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ADVANCED MECHANICS OF COMPOSITE MATERIALS Episode 9

Tham khảo tài liệu 'advanced mechanics of composite materials episode 9', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Chapter 5. Mechanics of laminates 267 h3 eh e2 Amn Dmn 3 5.2. Stiffness coefficients of a homogeneous layer Consider a layer whose material stiffness coefficients Amn do not depend on coordinate z. Then Amn h 1. 7 01 Am h 5.33 7mn r 1 mn Amn . and Eqs. 5.28 5.30 and 5.31 yield the following stiffness coefficients for the layer Bmn Amnh Cmn Amn e 5.34 Smn Amnh Both Eqs. 5.30 and 5.31 give the same result for Smn. It follows from the second of Eqs. 5.34 that the membrane-bending coupling coefficients Cmn become equal to zero if we take e h 2 i.e. if the reference plane coincides with the middle-plane of the layer shown in Fig. 5.9. In this case Eqs. 5.5 and 5.15 take the following de-coupled form Nx B11S0 B12Ẽ0 B14Y0 Ny B21S0 B22S0 B24Y0 x x y xy y x z-z- y xy Nxy B41 x B42fi B44Yxy Mx D11Kx D12Ky D14Kxy My D21Kx D22Ky D24Kxy 5.35 Mxy D41Kx D42Ky D44Kxy Vx S55Yx S56Yy Vy S65Yx S66Yy As can be seen we have arrived at three independent groups of constitutive equations i.e. for in-plane stressed state of the layer bending and twisting and transverse shear. Fig. 5.9. Middle-plane of a laminate. 268 Advanced mechanics of composite materials The stiffness coefficients Eqs. 5.34 become Bmn Amnh Dmn 12 h Smn Amnh 5.36 For an orthotropic layer there are no in-plane stretching-shear coupling B14 B24 0 and transverse shear coupling S56 0 . Then Eqs. 5.35 reduce to Nx B11 s 0 B12 x Ny B2 S 0 B22S0 Nxy B44Y0 x x y y L x y xy xy Mx D11Kx D12Ky My D21Kx D22Ky Mxy D44Kxy 5.37 Vx S55Yx Vy S66Yy In terms of engineering elastic constants the material stiffness coefficients of an orthotropic layer can be expressed as A11 Ex A12 vxyEx A22 Ey A44 Gxy A55 Gxz A66 Gyz 5.38 where Ex y Ex y 1 - VxyVyx . Then Eqs. 5.36 yield B11 Exh B12 vxxExh B22 Eyh B44 Gxxh D11 -1 Exh3 D12 VxyExh3 D22 1 Eyh3 Ấ Ấ Ấ D44 12 Gxyh .3 5.39 S55 Gxzh S66 Gyzh Finally for an isotropic layer we have E Ex Ey E vxy vyx v Gxy Gxz Gyz G 2 1 I v and B11 B22 Eh B12 vEh B44 S55 S66 Gh D11 D22 Eh3 D12 Eh3 D44 Gh3 5.40 where