Abstract
An extension of classical laminate theory to the anelastic domain is proposed. It is based on numerical methods which describe the laminate's response to mechanical loading not analytically but step by step. This approach is applied to a behaviour model of a non linear viscoelastic layer and its application to multiaxiaI creep tests on filament wound tubes ± 55°is discussed.
Abstract
The composite laminate materials under study are composed of layers which consist of unidirectional glass fibre-reinforced epoxy resin. The behaviour of the laminate depends on the fibre direction in each layer in relation to a laminate reference axis and on the position of each layer in the thickness of the laminate. The value of laminate theory lies in the fact only the behaviour of one layer need be identified in order to predict the behaviour of any laminate. The problem of the behaviour modelization is then situated at the level of an unidirectional fibre-reinforced composite. Initially, we limit our study to a viscoelastic behaviour. In a previous study [1] based on the extension ofa self-consistent method from [2], a non linear viscoelastic model of unidirectional composite is developed. This model expresses the strain resulting from creep tests by :
Abstract
where §e is the tensor of elastic compliances and §v the tensor of viscoelastic compliances which is similar to §e but depends on the stress so as to describe the non linearity of the viscoelasticity. The symbol" : " denotes the tensorial product contracted on two indices.
Abstract
n Laminate theory
Abstract
In analogy with classical laminate theory, the resulting stresses and moments in the laminate h/2 are defined by : N = J~. dz and -ht2
Abstract
They can be rewritten in the following form :
Abstract
where n is the number of layers, Zk_1 and z, the minimal and maximal coordinates of the k layer, eo and p the mean plane strains and the curvatures of the plate.
I Introduction
The composite laminate materials under study are composed of layers which consist of unidirectional glass fibre-reinforced epoxy resin. The behaviour of the laminate depends on the fibre direction in each layer in relation to a laminate reference axis and on the position of each layer in the thickness of the laminate. The value of laminate theory lies in the fact only the behaviour of one layer need be identified in order to predict the behaviour of any laminate. The problem of the behaviour modelization is then situated at the level of an unidirectional fibre-reinforced composite. Initially, we limit our study to a viscoelastic behaviour. In a previous study [1] based on the extension ofa self-consistent method from [2] , a non linear viscoelastic model of unidirectional composite is developed. This model expresses the strain resulting from creep tests by :
I Introduction
where §e is the tensor of elastic compliances and §v the tensor of viscoelastic compliances which is similar to §e but depends on the stress so as to describe the non linearity of the viscoelasticity. The symbol" : " denotes the tensorial product contracted on two indices. They can be rewritten in the following form :
I Introduction
where n is the number of layers, Zk_1 and z, the minimal and maximal coordinates of the k layer, eo and p the mean plane strains and the curvatures of the plate.
I Introduction
In contrast to the elastic case, we can no longer integrate analytically ( §e + §J-1. The tensor §v depends on the stress and thus can possibly depend on the z coordinate , for example in the case of bending or in tension when the composite material is not symmetrical.
I Introduction
ill Solution procedure
I Introduction
Two possibilities can be envisaged :
I Introduction
-The mean plane strains and the curvatures of the plate are imposed globally on the laminate.
I Introduction
The strain is given at every point z by the relation~=~o + z 0 p. Then equation (1) allows the stresses to be found in each layer and by integration to determine the stress state of the laminate.
I Introduction
-The external stresses are prescribed for the laminate.
I Introduction
Since it is impossible to deduce the stresses in each layer given the stresses applied to the laminate, the idea is to estimate the mean plane strains and curvatures in order to find ourselves in the same situation as the previous case. As far as the imposed and calculated external stresses are not equal within a given tolerance, a new estimation iteration is performed. In both cases, the solution of the behaviour model of a single layer with imposed strains and the integration of the stresses over the thickness are required. Moreover, the study of the second case requires the optimal and gradual estimation of the mean plane strains and curvatures.
I Introduction
• Solution of the behaviour model ofa single layer In order to accomplish this, equation (1) is differentiated with regard to time and a fourth-order Runge-Kutta method is chosen to solve the resulting differential equation.
I Introduction
• Integration of the stresses over the thickness of the laminate This is possible owing to the numerical method of Gauss -Legendre [3] . This method takes into account irregularly spaced points, and thus requires the discretization of each layer into elementary sub-layers.
I Introduction
• Optimal and gradual estimation of the mean plane strains and curvatures The Newton -Raphson method, whose convergence is fast and whose possible divergence is almost immediate, allows this problem to be solved. For this type of loading, as long as the stresses are relatively small, the model gives totally satisfactory results. Otherwise certain differences reveal themselves. These divergences can be explained by the particular nature of the model layer itself and do not imply a weakness in the present laminate theory. However, an interesting perspective for the final validation of this procedure is obtained by performing creep tests on other laminates with different angles and stackings.

{'paper_id': '139840561', 'title': 'Extension of Classical Laminate Theory to the Anelastic Domain', 's2_url': 'https://api.semanticscholar.org/CorpusID:139840561', 'source': 'S2ORC'}