Determination of in-plane elastic constants of angle ply laminates from tensile measurements

Abstract

This paper presents a new method to completely characterise the in-plane elastic properties of an important type of angle ply laminates using only unidirectional tests. The angle ply laminates having the same number of identical plies are considered. Some new results for orthotropic laminates are found using this technique, namely, the conditions of existence of a particular direction in which there is no shear extension coupling. The laminates are subjected to three tensile tests: two in the orthotropic axes and the third one in a particular direction. The results allow a complete semiexperimental characterisation of the angle ply laminate, which do not need a priori knowledge of the elastic properties of the constitutive layer. Comparisons are made between numerical predictions and experimental results. Finally, the computational results were compared to the experimental results, verifying the capability of the method in order to predict the in-plane elastic properties of composite laminates.

Keywords

Angle ply Elastic properties Tension Laminate

Introduction

Composite materials, in the context of high performance structures, have been increasingly used since the early 1960s, although materials such as glass fibre reinforced polymers were already being studied 20 years earlier. Initially, conventional test methods originally developed for determining the mechanical properties of metals and other homogeneous and isotropic materials were used. It was soon recognised, however, that these new materials, which are non-homogeneous and anisotropic, require different considerations for determining mechanical.^1–4

The experimental characterisation of composite materials has been a complex issue because it has been a continually evolving one. As new types of composites have been developed and new applications found, new testing challenges have continually evolved. In fact, the term composite material is not limited to a specific class of fabrication materials, rather a wide spectrum of materials of widely varying properties. Therefore, it can be expected that various test methods will be needed for different types of composite materials. Thus, there is little possibility of standardisation.⁵ Various test techniques were developed for evaluating the same properties; some were easy to perform but provided only limited results or data of doubtful quality. Other test methods were so complex, operator dependent and, in some cases, maybe also of disputable quality. Many of the conventional test methods have been reviewed by Carlsson et al., ^{1, 6} Hodgkinson,⁵ Lubin⁷ and Daniel and Ishai.⁸ Moreover, consensus organisations, such as ASTM and ISO, have made great progress in developing test methods suitable for composite materials; numerous test methods are described in Vol. 15 of The ASTM annual books of standards.⁹

The complete characterisation of anisotropic elastic materials is based on the use of as many independent tests as there are elastic constants to be measured. To date, several approaches have been explored: non-destructive characterisation techniques such as ultrasound¹⁰ and also numerical–experimental models.¹¹ However, in case of laminates, the practical study of Marin et al.^{12, 13} showed that the most common technique consisted of a complete characterisation only by tension tests.

For orthotropic laminates, two single tensile tests can be performed in the orthotropic axes, but they can provide only 75% of the characteristics. For complete characterisation, it is therefore essential to perform off-axis tension test, too. However, in general, there is an off-axis tensile–shear coupling. Thus, to have homogeneous tensile state, it is necessary to use pivoting jaws or an equivalent system.^{9, 11} To avoid the use of such complex equipment, an off-axis direction must be found for which this coupling is zero before the tensile test. Verchery and Gong¹⁵ discussed the conditions of existence of such directions ± ω for orthotropic materials developing the formula expressing this angle ± ω according to the elastic constants of the laminate. A useful method using the polar representation of elasticity tensors proposed by Verchery and Vannucci.^{16, 17} This method is so called the polar method and has been implemented successfully in tests, using an approximate value of the measured angle ω from a prior estimation of the elastic constants. However, in general, an unknown prior angle makes it difficult to use the off-axis tension test without coupling. Therefore, this study is concerned with solving this problem.

In this work, it is shown that, for some angle ply orientations ( ), there is a simple function between ω and α with excellent approximation. The inequality was simply considered using the polar method to determine the ranges α for which ω exists. The function f(α) depends on the elastic constants of the base ply. The function also depends on these constants, but presents a very particular form even in a wide variety of highly anisotropic unidirectional plies (reinforcement such as carbon, boron, aramid, etc., polymer matrixes, usual volume fractions).

For a certain stacking angle α₀, ω is practically equal to α. In addition, the limit angle α₀ slightly depends on the used material, usually 30–33°. It thus becomes possible to completely characterise these angle ply laminates by three tensile tests with traditional jaws in directions 0, 90° and α (or − α).

In order to validate this property experimentally, we performed tensile tests with specimens with rosettes, cut at an angle ply of 12-ply unidirectional plies of carbon–epoxy. The results confirm that off-axis specimens were well in simple traction condition. The set of elastic constants is determined from tensile in-plane tests in three directions and is very close to the estimations found out by the classical laminated plate theory (CLPT).

Determining direction of off-axis test

By the polar formalism,^16–21 the coefficients of the compliance matrix S of an orthotropic material,²¹ which is rotated by an angle θ, relative to orthotropic axes are defined by 1 where parameters t₀, t₁, r₀ and r₁ are the polar invariants, and integer index k is a parameter that defines the type of orthotropic: for odd k, the graph of E₁₁(θ) has two maxima along the axes of orthotropic, while for even k, the graph has two minima. Moreover, equation (1) determines the behaviour of in-plane or flexural of any orthotropic laminate. To be able to carry out an uncoupled off-axis test, the coupling component S₁₆(θ) must be null i.e. 2 3 Equation (3) gives directions of the orthotropic axis 0 and 90°. The positive angle from this equation is called ω; when k is odd, 0 ≤ ω ≤ π/4, and when k is even, π/4 ≤ ω ≤ π/2. Thus, the necessary and sufficient condition for achieving an off-axis test without coupling is to have 4 Finally, it is possible to show that the shear modulus G₁₂ at 0° may be defined by the Young's modulus and Poisson's ratio found at angle ω as follows 5

Polar formulation of angle Ω for angle ply laminate

The CLPT²² defined Cartesian formulae for determining stiffness coefficients; these formulae are also valid for the polar coefficients proposed by Verchery;¹⁶ particularly for in-plane behaviour, four complex equations can be found as follows 6 where the capital letters refer to the polar components; superscript ‘^-’ indicates in-plane behaviour of the laminate and those without superscript indicate coefficients of the basic ply. In addition, ‘h’ refers to the total thickness of the laminate, and ‘n’ refers to the number of layers. If the laminate is an angle ply, with the same number of plies oriented to +α and − α, then equation (6) is simplified 7 The orthotropic conditions expressed in polar formalism are for the based ply and laminate 8 The value is usually chosen for the basic ply, which corresponds to take the direction of the unidirectional fibres confused with the x axis of not turned frame.

In these conditions, for 0 ≤ α ≤ π/4, which is always the case (the angle ply with π/4 ≤ α ≤ π/2 is the same laminate but rotated π/2), equation (7) is simplified to 9 In the case of the uncoupled angle ply laminate, polar coefficients of the compliance matrix can be related to those of in-plane components as follows 10 where is invariant If , then , and consequently, for the orthotropic materials, then two last equations of equation (10) can be rewritten as 12 Finally, by dividing the above two equations and introducing equations (9), we obtain 13 Finally, equation (4) is verified for 14 Substituting equation (13) in equation (14) leads to 15 The angle will exist if and only if f(α) < 0. Under these conditions, is defined by equation (3) as follows 16 Equations (15) and (16) show that the value of depends only on the mechanical characteristics of based ply and the angle α of fibre orientation. Thus, for a given based ply, it is possible to vary α to verify equation (15).

Equations (13) and (14) show that, for α = 45°, always exists (case of cross-ply laminates): therefore, it is always possible to find an angle α from which an off-axis test without tensile–shear coupling can be achieved.

However, the above procedure only affects the in-plane behaviour of the laminate. In fact, in almost all cases, the flexural behaviour is completely anisotropic. There, however, exists a case for which the results can extend to the bending behaviour.

If the material is quasi-homogeneous,^{17–18, 23, 24} then the homogenised tensor A* and D* are identical, that is to say the angle of the off-axis bending test is equal to that of the in-plane off-axis test.

Determining for various based plies

According to the previous equations developed in the previous section, can be determined only by knowing perfectly the mechanical characteristics of based ply as well as fibre orientation angle α

The rest of this section is devoted to the determination of as a function of α for some practical cases of unidirectional based plies. The elastic properties of plies are given in Table 1; also Table 2 provides the results of calculating the polar coefficients based on elastic properties.

Table 1

Elastics properties of unidirectional plies

Number	Fibre	Matrix	E ₁₁	E ₂₂	ν₁₂	G ₁₂
1	HM carbon	Epoxy	230	14.4	0.32	4.9
2	HR carbon	Epoxy	132	9.5	0.33	5.3
3	Boron	Epoxy	204	18.5	0.23	5.6
4	Kevlar 49	Epoxy	76	9.8	0.34	3.7
5	E glass	Epoxy	38.6	8.3	0.26	4.1
6	Boron	Epoxy	235	138	0.23	66
7	Steel	Rubber	105	0.006	0.40	0.0015

Table 2

Polar parameters of plies

Number	T ₀	T ₁	R ₀	R ₁	K	t ₀	t ₁	r ₀	r ₁	k
1	32.0	31.9	27.2	27.1	0	35.1	8.9	15.9	8.1	1
2	19.7	18.6	14.4	15.4	0	38.3	13.5	8.9	12.2	1
3	29.7	29.0	24.1	23.3	0	30.0	7.1	14.7	6.1	1
4	11.9	11.7	8.2	8.4	0	49.3	13.3	18.3	11.1	1
5	7.5	6.5	3.4	3.8	0	50.5	16.6	10.5	11.8	1
6	72.9	56.3	6.9	12.5	0	3.6	1.2	0.2	0.4	1
7	13.1	13.1	13.1	13.1	0	10469	20834	62498	20832	1

Figure 1 shows variation of , α and function f(α), which is multiplied by a constant number to fit on the same chart. The first five graphs clearly show that and α have significantly identical values when α ≥ 30° apart from the elastic properties. Therefore, in practice, for all values of α belonging to [30°,45°], the elastic characterisation of angle ply can be completely determined through tensile tests.

a HM carbon/epoxy laminate; b HR carbon/epoxy laminate; c boron/epoxy laminate; d Kevlar 49/epoxylaminate; e E glass/epoxy laminate; f boron/epoxy laminate

The curves of Fig. 1f (boron/epoxy) show that this empirical rule is not applicable to all cases. In the case of metallic matrices, and α coincide when α becomes >40°, and for balanced tissue, which have all R₁ = r₁ = 0, this mapping occurs only for α = 45°. The common point of these two examples is that Young's moduli E₁₁ and E₂₂ have very close values, even identical.

Consequently, the condition under which the empirical rule discussed above is valid is that the ply must have a strong anisotropy, i.e. E₁₁/E₂₂ ≥ 4 approximately; this condition is fulfilled for all polymer matrix plies, as well as a number of other materials, such as rubber–steel plies for example (Fig. 2). Moreover, Fig. 1 shows that, in some cases, the condition of existence of is also verified for low values of α; this occurs when the based ply itself verifies equation (4). For these materials, when α is low, the equation (15) is respected, and for laminated plate, minimum values of E₁₁(θ) obtain in the direction and its minimum in the directions 0 and 90°.

Curves for steel/rubber angle ply laminate

When α increases, equation (15) is not verified anymore and the direction disappears; E₁₁(θ) has two extreme directions: a maximum at 0° and a minimum at 90°.

Finally, for α ≥ 30°, equation (15) is satisfied again, and E₁₁(θ) has a maximum at and two minima at 0 and 90°. These three possibilities are shown in Fig. 3; in this work, only the third situation is noteworthy.

Polar variations of E₁₁(θ) for boron/epoxy angle ply laminate

Experimental results

Tests were carried out on a laminate composed of identical angle ply plies in carbon/epoxy. A quasi-homogeneous laminate of 12 plies was used, whose lay-up was [a/ − a/a/ − a₃/a₃/ − a/a/ − a]; therefore, the laminate was decoupled and had the same in-plane and flexural properties regardless of the value of imposed α. The elastic characteristics of ply and the laminate are given in Tables 3 and 4. The value of α is 33°, and equation (16) gives an approximate value of 32.7° for (Fig. 4). Practically, these values may be considered identical, and Fig. 5 shows that, throughout the off-axes tensile tests, the shear value is very low. In addition, due to the quasi-homogeneity behaviour of the laminate, = .

Table 3

Elastic constants of ply and laminate/GPa

	E ₁₁	E ₂₂	ν₁₂	G ₁₂
Based ply	118	9.2	0.34	4.8
Laminate (CLPT)	35.3	11.1	1.21	26.2
Tensile tests	32.1	10.4	1.19	29.4
Bending tests	31.6	11.8	1.13	30.6

Table 4

Polar stiffness and compliance parameters

	T ₀	T ₁	R ₀	R ₁	K	t ₀	t ₁	r ₀	r ₁	k
Ply	17.6	16.8	12.8	13.7	0	41.4	13.9	10.7	12.5	1
CLPT	17.6	16.8	8.6	5.6	1	28.1	6.3	18.6	7.7	0
Tensile	18.8	15.6	10.6	5.0	1	29.5	6.6	20.9	8.1	0
Bending	19.3	16.8	11.3	4.7	1	27.5	5.6	19.3	6.6	0

Experimental curves for carbon/epoxy angle ply laminate

Strains variations thorough off-axis tensile test

Consequently, the bending tests were also carried out on the same test tubes in order to be able to compare the elastic characteristics obtained by bending and tensile tests. Figure 6 shows the curves of E₁₁(θ), of the ply and the laminate as calculated on the basis of CLPT.

Polar variations of E₁₁(θ) in GPa, calculated by CLPT

The tests results are summarised in Tables 3 and 4.

Figure 7 shows the three obtained curves. Three specimens were tested in all three directions. The experimental results agree well with theoretical predictions obtained by the CLPT. The angle found with the tensile tests gives 33.6°, whereas with the flexion = 35.1°.

Polar variations of E₁₁(θ) of angle ply laminate

Conclusions

This study showed that it is possible to completely determine the elastic behaviour of angle ply laminates, composed of unidirectional identical polymer matrix plies, using simple tensile tests. Therefore, it is vital that the lamination angle α be in the range of 30–45°; the three directions of tests are then 0 and 90° and α, which are the two axes of orthotropic and the fibre direction respectively. This constitutes a simple empirical rule, which does not require any preliminary calculation, and the precision of this rule can be directly verified experimentally.

The mechanical properties of unidirectional plies could be easily determined using this method. Since the three test directions are defined even if the elastic characteristics of the ply are not known, so it becomes possible to characterise the angle ply.

The advantage of this method is that all these properties are determined from tests on the same laminated plate; all specimens could also be cut from the same plate, which could avoid some experimental issues.

References

Carlsson

L. A.

, Adams

D. F.

and Pipes

R. B.

: ‘Experimental characterization of advanced composite materials’, 2014, Boca Raton, FL, CRC Press.

Buchanan

, Archer

, Townsend

, Jenkins

, McIlhagger

A. T.

and Quinn

J. P.

: ‘Determination of in-plane shear modulus of 3D woven composites with large repeat unit cells’, Plast. Rubber Compos., 2012, 41, 194–198.

X. -K.

and Bai

S. -L.

: ‘Tensile properties of multilayered biaxial weft knitted fabric reinforced composite material’, Plast. Rubber Compos., 2010, 39, 91–98.

Khondker

O. A.

, Ishiaku

U. S.

, Nakai

and Hamada

: ‘Tensile, flexural and impact properties of jute fibre-based thermosetting composites’, Plast. Rubber Compos., 2005, 34, 450–462.

Hodgkinson

M. J.

: in ‘Mechanical testing of advanced fibre composites’, (ed.), 2000, Boca Raton FL, USA, Wood head publishing Ltd & CRC Press LLC. Elsevier.

Carlsson

L. A.

, Adams

D. F.

and Pipes

R. B.

: ‘Basic experimental characterization of polymer matrix composite materials’, Polym. Rev., 2013, 53, (2), 277–302.

Lubin

: ‘Static test methods for composites’, 1985, New York, Van Nostrand Reinhold Company Inc.

Daniel

I. M

and Ishai

: ‘Engineering mechanics of composite materials’, 2nd edn; 2006, New York, Oxford University Press.

American Society for Testing and Materials: ‘‘ASTM annual book of standards, Vol.15.03, ‘Space stimulation; aerospace and aircrafts composite materials’, 2010, West Conshohocken, PA, ASTM.

10.

Gommers

, Verpoest

and van Houtte

: ‘Determination of the mechanical properties of composite materials by tensile tests. Part I: elastic properties’, J. Compos. Mater., 1998, 32, 310–334.

11.

Araujo

A. L.

, Soares

C. M.

, De Freitas

M. M.

, Pedersen

and Herskovits

: ‘Combined numerical-experimental model for the identification of mechanical properties of laminated structures’, Compos. Struct., 2000, 50, (4), 363–372.

12.

Marın

J. C.

, Cañas

, París

and Morton

: ‘Determination of G₁₂ by means of the off-axis tension test. Part I: review of gripping systems and correction factors’, Composites A, 2002, 33A, (1), 87–10.

13.

Marın

J. C.

, Cañas

, París

and Morton

: ‘Determination of G₁₂ by means of the off-axis tension test. Part II: a self-consistent approach to the application of correction factors’, Composites A, 2002, 33A, (1), 101–111.

14.

Boehler

J. P.

, El Aoufi

and Raclin

: ‘On experimental testing methods for anisotropic materials’, Res. Mech., 1987, 21, (1), 73–95.

15.

Verchery

and Gong

X. J.

: ‘Pure tension with off-axis tests for orthotropic laminates’ Proc. 12th Int. Conf. on ‘Composite materials’, Paris, France, July 1999, Woodhead Publishing, Paper 752..

16.

Verchery

: “Les invariants des tenseurs d'ordre 4 du type de l'élasticité”, In ‘Mechanical behavior of anisotropic solids/Comportment Méchanique des Solides Anisotropes’, (ed. Jean-Paul Boehler), 1982, Netherlands, Springer.

17.

Vannucci

and Verchery

: ‘A special class of uncoupled and quasi-homogeneous laminates’, Compos. Sci. Technol., 2001, 61, (10), 1465–1473.

18.

Verchery

: ‘Design rules for the laminate stiffness’, Mech. Compos. Mater., 2011, 47, (1), 47–58.

19.

Vincenti

, Vannucci

and Ahmadian

M. R.

: ‘Optimization of laminated composites by using genetic algorithm and the polar description of plane anisotropy’, Mech. Adv. Mater. Struct., 2013, 20, (3), 242–255.

20.

Vannucci

: ‘Plane anisotropy by the polar method’, Meccanica, 2005, 40, (4-6), 437–454.

21.

Vannucci

: ‘Designing the elastic properties of laminates as an optimisation problem: a unified approach based on polar tensor invariants’, Struct. Multidiscip. Optim., 2006, 31, (5), 378–387.

22.

Reddy

J. N.

: ‘Mechanics of laminated composite plates and shells: theory and analysis’, 2004, Boca Raton, FL, CRC Press.

23.

K. M.

and Avery

B. L.

: ‘Fully isotropic laminates and quasi-homogeneous anisotropic laminates’, J. Compos. Mater., 2117, 26, (14)–2107.

24.

Vannucci

, Barsotti

and Bennati

: ‘Exact optimal flexural design of laminates’, Compos. Struct., 90, (3), 337–345.