**1. Introduction**

122 Continuum Mechanics – Progress in Fundamentals and Engineering Applications

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With the purpose to create high strength advanced structures, new materials are being developed presenting favorable characteristics for specific applications. Composite Materials are examples of these developments. They can be formed by high strength long fibers, conveniently oriented in a matrix, to form a lamina of composite material. The lamina presents high strength in the fiber direction, but, since it is slender, does not have enough rigidity, what makes impossible the use of an isolated lamina. Piling up and gluing a set of laminas, a laminate is formed which one presents better characteristics than original and isolated materials. The main strength of each lamina is oriented according to the fiber directions. Thus, micro cracks can be produced if sufficient tension is applied in the transverse direction of the fibers, as shown in figure 1, since the resistance of the lamina in these directions depends only on the matrix material. The rise of several transverse cracks produces loss of stiffness in the laminate.

Several papers are found in technical literature dealing with the behavior of composite materials with transverse cracks. Vejen & Pyrz (2002) investigated the transverse crack growth in long fiber composites using the finite element method. Three criteria concerning pure matrix growth, fiber/matrix interface growth and crack kinking out of a fiber/matrix interface were implemented to form a software package for crack propagation calculus.

Cain and colleagues (2003) have studied unidirectional graphite bismaleimide composites to determine the effect of the matrix dominant properties on the failure of the material. The authors showed that the final fracture was caused by the development of a dominant matrix shear crack parallel to the fibers. They also concluded that the decrease in shear modulus of the composite was the most sensitive and best represented by damage evolutions.

Ogihara et al (1998) have proposed a two-dimensional model which considers that, in the case of displacements and stress fields in the interlaminar cross-play laminates, there is a prevalence of plane-strain case, even in the presence of transverse cracks. They have also commented that the failure process of cross-ply laminates is due an accumulation of transverse cracks and delamination.

Progressive Stiffness Loss Analysis

environment, jeopardizing its service life.

(1987 a,b), Hashin (1987), Talreja (1984) and Lim & Tay.

**2.1 Constitutive relations**

Method (MLGFM), which one will be briefly descript on this paper.

a damaged material at the level of a lamina is assumed as:

**2. Representation of the generalized damage in symmetric laminates**

of Symmetric Laminated Plates due to Transverse Cracks Using the MLGFM 125

One of the first damages that occurs in composite laminate are the transverse cracks, as mentioned by Allen and colleagues (Allen et al 1987 a,b) and Tay and Lim (Tay & Lim, 1996). The cracks appear in a layer where the highest stress values act transversally to the fiber, exceeding the matrix resistance. With loading increment, the increase on the number of transverse cracks may happen in a diffuse way reducing the structural rigidity. The accumulation of this damage can accelerate the beginning of delamination, changing the natural frequency of the composite structure and causing a greater degradation in severe

After the initiation and development of micro cracks, there is a process of accumulation of damage that reduces the structural stiffness.The tolerance to the damage is related to the stiffness of the structure which, in turn, is affected by the accumulation of micro defects during loading. The process of damage evolution in composite laminate is generally very complex due to the multiplicity of failure modes such as transverse cracks, delamination, decoupling fiber-matrix interface, and fiber breakage. The characterization of this process is generally possible when single cases are analyzed, where each failure mode can be separated and studied individually. The use of Fracture Mechanics, especially in terms of linear elastic fracture, has presented good results for isotropic material because, in this case, can be adequately characterized by a single parameter (the stress intensity factor). However, attempts to apply this method in composite laminates, whose behavior is orthotropic, have met unsatisfactory results, mainly when transverse cracks in the matrix are studied. Therefore, to determine changes of the mechanical properties in a laminate, the total number of cracks formed in the transverse layers must be taken to account, or, under a generalized crack distribution, the most appropriated methodology is based on Damage Mechanics.

Many researchers have developed studies to evaluate the properties of laminates subjected to generalized cracks in the matrix. Among these ones, can be cited the papers of Allen et. al.

The present paper has the objective to apply the Continuum Damage Mechanics Theory to long fiber laminate composites. The transverse cracks appearance in the matrix implies in a rigidity loss due to damage accumulation. The increase of the load is considerate monotonically. Several failure criterions are presented and implemented such as, the Maximum Stress Criterion, the Maximum Strain Criterion, Tsai-Hill and Tsai-Wu Criterion. The proposed methodology is restrict to the case of symmetric laminate and it is evaluated by a numerical approximation technique known as Modified Local Green's Function

The models developed by Talreja & Boehler (1990), Allen et. al. (1987 a,b) and Lim and Tay (1996) to describe the damaged composite laminates were based on the Continuum Damage Mechanics using internal state variables. In the presente paper, the model proposed by Allen et al (1987 a,b) will be used, which describes the damage through a set of internal state variables. The final result of the distributed damage is built in the constitutive equations through these variables. Thus, the stress-strain relationship of the representative volume of

Fig. 1. (a) A composite [0o/90o/0o]s laminate plate with transverse micro-cracks in the matrix; (b) extension of pre-micro cracks; (c) formation of new micro crack.

An analytical model based on the principle of minimum potential energy was developed by Ji *et al* (1998) and applied to determine the two-dimensional thermoelastic stress state in cross-ply composite laminates containing multiple equally spaced transverse cracks in the 90o plies subjected to tensile loading in the longitudinal direction. The criterion of strain energy release rate was employed to evaluate the critical applied stresses for two of the possible fracture modes. After some numerical experiments, they have concluded that the formation of new cracks never takes place until pre-existing cracks extends through the entire thickness of the 90o plies.

Wada *et al* (1999) have presented a damage mechanics model to predict the nonlinear behavior of laminated composites due to crack evolution. A new concept of cracking layer is proposed by a technique based on uniform work-softening layer. With this concept, the constitutive equations for a cracking layer are constructed according to modern plasticity theory. So, the lamina damage surface is defined in the stress space and the constitutive equations for a cracking layer are constructed by applying the defined damage surface to the associated flow rule.

**2 L**

**' <sup>o</sup> '**

**'o 'o**

**L L** 

**2 L**

lamina with longitudinal Transversal crack

**2a'** 

**2a'**

**2a**

**o**

**a' > a** 

**'o > <sup>o</sup>**

lamina with transversal fibers

Fig. 1. (a) A composite [0o/90o/0o]s laminate plate with transverse micro-cracks in the

An analytical model based on the principle of minimum potential energy was developed by Ji *et al* (1998) and applied to determine the two-dimensional thermoelastic stress state in cross-ply composite laminates containing multiple equally spaced transverse cracks in the 90o plies subjected to tensile loading in the longitudinal direction. The criterion of strain energy release rate was employed to evaluate the critical applied stresses for two of the possible fracture modes. After some numerical experiments, they have concluded that the formation of new cracks never takes place until pre-existing cracks extends through the

Wada *et al* (1999) have presented a damage mechanics model to predict the nonlinear behavior of laminated composites due to crack evolution. A new concept of cracking layer is proposed by a technique based on uniform work-softening layer. With this concept, the constitutive equations for a cracking layer are constructed according to modern plasticity theory. So, the lamina damage surface is defined in the stress space and the constitutive equations for a cracking layer are constructed by applying the defined damage surface to

matrix; (b) extension of pre-micro cracks; (c) formation of new micro crack.

entire thickness of the 90o plies.

lamina with longitudinal fibers

**(a)** 

**(b)** 

**(c)** 

**2t**

**2t**

**2t**

the associated flow rule.

One of the first damages that occurs in composite laminate are the transverse cracks, as mentioned by Allen and colleagues (Allen et al 1987 a,b) and Tay and Lim (Tay & Lim, 1996). The cracks appear in a layer where the highest stress values act transversally to the fiber, exceeding the matrix resistance. With loading increment, the increase on the number of transverse cracks may happen in a diffuse way reducing the structural rigidity. The accumulation of this damage can accelerate the beginning of delamination, changing the natural frequency of the composite structure and causing a greater degradation in severe environment, jeopardizing its service life.

After the initiation and development of micro cracks, there is a process of accumulation of damage that reduces the structural stiffness.The tolerance to the damage is related to the stiffness of the structure which, in turn, is affected by the accumulation of micro defects during loading. The process of damage evolution in composite laminate is generally very complex due to the multiplicity of failure modes such as transverse cracks, delamination, decoupling fiber-matrix interface, and fiber breakage. The characterization of this process is generally possible when single cases are analyzed, where each failure mode can be separated and studied individually. The use of Fracture Mechanics, especially in terms of linear elastic fracture, has presented good results for isotropic material because, in this case, can be adequately characterized by a single parameter (the stress intensity factor). However, attempts to apply this method in composite laminates, whose behavior is orthotropic, have met unsatisfactory results, mainly when transverse cracks in the matrix are studied. Therefore, to determine changes of the mechanical properties in a laminate, the total number of cracks formed in the transverse layers must be taken to account, or, under a generalized crack distribution, the most appropriated methodology is based on Damage Mechanics.

Many researchers have developed studies to evaluate the properties of laminates subjected to generalized cracks in the matrix. Among these ones, can be cited the papers of Allen et. al. (1987 a,b), Hashin (1987), Talreja (1984) and Lim & Tay.

The present paper has the objective to apply the Continuum Damage Mechanics Theory to long fiber laminate composites. The transverse cracks appearance in the matrix implies in a rigidity loss due to damage accumulation. The increase of the load is considerate monotonically. Several failure criterions are presented and implemented such as, the Maximum Stress Criterion, the Maximum Strain Criterion, Tsai-Hill and Tsai-Wu Criterion. The proposed methodology is restrict to the case of symmetric laminate and it is evaluated by a numerical approximation technique known as Modified Local Green's Function Method (MLGFM), which one will be briefly descript on this paper.
