NONLOCAL THEORY OF ERINGEN

Transcription

1 NONLOCAL THEORY OF ERINGEN Accordig to Erige (197, 1983, ), the stress field at a poit x i a elastic cotiuum ot oly depeds o the strai field at the poit (hyperelastic case) but also o strais at all other poits of the body. Erige attributed this fact to the atomic theory of lattice dyamics ad experimetal observatios o phoo dispersio. Thus, the olocal stress tesor at poit x is expressed as K( xx, ) tx ( ) dx V where t(x) is the classical, macroscopic stress tesor at poit x ad the K( x x, ) kerel fuctio that represets the olocal modulus, x x beig the distace (i Euclidea orm) ad is a material costat that depeds o iteral ad exteral characteristic legths (such as the lattice spacig ad wavelegth, respectively).

2 NONLOCAL THEORY OF ERINGEN where e ea ( 1 ) t, is a material costat, ad aad ad are the iteral ad exteral characteristic legths,respectively. I particular, we have the followig olocal stress-strai relatios for beams: xx E xx xx x G ea xz x xz ( ) xz

3 NONLOCAL THEORIES OF BEAMS For example, the axial stress resultat (i all theories) is N xx N x xx EA Bedig momet i the olocal Euler-Beroulli beam theory is xx M E xx M x E xx EI E Bedig momet ad shear force i the olocal Timosheko beam theory becomes M Q M EI Q GAK x x T T T T T T, s

4 NONLOCAL THEORIES OF BEAMS (cotiued) The stress resultats i the olocal Reddy beam theory are R R R M R R R P R R M EI EJ, P EJ EK, x x R R R Q R R R R R R Q GA GI, R GI GJ, x x ( A, I, J, K ( z, z, z da 4 6 ) 1, ) A The axial force ca be expressed as N EA m x xt x 3 u u f ad the axial equatio of motio becomes 4 u f u u EA f m x x x t x t

5 The Nolocal Euler-Beroulli Beam Theory The bedig momet i the EBT is w w w w x x x t x t E E E 4 E E E M EI N q m m ad the equatio of motio becomes w w w w x x x x x t x t E E 4 E E w w w q N m m x x t x t E E E 4 E E EI N q m m

6 The Nolocal Timosheko Beam Theory The stress resultats i the TBT are T T T 3 T T T w w M EI q N m m x x x t xt T T w T w w Q GAK s q N m x x x x t ad the equatios of motio becomes T T T w T w GAK q N s x x x x T T 4 T T w w w q N m x x x t x t T T T 4 T w T EI GAK m m s x x x t x t T T T

7 The Nolocal Reddy Third-Order Beam Theory The stress resultats i the RBT are R R R R R ˆ R ˆ ˆ w P w M EI c EJ 1 c q m 1 x x x x t R 3 R 3 R 4 R R w w N m cm 1 4 x x xt xt x t R R R ˆ R R w P R w Q GA c N 1 x q x x x x R 3 R 3 R 4 R w w m cm c m x t x t x t x t R 3 R 3 R 4 R R P w c P c EJ c EK x x x x x

8 The Nolocal Reddy Third-Order Beam Theory (cotiued) The first equatio of motio is R R R R w R w R w GA N q N q x x x x x x x 3 R 3 R 4 R w c EJ c EK x x x R 3 R R R w w m cm c m t 1 4 x t 1 6 x t t x t w w m cm c m x t x t x t x t 4 R 5 R 5 R 6 R ()

9 The Nolocal Reddy Third-Order Beam Theory (cotiued) The secod equatio of motio is R R 3 R R ˆ ˆ w R w EI c EJ GA 1 3 x x x x w mˆ t t xt R R 3 R cmˆ R 4 R 5 R w mˆ cmˆ ( ) x t x t x t

10 ANALYTICAL SOLUTIONS Solutio form x x wxt (,) Wsi e, ( xt,) cos e i t i t Φ 1 L 1 L Bedig Solutio -EBT w 4 QL x ( ) si EI L E x L x Q q( x) si dx L L

11 ANALYTICAL SOLUTIONS (CONTINUED) Bedig Solutio -TBT w T T 4 QL x ( x) Λ si 4 4 EI L 1 3 QL x ( x) cos 3 3 EI L 1 ( 1 ), Λ Ω Ω 1 L EI GAK L s

12 ANALYTICAL SOLUTIONS (CONTINUED) Bedig Solutio -RBT 4 R ˆ 1 QL x w ( x) B EI si 4 4 ˆ 1 L J EI L A cb 1 I ˆ 3 R B QL x ( x) cos, ˆ 1 J EI L L A cb 1 I ˆ

13 ANALYTICAL SOLUTIONS (CONTINUED) Bucklig Solutio - EBT E ( ) N m m EI 1 EI N E, 1 L L 4 L L L 1 Bucklig Solutio - TBT N T 1 EI, Λ ( 1 Ω), Ω Λ L 1 1 EI GAK L s

14 ˆ ˆ ˆ R J A cb EI I N L B EI L Bucklig Solutio - RBT Vibratio Solutio - EBT 4 1, EI M m m M L L ANALYTICAL SOLUTIONS (CONTINUED)

15 ANALYTICAL SOLUTIONS (CONTINUED) Vibratio Solutio - TBT 4 mm 4 m Λ m EI GAK L L s 4 1 EI, m m Σ Λ Σ L L Vibratio Solutio - RBT ˆ ˆ 4 EI c EJB A EI 1 m L ˆ EI B L

16 NUMERICAL RESULTS - BENDING 6 E L 1 h,.,, varied ea EBT for all L/h ratios ww 1 EI 4 ql All shear deformatio theories for L/h = 1 w

17 NUMERICAL RESULTS BUCKLING AND VIBRATION ea L N N L EI m, cr 1 EBT for all L/h ratios EI Bucklig (ope symbols) Vibratio (dark symbols) All shear deformatio theories for L/h =1 ( N, ) cr

18 SUMMARY - REMARKS The olocal theory of Erige has softeig effect o the stiffess characteristics of beams. Cosequetly, the deflectios are larger tha those of the covetioal beams bucklig loads ad frequecies are smaller The equatios of equilibrium of beams based o the olocal theory of Erige caot be derived from a eergy or a fuctioal; it is possible to costruct a fuctioal usig the iverse method from kow goverig equatios, but the resultig boudary coditios are ot the same as those derived. The differetial olocal model of Erige eed to be examied further for its validity.