Major Poisson s ratio

The convention normally used is that direct stresses and strains have one suffix to indicate the direction of the stress or strain. Shear stresses and strains have two suffices. The first suffix indicates the direction of the normal to the plane on which the stress acts and the second suffix indicates the direction of the stress (or strain). Poisson s Ratio has two suffices. Thus, vi2 is the negative ratio of the strain in the 2-direction to the strain in the 1-direction for a stress applied in the 1-direction (V 2 = — il for an applied a ). v 2 is sometimes referred to as the major Poisson s Ratio and U2i is the minor Poisson s Ratio. In an isotropic material where V21 = i 2i. then the suffices are not needed and normally are not used. 

It may be seen that when the moment is applied, the major Poisson s ratio v y corresponds as it should to the value when the in-plane stress, Ox, is applied. 

The so-called major Poisson s ratio, 2, is obtained by an approach similar to the analysis for E. First, the major Poisson s ratio is... 

The major Poisson s ratio is vl2 and E = E2/El is the ratio of the transverse to longitudinal modulus. Equation 8.24 gives the induced curvature for anticlastic deformation of an unsymmetric cross-ply laminate. The curvature is dependent on the thermal and chemical strain mismatch (e, — e2), lamina mechanical properties (v12, E) and the half-thickness, h. 

Extensive experimental testing was performed on IM6/3100 to obtain longitudinal and transverse stiffnesses and major Poisson s ratio as a function of degree of cure. A detailed explanation of the test procedure can be found in [5], The model predictions and experimental results for this material system are shown in Figures 8.11-8.13. From these relations the lamina stiffnesses Qy can be obtained from... 

Figure 8.13 Major Poisson s ratio development during cure for IM6/3100...

Figure 4 shows the variation of the laminate axial modulus E, transverse modulus E., shear modulus, major Poisson s ratio (normalised by their undamaged value) as a function of the relative delamination area D " for the [Oj /30j /- 302], laminate Matrix crack density was... 

Note that, as Eqn (6.2c) implies, the minor Poisson s ratio is related to the major Poisson s ratio via the relation ... 

V12 = major Poisson s ratio (load in 1 -direction, strain in 2-direction)(please note that this is different from V21, which is the minor Poisson s ratio)... 

The engineering properties of interest are the elastic constants in the principal material coordinates. If we restrict ourselves to transversely isotropic materials, the elastic properties needed are Ei, Ei, v, and G23, i.e. the axial modulus, the transverse modulus, the major Poisson s ratio, the in-plane shear modulus and the transverse shear modulus, respectively. All the elastic properties can be obtained from these five elastic constants. Since experimental evaluation of these parameters is costly and time-consuming, it becomes important to have analytical models to compute these parameters based on the elastic constants of the individual constituents of the composite. The goal of micromechanics here is to find the elastic constants of the composite as functions of the elastic constants of its constituents, as... 

For the transverse shear modulus, the approach designated self-consistent was based on the formula obtained by the self-consistent method for the plane-strain bulk modulus (11.61), on the transverse modulus calculated using the Chamis approach (11.49b) and the in-plane Poisson s ratio given by the rule of mixtures. Except when used to predict the axial modulus and the major Poisson s ratio, the rule of mixtures underestimates the remaining composite elastic properties. The Bridging Model proved to be a very effective theory to account for all five elastic properties for unidirectional composites that are transversely isotropic. 

The properties shown in Table 5.3 are axial, transverse and shear moduli, Poisson s ratio, tensile and compressive strengths in the axial and transverse directions, and in-plane shear strength. The Poisson s ratio presented is called the major Poisson s ratio. It is defined as the ratio of the magnitude of transverse strain divided by the magnitude of axial strain when the composite is loaded in the axial direction. Note that transverse moduli and strengths are much lower than corresponding axial values. 

E] = Young s modulus of ply along fiber direction Ef = Young s modulus of ply transverse to fiber direction Plt = Major Poisson s ratio Ptl = Minor Poisson s ratio Glt = Shear modulus of ply... 

Determination of Major Poisson s Ratio of Unidirectional Lamina... 

The major Poisson s Ratio of an orthotropic lamina is defined as the negative of the ratio of transverse strain to longitudinal strain when the representative element is subjected to uniaxial loading in the longitudinal direction such that... 

Putting the definitions of Equations 8.11 and 8.12 into Equation 8.15 gives the desired expression relating the macroscopic major Poisson s ratio to the fiber and matrix Poisson s ratios as a function of the fiber volume fraction ... 

Hence, the major Poisson s ratio also obeys the rule of mixtures, predicting a linear variation in Plt ranging in value from to Pf as the liber volume fraction goes... 

Major Poisson s ratio (rule of mixtures) p-Lx + PniKi... 

Thus, it is apparent that the composite longitudinal Young s modulus and major Poisson s ratio are strongly influenced by the fiber elastic response whereas the composite transverse Young s modulus and shear modulus behavior is dominated by the matrix elastic response, except at large fiber volume fractions. 

The major Poisson s ratio Plt is obtained from the same tensile specimen by measuring transverse and longitudinal strains and taking their ratio, such that... 

In practice, the major Poisson s ratio is measured using a longitudinal tensile specimen (Equation 8.45), and the minor Poisson s ratio is computed using Equation 8.49. Finally, a pure shear stress is applied to a 0° test specimen, such that... 

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