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Ij = σji ) [3]. 9) ✐ ✐ ✐ ✐ ✐ ✐ “K15072” — 2013/3/21 — 10:58 ✐ Mechanics of Orthotropic Materials ✐ 5 Fig. 4: Stress components. 1: Contracted notation convention used by various FEA software packages. 7). 6). Note that the six components of stress σα with α = 1 . . 10), but {σ} is not a vector, but just a convenient way to arrange the six unique components of a symmetric second-order tensor. 1. 10) where the subscript ()A denotes a quantity in ABAQUS notation. 14) ∂ . 15) ✐ ✐ ✐ ✐ ✐ ✐ “K15072” — 2013/3/21 — 10:58 ✐ Mechanics of Orthotropic Materials ✐ 7 The principle of virtual work (PVW) provides an alternative to the equations of equilibrium [4].

9 The following data has been obtained experimentally for a composite based on a unidirectional carbon-epoxy prepreg (MR50 carbon ﬁber at 63% by volume in LTM25 Epoxy). Determine if the restrictions on elastic constants are satisﬁed. 10 Explain contracted notation for stresses and strains. 11 What is an orthotropic material and how many constants are needed to describe it? 12 What is a transversely isotropic material and how many constants are needed to describe it? 48) numerically. 91). 15 Demonstrate that a material having two perpendicular planes of symmetry also has a third.

12). 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.