Numerical study of appearence and development of damages in composite structures.

Pastrama, Stefan Dan ; Hadar, Anton ; Jiga, Gabriel 等

1. INTRODUCTION

Post-critical calculus of a laminate reinforced structure starts together with the appearance of the first delaminations in the composite structure. The main damages that could be produced in the composite material are matrix and fiber breaking and delaminations. In this case, the numerical methods are the only able to offer palpably results.

Post critical calculus is nonlinear and can emphasize both the failure mode of a composite structure and the mechanism of damage propagation (Hadar et al., 2007). In this way, the type of damage and the place where it appears can be detected.

After applying the failure criteria on each element, the finite element analysis can be continued in two ways: i. by keeping the initial load and performing the post critical calculus till the total failure of the structure and ii. by lowering the applied loads in order to determine the carrying capacity of the damaged structure.

2. IDENTIFICATION OF THE DAMAGES

The authors developed a finite element code in order to identify the damages and their propagation by an interactive calculus.

The application of failure criteria for plies is generally a difficult process. The authors show three possible failure modes of the fiber reinforced composite. The fibers are considered to be in a broken stage if (Fig. 1):

[[sigma].sub.l] [greater than or equal to] [[sigma].sub.af] or [square root of [[tau].sup.2.sub.lt]] + [[tau].sup.2.sub.lz]] [greater than or equal to] [[tau].sub.af] (1)

where [[sigma].sub.af] and [[tau].sub.af] and are the normal and shear allowable stresses for the fibers. If one of the restrictions (1) is satisfied, the stiffness matrix of the element is eliminated.

A matrix breaking is considered to appear when the stresses in the center of each element satisfy the conditions (Fig. 1):

[[sigma].sub.t] [greater than or equal to] [[sigma].sub.am] or [square root of [[tau].sup.2.sub.tl] + [[tau].sup.2.sub.tz]] [greater than or equal to] [[tau].sub.am] (2)

where [[sigma].sub.am] and [[tau].sub.am] are the allowable stresses for the matrix.

The elasticity matrix [[bar.D]] corresponding to the elements where one of the conditions (2) is satisfied, has another shape with respect to the equivalent one for transversely isotropic materials, (Wei, 1991).

[FIGURE 1 OMITTED]

The finite element analysis is resumed using an iterative procedure; in this case, the elasticity matrix corresponding to the elements where matrix breakings appear, will modify its shape.

In order to discover the appearance of delaminations, the stresses in the center of the adjacent area between two layers are compared with the allowable contact stresses. The separation surface between two adjacent elements is considered to be a possible zone for delaminations, when one of the following relationships is satisfied (Fig. 1):

[[sigma].sub.z] [greater than or equal to] [[sigma].sub.ad] or [square root of [[tau].sup.2.sub.lz] + [[tau].sup.2.sub.tz]] [greater than or equal to] [[tau].sub.ad] (3)

where [[sigma].sub.ad] and [[tau].sub.ad] represent the allowable normal and shear stresses for delamination. If equation (3) shows the appearance of delamination, the calculus is continued by modifying the shape of the elasticity matrix [[bar.D]] (Finn, 1993).

The stresses [[sigma].sub.af], [[tau].sub.af], [[sigma].sub.am], [[tau].sub.am], [[sigma].sub.ad] and [[tau].sub.ad] can be experimentally determined and assigned as input data.

3. CALCULUS OF STRESS AND STRAIN STATE USING IMPOSED NODAL DISPLACEMENTS

Due to the fact that generally only the areas with high stress gradients are of interest, the study of a complex structure made from fiber reinforced layered composites is recommended to be done in two steps:

Step 1: a. Analysis of the stress and strain state for the whole structure using a commercial numerical code; b. Detection of areas with stress concentrators; c. Delimitation of these areas from the whole structure, retaining the displacements of the nodes situated on the contour of the isolated part, the displacements being obtained as a result of the sue of the numerical code (Flanagan, 1993);

Step 2: a. Analysis of the isolated areas with specialized finite elements destined to the study of local problems in multilayered composite materials (Hadar et al., 2006); in this case, the input data are the displacements obtained in the previous analysis; b. achievement of a post-critical calculus in order to determine the bearing capacity of the structure.

3.1 Methodology and algorithm for application of failure criteria and post-critical calculus

The flow chart of the algorithm is presented in Fig. 2 and is used for the application of the composite failure criteria, for the localization of the type of damage as well as for a post-critical calculus under an unmodified or modified (diminished) load, in order to pursue the way of damage propagation, up to the total failure of the structure.

In order to realize a post-critical calculus, the matrices of elasticity in a local system of coordinates, where the apparition of one or several damages has been observed, are modified:--for the finite elements characterized by breaking of fibers, due to the fact that the strength element (in our case, the fiber) disappears, the corresponding matrices of elasticity are canceled; in order to continue the calculus with finite elements without any supplementary modeling of the structure, very small values will be assigned for these elements of the elasticity matrix ([10.sup.4]-[10.sup.5] times less than for the initial ones);

--for the finite elements characterized by a matrix failure, the elasticity matrix form will be modified, corresponding to (Wei, 1991); for the disappearing elements of the elasticity matrix very small values will be also assigned;

--if between two adjacent layers a delamination will occur, the elasticity matrix corresponding to these elements will be also replaced (Finn, 1993), taking into consideration that the disappearing elements will acquire very small values;

--if in the structure one can find finite elements having multiple damages, the main flaw will be considered the fiber breaking, then the matrix failure and finally the delamination. For example, if in the same finite element delaminations and fiber breakings appear simultaneously, for this element, the matrix of elasticity will be replaced (Wei, 1991), the main flaw being considered to be the matrix failure.

3.2 Flow chart of the applicability of stop criteria for the post-critical calculus

In conformity with the code, the post-critical calculus continues till one of the following criteria is accomplished:

a) The criterion of the number of damaged elements.

In this case the program calculates the number of elements having damages as matrix breakings ([N.sub.01]), fiber breakings ([N.sub.02]) and delaminations ([N.sub.03]) and compares this number with the total number of elements of the whole structure ([N.sub.0]). According to Finn (1993), Hadar et al. (2006) and Flanagan (1993), the numerical code should warn that the structure fails if one of the following restrictions is accomplished:

[N.sub.01]/[N.sub.0] > 0.3 ; [N.sub.02]/[N.sub.0] > 0.15 ; [N.sub.03]/[N.sub.0] > 0.2 (4)

b) The criterion of the limit nodal displacements.

The displacement of each node of the structure is calculated with the equation [delta] = [square root of [u.sup.2] + [v.sup.2] + [w.sup.2]], where u, v and w are the nodal displacements on the global system of axes Oxyz. The program considers that the structure fails when:

[delta] > [[delta].sub.lim] (5)

c) The criterion of the volume of damaged elements.

Let us consider [V.sub.01], [V.sub.02], [V.sub.03] the volumes of the elements where damages (fiber or matrix breakings, delaminations) occur and let [V.sub.0] be total volume of the structure. The code proposed by the authors stops the post-critical calculus when one of the following restrictions is accomplished:

[V.sub.01]/[V.sub.0] > 0.3 ; [V.sub.02]/[V.sub.0] > 0.15 ; [V.sub.03]/[V.sub.0] > 0.2 (6)

All these criteria can be applied only if damages occur in the composite structure. The stopping criteria of post-critical analysis may be modified in function of the analyzed structure. In this case reference values can be established for the failure of the reinforced composite structure.

[FIGURE 2 OMITTED]

4. CONCLUSIONS

The computer code calculates and displays the displacements [u.sub.x], [u.sub.y], [u.sub.z] of each node of the whole structure. Through this program, the stresses in the center of each element can be calculated. The obtained stresses are related as well to the global system of axes Oxyz as to the local system Oltz. After applying the failure criteria, the computer code displays the number of elements where damages occur as well as the type of damage. The finite element calculus is resumed in a non-linear regime, in an iterative manner, the stiffness matrices of damaged finite elements being modified in function of the type of flaw. When one of the conditions referable to the structure failure is accomplished, the code displays it.

5. REFERENCES

Finn, S.C., Springer, G.S. (1993), Delaminations in Composite Plates Under Transverse Static or Impact Loads--a Model, Composite Structures, Vol.23.

Flanagan, G. (1993), A Sublaminate Analysis Method for Predicting Disband and Delamination Loads in Composite Structures", Journal of Reinforced Plastics and Composites, Vol.12.

Hadar, A., Constantinescu, I.N., Jiga, G., Ionescu, D. S., (2007), Some Local Problems in Laminated Composite Structures, Materiale Plastice, Vol. 44, nr 4, pp. 354-360.

Hadar, A., Nica, M.N., Constantinescu, I. N., Pastrama, S.D., (2006), The Constructive and Geometric Optimization of the Junctions in the Structures Made from Laminated Composite Materials, Journal of Mechanical Engineering, Vol. 52, No. 7-8/06, Ljubljana, p. 546-551.

Wei, J., Zhao, J.H. (1991), Three-dimensional Finite Element Analysis on Interlaminar Stresses of Symmetric Laminates, Computers and Structures, Vol.41, No.4.