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Introduction to Elasticity Part 11

Tham khảo tài liệu 'introduction to 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 | a t ai t a2 t Erei t - i Aei Erei t - 2 Aễ2 As the number of applied strain increments increases so as to approach a continuous distribution this becomes a t 12 ơj t 12 Erei t j V j j --- a t Ị Erel t de Ị Erel t d d 44 Example 9 In the case of constant strain rate e t Ret we have d d Re R d d e For S.L.S. materials response Erel t ke k-1 exp t r - t-j Erel t ke ki e T Eqn. 44 gives the stress as ơ t ke k1e - t-j T Re d Maple statements for carrying out these operations might be define relaxation modulus for S.L.S. Erel k e k 1 exp -t tau define strain rate eps R t integrand for Boltzman integral integrand subs t t-xi Erel diff subs t xi eps xi carry out integration sigma t int integrand xi 0.t which gives the result ơ t keRet ki R t 1 exp t T This is identical to Eqn. 37 with one arm in the model. The Boltzman integral relation can be obtained formally by recalling that the transformed relaxation modulus is related simply to the associated viscoelastic modulus in the Laplace plane as stress relaxation e t eou t ẽ ỡ Sẽ S s Erel s 1S s Co s 18 Since sf f the following relations hold ỡ E sẼrelẽ Ẽ relẽ Ẽrel é The last two of the above are of the form for which the convolution integral transform applies see Appendix A so the following four equivalent relations are obtained immediately a t Ị Ẽrel t - 5 C d Jo f Ẽrel ẽ t - d Jo f Erel t - e d Jo f Ẽrel e t - d 45 o These relations are forms of Duhamel s formula where Ẽrel t can be interpreted as the stress ơ t resulting from a unit input of strain. If stress rather than strain is the input quantity then an analogous development leads to e t Ccrp t - j d 46 o where Ccrp t the strain response to a unit stress input is the quantity defined earlier as the creep compliance. The relation between the creep compliance and the relaxation modulus can now be developed as ơ sErelC sCcrpơ 7 s ẼrelCcrptt ẼrelCcrp s 2 Erel t - Ccrp d Ẽrel Ccrp t - d t It is seen that one must solve an integral equation to obtain a creep function from a