Parametric elastic constitutive model of plain-weave fabric reinforced composite ply

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The objective of this research is to express anisotropic elastic constants of plain-weave fabric reinforced (PWFR) composite ply (at the macroscopic continuum scale) 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.

[pwfr cell]

Introduction

The figure above presents the schematic geometry of a representative volume element (RVE) of the PWFR composite. It consists of two families of interlaced yarns immersed in matrix. Yarns are assumed transversely isotropic with the principal direction aligned with curved yarn axis, while matrix is assumed isotropic. The microstructure is described by the following parameters:

the matrix elastic constants, E^m, ν^m,
the yarn elastic constants, E^y_t , E^y_p , G^y_pt , ν^y_pt , ν^y_tt (where p and t denote, respectively, the directions locally parallel and transversal to the curved yarn axis),
geometric parameters, s, w₁ , w₂ , φ, t, (where φ = t₁/w₁ = t₂/w₂)

The yarn elastic constants should not be confused with fiber elastic constants. If fiber and matrix properties are only known, the yarn elastic constants may be estimated with one of known analytical methods or with the author's numerical method presented here.

The geometric assumptions lead to orthotropic resultant composite properties, i.e., the following 9 elastic constants are going to be computed and expressed as functions of the above parameters:

E₁ , E₂ , E₃ , G₁₂ , G₁₃ , G₂₃ , ν₁₂ , ν₁₃ , ν₂₃ .

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 and periodic boundary conditions) 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^y_p , h₂ = ν^m , h₃ = E^y_t / E^y_p , h₄ = G^y_pt / E^y_t , h₅ = ν^y_pt √(E^y_t / E^y_p) , h₆ = ν^y_tt , h₇ = w₁/s , h₈ = w₂/s , h₉ = φ , h₁₀ = (t₁+t₂)/t.

The yarn longitudinal stiffness modulus E^y_p has been assumed unit in the computations – hence, the stress-dimensioned resulting moduli E₁ , E₂ , E₃ , G₁₂ , G₁₃ , G₂₃ , 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 4²•3⁸ = 104796. (In fact, the actual number of analysed instances was only 69984, since the results for switched values of h₇ and h₈ appear to be the same with only indices 1 and 2 switched.)

parameter no. values values

h₁ 4 0.01 0.07 0.31 1.00

h₂ 3 0.24 0.36 0.48

h₃ 4 0.01 0.07 0.31 1.00

h₄ 3 0.1 0.5 2.0

h₅ 3 0.0 0.23 0.46

h₆ 3 0.0 0.23 0.46

h₇ 3 0.3 0.6 0.9

h₈ 3 0.3 0.6 0.9

h₉ 3 0.06 0.15 0.30

h₁₀ 3 0.65 0.75 0.90

Results

Computations were performed with the use of the finite element code ABAQUS, v. 6.9. An enhanced geometric model of the RVE and a dedicated structured mesh generator were employed. Details of this model and comparison of its results with other author's data can be found in [1]. 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 10 data in the record are values of the dimensionless parameters h_k, k = 1,…,10.
Subsequent 2 data in the record are values of the warp and weft volume fractions (warp yarns are those aligned with the axis x₁), computed from the FE model of RVE.
Further, the orthotropic elastic constants of the composite are listed in the following order: E₁ , E₂ , E₃ , G₁₂ , G₁₃ , G₂₃ , ν₁₂ , ν₁₃ , ν₂₃ .

Since the yarn 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_i , G_ij ) should be eventually scaled by the actual value of the modulus.

Discussion

Detailed discussion of results can be found in [2]. To summarize, derivation of constitutive properties of PWFR ply from the equivalent microstructure models gives satisfactory results. Anisotropic elastic constants of PWFR composite 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. Enhanced geometric model for numerical microstructure analysis of plain-weave fabric reinforced composite. Advanced Composite Materials, 2014, DOI: 10.1080/09243046.2014.898439.
[2] P. Kowalczyk. Parametric constitutive model of plain-weave fabric reinforced composite ply. (submitted for publication)

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 [2]. 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 [2] 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 3, 2014

parameter	no. values	values
h₁	4	0.01	0.07	0.31	1.00
h₂	3	0.24	0.36	0.48
h₃	4	0.01	0.07	0.31	1.00
h₄	3	0.1	0.5	2.0
h₅	3	0.0	0.23	0.46
h₆	3	0.0	0.23	0.46
h₇	3	0.3	0.6	0.9
h₈	3	0.3	0.6	0.9
h₉	3	0.06	0.15	0.30
h₁₀	3	0.65	0.75	0.90