Kirchhoff Hypothesis. MER452: Composite Materials

Transcription

1 Kirchhff Hpthesis MER45: Cmpsite Materials 1

2 Cmpsite Design Eperimental Data STRUCTURE CONSTITUENTS COMPOSITE Micr Mechanics STRUCTURAL ELEMENT ELEMENTARY STRUCTURE Finite Element Analsis E E G

3 Classical Laminatin Ther The influence f fiber directin stacking arrangements material prperties and mre n structural respnse 3

4 Laminate Crdinate Sstem Laminate thickness H Laer thickness h nt all laers same h k th laer - h k -ais dwnward frm gemetric midplane can be between laers can be within a laer Fiber angles identified relative t ais 4

5 Laminate Nmenclature 45 / 45 / 45 / 0 45 / 0 / 45 / 0 / 0 / s 45 / 45 / T T Laer 1 is the mst - Laer N is the mst + T categrie a laminate as smmetric a mirrr abut the gemetric midplane material prperties fiber rientatin thickness f laer 5

6 The Kirchhff Hpthesis Mid 1800 s simplified analsis Beams plates shells metal wd cncrete and ther materials 6

7 Initiall Flat Laminated Plate Acted Upn b Varius Lads Lads applied mments M distributed lads q inplane lads N pint lads P Multiple laers f fiber reinfrced material Fibers parallel t the plane f the plate Laers are perfectl bnded 7

8 Defrmatin f Lines Nrmal t Gemetric Midplane Befre defrmatin are straight Despite the defrmatins caused b the applied lads line AA remains straight and nrmal t the defrmed gemetric midplane and des nt change length 8

9 Cnsequences f Kirchhff Hpthesis 9

10 Implicatins f the Kichhff Hpthesis in X-Z Plane N thrugh-thickness strain Small defrmatins Tw cmpnents f translatin u in directin hrintal translatin w in directin vertical translatin One cmpnent f rtatin abut -ais w 10

11 Resulting Displacement Field in XZ plane u u w w w 11

12 Resulting Displacement Field in YZ plane v v w w w 1

13 1-D Strain-Displacement 13

14 -D Strain-Displacement 14

15 Strain-Displacement Relatins frm Ther f Elasticit u v u v v w w w u 15

16 16 Laminate Strains w u v u v w w u w w w v w w w w v v w u u ) ( ) ( ) (

17 Laminate Strains Cmpsed f Tw Parts Etensinal Stain f the Reference Surface Curvature f the Reference Surface inverse f the radius f curvature invlves mre than just secnd derivative Fr small strains secnd derivative and curvature identical 17

18 18 Strain Ntatin w u v w v w u

19 Laminate Strains using Revised Ntatin The and are er because the Kirchhff hpthesis assumes that lines perpendicular t the reference surface befre defrmatin remain perpendicular after the defrmatin; right angles in the thickness directin d nt change when the laminate defrms 19

20 0 Laminate Stresses Q Q Q Q Q Q Q Q Q

21 [0/90] s Laminate Aial 1000 Laminate Stress & Strain 1

22 [0/90] s Laminate Aial 1000 Material Stress & Strain

23 Aluminum Aial

24 [0/90] s Laminate 3.33 m -1 Laminate Stress & Strain 4

25 [0/90] s Laminate 3.33 m -1 Material Stress & Strain 5

26 Aluminum 3.33 m -1 6

27 Anticlastic Curvature [0/90] s =3.33m -1 7

28 Anticlastic Curvature [0/90] s =3.33m m

29 Anticlastic Curvature [0/90] s ε =0.1mm mm

30 30 Stress Resultants / / / / / / / / / / / / H H H H H H H H H H H H d M d N d M d N d M d N

31 [0/90] s Laminate Aial 1000 Laminate Stress & Strain 31

32 [0/90] s Laminate Aial 1000 Material Stress & Strain 3

33 [0/90] s Laminate Aial 1000 Stress Resultants and Defrmatin 33

34 [0/90] s Laminate 3.33 m -1 Laminate Stress & Strain 34

35 [0/90] s Laminate 3.33 m -1 Material Stress & Strain 35

36 [0/90] s Laminate 3.33 m -1 Stress Resultants and Defrmatin 36

37 Aluminum 3.33 m -1 37