**3. Approximation by computational methods**

In this paper, the maximum crack opening displacement *u2* is determined by the expressions (18) and (20). The crack density *ρ* depends on the distance between two adjacent transverse cracks *l*, and its values are successively modified by the verification of the composite material rigidity loss.

Progressive Stiffness Loss Analysis

contour elements) by the MLGFM.

matrices **G(**P,Q**), G(**p,Q**), G(**P,q**)** and **G(**p,q**).** 

**[GT(**P,Q**) a(**P**)]**d**<sup>+</sup>**

**[GT(**P,q**) a(**P**)]**d**<sup>+</sup>**

boundary (equation (24)):

 **u(**Q**)** =

 **u(**q**)** =

of Symmetric Laminated Plates due to Transverse Cracks Using the MLGFM 131

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(a) Finite Elements Mesh (b) Contour Elements Mesh Fig. 4. Symmetric boundary conditions and plate for a 2x2 mesh (4 finite elements and 8

The most important steps of the MLGFM are detailed in the work sof Barbieri *et al* (1998a) and Machado *et al* (2008). It is possible to show that through the MLGFM two sets of equations are formed, the first one in the domain (equation (23)) and the other one on the

generalized force is applied over the point j, in the direction of a unitary vector **n**j.

where Q, P are two points in the domain; q, p are other two points on the boundary; **a**(P) is the vector of independent terms for the original problem; f(p) is the vector associated to the fluxes on the boundary; **G**(i,j) are the Green's functions which may be understood as the generalized displacement in the point i in the direction of an unitary vector **ni**, when a

Equations (23) and (24) describe completely the problem. Since these equations involve domain and boundary integrals, two types of meshes are necessary, one in the domain and the other on the boundary, using FE and BE methods, respectively. The FE domain approximation is also used to develop the Green's functions which are associated to the

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**[GT(**p,Q**) f(**p**)]**d ; P,Q ; p (23)

**[GT(**p,q**) f(**p**)]**d ; P ; p, q (24)

Considering a conventional structural approximation by conventional Finite Element Method, the problem can be expressed by a system of algebraic equations, representing a typical element:

$$
\begin{bmatrix} K\_{11} & K\_{12} & K\_{16} \\ K\_{21} & K\_{22} & K\_{26} \\ K\_{61} & K\_{62} & K\_{66} \end{bmatrix} \begin{bmatrix} d\_x \\ d\_y \\ d\_z \end{bmatrix} = \begin{bmatrix} F\_1^a \\ F\_2^a \\ F\_6^a \end{bmatrix} + \begin{bmatrix} F\_1^d \\ F\_2^d \\ F\_6^d \end{bmatrix} \tag{22}
$$

where *Kij* are the element stiffness matrix, (*dx*, *dy*, *dz*) are the components of the element displacement vector and {*Fa*} and {*Fd*} are the applied force vector and the element damage force vector.

The procedure used in this paper to obtain the expected results is a little different because it uses a different computational method known as Modified Local Green's Function Method (MLGFM), in witch the system defined in expression (22) is not directly applied in a conventional FEM. A detailed explanation of this procedure can be found in Barbieri *et al* (1998a,b) and Machado *et al* (2008). The MLGFM is an integral method that determines the unknowns on the boundary, similarly to the Boundary Element Method, but the fundamental solutions are generated automatically by projections of the Green's Functions developed from de field, as in the Finite Element Method. The matrix and the vectors indicated in (22) will be used to produce values at the boundary, as explained in the next topic.
