Stress transformation matrix

The stress-strain relations in this book are typically expressed in matrix form by use of contracted notation. Both the stresses and strains as well as the stress-strain relations must be transformed. First, the stresses transform for a rotation about the z-axis as in Figure A-1 according to... 

The next level of complexity in the treatment is to orient the apphed stress at an angle, 6, to the lamina fiber axis, as illustrated in Figure 5.119. A transformation matrix, [T], must be introduced to relate the principal stresses, o, 02, and tu, to the stresses in the new x-y coordinate system, a, ay, and t j, and the inverse transformation matrix, [T]- is used to convert the corresponding strains. The entire development will not be presented here. The results of this analysis are that the tensile moduli of the composite along the x and y axes, E, and Ey, which are parallel and transverse to the applied load, respectively, as well as the shear modulus, Gxy, can be related to the lamina tensile modulus along the fiber axis, 1, the transverse tensile modulus, E2, the lamina shear modulus, Gu, Poisson s ratio, vn, and the angle of lamina orientation relative to the applied load, 6, as follows ... 

Referring to Fig. A.2, assume that the principal coordinates align with z, r, and O. The unit vectors (direction cosines) just determined correspond with the row of the transformation matrix N. Thus, if the principal stress tensor is... 

At this stage of the derivation, emphasis must be stressed on the positive definiteness of both quadratic forms involved. As a consequence of this positive definiteness, the transformation matrix must be real and the values t, must all be real and positive. 

Equation 10.1 is a second-rank tensor with transpose symmetry. The normal components of stress are the diagonal elements and the shear components of stress are the nondiagonal elements. Although Eq. 10.1 has the appearance of a [3 x 3] matrix, it is a physical quantity that, for one set of axes, is specified by nine components, whereas a transformation matrix is an array of coefficients relating two sets of axes. The tensor coefficients determine how the three components of the force vector, /, transmitted across a small surface element, vary as different values are given to the components of a unit vector / perpendicular to the face (representing the face orientation) ... 

If Te and Ts are respectively the transformation matrices for the strain and stress vectors the transformation matrix which allows writing the vectors in the cylindrical coordinate, we get ... 

Stresses transform from one coordinate-axis system to another according to well-defined transformation laws that utilize direction cosines of the angles of rotation between the final and initial coordinate-axis systems. Matrixes that obey such transformation laws are referred to as tensors (McClintock and Argon 1966). There are three sets of stress relations that are scalar and invariant in coordinate-axis transformations. The first such stress invariant of particular interest is the mean normal stress o- , defined as. 

As with stresses, transformations of strain elements from one coordinate-axis system into another obtained by rotations of axes obey the same transformation laws as those of stresses, utilizing the same direction cosines and making the symmetrical strain matrix also a tensor (McClintock and Argon 1966). 

In the most general case, aU nine components of the strain tensor are linearly related to all nine components of the stress tensor. This would yield Hooke s law with 81 parameters Ey = Sgitiaw = Sgucrid with the summation taking place over the repeating indices and with the inverse transform matrix of Ojj = CijyEy. Obviously, there is no inverse proportionality between individual s and c. The 81 s and c parameters transform linearly upon turning the coordinate system, that is. 

These transformation coefficients may be summarized in the transformation matrix T for the subsequent transformation of stresses, respectively strains, in matrix representation ... 

Accordingly, we use the stress and strain transformations of Equations (2.74) and (2.75) along with Reuter s matrix. Equation (2.77), after abbreviating Equation (2.80) as... 

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. 

The model of a reacting molecular crystal proposed by Luty and Eckhardt [315] is centered on the description of the collective response of the crystal to a local strain expressed by means of an elastic stress tensor. The local strain of mechanical origin is, for our purposes, produced by the pressure or by the chemical transformation of a molecule at site n. The mechanical perturbation field couples to the internal and external (translational and rotational) coordinates Q n) generating a non local response. The dynamical variable Q can include any set of coordinates of interest for the process under consideration. In the model the system Hamiltonian includes a single molecule term, the coupling between the molecular variables at different sites through a force constants matrix W, and a third term that takes into account the coupling to the dynamical variables of the operator of the local stress. In the linear approximation, the response of the system is expressed by a response function X to a local field that can be approximated by a mean field V ... 

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