Elastic compliance 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 ... 

Where, [c°] is undamaged elastic compliance matrix of rock, [g,] is the inverse matrix of coordinate-transfer matrix, is the element volume, S is the area of fractures of the i -th fracture, respectively. 

The best estimate of the elastic compliance matrix S at P = 1 atm and T = 435 K is obtained by inversion of the stiffness matrix and using standard propagation of errors. One obtains for S (in GPa )... 

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). 

The elastic midplane strains and curvatures are related to the resultant forces (AT,) and moments (AQ through the laminate compliance matrix... 

Matrix cracks increase the elastic compliance. Numerical calculations indicate that the unloading elastic modulus, E, is given by21... 

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... 

Concentrating initially on time dependence for infinitesimal strains, the correspondence principle relating viscoelastic and elastic behaviour, well established for isotropic systems, may be simply extended to apply to the anisotropic case. There is, however, a difficulty in showing that the compliance matrix Sy will necessarily have the same symmetry properties in the viscoelastic case as in the classically elastic case. This difficulty arises from the thermodynamic nature of part of the argument used in proving symmetry. In the viscoelastic case the proof would depend upon the less well established principles of irreversible thermodynamics. No discussion on this point will be attempted the symmetry properties of Sij as determined in elastic theory will be accepted and its validity examined in the light of the experimental data available. This data shows that there may be systematic deviations from the assumptions in work at finite strains and further work is needed in this area. However, the manner in which these deviations occur does not detract significantly from the utility of the simple formalism in many cases. 

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 ... 

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 the matrix notation for the elastic compliance and stiffness, we have... 

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) ... 

Similarly, the flexural elastic moduli of symmetric laminates are readily obtained from the bending compliance matrix. 

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]. 

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

S is the compliance tensor, the inverse of the elasticity tensor C Because in-verting a matrix is an awkward calculation, the components of the compliance matrix are written explicitly here ... 

If we take a closer look at the elasticity matrix Cap), equation (2.22), and the compliance matrix Sap), equation (2.32), we realise the following pattern Both are of the form... 

In the elasticity matrix Cap), the same components are occupied as in the compliance matrix Sap)A Both matrices can be converted using the following equations which are also valid for an isotropic material ... 

The unit cell of the orthorhombic crystal is brick-shaped. The elastic properties are therefore symmetric with respect to three perpendicular planes. In a coordinate system that is parallel to the edges of the unit cell, the compliance matrix (equation (2.31)) takes the form... 

In this case, we have five independent elastic parameters since there is a relation between the Vij due to the symmetry of the compliance matrix, similar to that for orthotropic materials V2i = vn and u i/E = vy-ijE-i. 

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... 

Equation 7 includes the prediction that the ratio of the piezoelectric coefficient and the remanent polarization P3 = Pr should be approximately equal to the elastic compliance /Ym of the matrix phase for the dipole-density effect or to the elastic compliance HYd of the dipole phase for the dipole-moment effect, respectively (or inversely proportional to the respective elastic modulus). First results assembled from the literature and from our own experimental data on PVDF and on cellular-foam PP and tubular-channel FEP ferroelectrets (Altafim et al. 2009) are shown in Fig. 5 (Qiu et al. 2013, 2014). They provide experimental evidence that the direct piezoelectric thickness coefficient of polymer materials can indeed be roughly approximated by the product of the remanent polarization in the poled material and of its overall elastic compliance. Additional data from the literature on other... 

It is, however, not surprising that the prediction is in general only approximately Mfilled for the polymeric piezoelectrics and seems to break down completely for the inorganic piezoelectrics. The model apparently works best for polymers in which the elastic compliance of the phase that is essential for piezoelectricity clearly dominates the overall compliance (i.e., the soft amorphous matrix in semicrystalline polymers above their respective glass-transition temperatures or the gas-filled cavities in polymer ferroelectrets, respectively). When looking at the spring model and its comparison with selected experimental data from the literature (cf. Fig. 5), the following points should be taken into accormt ... 

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... 

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