One parameter of importance when analysing a non-regularized problem is the regularization distance (RD). The RD is defined as the distance from the non-regularized focus necessary to reduce the perturbations of the non- regularized parameters to a 5% of its maximum value.
It can be considered that stresses and other parameters at points located at higher distances than the RD from all the non-regularized focuses are very close to their regularized values. Therefore, two non-regularized focuses distanced more than the summation of both RD associated to both focuses can be considered independent, and one of those focuses do not affect to the parameters in the closer area of the other focus.
That distance may be used to divide a section in several problems to analyse. E.g., in a L-shaped beam with a high R/t ratio the curved part
3.11. Regularization distance 126
may be long enough so one change of curvature does not affect to the other one and, therefore, each change of curvature may be analysed independently considering the curved part as a semi-infinite beam. This is important in sections having many changes of curvature, where the complexity may be reduced by dividing the section in several chains of beams.
The perturbations have been obtained as a summation of exponential functions according to (3.136) and (3.137) (notice that the hyperbolic sine is equivalent to an exponential function). The decay rate of those functions are determined by the coefficients of the exponents of the exponentials, which, at the same time, are given by the square root of the eigenvalues λi of the
matrix G. Typically, the eigenvalues are complex numbers causing that the exponentials are damped oscillations. The decay rate is given only by the real part of the square root of λi, associated to the damping, the complex
part being associated to the oscillations. Therefore, the decay rate of the perturbations may be approximated by the exponential with the coefficient in the exponent with a lower real part, √λmin:
Repλmin = min i Repλi . (3.164)
Notice that the coefficient with a lower real part of √λmin does not
necessarily coincide with the eigenvalue λi with the minimum real part, as
the coefficient is defined as the square root of the eigenvalue and the real part of the square root of λi depends also on the imaginary part of λi.
Therefore, the regularization distance, Lregmay be approximated by the
value of s where the exponential associated to√λmin has declined to the 5%
of its maximum value (which is given in s = 0): e− Re(
√
λmin)Lreg2t ' 0.05, (3.165)
where the inverse of one half of the thickness has been added in the expres- sion to consider the dimensional value of the regularization distance Lreg
instead of the non-dimensional ones used in the development of the MBM and the LPBM.
Hence, the approximation of the RD yields: Lreg '
3t 2 Re √λmin
. (3.166)
Equation (3.166) may be used to obtain the RD numerically by using the LPBM with a model order high enough. However, a closed-form equation is desired to evaluate the RD before running the non-regularized model. This closed-form equation may be obtained by approximating √λmin by the
127 Bi-dimensional models for evaluating interlaminar stresses
corresponding parameter in a model with order n = 2, since generally the eigenvalues associated to higher orders are lower. A model with order n = 2 applied to a homogeneous material has two eigenvalues which are given by the following expressions:
λ1,2 = 3(Q11Q33− Q213) 2Q11Q55 ± 3 2 s (Q2 13− Q11Q33)(Q213− Q11Q33+ 20Q255) Q2 11Q255 . (3.167) For a composite material the corresponding eigenvalues may be approx- imated by equation (3.167) by considering the homogeneous equivalent ma- terial and calculating the corresponding stiffnesses Qij. The eigenvalue with
the minimum real part is obtained by choosing the minus sign in equation (3.167): λmin = 3(Q11Q33− Q213) 2Q11Q55 −3 2 s (Q2 13− Q11Q33)(Q213− Q11Q33+ 20Q255) Q2 11Q255 . (3.168) Notice that the eigenvalue λmin is highly influenced by the parameter
Q55, the RD being highly influenced by the shear stiffness of the material.
Summarizing, the RD may be evaluated by equation (3.166) calculating numerically √λmin by using an order high enough in the LPBM. Notwith-
standing,√λmin can be approximated by using the homogeneous equivalent
material and using equation (3.168) for a quick estimation of the RD. Considering a single-ply laminate with ply properties given in Table 3.2 and changing the orientation of the ply, the square root of the eigenvalue whose square root has the minimum real part is depicted in the complex plane in Figure 3.24 for the approximation given in equation (3.168), a straight beam and a curved beam with R = t.
The 0oply and the 90oply cases are marked in the Figure. The eigenvalue square root in low orientations has a null imaginary part until an orientation between 45o and 55o where the imaginary part becomes not null. Notice a
discontinuity in the curved beam case when the imaginary part becomes not null. That discontinuity is due to a change in the eigenvalue with the minimum real part in its square root.
The RD yields as depicted in Figure 3.25, where a discontinuity in the slope is observed when the imaginary part of √λmin becomes not null.
Therefore, the approximation results slightly anti-conservative, specially in the straight beam case. Furthermore, it can be observed that the RD are, for the present material properties and a single-ply laminate, delimited