Stress-strain relations orthotropic

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. 

The transformed reduced compliance and stiffness matrices [5] and [G1 relate off-axis stress and strain in an orthotropic lamina. Since these matrices are fully populated, the material responds to off-axis stresses as though it was fully anisotropic. Some consequences of the anisotropic nature of a unidirectional lamina are discussed in the following. 

STRESS-STRAIN RELATIONS FOR PLANE STRESS IN AN ORTHOTROPIC MATERIAL... 

However, as mentioned previously, orthotropic laminae are often constructed in such a manner that the principal material coordinates do not coincide with the natural coordinates of the body. This statement is not to be interpreted as meaning that the material itself is no longer orthotropic instead, we are just looking at an orthotropic material in an unnatural manner, i.e., in a coordinate system that is oriented at some angle to the principal material coordinate system. Then, the basic question is given the stress-strain relations In the principal material coordinates, what are the stress-strain relations in x-y coordinates ... 

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. 

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... 

Derive the thermoelastic stress-strain relations for an orthotropic lamina under plane stress, Equation (4.102), from the anisotropic thermoelastic stress-strain relations in three dimensions. Equation (4.101) [or from Equation (4.100)]. 

The analysis of such a laminate by use of classical lamination theory revolves about the stress-strain relations of an individual orthotropic lamina under a state of plane stress in principal material directions... 

Rather than a plane-stress state, a three-dimensional stress state is considered in the elasticity approach of Pipes and Pagano [4-12] to the problem of Section 4.6.1. The stress-strain relations for each orthotropic layer in principal material directions are... 

The basic approaches as summarized by Ashton and Whitney [6-31] will now be discussed. First, a symmetric laminate with orthotropic laminae having principal material directions aligned with the plate axes will be treated. The transverse normal strain can be found from the orthotropic stress-strain relations, Equation (2.15), as... 

The elastic stress-strain relations for an orthotropic lamina under plane stress conditions are... 

For a unidirectional laminate the elastic stress-strain relations define an orthotropic material for which the generalized form of Hooke s Law, relating the stress o to the strain e,... 

Orthorhombic crystals have three mutually perpendicular principal symmetry axes. Since a 180° rotation about each principal axis results in no change, there can be no linear relations between shear stresses and normal strains or between shear stresses and shear strains with different subscripts. This can be proved immediately by observing that, if this were not so, the stated symmetry would not be present. This establishes that in such materials only nine independent elastic compliances (or constants) remain, namely sn, S22, 33, 12, 13, 23, 44, 55, and See- Many technologically important materials, such as rolled metal plates, unidir-ectionally produced polymer films and paper, composite sheet materials, and even wood, have such symmetry, which is referred to as orthotropic symmetry, when it relates to materials rather than crystals. 

The usual stress-strain relations of orthotropic materials is ... 

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