13 ELASTIC COEFFICIENTS The definition of a linear elastic anisotropic medium was given in Chapter 9.We have also given,without justification,the behavior relations characterizing the particular case of orthotropic materials.Now we propose to examine more closely the elastic constants which appear in stress-strain relations for these materials.In the case of transversely isotropic materials,we will study also the manner in which the constants evolve. 13.1 ELASTIC COEFFICIENTS IN AN ORTHOTROPIC MATERIAL Recall:Consider the relation for elastic behavior written in Paragraph 9.1.1 in the form: Emn Omnpqx Opq Recall that the components of a tensor expressed in the coordinate system 1,2,3 are written as du in a coordinate system I,II,II using the relation: UKL=cosT coSf cosk cos?Pmnpa (13.1) in which: cos”=cos(m,7 By definition,'for mechanical behavior,an orthotropic medium has at any point two orthogonal planes of symmetry.Consider here two coordinate systems 1,2,3 and I,II,III,constructed on these planes and their intersection.One plane can be obtained from the other by a 1800 rotation about the 3 axis as shown in Figure 13.1.One can deduce [-10 [cos"] 0-1 0 001 TSee Section 9.2. 2003 by CRC Press LLC13 ELASTIC COEFFICIENTS The definition of a linear elastic anisotropic medium was given in Chapter 9. We have also given, without justification, the behavior relations characterizing the particular case of orthotropic materials. Now we propose to examine more closely the elastic constants which appear in stress–strain relations for these materials. In the case of transversely isotropic materials, we will study also the manner in which the constants evolve. 13.1 ELASTIC COEFFICIENTS IN AN ORTHOTROPIC MATERIAL Recall: Consider the relation for elastic behavior written in Paragraph 9.1.1 in the form: Recall that the components jmnpq of a tensor expressed in the coordinate system 1,2,3 are written as FIJKL in a coordinate system I,II,II using the relation: (13.1) in which: By definition,1 for mechanical behavior, an orthotropic medium has at any point two orthogonal planes of symmetry. Consider here two coordinate systems 1,2,3 and I,II,III, constructed on these planes and their intersection. One plane can be obtained from the other by a 180∞ rotation about the 3 axis as shown in Figure 13.1. One can deduce 1 See Section 9.2. emn = jmnpq ¥ spq FIJKL cosI m cosJ n cosK p cosL q = jmnpq cosI m = cos( ) m, I cosI m [ ] –1 0 0 0 1– 0 0 01 = TX846_Frame_C13 Page 259 Monday, November 18, 2002 12:29 PM © 2003 by CRC Press LLC