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Tham khảo tài liệu 'introduction to continuum mechanics 3 episode 7', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 226 The Elastic Solid It can be shown that any isotropic fourth order tensor can be represented as a linear combination of the above three isotropic fourth order tensors we omit the rather lengthy proof here. In part B of this chapter we shall give the detail reductions of the general Cyki to the isotropic case . Thus for an isotropic linearly elastic material the elasticity tensor can be written as a linear combination of Aijki Byki and Hịjki. CịjkỊ A-Ăịjkl a Eijkỉ p Hijkl 5.3.5 where Ấ a and p are constants. Substituting Eq. 5.3.5 into Eq. i and since Aiịkl Ekl ôìj ồkl Ekl ỗij Ekk ôij e iii fyjkl Ekl ôik ôjl Ekl E j iv Hijki Ekl ỏii ồjkEkl Eji Eịj v we have Tịj CijkỊ Eki Ảe ôịj Eịj vi Or denoting a p by 2 z we have Tỹ Ấ e ôịj 2 fl Ey 5.3.6a or in direct notation T ẤeI 2 E 5.3.6b where e Ekk first scalar invariant of E. In long form Eqs. 5.3.6 are given by 11 11 E22 Eĩì 2 A EU 5.3.6c 22 - I 11 e22 33 2 a E22 5.3.6d 33 11 22 33 2a 33 5.3.6e Tỉ2 2fiEỉ2 5.3.6Í 13 2 z E13 5.3.6g T23 2fiE23 5.3.6h Equations 5.36 are the constitutive equations for a linear isotropic elastic solid. The two material constants Ấ and are known as Laine s coefficients or Laine s constants. Since Ey are dimensionless Ấ and are of the same dimension as the stress tensor force per unit area. For a given real material the values of the Lame s constants are to be determined from suitable experiments. We shall have more to say about this later. Linear Isotropic Elastic Solid 227 Example 5.3.1 -6 Find the components of stress at a point if the strain matrix is 30 50 20 E 50 40 0 X10 20 0 30 and the material is steel with Ấ 119.2 GPa 17.3 X106 psi and 4 79.2 GPa 11.5 X106 psi . Solution. We use Hooke s law Tịj Ấ e ôịj 4- 2 4 Eij by first evaluating the dilatation e lOOx 10 6. The stress components can now be obtained Til Ấ e 4- 2 4 n 1.67 X IO 2 GPa 722 e 4- 2 4 E22 1.83 X 10 2 GPa T33 Ả e 4- 2 4 E33 1.67 X IO-2 GPa r12 T21 2 4 12 7.92 X 10-2 GPa 713 T31 2 4 13 3.17 X IO-3 GPa 723 732 0 GPa Example
