Parametric elastic constitutive model of uni-directional fiber-matrix composite

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The objective of this research is to express anisotropic elastic constants of uni-directional fiber reinforced composite (treated as continuum) as direct functions of microstructure parameters. An extensive data base has been created from results of multiple numerical analyses, in which values of elastic constants for different microstructure instances are gathered. Now, elastic constants of a composite characterized by arbitrary values of microstructure parameters can be determined, without additional costly computations, by merely an interpolation of the data base results.

Introduction

The microstructure of the UD composite is described by the following parameters:

the matrix elastic constants (isotropic), E^m, ν^m,
the fiber elastic constants (transversely isotropic), E^f_t , E^f_p , G^f_pt , ν^f_pt , ν^f_tt (where p and t denote, respectively, the directions parallel and transversal to the fiber axis),
the fiber volume ratio φ.

The further introduced geometric assumptions lead to the transversely isotropic composite properties, i.e., the following 5 elastic constants are going to be computed and expressed as functions of the above parameters:

E_p , E_t , G_pt , ν_pt , ν_tt .

There are approximate analytical formulae cited in the literature, that formulate the relationships of our interest. Among most advanced and precise one can mention those of Halpin & Tsai [2] and Hashin & Rosen [3]. These will be, however, disregarded in this study since they are limited to isotropic microstructure constituents (which in the case of fibers is often not true) and they still do not provide the full set of composite material constants. Thus, the numerical models will only be considered.

It is assumed that fibers are laid parallel to each other (no twist) and that they are of approximately the same diameter. The well known homogenization methods based on the concept of a repeatable representative volume element (RVE) are employed. The methods have been widely exploited in the case of UD composites (see e.g. [4],[5]), however, no parametric dependence of the resulting elastic constants on the input parameters has been investigated. Besides, the concept in its original shape enforces assumption of a predefined regular layout pattern of fibers in the tow cross-section. In order to liberate from this limitation, the analysis is done here for different patterns at the same fiber volume ratio and the results are averaged. These different patterns are defined by the values of two angles: α (60°<α<120°) and β (0°<β<180°), see figure below.

[prismatic cell] [qcs cell]

regularized and parametrized cross-section RVE

Methods

As mentioned above, the main idea of the parametric constitutive model presented here consists in computation of the full set of homogenized material constants for a sufficiently large number of microstructures (defined by different value sets of microstructure parameters) and construction of a data base of composite material constants. Then, elastic constants of a composite characterized by arbitrary values of microstructure parameters can be determined by merely an interpolation of the data base results. Besides, sensitivity of the composite material constants with respect to the parameters can be easily computed, too.

The material constants for a given microstructure parameter set are computed by FE analysis of a RVE for six load cases. Comparison of averaged stresses and strains (6-component arrays) allows to determine the full 6x6 elastic stiffness matrix of the composite, and further — the homogenized engineering material constants. Detailed geometric formulation and finite element model (including the load cases, periodic boundary conditions and details of the above mentioned angle averaging) is described in the author's publication [1] and it will not be presented here.

To ensure thermodynamical correctness of the input material constants and to eliminate dimensionality from the microstructure parameter sets, the following dimensionless parameters have been defined:

h₁ = E^m / E^f_p , h₂ = ν^m , h₃ = E^f_t / E^f_p , h₄ = G^f_pt / E^f_t , h₅ = ν^f_pt √(E^f_t / E^f_p) , h₆ = ν^f_tt , h₇ = φ .

The fiber longitudinal stiffness modulus E^f_p has been assumed unit in the computations – hence, the stress-dimensioned resulting moduli E_p , E_t , G_pt , saved in the results data base, must be eventually scaled by the actual value of this modulus. Each of the parameters was assigned a set of characteristic values for which the computations were done; these values are listed in the table below. Each parameter was independently running through its set of values so that the total number of different microstructure instances analysed was 6•4•6•5•5•5•5 = 90000.

parameter no. values values

h₁ 6 0.01 0.0324 0.0961 0.3136 0.64 1.00

h₂ 4 0.24 0.32 0.40 0.48

h₃ 6 0.01 0.0324 0.0961 0.3136 0.64 1.00

h₄ 5 0.1 0.4 1.0 2.0 4.0

h₅ 5 0.0 0.1 0.2 0.3 0.4

h₆ 5 0.0 0.1 0.2 0.3 0.4

h₇ 5 0.24 0.36 0.48 0.60 0.72

Results

Computations were performed with the use of the finite element code ABAQUS, v. 6.9. Tabularized results of static computations in the form of a formatted ASCII file can be downloaded here.

This is an ASCII file in the CSV format, with comma as the column separator. First lines contain description text. Further lines contain numerical data. Each data record (line) corresponds to one microstructural instance (a set of microstructure parameter values). The meaning of data entries in each record is as follows:

First 7 data in the record are values of the dimensionless parameters h_k, k = 1,…,7.
Subsequent 5 data in the record are values of the transversely isotropic elastic constants of the composite: E_p , E_t , G_pt , ν_pt , ν_tt .

Since the fiber longitudinal modulus has been assumed unit in the computations, and the other stress-dimensioned moduli of the microstructure components are expressed in proportion to this modulus (see definition of the parameters h_k), the resulting stress-dimensioned moduli (E_p , E_t , G_pt ) should be eventually scaled by the actual value of the modulus.

Discussion

Detailed discussion of results with comparison to other author's data can be found in [1]. To summarize, derivation of constitutive properties of UD tow from the equivalent microstructure models gives satisfactory results. Anisotropic elastic constants of UD composite tow have never been so far computed as explicit functions of microstructural parameters. Such constitutive model enables e.g. sensitivity analysis of structural response of composite elements with respect to the microstructural parameters.

References

[1] P. Kowalczyk. Parametric constitutive model of uni-directional fiber-matrix composites. Finite Elements in Analysis and Design, 50: 243–254, 2012 (link).
[2] J.C. Halpin, J.L. Kardos. The Halpin–Tsai equations: A review. Polymer Engineering and Science, 16(5):344–352, 1976. (link)
[3] Z. Hashin, W. Rosen. The elastic moduli of fiber-reinforced materials. Journal of Applied Mechanics, 6:223–232, 1964.
[4] C.T. Sun, R.S. Vaidya. Prediction of composite properties from a representative volume element. Composites Science and Technology, 56:171–179, 1996.
[5] E. Barbero. Finite Element Analysis of Composite Materials. CRC Press, 2008.

Note: The material published above is subject to copyright. It is a summary of the author's research whose extended description can be found in [1]. You may refer to the contents as well as use numerical data downloaded from this page for your non-commercial research purposes provided that, whenever you publish your results related to this material, you make reference to [1] and to this site.
Financial support of EC within the MAAXIMUS Project is gratefully acknowledged.

Page created and maintained by Piotr Kowalczyk ( Piotr.Kowalczyk(-at-)ippt.pan.pl )

My Institute, [SPMKM]

My Department

Last updated: July 9, 2013

parameter	no. values	values
h₁	6	0.01	0.0324	0.0961	0.3136	0.64	1.00
h₂	4	0.24	0.32	0.40	0.48
h₃	6	0.01	0.0324	0.0961	0.3136	0.64	1.00
h₄	5	0.1	0.4	1.0	2.0	4.0
h₅	5	0.0	0.1	0.2	0.3	0.4
h₆	5	0.0	0.1	0.2	0.3	0.4
h₇	5	0.24	0.36	0.48	0.60	0.72