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Elasticity Part 11

Tham khảo tài liệu 'elasticity part 11', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Sadd Elasticity Final Proof 3.7.2004 2 56pm page 289 Axis of Symmetry x FIGURE 11-4 Axis of symmetry for a transversely isotropic material. cos Ớ sin Ớ 0 Qij sin Ớ cos Ớ 0 0 0 1 11.2.11 Using this transformation and invoking symmetry for arbitrary rotations corresponds to the case of n 1 and such materials are called transversely isotropic. The elasticity stiffness matrix for this case reduces to C11 C12 C13 0 0 0 C11 C13 0 0 0 Cj C33 0 C44 0 0 0 0 C44 0 C11 - C12 2_ 11.2.12 Thus for transversely isotropic materials only five independent elastic constants exist. 11.2.4 Complete Symmetry Isotropic Material For the case of complete symmetry the material is referred to as isotropic and the fourth-order elasticity tensor has been previously given by Cijki Ằỗiíỏki m ỏikỗji ỏiiỏjk 11.2.13 This form can be determined by invoking symmetry with respect to two orthogonal axes which implies symmetry about the remaining axis. In contracted matrix form this result would be expressed Anisotropic Elasticity 289 TLFeBOOK Sadd Elasticity Final Proof 3.7.2004 2 56pm page 290 1 2m 1 1 0 0 0 1 2m 1 0 0 0 Q 1 2m 0 m 0 0 0 0 m 0 m 11.2.14 Thus as shown previously only two independent elastic constants exist for isotropic materials. For each of the presented cases a similar compliance elasticity matrix could be developed. EXAMPLE 11-1 Hydrostatic Compression of a Monoclinic Cube In order to demonstrate the difference in behavior between isotropic and anisotropic materials consider a simple example of a cube of monoclinic material under hydrostatic compression. For this case the state of stress is given by ơịj pôịj and the monoclinic Hooke s law in compliance form would read as follows ex x Sii S12 S13 0 0 Si6_ -p ey S22 S23 0 0 S26 -p ez S33 0 0 S36 -p 2eyz S44 S45 0 0 2e zx S55 0 0 2exy S66 0 11.2.15 Expanding this matrix relation gives the following deformation field components ex S11 S12 S13 p ey - S12 S22 S23 p ez - S13 S23 S33 p eyz 0 11.2.16 ezx 0 exy - 2 S16 S26 S36 p The .