Shear-extension coupling

The problem areas in composite structures design are related to some of the following observations. One, the behavioral characteristics of composite materials are much more complicated than those of metals. Bending-extension coupling, shear-extension coupling, and bend-twist coupling are all responses that are typically not encountered in a metal structure but are in a composite structure, so you must know how to deal with them. However, that circumstance is a somewhat intimidating situation. 

Note that an orthotropic material that is stressed in principal material coordinates (the 1, 2, and 3 coordinates) does not exhibit either shear-extension or shear-shear coupling. Recall that an orthotropic material has nine independent constants because... 

The term Cg3 is zero because no shear-extension coupling exists for an orthotropic lamina in principal material coordinates. For the orthotropic lamina, the Qn are... 

Note that the transformed reduced stiffness matrix Qy has terms in all nine positions in contrast to the presence of zeros in the reduced stiffness matrix Qy. However, there are still only four independent material constants because the lamina is orthotropic. In the general case with body coordinates x and y, there is coupling between shear strain and normal stresses and between shear stress and normal strains, i.e., shear-extension coupling exists. Thus, in body coordinates, even an orthotropic lamina appears to be anisotropic. However, because such a lamina does have orthotropic characteristics in principal material coordinates, it is called a generally orthotropic lamina because it can be represented by the stress-strain relations in Equation (2.84). That is, a generally orthotropic lamina is an orthotropic lamina whose principai material axes are not aligned with the natural body axes. 

Because of the presence of Q g and Q2e in Equation (2.84) and of 3 g and 326 f Equation (2.87), the solution of problems involving so-called generally orthotropic laminae is more difficult than problems with so-called specially orthotropic laminae. That is, shear-extension coupling complicates the solution of practical problems. As a matter of fact, there... 

Compare the transformed orthotropic compliances in Equation (2.88) with the anisotropic compliances in terms of engineering constants in Equation (2.91). Obviously an apparenf shear-extension coupling coefficient results when an orthotropic lamina is stressed in non-principal material coordinates. Redesignate the coordinates 1 and 2 in Equation (2.90) as X and y because, by definition, an anisotropic material has no principal material directions. Then, substitute the redesignated Sy from Equation (2.91) in Equation (2.88) along with the orthotropic compliances in Equation (2.62). Finally, the apparent engineering constants for an orthotropic iamina that is stressed in non-principal x-y coordinates are... 

An important implication of the presence of the shear-extension coupling coefficient is that off-axis (non-principal material direction) tensile loadings for composite materials result in shear deformation in addition to the usual axial extension. This subject is investigated further in Section 2.8. At this point, recognize that Equation (2.97) is a quantification of the foregoing implication for tensile tests and of the qualitative observations made in Section 1.2. 

Extension-extension coupling coefficients (Poisson s ratios) Shear-extension coupling coefficients (coefficients of mutual influence) Shear-shear coupling coefficients (Chentsov coefficients)... 

Alternatively to (3), account for certain end and edge effects (e.g., shear-extension coupling) in the data-reduction process. 

The stiffnesses of an antisymmetric laminate of anisotropic laminae do not simplify from those presented in Equations (4.22) and (4.23). However, as a consequence of antisymmetry of material properties of generally orthotropic laminae, but symmetry of their thicknesses, the shear-extension coupling stiffness A.,6,... 

For cross-ply laminates, a knee in the load-deformation cun/e occurs after the mechanical and thermal interactions between layers uncouple because of failure (which might be only degradation, not necessarily fracture) of a lamina. The mechanical interactions are caused by Poisson effects and/or shear-extension coupling. The thermal interactions are caused by different coefficients of thermal expansion in different layers because of different angular orientations of the layers (even though the orthotropic materials in each lamina are the same). The interactions are disrupted if the layers in a laminate separate. 

Specially orthotropic plates, i.e., plates with multiple specially orthotropic layers that are symmetric about the plate middle surface have, as has already been noted in Section 4.3, force and moment resultants in which there is no bending-extension coupling nor any shear-extension or bend-twist coupling, that is,... 

A specially orthotropic laminate has either a single layer of a specially orthotropic material or multiple specially orthotropic layers that are symmetrically arranged about the laminate middle surface. In both cases, the laminate stiffnesses consist solely of A, A 2> 22> 66> 11> D 2, D22, and Dgg. That is, neither shear-extension or bend-twist coupling nor bending-extension coupling exists. Thus, for plate problems, the transverse deflections are described by only one differential equation of equilibrium ... 

Symmetric angle-ply laminates were described in Section 4.3.2 and found to be characterized by a full matrix of extensional stiffnesses as well as bending stiffnesses (but of course no bending-extension coupling stiffnesses because of middle-surface symmetry). The new facet of this type of laminate as opposed to specially orthotropic laminates is the appearance of the bend-twist coupling stiffnesses D. g and D2g (the shear-extension coupling stiffnesses A. g and A2g do not affect the transverse deflection w when the laminate is symmetric). The governing differential equation of equilibrium is... 

SHEAR-EXTENSION COUPUNG AND BEND-TWIST COUPLING... 

The first problem area of the so-called anisotropic analysis will be broken down into two subareas shear-extension coupling and bend-twist coupling. We have already observed for the most complicated laminate in the design philosophy proposed earlier that the A,q and A26 stiffnesses are both zero. There is no shear-extension coupling in the context of that philosophy. However, in contemporary composite structures analyses, it is relatively easy to include the treatment of shear-extension coupling, so you should not be overwhelmed by that behavioral aspect or by the calculation of its influence. 

Shear-extension coupling, 26 782 Shear flow-induced coalescence, of droplets, 20 331-332, 333... 

Matrix cracking in angle-ply laminates also introduces the coupling between extension and shear. The axial/transverse shear-extension coupling coefficients [5] that characterise shearing in the xy plane caused by respectively axial/transverse stress are plotted in Fig. 5 as a function of the relative delamination area D . There is no experimental data to compare our analytical predictions and this is the topic of current work. Results will be presented at a future event. 

Fig. 5. Shear-extension coupling coefficients of a [Oj /SO /-30j], AS4/3506-I laminate as a function of relative delamination area, %. Crack density 3cm ...

The remaining beam stiffness coefficients depend on the shell stiffnesses that result from coupling on the laminae level. This concerns the coupling between shear and extension, Ai3(s), between extension and lengthwise curvature, 13(5) and 631(5), as well as between lengthwise curvature and twist, 613(5). So, the beam stiffness coefficients responsible for the coupling of extension with shear and torsion read ... 

Remark 9.1. The sought analytic solution will consider the constitutive coupling of extension and torsion as well as of shear and bending in combination with actuation of extension, torsion, and warping as well as constant and linear external loads in view of the line forces and torsional line moment. 

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