Technical University of Łdź

Department of Technical Mechanics and Informatics

ul. Żeromskiego 116, 90-924 Łdź,Poland

Phone: (48) (42) 636 14 29

E-mail: piotr.szablewski@p.lodz.pl

Abstract

This paper describes a simple sinusoidal geometrical model of textile composite. On the basis of this model it will be presented howto obtain certain geometrical parameters which fully characterize the geometry of composite structure. Using geometrical considerations it is possible to obtain from this model basic mechanical parameters very useful for further strength analysis. Such mechanical considerations and method for calculating mechanical parameters will be presented in future article.

On the basis of above‑mentioned theoretical considerations a special computer program was developed. The method of calculation presented in this paper can be applied to more complicated models of textile composites.

Key words: textile composites, textile mechanics, numerical methods, woven fabric, unit cell model.

1.Introduction

Textile composites represent a class of advanced materials which are reinforced with textile preforms for structural or load bearing applications. In general, composites can be defined as a select combination-of dissimilar materials with a specific internal structure and external shape.The unique combination of two material components leads to singular mechanical properties and superior performance characteristics not possible with any of the components alone (see ref. [1], [2], [3]). The range of applications for composite materials appears to be limitless. Textile composites can be defined as the combination of a resin system with a textile fiber, yarn or fabric system. They may be either flexible or quite rigid. In this paper will be presented how to build a simple geometrical model of textile composite and how to get from this model certain geometrical and mechanical parameters, very useful for further strength analysis.

2. The geometrical model of textile composite

Let us consider a balanced plain weave textile composite in which the warp and fill yarns contain the same number of fibers n with all filaments having the same diameter , and with the warp and fill yarn shaving the same yarn packing density . The cross-sectional area of the yarns is given by .

The representative unit cell (RUC) is a rectangular consisting of two warp yarns interlaced with two fill yarns with resin matrix filling the remaining portion of the volume.

Its dimensions are and its thickness is denoted by H.The other parameters are denoted and shown in Figure 1. The thickness ofthe yarn along the center line of the yarn path is denoted by t. If the fiber volume fraction specified for the unit cell is too small, it is necessary to add an additional resin layer of thickness to the unit cell. The geometry of the path of the warp or fill yarn is modeled using two assumptions:1 - the center line of the yarn path consists of undulation portions and straight portions, with the center line of the undulating portions described by the sine function, 2 - the cross‑sectional area and the thickness of the yarn normal to its center line are uniform along the arc-length of the center line. The center line of the warp yarn path in an undulating regions specified by

, . (1)

The trigonometric functions of the angle in terms of function are

, , . (2)

, but ,

where is so-called crimp angle (see fig.1.).

. (3)

Figure 1. Cross section of the representative unit cell (RUC) along the warp yarn.

The volume fraction of the fibers in the RUC is given by . For the warp yarns the arc-length is given by .

Using it and integrating we have relationship for volume fraction

, (4)

, , (), (5)

where ( is the thickness of resin layer) and E(m) is complete elliptic integral of the second kind.

2.1. The geometry of the undulation region

On the basis of a constant thickness normal to the centerline path, we can determine the equations of the lower and upper curves of the warp yarn in the cross section of the RUC along the undulating portion of the path. These lower and upper curves are depicted in the detail of the undulation region shown in Figure 2. The parametric equations of the lower and upper curve are:

, , (7)

, . (8)

Using eqs. (1) to (3) and setting in the first of eqs. (7) , it can be shown that the value of is the root of the equation

, (9)

where and .

Figure 2. Geometry of an undulation region along the warp yarn.

Using complicated mathematical considerations we can get the relationship for the cross-sectional area of a tip of the fill yarns in the undulating region, determined by integration using the functions given in eqs. (7) to (9)

.

Now the cross-sectional area of the fill yarn, which is the same as the warp yarn, is .

Finally

, where . (10)

Substituting eq. (10) for A in eq. (4), we can rearrange the result as a quadratic equation in the ratio of .

. (11)

Equation (11) is used in the iterative procedure to determine parameters .

The function of the crimp angle is defined by

,

where

(12)

and are the incomplete elliptic integrals of the first and second kind.

Assume that , we obtain approximations to the volume fraction equation (4), and the yarn shape equation (10). We neglect terms of order and higher in the series expansions to get the final results:

, . (13)

Small crimp angle approximations (13) are used in the iterative procedure at the beginning of calculations.

2.2. Algorithm to determine the architecture parameters

Input parameters: n, , , a, , tol, imax. Output parameters: t, , , , - new yarn spacing only if , V - vol. of of the RUC, - vol. of the yarns in of the RUC, - vol. of the resin in of the RUC.

1. Compute the cross-sectional area A of the yarn from . Set .

From the small crimp angle equations (13) compute the initial values of the thickness t and length . The initial crimp angle is computed from eq. (3).

2. Begin iteration loop: for to imax in steps of one.

2a. Set or and for the crimp angle solve the quadratic equation (11) for the two roots of the ratio .

2b. If no positive real root exists, then STOP the program and write a warning message.

2c. Take the smallest positive real root, and multiply the root by the yarn spacing a to get a new value of the undulation length .

2d. If , then go to step 2e. else, if , then set and solve eq. (11)

for .

2e. Determine a new thickness t from the fiber volume fraction equation (4).

2f. Calculate a new crimp angle from eq. (3) using the new and t determined in steps 2c to 2e.

2g. Calculate the difference in the iterates using the measure .

2h. If , then increase the index .

2i. If , then go to step 2a. else, if , then STOP and print non-convergence message.

2j. If then the fixed point iteration is judged to have converged to the crimp angle .

3. Check the constraint on the crimp angle.

3a. If , then STOP. Convergence of the geometric parameters t, , and has been achieved and the geometry of the RUC is establish.

3b. If , then set and it is assumed that the specified value of the yarn spacing a is incorrect. Solve for t, , and a in steps from 3c to 3g.

3c. Set or and solve the quadratic equation (11) for the two roots .

Take the positive real root for the ratio of .

3d. If , then go to step 3e. else, if , then set and solve eq. (11) for .

3e. Because , we determine from eq. (3) that . Substitute this expression for t into the volume fraction equation (4) and solve for the new yarn spacing to get , where .

3g. For from step 3f, compute , , and .

Numerical example. Input data: , , , , , imax = 30.

Output data:

For n = 14000 a maximum crimp angle condition. The yarn spacing a is re-computed.

In Figure 3 it is shown numerically calculated shape of the undulation region along the warp yarn for two cases: a) model is valid (), b) model is not valid because a cusp forms in the filament in a region close to the end of the undulating region ().

Figure 3. Geometry of an undulation region along the warp yarn calculated numerically.

3. Conclusions

Presented method of analysis was very useful for modelling of textile composites. It can be applied of course for more complicated models (not only with sinusoidal centreline of the yarn). On the basis of this model certain geometrical parameters were obtained. These parameters fully characterize the geometry of considered composite structure. The method has shown that presented model is valid only for crimp angle less than 45 degree (). If the crimp angle the model is not valid because a cusp forms in the filament in a region close to the end of the undulating region. Using geometrical considerations it is possible to obtain from this model basic mechanical parameters very useful for further strength analysis. Such mechanical considerations and method for calculating mechanical parameters will be presented in future article. Using Mathematica environment a special computer program was developed, to calculate the geometrical parameters.

References

1. Cox B. N., Flanagan G., Handbook of Analytical Methods for Textile Composites,

Nasa Contractor Report 4750, Hampton, Virginia, March 1997.

2. Hull D., Clyne T. W., An Introduction to Composite Materials, Cambridge University Press, 1997.

3. Jones R. M., Mechanics of Composite Materials, McGraw-Hill Book Company, Washington, D. C., 1975.

4. Tsai S. W., Hahn H. T., Introduction to Composite Materials, Technomic Publishing Co., Westport, CT, 1980.

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