Mechanical behavior of unidirectional composites according different failure criteria

A BSTRACT . This work is about study of the mechanical behaviour of unidirectional Kevlar / Epoxy composite laminates according to different failure criteria. Varying strength parameters values, makes it possible to compare the ultimate mechanical characteristics obtained by the criteria of Tsai-Hill, Norris, Fisher, Ashkenazi and Tsai-Wu. The epoxy matrix of the material in question is reinforced with up to 60  of its volume by aramid fibers. The stack of four layers composing the arbitrarily oriented and alternating [+  /-  ] S materials results in balanced symmetrical laminates. The laminate is subjected to uniaxial tensile membrane forces. Estimate of their ultimate strengths and the plotting of the failure envelope constitute the principal axis of this study. Using the theory of maximum stress, we can determine the various modes of damage of the composite. The different components of the deformation are presented for different


INTRODUCTION
omposite materials, in the most common sense of the term, are a set of synthetic materials designed and used mainly for structural applications; the mechanical function is dominant. The mechanical behaviors of the composite, as well as the degradation mechanisms leading to its rupture depend on the nature of the constituents and on the architecture of the fiber preform [1]. C The profile is required because it guides the engineer in designing structures with precise properties in relation to the needs. The study of the stability of these structures requires, among other things, knowledge of the limiting behavior of the material. This behavior is expressed by a failure criterion. Which expresses the relationships between the components of

PREDICTION OF MATERIAL FAILURE
failure criterion is characterized by the knowledge of a scalar function    σ . There is no rupture of the material if prevailing stresses do not exceed the ultimate stress value; that is to say, as long as the following inequality is satisfied: A When the equality is satisfied, we obtain the failure envelope or the limiting surface [25].

Tsai-Hill Criterion
The evaluation of the resistance of the composite material working in tension, is ensured by the rupture criterion of Tsai-Hill. It allows us to predict the ultimate resistance of the least resistant ply, in the case of the plane stress [4][5][6]25]: With:

Norris Criterion
Norris [25 -29] assumes that in the constraint field, the point: lies on the fracture surface. So after substitution of (3) in relation (2) we find: K = 1.

Fisher's criterion
Fisher's criterion is applied to orthotropic materials and is based on Norris analysis.

Ashkenazi Criterion
This criterion is used for unidirectional composite materials. Ashkenazi [25,29,31] assumes that the points: lie on the failure surface. T: Ultimate tensile strength at 45 ° from the direction of the fibers. It must satisfy the stability condition: The coefficient K is then:

Tsai-Wu criterion
We will retain the Tsai-Wu criterion where the constants i F and ij F are the components of two tensors, respectively of rank 2 and 4, with the parameters ij F [32,33]: S: Stress at failure in shear, in the plane of the layer.

Maximum stress criterion
This criterion states that the structure withstands the conditions of use if the calculated stresses meet the conditions below.
If one of the inequalities is no longer true, the limit state is reached; the failure is then attributed to the stress present into this inequality. A failure envelope [34] is a three-dimensional plot of the combinations of the normal and shear stresses that can be applied to an angle laminate just before failure. One may develop failure envelopes for constant shear stress  12 and then use the two normal stresses  11 and  22 as the two axes. Then, if the applied stress is within the failure envelope, the laminate is safe; otherwise, it has failed.

METHODS
n the case of stresses planes (thin plate), the tensor {σ} as a function of that of the strains for a unidirectional composite layer, is given by the following relation:  Values of the reduced stiffness matrix in membrane [ ij Q ]as a function of the elastic constants [35]are as follows:

I
The indices 1 and 2 refer to the direction longitudinal and perpendicular to the fibers. This constitutive law for a layer of stacking order k in the laminate ( Fig. 1) ,is not ,in general, that of the structure. When the orientation of the fibers changes, the matrices base change, makes it possible to express the tensor of the stresses in the reference mark of the plate (x, y) according to the transformed reduced stiffness matrix: In the case of a symmetrical composite plate working only as a membrane, its forces will be expressed in the form [6]: with: In the case of the balanced symmetrical laminate, working in uni-axial tension we have: and: By substituting Eqns. (16) in the matrix form (12), and for a layer of order K, we find that: In the (natural) orthotropic plane we have: with: where   cos C and   sin S We put: So, formula (18) can be rewritten in the following form: and by the computation of the tensor of stresses in the orthotropic coordinate system, one can use the energy criteria to predict the limits of membrane forces that the laminate can withstand. These criteria must be applied successively to each ply constituting the laminate, for orientations from 0° to 90° with a step of 1. The membrane force applied to each ply constituting the laminate will be obtained as follows: The use of the Ashkenazi criterion is possible if the breaking stress of the bend oriented at 45° is introduced.
To determine the ultimate membrane force either in tension or in compression, we can use the tensor criterion of Tsai-Wu and find the solution to the following equation: Using the theory of the maximum stress, we determine the various modes of failure by the substitution of the components of the tensor of the stresses in the equation defining the criterion of the maximum stress theory: We have the rupture forces as: Therefore, the membrane force applied to each layer,   t x k N , is the minimum of the forces obtained by the last three modes. Then, the limit force capable of avoiding the breaking of the least resistant layer, is determined for each failure criterion.

Material used
he material examined is a unidirectional composite of Kevlar fiber and Epoxy resin. It is currently one of the most industrially developed composite families, especially in the production of high-performance parts. Its advantages are: low density, high tensile strength, low cost, and high impact resistance. Its drawbacks include low compressive properties and degradation in sunlight [34] The elastic constants and the mechanical characteristics obtained experimentally for unidirectional Kevlar / Epoxy composites (h: The total thickness of the composite plate , Vf = 0.6 is the volume fraction of the fiber) are [33]:  In the interval of 0 ° <  <28 °, we notice that there are different spectra of curves which represent the variation of the ultimate tensile force for each criterion. With this interval of angle of orientation, we notice on Fig. 3 a very fast reduction of the forces of membrane; and with breaks of the fibers and shears of the matrix, the criteria of rupture considered, do not produce the same ultimate values and become very different as the degree of anisotropy increases. Once  reaches the value 28 °, and with the exception of the theory of maximum stress, the curves tend to have the same values.

Ultimate tensile strengths of laminates
As one moves away from  = 28 ° and approaches 90 °, the membrane forces become more and more equal, monotonous and take more or less the form of a straight line. They then converge parallel to the axis of the abscissa. At this stage, the tensile breaking stresses of the matrix, are responsible. T Moreover, according to Fig. 2, the energy failure criteria provide continuous curves, unlike the maximum stress theory (Fig. 3) based on three distinct failure equations, where we have a discontinuity. On the other hand, the tensor criterion and the energy criteria have only one equation, therefore only one aspect of the curves. Nevertheless, it is the Tsai-Hill criterion which gives good results comparing to experimental results for unidirectional composites in tension [22,23,35,36,37]. These three equations (Formula 9) represent the three failure modes of the material. It is the outer layers oriented at + that rupture. The first mode is concerned with the orientation angles between 0 ° and 10 °, where we have the tensile breakage of the fibers. When the arrangement of the layer changes to an orientation equal to or less than 38 °. We have a shear failure of the matrix. As we approach the 90 ° angle, the low strength of the Epoxy resin, causes it to rupture by tension; and this is the third mode of rupture of the material [35].

Curves of boundary surfaces
The boundary surface curve (failure envelope) of our composite, allows us to determine the surface where one of the stresses can be applied without breaking the material. In Fig. 5, we have the failure envelopes obtained by the Tsai-Hill criterion in a plane form for shear stresses   12 0 , 20 GPa and 48.50 GPa ; they have elliptical shapes as can be seen (Interaction between the normal stresses).
The transverse stress is obtained as a function of the longitudinal stress and the different values of  12 . The boundary surface curve depends on the orientation of the fibers of the broken layer. It can be noticed that the increase in the shear stress causes the reduction of the surface of the rupture envelope disappears and when it reaches the ultimate shear stress. In addition, and unlike its configuration that obtained by the theory of maximum stress, indicates that the behavior of the material is not asymmetrical. The failure envelope (Fig. 6) determined by the theory of maximum stress is characterized by the absence of the interaction between the two stresses  11 and  22 ,which clearly means that it is not a function of the shear stress. The curve consists only of horizontal lines and verticals exhibiting the shape of a rectangle.   he examined composite material is unidirectional Kevlar / Epoxy reinforcement. It is composed, in our case, of four layers whose arrangement constitutes symmetrical laminates [/-] S . Our study allowed us to observe an elastic phase of the material until rupture, with the absence of the plastic phase. In addition, we noticed the absence of coupling between the tensile membrane force and the angular distortion of the material due to the balanced arrangement of the laminates. The study of the deformations for different stacking sequences made us choose the cross laminate as the best configuration. Among variety of failure criteria, we have chosen that of Tsai-Hill for unidirectional composites, especially when the orientation of the layers is in the vicinity of 0° and smaller than 28°. In addition, results obtained with then different criteria, are similar when moving forward 90°. The maximum stress theory makes it possible to determine three modes of rupture of the outer plies which are resistant. The curves of the envelopes of failure obtained by the criterion of Tsai-Hill, present elliptical shapes; on the other hand the theory of the maximum stress theory allowed us to obtain rectangles which are independent of the influence of the tangential component of the tensor of the stresses applied to the external layer.