One of the most important parameters to be considered for studying anisotropic materials (i.e. fiber reinforced composite materials, natural wood and wood products, and mammal bones, among others) are the inplane shear properties: inplane shear modulus and inplane shear strength. On that sense, several test methods are currently available for measuring inplane (or intralaminar) shear properties of anisotropic materials. Those methods consider a variety of specimen geometries, applied loading, and material configurations [1]. The more accepted methods for determining these inplane shear properties are:
All these tests have been surveyed, described in detail, and compared in terms of advantages and drawbacks, which make them nonideal methods but complementary ones.
In recent years, other methods for determining the inplane shear properties under flexural loading have been proposed, being the most important:
The last method considers a threepoint bending (3PB) test stating that normal and shear stresses change from point to point in the specimen, depending on fiber orientation angle and specimen geometry. It has been studied for two loading contact conditions, depending on either liftoff between specimen and fixture supports takes place due to the bendingtwisting coupling effects. Such method considers the condition of small displacements and that liftoff between the specimen and the fixture supports occurs. The most important advantages of the offaxis 3PB method are its simplicity of flexure test configuration, its absence of endconstraint effects on the specimen, there is no need of strain gages to determine the inplane shear modulus, it considers the deformation of the whole specimen, and the failure can be supposed to start at a precise point. Suitability of the method for carbonfiber composites has been already done in previous works [8,9]. The aim of this work is to study its suitability for other materials, such as wood.
Due to their anisotropy, unidirectional offaxis composite laminates under flexure show bending–twisting coupling. In the case of a 3point bending flexure test, these couplings lead to liftoff of the specimen on the fixture supports. This effect was analyzed and proposed as an experimental method in the works of Mujika et al. [6,8,9]. The 3point bending test for a unidirectional offaxis laminate, as proposed by Mujika et al. [6,8,9], is based on the experimental displacement of the middle point. The following assumptions are made in the analysis in [6], and are thus required conditions:
Under these assumptions, uniform distributions of moments and shear forces per unit length along cross sections satisfy equilibrium conditions, except in the vicinity of reactions and load application zones. A typical offaxis bending test, where liftoff is observed, as well as the assumed load configuration and reactions are shown in Figure 1 [1].
Liftoff on the supports has been considered critical for the validity of the three point offaxis bending test. If liftoff in the support does not occur, the case is statically indeterminate, while if liftoff occurs the problem is statically determinate. All the existing analytical formulation has assumed liftoff on the supports. The frontier between both cases has been established by determining the critical value of the spantowidth ratio, c, for the lift off to occur [8]. Liftoff occurs for c values larger than the critical one, so it is indeed a geometrical condition which could theoretically be accomplished for any material.
The value of the critical spantowidth ratio for liftoff, c_{LO}, is:

(1) 
where S_{ij} with i, j = x, y, s are the inplane compliance coefficients related to bending–twisting effects; L’ is the length of the specimen; and b is the width of the specimen (Figure 1).
The assumed load application condition has not been verified in previous works dealing with the 3point offaxis bending test [6,8,9]. However, and as previously explained, the existing analytical formulations are based on the hypothesis of an applied point load in the centre of the specimen. In order to satisfy the condition of a point load in the middle load line, the specimen has to separate from the load application fixture. A downwards liftoff, similar to that in the support lines, is thus required. In this case, due to symmetry in the middle line of the specimen, it is a symmetrical deformation: both extreme zones separate from the load application fixture, and only the central zone remains in contact. In this way, the assumed point load hypothesis is valid.
In order to verify the downwards displacement condition, the resulting relative displacement w_{M} must be positive for the load application line. As a simplified verification condition, only the relative displacement of the extreme lateral point (^{1}/_{2}, 0) to the centre point in the middle line may be considered, as can be seen in Figure 2, and should be as well positive.
On that sense, based on the displacement of the extreme point located in the middle (load application) line (^{1}/_{2} , 0), another spantowidth ratio condition can be derived. In this case, span values higher than the limit value c_{mid} deform according to the assumptions in previous works [6], and as a result the point load assumption is adequate. Such limit c value is obtained from the expression]:

(2) 
Therefore for the 3point offaxis bending test two spantowidth ratio conditions are required [1]:
c > c_{LO} (1) and c > c_{mid} (2).
The value of the parameter c_{mid} must be in the real domain, so the square root term must be positive. This condition may be expressed then as:

(3) 
Then, the limit c_{mid} span length is not related to any geometrical parameter, only to material properties. It proves the possibility that for a certain material, the required point load application condition could not be achieved. Consequently, the offaxis bending method, with its assumptions would consider not valid.
The condition for the suitability of the 3point offaxis bending method for the analyzed material is defined, from (3), with the γ_{mid} parameter as:

(4) 
If γ_{mid} is in the range between zero and one, 0 < γ_{mid} < 1, the offaxis bending method is not valid. It would be appropriate, in its present formulation, in any other case [1].
A plot of the two c limit values, c_{LO} [9] and c_{mid} [1], for two different materials, carbon fiber and beechwood, is shown in Figure 3a and 3b respectively. In those cases where no real solution exists, c_{mid} is plotted as zero. Both parameters show a similar trend. Besides, the value for the c_{LO} parameter is usually higher than the c_{mid}. Hence, the verification of the c_{LO} condition is usually enough to verify the point load application condition as well. Nevertheless, for some materials (e.g. beechwood), in a certain orientation range, from 20º to 60º, the existing 3point offaxis bending method is not valid [1].
As stated in the previous Section, for certain materials the load hypotheses assumed in the 3point bending offaxis method may not be verified. In this Section, a parametric analysis is accomplished in order to verify the suitability of the offaxis bending method to existing composite materials.
The elastic constants of several manmade composites (glass fiber, carbon fiber, kevlar fiber) and natural composites: spruce (softwood) and beech (hardwood), are presented in Table 1.
Two parameters, based on the elastic moduli ratio, are defined:

(5) 

(6) 
where E_{L} and E_{T} are the longitudinal and transverse elastic moduli, and G is the shear modulus.
Table 1. A representative set of composite materials, both manmade and natural [1].
Composite material  E_{L} [GPa]  E_{T} [GPa]  G [GPa]  Τ  
Boron/epoxy [10]  201  21.7  5.4  9.26  4.02 
Carbon/epoxy [10]  142  10.3  7.2  13.78  1.43 
Kevlar/epoxy [10]  87  5.5  2.2  15.82  2.50 
Eglass/epoxy [10]  39  8.6  3.8  4.54  2.26 
Beech (clear wood) [11]  11.82  0.59  1.36  20.03  0.43 
Spruce (clear wood) [11]  18.28  0.23  0.62  79.48  0.37 
D40 Timber (hardwood) [12]  11.00  0.75  0.70  14.67  1.07 
C24 Timber (softwood) [12]  11.00  0.37  0.69  29.73  0.54 
As can be seen in Table 1, ratio (5) becomes higher for natural composites, and lower for manmade. Conversely, the Τ ratio (5) is high for manmade materials, and usually lower than one for natural materials such as softwood [1].
In order to analyze the validity of the 3point offaxis bending method, the defined material ratios, and Τ, for different fiber orientations and the derived verification condition γ_{mid} (4), is plotted in Figures 4 to 8. As it may be seen, the Τ ratio plays a major role in the validity of the offaxis bending method. As a rule of thumb, it may be established that those materials with a Τ ratio lower than 0.5 are usually not suitable for the 3point offaxis bending test.
When the influence of the fiber orientation is analyzed, several differences arise, mainly related to the Τ/ slope of the limit line between the valid and nonvalid materials zones. For the 15º fiber orientation (Figure 4), the slope is positive, that is, Τ value increases with increasing . In the 30º fiber orientation (Fig. 5), such limit line becomes horizontal when Τ ≈ 0.6. For higher fiber orientations (i.e. 45º, 60º and 75º), the slope is negative, and Τ value diminishes with increasing , as can be seen in Figures 6, 7 and 8 respectively. The 45º fiber orientation is the most restrictive (Figure 6), that is, the orientation for which a higher number of materials would estimate not valid. The nonvalid zone diminishes for higher angles, such as 60º and 75º, (Figures 7 and 8) [1].
In addition, for those material ratios in which the load line displacement condition is verified, a comparison between the c_{LO} (support liftoff) and c_{mid} (load line) limits is also plotted in Figures 68. For a number of materials the previously found c_{LO} condition is lower than the c_{mid} parameter required for the load displacement condition to be verified. Consequently, in those cases, c_{mid} is the minimum spantowidth ratio required for the test to be valid [1].
In this paper the offaxis threepoint bending test for characterizing inplane shear properties of anisotropic materials (i.e. fiber reinforced composite materials and natural wood) has been analyzed theoretically. In this method normal and shear stresses vary point to point through the specimen depending on both material configuration (i.e. fiber orientation angle) and geometrical parameters (i.e. spantowidth ratio, c). Bending–twisting coupling due to the marked anisotropy of unidirectional fiber reinforced materials is featured on offaxis laminates under flexure loading. Such coupling effect gives rise to two possible situations on both support and loading application conditions: a liftoff of the specimen on the fixture supports, and a point contact in load application line.
Besides to support and loading restrictions, material elastic constants (i.e. longitudinal, transversal and shear elastic moduli: E_{L}, E_{T}, G) have an effect on the application of mentioned method in order to determine inplane shear properties of anisotropic materials. Thereby, this work presents an analytical approach for investigating the conditions to assure the applicability of the offaxis 3 pointbending test. Main results reveal that, regarding the spantowidth ratio, liftingoff condition c_{LO}, is usually higher than the cylinder line load displacement condition c_{mid}. Therefore assessing the c_{LO} condition is generally enough to verify also the load application c_{mid} condition. As a consequence, usually the minimum spantowidth ratio required for the test to be valid is c_{LO}.
Regarding the ratios between elastic moduli, = E_{L}/E_{T} and Τ = E_{T}/G, results indicate that ratio is higher for natural wood than for fiber reinforced plastics, and on the contrary Τ ratio is high for manmade composite materials, and even lower than one for some natural materials. As Τ ratio has a greater influence than ratio in the validity of the offaxis threepoint bending test, materials with a Τ ratio lower than 0.5 are not usually suitable for such method (i.e. wood, as beech and spruce). In addition, concerning material configuration, it has been found that 45º is the most restrictive fiber orientation angle for which the analyzed test has a larger number of materials that are not considered valid.
Currently, since the analyzed model is not suitable for some composite materials as natural wood, another model is being developed. The authors of this paper are in the process of verification of the new formulation in the light of experimental results.
The support of the Wood Chair, joint initiative of the Government of Navarre and the University of Navarra, is greatly acknowledged. The authors wish to express their gratitude to Miguel Yurrita, who helped in a preliminary phase of this research.
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Published on 19/10/17
Accepted on 19/10/17
Submitted on 19/10/17
Volume 01  Comunicaciones Matcomp17 (2017), Issue Núm. 1  Comportamiento en Servicio de los Materiales Compuestos, 2017
DOI: 10.23967/r.matcomp.2017.10.016
Licence: Other
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