Stiffness compliance transformations

This chapter began by describing briehy the elasticity of anisotropic materials, providing the fundamental relationships and the allowed simplihcations by the existence of material planes of symmetry. The current unidirectional composites are usually classihed as transversely isotropic materials. In this case, only hve independent elastic constants are necessary to fully characterize unidirectional composites. The micromechanics provides the analytical and numerical approaches to predict the elastic constants based on the elastic properties of the composite constituents. Several classical closed formulas are revisited and compared with experimental data. Finally, stiffness and compliance transformations are given in the context of unidirectional composites. Experimental data are used to assess theoretical predictions and illustrate the off-axis in-plane elastic properties. 

Thus, the Tsai-Wu tensor failure criterion is obviously of more general character than the Tsai-Hill or Hoffman failure criteria. Specific advantages of the Tsai-Wu failure criterion include (1) invariance under rotation or redefinition of coordinates (2) transformation via known tensor-transformation laws (so data interpretation is eased) and (3) symmetry properties similar to those of the stiffnesses and compliances. Accordingly, the mathematical operations with this tensor failure criterion are well-known and relatively straightforward. 

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... 

Stress and strain are second-order tensors while stiffness and compliance are fonrth-order tensors [14]. Hence these entities are mled by the tensor transformation laws that establish the relationships between the components... 

The transformation eqnations for the stiffnesses and the compliances then read ... 

Similarly if the compliances and stiffnesses are defined for one set of axes (usually the S3mimetry axes of the polymer) their values can be determined in another set of axes by the tensor transformation rule... 

Due to the symmetry of T and E, the number of components of the stiffness and compliance tensors is reduced from a total of 81 to 36 independent ones. Thus, it is possible to represent these fourth-order tensors alternatively in the form of (6 x 6) matrices (which, of course, do not have the transformation properties of a tensor), and to express Hooke s law and inverse Hooke s law in direct matrix notation (engineering notation) as... 

The matrices [5] and [Q] appearing in Equations 8.42 and 8.43 are called the reduced compliance and stiffness matrices, while [5 ] and Q in Equations 8.54 and 8.55 are known as the transformed reduced compliance and stillness matrices. In general, the lamina mechanical properties, from which compliance and stiffness can be calculated, are determined experimentally in principal material directions and provided to the designer in a material specification sheet by the manufacturer. Thus, a method is needed for transforming stress-strain relations from off-axis to the principal material coordinate systan. 

The purpose of introducing the concepts developed in Equation 8.56 through Equation 8.63 has been to provide the tools needed to present a straightforward derivation of the transformed reduced compliance and stiffness matrices [5] and [Q based on matrix algebra. The derivation makes use of the following sequence of operations to obtain stress-strain relations in the reference (x, y) coordinate system ... 

If the matrix terms defining [5] and [Q in Equations (8.64) and (8.65) are combined, then the following individual transformed compliance and stiffness components... 

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. 

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