**6. Conclusion**

138 Continuum Mechanics – Progress in Fundamentals and Engineering Applications

To evaluate the stiffness loss of laminated plates due to micro-crack accumulation under increasing monotonic loading using the MLGFM, the following conditions were considered: a. Stress-strain relations of a thin orthotropic laminate are considered in plain stress state; b. Dimensions of the squared plate are 2,0 m x 2,0 m, but only a ¼ was modeled due to its

e. The value of *GIc* adopted is 250 J/m2 for the glass/epoxi laminate (Tay & Lim, 1993).

**Material E11 (GPa) E22 (GPa) G12 (GPa) G23 (GPa) υ<sup>12</sup>**

Glass / Epoxy (Gl/Ep) 41.70 13.00 3.40 3.40 0.3

*Xt* **(MPa)** *Yt* **(MPa)** *Xc* **(MPa)** *Yc* **(MPa)** *C* **(MPa)**  1170.00 32.00 53.00 18.00 45.00

In order to compare the failure criterion, a [0o/90o3]s glass/epoxy symmetric laminated with total thickness of 1,624mm was used. All layers on the laminated have the same thickness. The results are presented in figure 8. All criterions were implemented in the same program

Table 4. Strength limits for glass/epoxy laminate (Highsmith & Reifsnider, 1982)

**5.2 Progressive stiffness loss of laminate** 

c. Axial tension loading in "x" direction;

to facilitate the comparison.

double symmetry: <sup>2</sup> {( , ) : (0 1,0;0 1,0} *xy R x y* ;

Table 3. Mechanical properties (Highsmith & Reifsnider, 1982)

Fig. 8. Gl/Ep [0º/90º3]s Laminated – Failure Criterion comparison.

d. The properties of the material used are listed in the tables 2 and 3;

The present paper deals with damage composite laminate with transverse cracks in the matrix applying Continuous Damage Mechanics Theory, which was initially proposed by Kachanov (Kachanov, 1958) and than adapted by Allen (Allen et al., 1987a,b) for orthotropic laminated composites. This theory was also applied by Lim and Tay (Lim & Tay,1996) in laminates with transverse cracks to describe the stiffness loss of the structure. The adapted Damage Theory considers the mechanism associated to the transverse cracks through the internal state variables inside the constitutive relations based on the Continuous Damage Mechanics.

The theoretical model was implemented in a computational program, developed in FORTRAN language, based on the Modified Local Green's Function Method (MLGFM). The approximated solution was obtained by the MLGFM. The damage evolution model, originally developed for FEM, can be applied also to MLGFM without substantial changes in the original code.

In the presented results, it can be observed that the conventional criterions catch only the moment when the 90º layers no longer influences the stiffness of the laminated. Most of the criterions were able to determine the loss of stiffness. The strain energy criterion is able to evaluate the damage evolution, identifying the moment when the transverse cracks starts to affect the laminated rigidity. However, during the strain increase, the efficacy of the method to evaluate the stiffness loss decreases. Even so, as shown in the figure 8, the implemented

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code is able to denote, for all criterions, the stiffness loss in laminated composites when transverse cracks are formed in the matrix.

It is important to note that the actual stage of damage of a laminated plate depends on the historical of loading. As the micro cracks rise by quantity, length and opening, the external load must be applied step by step. A tolerance and a stopping strategy must be decisive for the accuracy and approximation of the true solution.

#### **7. References**


code is able to denote, for all criterions, the stiffness loss in laminated composites when

It is important to note that the actual stage of damage of a laminated plate depends on the historical of loading. As the micro cracks rise by quantity, length and opening, the external load must be applied step by step. A tolerance and a stopping strategy must be decisive for

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the accuracy and approximation of the true solution.


**8** 

Yong X. Gan

*USA* 

**Energy Dissipation Criteria for Surface** 

*Department of Mechanical, Industrial and Manufacturing Engineering,* 

This chapter presents the energy dissipation approach for analyzing surface contact damages in various materials, including composite materials. As known, surface contact is a very common phenomenon, which can be found in daily life and many scientific and engineering problems. The contact of different bodies can be modeled as indentation. Analysis of indentation and modeling of the deformation states of indented materials are often difficult because of the complexity of stress distributions within indentation zones. It is also very difficult to evaluate stress states in regions underneath an indented zone. Instrumental indentation has been performed on various materials including composite materials. Experimental studies on indention of coatings and brittle materials have been reported extensively, but the criterion for evaluating the extent of damage is not unified. Ductile materials deform relatively stable in indentation processes. While brittle materials are sensitive to compressive contact loadings in view of the formation of surface cracks. Therefore, it is difficult to find a unified stress or strain based damage criterion to characterize the damage evolution. Energy dissipation analysis may be more accurate to describe the deformation behavior of such materials. Specifically, under wedge indentation, the analysis should be investigated because the stress field has the singularity which limits the applicability of the strength criterion. In this chapter, the load-displacement relations with elastic-plastic responses of the materials associated with the indentation processes will be obtained to calculate the hysteresis energy. Lattice rotation measurement using electron backscatter diffraction (EBSD) technique will be performed in the region ahead of the indenter tip to measure the dimension of the contact damage zone (CDZ) and the results will be used to define the length scales in contact deformation. A unified criterion using the

hysteresis energy normalized by the length scales will be established.

Damage evolution in composite materials is very sensitive to the interaction of reinforcements and matrices in interface regions. For example, the development of damage in glass particle and fiber reinforced epoxy composite materials is strongly influenced by the interface debonding conditions [1]. However, the exact effect of bonding conditions on the performance of particle filled composite materials is still not fully understood. Kawaguchi and Pearson [2] reported that strong matrix-particle adhesion may lower the fatigue crack propagation resistance. While the studies on Si3N4 nanoparticle filled epoxy composites

**1. Introduction** 

**Contact Damage Evaluation** 

*College of Engineering, University of Toledo,* 

Wada, A.; Motogi, S.; Fukuda, T. (1999). Damage mechanics approach to nonlinear behavior of FRP laminates with cracking layers. *Adv. Composite Mater*., Vol 8, No. 3, pp. 217- 234
