Stress-strain behavior lamina

The stress-strain reiations in principal material coordinates for a lamina of an orthotropic material under plane stress are 

The reduced stiffnesses, Qy, are defined in terms of the engineering constants in Equation (2.66). In any other coordinate system in the plane of the lamina, the stresses are 

The stress-strain relations in arbitrary in-plane coordinates, namely Equation (4.5), are useful in the definition of the laminate stiffnesses because of the arbitrary orientation of the constituent laminae. Both Equations (4.4) and (4.5) can be thought of as stress-strain relations for the k layer of a multilayered laminate. Thus, Equation (4.5) can be written as 

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The properties of the lamina constituents, the fibers and the matrix, have been only briefly discussed so far. Their stress-strain behavior is typified as one of the four classes depicted in Figure 1-8. Fibers generally exhibit linear elastic behavior, although reinforcing steel bars in concrete are more nearly elastic-pertectly plastic. Aluminum, as well as... 

First, the stress-strain behavior of an individual lamina is reviewed in Section 4.2.1, and expressed in equation form for the k " lamina of a laminate. Then, the variations of stress and strain through the thicyiess of the laminate are determined in Section 4.2.2. Finally, the relation of the laminate forces and moments to the strains and curvatures is found in Section 4.2.3 where the laminate stiffnesses are the link from the... 

In cross-ply laminates, the stress-strain behavior is slightly nonlinear, as illustrated in Figure 5.123. The stress-strain behavior of a unidirectional lamina along the fiber axis is shown in the top curve, while the stress-strain behavior for transverse loading is illustrated in the bottom curve. The stress-strain curve of the cross-ply composite, in the middle, exhibits a knee, indicated by strength ajc, which corresponds to the rupture of the fibers in the 90° ply. The 0° ply then bears the load, until it too ruptures at a composite fracture strength of ct/. 

In Section 2.2, the stress-strain relations (generalized Hooke s law) for anisotropic and orthotropic as well as isotropic materials are discussed. These relations have two commonly accepted manners of expression compliances and stiffnesses as coefficients (elastic constants) of the stress-strain relations. The most attractive form of the stress-strain relations for orthotropic materials involves the engineering constants described in Section 2.3. The engineering constants are particularly helpful in describing composite material behavior because they are defined by the use of very obvious and simple physical measurements. Restrictions in the form of bounds are derived for the elastic constants in Section 2.4. These restrictions are useful in understanding the unusual behavior of composite materials relative to conventional isotropic materials. Attention is focused in Section 2.5 on stress-strain relations for an orthotropic material under plane stress conditions, the most common use of a composite lamina. These stress-strain relations are transformed in Section 2.6 to coordinate systems that are not aligned with the principal material... 

The macromechanical behavior of a lamina was quantitatively described in Chapter 2. The basic three-dimensional stress-strain relations for elastic anisotropic and orthotropic materials were examined. Subsequently, those relations were specialized for the plane-stress state normally found in a lamina. The plane-stress relations were then transformed in the plane of the lamina to enable treatment of composite laminates with different laminae at various angles. The various fundamental strengths of a lamina were identified, discussed, and subsequently used in biaxial strength criteria to predict the off-axis strength of a lamina. 

Shear-stress-shear-strain curves typical of fiber-reinforced epoxy resins are quite nonlinear, but all other stress-strain curves are essentially linear. Hahn and Tsai [6-48] analyzed lamina behavior with this nonlinear deformation behavior. Hahn [6-49] extended the analysis to laminate behavior. Inelastic effects in micromechanics analyses were examined by Adams [6-50]. Jones and Morgan [6-51] developed an approach to treat nonlinearities in all stress-strain curves for a lamina of a metal-matrix or carbon-carbon composite material. Morgan and Jones extended the lamina analysis to laminate deformation analysis [6-52] and then to buckling of laminated plates [6-53]. 

Fibers are linearly elastic up to fracture (ct = Ee), but the stress-strain relations of typical polymer matrices are nonlinear, as a consequence of their viscoelastic behavior (see Fig. 15.10) therefore, ci cannot be replaced by E e . Furthermore, the fibers and the matrix in the laminae are assumed to fail independently, as if they were each tested alone. The behavior of the... 

Other anisotropic elasticity relations are used to define Chentsov coefficients that are to shearing stresses and shearing strains what Poisson s ratios are to normal stresses and normal strains. However, the Chentsov coefficients do not affect the in-plane behavior of laminaeS under plane stress because the coefficients are related to S45, S46, Equation (2.18). The Chentsov coefficients are defined as... 

Let us first consider the case of an isotropic material, then simplify it for the case of an orthotropic material (same properties in the two directions orthogonal to the fiber axis—in this case, directions 2 and 3), snch as a nnidirectionally reinforced composite lamina. Eqnation (5.128) can be written in terms of the strain and stress components, which are conpled dne to the anisotropy of the material. In order to describe the behavior in a manageable way, it is cnstomary to introdnce a reduced set of nomenclature. Direct stresses and strains have two snbscripts—for example, an, 22, ti2, and Y2i, depending on whether the stresses and strains are tensile (a and s) or shear (t and y) in natnre. The modnli should therefore also have two subscripts En, E22, and G 2, and so on. By convention, engineers nse a contracted form of notation, where possible, so that repeated snbscripts are reduced to just one an becomes a, En becomes En but Gn stays the same. The convention is fnrther extended for stresses and strains, such that distinctions between tensile and shear stresses and strains are... 

The orthotropic stress and strain relationships of Equations 8.42 and 8.43 were defined in principal material directions, for which there is no coupling between extension and shear behavior. However, the coordinates natural to the solution of the problem generally will not coincide with the principal directions of orthotropy. For example, consider a simply supported beam manufactured from an angle-ply laminate. The principal material coordinates of each ply of the laminate make angles 0 relative to the axis of the beam. In the beam problem stresses and strains are usually defined in the beam coordinate system (jc,y), which is off-axis relative to the lamina principal axes (L, T). 

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