An approach of delamination energy calculation for a composite material.

Sabau, Emilia ; Iancau, Horatiu ; Popescu, Constantin 等

Abstract: The paper gives a mathematical model of the fibers sliding with respect to the matrix in composite materials. Also, it is present some results regarding interlaminar defects. The onset and propagation of interlaminar defects is one of the main damage. Delamination is an insidious kind of failure as it develops inside of the material, without being obvious on the surface. It is very important in industrial practice to know the principal causes of "delamination" appearance, and these to can be eliminated. Composite materials have applicability almost in all domains: electronics, electrotechnics, civil constructions, automotive transports, railway transports, naval transports, transport on cable, aerial and spatial transport, sport, and so on.

Key words: composite, delamination, fracture energy, interfaces, sliding friction.

1. INTRODUCTION

A composite material is a combination of two or more materials, differing in form or composition on a macro scale, (Inacau & Nemes, 2003). Delamination is interlaminar damage; it is the separation on a certain length of the sheets from the interior composite material (figure 1). This phenomenon can be produced locally or can cover a large area that leads in the end at the breaking up of the composite structure.

This problem is met between superior part and the core of the composite structure or local compression of the first layer, due to mechanical solicitations or special causes, (Kachanov, 1990).

"Delamination" can be produced during production or exploitation of the composite structure and can have a great variety of causes (the unsuitable choice of the component materials, technological imperfections, stress solicitation, and so on).

In the solicitation process of a composite, after a time, unavoidable appears the sliding phenomenon at the interface between fiber and matrix. This phenomenon is undesirable and leads to the breaking of the composite structure, (Kollar & Springer, 2003).This can have multiple causes: technological imperfections, the selection and inadequate quality of the component materials of the composite as well as the over solicitation in time of the composite structure.

The sliding phenomenon is governed by a friction (abrasion) law of Coulomb, based on the apposed resistance at the contact interface of the two corps (fiber-matrix), the movement or tendency of movement of one of the two corps on the contact surface of the other's.

According to the nature of the relative movement there are three types of friction:

a) Sliding type, when the displace is produced in the tangent plane at interface,

b) Rolling type, when the studied piece revolves (rolls) around an axis situated in tangent plane to the interface, respectively,

c) Swivel type, when the studied piece revolves around the normal to the contact interface of the two corps.

[FIGURE 1 OMITTED]

2. FORMULATION OF LIMIT PROBLEM ATTACHED TO SLIDING OF FIBRE IN COMPOSITE MATERIALS

In this paragraph we'll formulate the classical equilibrium problem of a composite material, where a sliding phenomenon of the fiber with respect of matrix appears. Let [OMEGA] [subset] [R.sup.3] be the domain occupied by a composite material. Let [??] be the displace vector, [??] the deformation and [??] the tensions vector. The following relation holds:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

Where [a.sub.ijkh] are the Lame' tensions components of the elastic composite [OMEGA]. If [??] = ([f.sub.1], [f.sub.2], [f.sub.3]) is the external forces vector, then the elastic equilibrium of the composite equations are:

- [3.summation over (j=1)] [partial derivative][[sigma].sup.[epsilon].sub.ij]/[partial derivative][x.sub.j] = [f.sub.i], I = 1 / 3 in [[OMEGA].sup.[epsilon].sub.1], where: (3)

[[sigma].sup.[epsilon].sub.ij] = [3.summation over (k=1)] [3.summation over (h=1)] [a.sup.[epsilon].sub.ijkh] [e.sub.kh] ([[??].sup.[epsilon]]) (4)

In the equations (3) and (4) the index [epsilon] shows that we have a periodical case of [epsilon]y period, where [epsilon] > 0 is a small parameter, and y is the cell volume repeating periodically.

[a.sup.[epsilon].sub.ijkh] (x) = [a.sub.ijkh] (1/[epsilon] x x) are the y periodical homogenized

coefficients and [[??].sup.[epsilon]](x) = [??](1/[epsilon] x x), (Iancau & al., 2005).

For the static case the boundary conditions are of Dirichlet type, namely:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

where [[GAMMA].sup.[epsilon]] is the boundary between two constituents (matrix, fiber), and [.] denotes the jump of the function to [[GAMMA].sup.[epsilon]] respective [[lambda].sub.0] the sliding friction coefficient. Denote by [[??].sup.[epsilon]] the versor of the normal to the transformed domain boundary [[OMEGA].sup.[epsilon].sub.1] by the change of variable 1/[epsilon] x x, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

The notation [T.sup.1] and [T.sup.2] refers to the Coulomb's law on the surface [[GAMMA].sup.[epsilon]] reported to the two cells of the composite having the interface [[GAMMA].sup.[epsilon]], (Gay & Gambelin, 1991). Finally, (3) + (5) represent the Dirichlet's boundary value problem. From all this mathematical equations we can emphasis the delamination phenomenon at composite material.

3. EXPERIMENTAL INSTALLATION

For experimental emphasis of the "delamination" phenomenon has been proposed the design and realization of a special installation, which has the components, represented in figure 2. The components of installation are: slice knife--role of sample slicing (composite structure); composite sample; sample holder--fixes the sample; cylindrical guide--guides the sample holder ensuring one degree of freedom; comparator with dial--indicates the deformation of elastic element; elastic element--role in the measuring of the forces which are applied on the probe; handle--ensures the sample displacement to knife 1.

3.1. Installation operating

The starting of the experimental installation is manually, and it is realized through rotation of the handle. This motion is transmitted farther through the screw to the nut that is ensured against the rotational motion, ensuring only the translational motion. Through the nut is performed the displacement of the ensample: elastic element--sample holder--sample, to the sliding knife (1). The spring deformation is indicated by the comparator with dial, whose pick-up is in contact with another fix surface of the elastic element point. The value of the applied force can be determinate following the position of the pointer of the comparator with dial (5), knowing the calibration of the elastic element (6). The sample splitting as result of longitudinal motion made by the probe is realized with the help of the sliding knife located on the knife holder, allowing to this a regulation on vertical direction and fixed in the desired position of the elastic element. The standardization was realized with the help of the "Instron 1196" test press.

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

4. EXPERIMENTAL METHODOLOGY

The samples are cut out from composite plates, realized from different processes: press, resin transfer molding (RTM), manually manufacturing, and so on, at suitable dimensions having parallelepiped form. The sample is simple fixed on the press table at the ends of the specimen. The samples have composite structures, created from different materials and grades of reinforcement. The used matrix is a special resin: RTM INJECTION Type "NORSODYNE-20282I.

The used samples are of three types: A: Fibers glass tissue RT 190 g/[m.sup.2], 60% reinforcing grade, 30 layers, with RTM injection in vibration mode; B: Fibers glass tissue RT 190 g/m2, 60% reinforcing grade, 30 layers, with RTM injection in simple mode; C: Fibers glass tissue RT 190 g/[m.sup.2], 60% reinforcing grade, 30 layers, with RTM injection with obvious defects;

5. THE EXPERIMENTAL RESULTS

The first phase consists in the sample penetration by the knife. It was done 7 measurements on the same sample. Considering the sample behavior that is of elastic linear type, we can estimate the critical force that causes the fissure advance. In this case, the necessary energy of the crack propagation is given of relation (1), (Iancau & al., 2005):

[K.sub.del] = 3E[e.sup.3][h.sup.5]/32[a.sup.4] [K/[mm.sup.2]] (7)

where E is the Young's module of the composite, a, e and h represent the geometrical parameters of the trial and they are defined in figure 3. The obtained results are shown in the Table 1. The shown values are the average of 7 trials on the same type of sample. After these experiments results that the plate of fiber glass tissue is more resistant in vibration mode.

6. CONCLUSIONS

The "delamination" phenomenon is undesirable and constitutes a major problem. It is very important in industrial practice to know the principal causes of "delamination" appearance, to can be eliminated. In the same time, it is useful to quantify energetically the resistance at "delamination" of the different composite structures. In the future, we will have in view, the extension of the experimental investigations with the help of suggested installation and verification in the practice of mathematical model.

7. REFERENCES

Gay, D. & Gambelin J. (1991), Une aproche simple du calcul des structures par la methode des elements finis, Edition III, Ed. Hermes, ISBN 2-86601-268-2, Paris

Iancau, H.; Crai, A.; Potra, T.Gh. & Sabau, E. (2005), Sliding fibers in composite materials with organic matrix, Proceedings of the 7th International Conference Modern Technologies in Manufacturing, Gyenge, Cs. (Ed.), pag. 229-232, ISBN 973-9087-83-3, Romania, October 2005, Editura Mures, Cluj-Napoca

Inacau, H. & Nemes, O. (2003), Materiale compozite. Conceptie si fabricatie, Composite materials. Manufacture and conception. Ed. Mediamira, ISBN 973-9357-24-5, Cluj-Napoca, Romania

Kachanov, L.M. (1990), Delamination Buckling of Composite Materials, Library of Congress in Publication Data, ISBN 90-247-3770-2, Brookline, Massachusetts, USA

Kollar, L. & Springer, G. (2003), Mechanics of Composite Structures, Cambridge University Press, ISBN 978-0-511-05703-8, New York, USA

Table 1. The experimental results

 The The The
Samples: Young's sample crack
 module thickness size [K.sub.del]
 [MPa] 2e [mm] a [mm] [J/[mm.sup.2]]

A 46000 4 5 1766
B 46000 3 6 1630
C 46000 4 10 46,57