Strain transformation matrix

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... 

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 ... 

F Transformation matrix for linear and circular basis sets, (1.57) strain tensor, (7.41). 

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 ... 

The strain tensor in the Xi/ coordinate system, rotated by 45°, can be found using the transformation matrix... 

Strain-displacement transformation matrix Vector of virtual displacements... 

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. 

R engineering strain correction matrix rotational transform, matrix... 

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

When these equations are utilized for the strain transformation relation on the right-hand side of Eqs. (3.22), the transformation matrix is multiplied from the left by the correction matrix and from the right by its inverse. For the rotation around a common base vector, it is straightforward to show that this results in a transposed and inverted transformation matrix ... 

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 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 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 ... 

---