants are the second derivatives of the energy with respect to the strains. The converged matrix, C, in Eq. (9) contains the second derivatives with resj ct to the unit cell parameters, a,b,d,a,Ry and others parameters such as a and the atom coordinates, v. The cell parameters are strains only in crystal systems with orthogonal basis vectors (cubic, tetragonal and orthorhombic). Thus two further operations are required, elimination of the extra parameters and transformation to Cartesian basis. The elimination of the extra parameters to find the elastic constants has been described [2] as has their elimination to find the compliance matrix [12]. The transformation to Cartesian basis has also been described [2],... 

In dealing with engineering problems, we often desire to convert Cyij or Syu to the engineering moduli (Young s moduli, shear moduli and Poisson s ratios). The engineering moduli are easily calculated from the components of the contracted compliance matrix. The formulas are as follows (There are 9 nonzero independent elastic constants for orthotropic materials) ... 

The compliance matrix for these axes involves nine independent elastic constants ... 

The Cauchy stress tensor cr and Green Lagrange strain tensor Cgl are of second order and may be connected for a general anisotropic linear elastic material via a fourth-order tensor. The originally 81 constants of such an elasticity tensor reduce to 36 due to the symmetry of the stress and strain tensor, and may be represented by a square matrix of dimension six. Because of the potential property of elastic materials, such a matrix is symmetric and thus the number of independent components is further reduced to 21. For small displacements, the mechanical constitutive relation with the stiffness matrix C or with the compliance matrix S reads... 

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... 

Oriented polymer films that are prepared by either rolling, rolling and annealing, or some commercial one-way draw processes, may possess orthorhombic rather than transversely isotropic symmetry. For such films, the elastic behaviour is specified by nine independent elastic constants. Choose the initial drawing or rolling direction as the 3 axis for a system of rectangular Cartesian coordinates the 1 axis to lie in the plane of the film and the 2 axis normal to the plane of the film (Figure 8.2). The compliance matrix is... 

Note that the stiffness matrix and the compliance matrix have the same number of independent components (called elastic constants and elastic coefficients, respectively) and zero elements at the same positions, cf. [Nye 1957, Hearmon 1961], The compliance matrix can be calculated from the stiffness matrix via matrix inversion as follows... 

Summarizing the results obtained above, the compliance matrix for an orthotropic lamina can be expressed in terms of engineering elastic constants as... 

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