tailieunhanh - ADVANCED MECHANICS OF COMPOSITE MATERIALS Episode 11

Tham khảo tài liệu 'advanced mechanics of composite materials episode 11', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Chapter 6. Failure criteria and strength of laminates 337 For tension in the directions of axes 1 and 2 in Fig. and for shear in plane 1 2 we can write Eq. in the following forms similar to Eqs. 77 45 _ -- 45 _ n T45 I F I Ơ1 Ơ 45 Ơ2 0 T12 0 1 p 45 _ n 45 _ _ _45 _ nA 1 F I Ơ1 0 Ơ2 Ơ45 T12 0 1 7-V 45 _ n 45 _ n 45 _ _ 1 F I Ơ1 0 Ơ2 0 T12 T45 I 1 Here Ơ45 and T45 determine material strength in coordinates 1 and 2 see Fig. . Then Eq. can be reduced to 77 -45 .45 _45 1 45Ỹ2 í 45VÌ i 1 2 1 45 45 i Tn 1 Ơ1 Ơ2 T12 ZJT Ơ1 I Ơ2 1 Z2--- Ơ1 Ơ2 I 22 I 1 2 2 2. Ơ45 Lv J V45 Ơ45 V45 where Ơ 45 and T 45 are given by 1 2 Ơ 45 1 4 T 45 1Ơ 2 Comparing Eq. with Eq. we can see that Eq. in contrast to Eq. includes a term with the product of stresses Ơ145 and Ơ245. So the strength criterion under study changes its form with a transformation of the coordinate frame from 1 and 2 to 1 and 2 in Fig. which means that the approximation polynomial strength criterion in Eq. and hence the original criterion in Eq. is not invariant with respect to the rotation of the coordinate frame. Consider the class of invariant strength criteria which are formulated in a tensorpolynomial form as linear combinations of mixed invariants of the stress tensor Ơij and the strength tensors of different ranks Sij Sijki etc. . SikƠik SikmnƠikƠmn 1 i k i k m n Using the standard transformation for tensor components we can readily write this equation for an arbitrary coordinate frame. However the fact that the strength components form a tensor induces some conditions that should be imposed on these components and not necessarily correlate with experimental data. To be specific consider a second-order tensor criterion. Introducing contracted notations for tensor components and restricting ourselves to the consideration of orthotropic 338 Advanced mechanics of composite materials materials referred to the principal material coordinates