Determination of shear modulus for fiber composite using experimental-calculation methods

. The article is devoted to the description and analysis of the results of the experiment on orthogonal fiberglass disk twisting. The possibilities of joint experiment and finite element method simulation are shown, in the case when the data obtained in the experiment are taken as initial data for the computational model. The proposed method is characterized by sufficient simplicity, the values obtained for the shear modulus show good agreement with the results of other researchers


Introduction
Fiber-reinforced composites have long been used as a structural material in various engineering and civil engineering projects [1].Therefore, the determination of mechanical properties is very important [2].The properties of composites depend on the reinforcement -in which direction there are more fibers, in that direction the strength is higher [3,4].If it is necessary to obtain a material that works well in tension or bending in a predetermined direction, the fibers should be laid along the lines of normal stresses action (Fig. 1a).If strength in two perpendicular directions is required, orthogonal laying is used (Fig. 1b).A transversally isotropic material is quite easy to obtain: reinforcement schemes with four or more fiber families at equal angular intervals have the required isotropy in the plane of fiber placement (Fig. 1c).
When using composites there are no difficulties with the design of structures intended for operation in tension, compression, bending (with some reservations).It is quite different with torsion, because here the shear characteristics should be taken into account [5,6], which are very low for composites: the shear strength is an order of magnitude lower than the tensile strength, for example, for steel this ratio is about two.The shear strength in any material depends on the shear modulus, which is very low for composites and its experimental determination is difficult [7,8].
The present work is an attempt to combine experimental and numerical methods to determine the shear modulus of a composite plate.

Methods
When calculating elements made of orthotropic composite materials, 9 independent constants are used: 3 elastic moduli, 3 shear moduli and 3 Poisson's ratios.
The determination of elastic moduli and Poisson's ratios (unlike the ultimate stresses) is easy, but shear moduli are a challenge.Shear modulus is the coefficient of proportionality between the angle of rotation of a cross section and the shear stresses forcing the section to twist by that angle (Fig. 2).

Fig. 2. Shear moduli for a laminate
For a package consisting of n layers, the determination of moduli in the xz and yz planes is not difficult -these are bending tests, but the determination of the shear modulus in the xy plane, responsible for the section twisting in the plane of the plate (Fig. 3) is a rather complex and nontrivial task.The existing method of a square plate twisting [9] for determining Gxy has limitations: first, the plate must be square, second, it must be orthogonally reinforced (Fig. 1.b), third, the plate faces must be strictly parallel and perpendicular to the material layers.There are more conditions -the deflection under load should be no more than 10% of the thickness, and the plate itself should be very thin to minimize the effect of shear on the deflection.
The work on determination of shear moduli is as follows: 2 types of specimens -strips and a square plate (Fig. 3) -are made of an orthogonal composite plate manufactured using the vacuum infusion method.The specimens -strips are tested for bending and Gyz and Gxz are determined, the plate is tested for torsion and Gxy is determined.
The authors aimed to determine the in-plane shear modulus Gxy using the finite element method.For this purpose, specimens of 2 types, strips and a disk, were cut from a plate with orthogonal fiber stacking (Fig. 4).The strips were tested in tension and bending to determine the elastic and shear moduli.The disk was tested for twisting with different arrangement of supports (Fig. 4b).The experimental data obtained: elastic moduli, shear moduli, and deflections in the tests of the round plate were taken as material characteristics for modeling the numerical experiment on torsion of the round plate in FEM, which resulted in the selection of the shear modulus in the plane of the round plate.Poisson's ratio and modulus of elasticity of epoxy resin were taken from literature and the data are summarized in Table 1.The results of the disk loading are shown in Fig. 5, it can be seen that the axes of symmetry of the plate do not coincide with the orthotropy axes of the material, the angle of deviation is about 5.5 degree.

Simulation
Numerical experiment simulation was carried out in the finite element complex in the Static Structural module of the ANSYS Mechanical software package (Fig. 6).The contact axisymmetric problem of interaction of the disk with the supports having a spherical shape at the ends was solved.In this case, after selecting the in-plane shear modulus Gxy for the loading angle φ=0º, it is necessary to check whether the test results for the other loading angles will coincide with the selected material model.The experimental and simulation results are shown in Table 2.

Conclusions
The analysis of the work shows that modeling with FEM gives a good enough approximation for determining the in-plane shear modulus of the material.In the first approximation, the material for the numerical experiment was modeled as orthotropic, but without taking into account the layers, i.e. homogeneous, which, of course, is a strong assumption.
The combined use of laboratory experiment and numerical simulation gives quite acceptable results.
From the results of the disk twisting, it is simple to determine the material anisotropy axes.

Fig. 1 .
Fig. 1.Different reinforcement systems and loading schemes for which they are predominantly designed: a) unidirectional, b) orthogonal, c) quadraxial

E3SFig. 3 .
Fig. 3. Schemes of tests for determination of shear moduli of composite laminate

Fig. 4 .
Fig. 4. a) scheme of cutting specimens from the orthogonal plate for testing top view, b) arrangement of supports for disk twisting

Fig. 5 .
Fig. 5. Dependence of the deflection ν on the loading angle φ.The deflection value starts from the center of the circle.The angle φ=0º corresponds to the symmetry axes of the disk

Table 1 .
Elastic constants for round plate material

Table 2 .
Comparison of the numerical and laboratory experiments