**4.2 Maximum strain criterion**

This theory is analogous to the Maximum Stress Criterion, but the fail criterion is controlled by deformation limits in the principal directions of material. In this theory the material will fail when one of the following limits are reached:


where X, X, Y, Y are the maximum deformation deformation values in the principal directions 1 and 2, for tensile and compressive loading, *C* , is the maximum angular distortion in the plane 1-2.

#### **4.3 Tsai-Hill Criterion**

134 Continuum Mechanics – Progress in Fundamentals and Engineering Applications

causing changes in the material properties and in its local stress distribution. In this way, the main difficulty in this kind of analysis is the adoption of a failure criterion, *Fa* , that

The theories introduced to prevent the failure of an orthotropic laminate are adaptations of failure criterion for isotropic materials, modified for biaxial stress cases, such as, Maximum Stress Criterion, Maximum Strain Criterion, Tsai-Hill Criterion and Tsai-Wu Criterion

A new criterion is presented, based on the strain energy release rate, to evaluate the formation of a new micro crack (Anderssen et al., 1998; Ji et al., 1998; Kobayashi et al.,2000). The released energy is used because it is practically independent from the crack length

According to the Maximum Stress Criterion, for orthotropic materials, while the stresses in the principal directions of the material are lower than strength of the material in this

where *Xt* is the longitudinal tensile strength, *Yt* is the transverse tensile strength, *Xc* is the longitudinal compressive strength, *Yc* is the transverse compressive strength and *C* is the

This theory is analogous to the Maximum Stress Criterion, but the fail criterion is controlled by deformation limits in the principal directions of material. In this theory the material will

where X, X, Y, Y are the maximum deformation deformation values in the principal

*t*

*c*


<sup>2</sup> *Yt*

<sup>2</sup> *Yc*

<sup>1</sup> *Xt* - longitudinal direction

<sup>1</sup> *Xc* - longitudinal direction



, is the maximum angular



<sup>2</sup> *Yt* - transverse direction

<sup>2</sup> *Yc* - transverse direction

<sup>12</sup> *C* - plane shear

conveniently describes the damage evolution due to a failure mode.

(Anderssen et al., 1998). Some of these criterions are presented here.

(Reddy, 1997; Vasiliev & Morozov, 2001; Mendonça, 2005).

**4.1 Maximum stress criterion** 

shear strength of the lamina.

**4.2 Maximum strain criterion** 

distortion in the plane 1-2.

direction, there are no fails, which means:

Tensile failure

Compressive failure

Shear failure

fail when one of the following limits are reached:

Tensile failure <sup>1</sup> *<sup>X</sup>*

Compressive failure <sup>1</sup> *<sup>X</sup>*

directions 1 and 2, for tensile and compressive loading, *C*

Shear failure 12 *C*

An adaptation made by Tsai in the Hill Criterion for transverse orthotropic laminate at plane stress condition, resulted in the expression (43):

$$\frac{\sigma\_1^2}{X^2} + \frac{\sigma\_2^2}{Y^2} - \frac{\sigma\_1 \sigma\_2}{X^2} + \frac{\tau\_{12}^2}{C^2} = 1\tag{43}$$

#### **4.4 Tsai-Wu Criterion**

A simple procedure was proposed by Tsai-Wu, changing the Tsai-Hill Criterion in equation (43). When the tensile and compression strength are similar, the expression (44) becomes the Tsai-Hill Criterion.

$$\frac{\sigma\_1^2}{X^2} + \frac{\sigma\_2^2}{Y^2} - \frac{\sigma\_1 \sigma\_2}{XY} + \frac{\tau\_{12}^2}{C^2} = 1\tag{44}$$

#### **4.5 Strain Energy Release Rate Criterion**

The Strain Energy Release Rate Criterion to describe the damage evolution was applied by Lim and Tay (Lim & Tay, 1996). Consider a symmetric laminated composite of width *b* and length *L*, with the configuration [0°l/90°m]s, where l and m are integers. When the laminated is loaded uniaxially in tension, the stress-strain curve is linear until the failure criterion is reached for the first time, at point A (figure 5). A transverse crack is introduced in 90° layer.

Fig. 5. Stiffness loss in composite laminates [0°l/90°m]s – Lim & Tay (1995).

The result is a reduction in the effective stiffness of the laminate in the loading direction, and this is represented by the OB segment in figure 5. Upon further loading, this reduction is verified by the segment BC. This process is repeated until line OF is reached. Note that

Progressive Stiffness Loss Analysis

50

() formulation.

E/Eo [%]

formulation

60

70

80

E/Eo [%]

90

100

case.

of Symmetric Laminated Plates due to Transverse Cracks Using the MLGFM 137

The loss of stiffness is observed in figure 6 for different meshes and compared with the results obtained by Lim & Tay (1996) and experimental results. As the crack density grows up, the stiffness diminishes. The results are better with finest meshes, but even with coarse meshes, the approximation is good. Figure 7 shows the loss E/Eo versus crack density ζ for the case Gl/Ep [0°/90°]s – using () formulation. The same considerations are made for this

> Mesh 1x1 Mesh 2x2 Mesh 3x3 Mesh 4x4 Mesh 5x5 Mesh 6x6 Lim e Tay Experimental

> > Mesh 1x1 Mesh 2x2 Mesh 3x3 Mesh 4x4 Mesh 5x5 Mesh 6x6 Lim e Tay Experimental

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8 2 ζ [mm]

Fig. 6. Stiffness loss E/Eo versus crack density for the Gr/Ep [0o./90o]s laminated, using

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8 2 ζ [mm]

Fig. 7. Stiffness loss E/Eo versus crack density ζ for the case Gl/Ep [0°/90°]s – using ()

this dotted line represents the stiffness of the laminate where the contribution of the 90° layers was neglected.

When the area BCHG reaches a critical value. An additional transverse crack is formed and the effective stiffness reduces again, as indicated by the segment OC. Denoting the area BCHG in figure 5 as*U*0*<sup>i</sup>* , where *i* indicates the lamina in analysis, the strain energy density is given by:

$$
\Delta U\_{0i} = \frac{1}{2} \left( \sigma\_{xi} \Delta \varepsilon\_{xi} + \sigma\_{yi} \Delta \varepsilon\_{yi} + \tau\_{xyi} \Delta \gamma\_{xyi} \right) \tag{45}
$$

Where *xi* , *yi* e *xyi* are the stress in *x* direction, *y* direction and *xy* plane shear of the lamina *i*, respectively, and *xi* , *yi* e *xyi* are the strain in *x* direction, *y* direction and *xy* plane of the lamina *i*, respectively. Therefore, the energy *Ui* , necessary to form a new crack, can be defined as:

$$
\Delta U\_i = \frac{t}{t\_2} L \Delta U\_{0i} \tag{46}
$$

Where *t* is the thickness of the laminated, *t2* is the thickness of the 90° plies an *L* is the length of the laminated.

In this way, a transverse crack is assumed to be formed when:

$$\mathcal{U}\_i \ge \mathcal{G}\_{lc} \tag{47}$$

Where *GIc* is the mode I energy release rate for the formation of a transverse crack. The process of determining the transverse crack density is repeated for each successive micro crack, using the same value for *GIc* . As seen in figure 5, a series of points (A, B, C, D, …, E) can be generated until the limit OI is reached. From this limit, matrix cracking in the 90º layers no longer influences significantly the laminate stress-strain behavior. It must be observed that in practice, the intervals between the points are very small, turning the curve smooth, rather than the curve shown in figure 5.

#### **5. Applications**

#### **5.1 Analysis of laminated plates by the MFLGM**

The first application refers to the analysis of a laminate plate, whose materials of its lamina are defined in table 2. The aim of this application is to determine the stiffness loss E/Eo due to the improvement of crack density for the Gr/Ep [0o./90o]s laminated, using () formulation


Table 2. Material Properties - Highsmith e Reifsnider (1982)

this dotted line represents the stiffness of the laminate where the contribution of the 90°

When the area BCHG reaches a critical value. An additional transverse crack is formed and the effective stiffness reduces again, as indicated by the segment OC. Denoting the area BCHG in figure 5 as*U*0*<sup>i</sup>* , where *i* indicates the lamina in analysis, the strain energy density

<sup>0</sup> <sup>1</sup>

plane of the lamina *i*, respectively. Therefore, the energy *Ui* , necessary to form a new crack,

2 *i i <sup>t</sup> U LU*

Where *t* is the thickness of the laminated, *t2* is the thickness of the 90° plies an *L* is the length

Where *GIc* is the mode I energy release rate for the formation of a transverse crack. The process of determining the transverse crack density is repeated for each successive micro crack, using the same value for *GIc* . As seen in figure 5, a series of points (A, B, C, D, …, E) can be generated until the limit OI is reached. From this limit, matrix cracking in the 90º layers no longer influences significantly the laminate stress-strain behavior. It must be observed that in practice, the intervals between the points are very small, turning the curve

The first application refers to the analysis of a laminate plate, whose materials of its lamina are defined in table 2. The aim of this application is to determine the stiffness loss E/Eo due to the improvement of crack density for the Gr/Ep [0o./90o]s laminated, using ()

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

Grafite / Epoxi (Gr/Ep) 142,00 9,85 4,48 3,37 0,3 Glass / Epoxi (Gl/Ep) 41,70 13,00 3,40 3,40 0,3

0

 

are the stress in *x* direction, *y* direction and *xy* plane shear of the

*yi yi xyi xyi* (45)

are the strain in *x* direction, *y* direction and *xy*

*<sup>t</sup>* (46)

*U G i Ic* (47)

 

*U i xi xi* 

> e *xyi*

2

 , *yi* 

In this way, a transverse crack is assumed to be formed when:

smooth, rather than the curve shown in figure 5.

**5.1 Analysis of laminated plates by the MFLGM** 

Table 2. Material Properties - Highsmith e Reifsnider (1982)

layers was neglected.

lamina *i*, respectively, and *xi*

can be defined as:

of the laminated.

**5. Applications** 

formulation

is given by:

Where *xi* , *yi* e *xyi* 

The loss of stiffness is observed in figure 6 for different meshes and compared with the results obtained by Lim & Tay (1996) and experimental results. As the crack density grows up, the stiffness diminishes. The results are better with finest meshes, but even with coarse meshes, the approximation is good. Figure 7 shows the loss E/Eo versus crack density ζ for the case Gl/Ep [0°/90°]s – using () formulation. The same considerations are made for this case.

Fig. 6. Stiffness loss E/Eo versus crack density for the Gr/Ep [0o./90o]s laminated, using () formulation.

Fig. 7. Stiffness loss E/Eo versus crack density ζ for the case Gl/Ep [0°/90°]s – using () formulation

Progressive Stiffness Loss Analysis

1,624mm. The results are presented in figure 9.

Fig. 9. Gl/Ep [0º/90º3]s Laminated – Results comparison.

**6. Conclusion**

Mechanics.

in the original code.

of Symmetric Laminated Plates due to Transverse Cracks Using the MLGFM 139

To demonstrate the model capability to prevent the laminate stiffness loss, a comparison between the results obtained by the strain energy criterion implemented in this paper and the results obtained by Talreja (Talreja,1984) and Tay (Tay & Lim, 1993) was made. This analysis also used the [0o/90o3]s glass/epoxy symmetric laminated with total thickness of

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

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 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

### **5.2 Progressive stiffness loss of laminate**

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:



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


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

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 to facilitate the comparison.

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

To demonstrate the model capability to prevent the laminate stiffness loss, a comparison between the results obtained by the strain energy criterion implemented in this paper and the results obtained by Talreja (Talreja,1984) and Tay (Tay & Lim, 1993) was made. This analysis also used the [0o/90o3]s glass/epoxy symmetric laminated with total thickness of 1,624mm. The results are presented in figure 9.

Fig. 9. Gl/Ep [0º/90º3]s Laminated – Results comparison.
