By Sir James H. Jeans, Physics
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Extra info for An elementary treatise on theoretical mechanics
Furthermore, since the bulk modulus must be positive, we have ν < 12 . Finally, the Poisson’s ratio of isotropic materials is constrained by −1 < ν < 12 . 13 Engineering Constants Please note from here forward () denotes the lamina coordinate system. Our next task is to write the components of the stiﬀness and compliance matrices in terms of engineering constants for orthotropic materials. For this purpose it is easier to work with the compliance matrix, which is deﬁned as the inverse of the stiﬀness matrix.
The diagonal terms C44 and C55 remain positive because they are multiplied twice. Therefore, Ci4 = −Ci4 with i = 4, 5, Ci5 = −Ci5 with i = 4, 5, with everything else unchanged. Since the material properties in a monoclinic material cannot change by a reﬂection, it must be C4i = Ci4 = 0 with i = 4, 5, C5i = Ci5 = 0 with ✐ ✐ ✐ ✐ ✐ ✐ “K15072” — 2013/3/21 — 10:58 ✐ 20 ✐ Finite Element Analysis of Composite Materials Fig. 11: Orthotropic material. i = 4, 5. 65) Orthotropic Material An orthotropic material has three planes of symmetry that coincide with the coordinate planes.
In this case, any plane containing the ﬁber direction is a plane of symmetry. A transversely isotropic material is described by ﬁve constants. When the axis of symmetry is the ﬁber direction (1-direction), 3D Hooke’s law reduces to ✐ ✐ ✐ ✐ ✐ ✐ “K15072” — 2013/3/21 — 10:58 ✐ 22 ✐ Finite Element Analysis of Composite Materials Fig. 12: Randomly distributed E-glass ﬁbers with 200X magniﬁcation. 71) Note the equations would be diﬀerent if the axis of symmetry is not the 1direction. 72) G12 = G13 In addition, any two perpendicular directions on the plane 2-3 can be taken as axes.