Kirchhff Hpthesis MER45: Cmpsite Materials 1
Cmpsite Design Eperimental Data STRUCTURE CONSTITUENTS COMPOSITE Micr Mechanics STRUCTURAL ELEMENT ELEMENTARY STRUCTURE Finite Element Analsis E E G
Classical Laminatin Ther The influence f fiber directin stacking arrangements material prperties and mre n structural respnse 3
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
Laminate Nmenclature 45 / 45 / 45 / 0 45 / 0 / 45 / 0 / 0 / s 45 / 45 / 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
The Kirchhff Hpthesis Mid 1800 s simplified analsis Beams plates shells metal wd cncrete and ther materials 6
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
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
Cnsequences f Kirchhff Hpthesis 9
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
Resulting Displacement Field in XZ plane u u w w w 11
Resulting Displacement Field in YZ plane v v w w w 1
1-D Strain-Displacement 13
-D Strain-Displacement 14
Strain-Displacement Relatins frm Ther f Elasticit u v u v v w w w u 15
16 Laminate Strains w u v u v w w u w w w v w w w w v v w u u ) ( 0 0 0 ) ( ) (
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 Strain Ntatin w u v w v w u
Laminate Strains using Revised Ntatin 0 0 0 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
0 Laminate Stresses 66 6 1 6 1 16 1 11 Q Q Q Q Q Q Q Q Q
[0/90] s Laminate Aial 1000 Laminate Stress & Strain 1
[0/90] s Laminate Aial 1000 Material Stress & Strain
Aluminum Aial 1000 3
[0/90] s Laminate 3.33 m -1 Laminate Stress & Strain 4
[0/90] s Laminate 3.33 m -1 Material Stress & Strain 5
Aluminum 3.33 m -1 6
Anticlastic Curvature [0/90] s =3.33m -1 7
Anticlastic Curvature [0/90] s =3.33m -1 0 3.33m 0 0 0 0 1 8
Anticlastic Curvature [0/90] s ε =0.1mm mm 0.1 0 0 0 0 0 9
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
[0/90] s Laminate Aial 1000 Laminate Stress & Strain 31
[0/90] s Laminate Aial 1000 Material Stress & Strain 3
[0/90] s Laminate Aial 1000 Stress Resultants and Defrmatin 33
[0/90] s Laminate 3.33 m -1 Laminate Stress & Strain 34
[0/90] s Laminate 3.33 m -1 Material Stress & Strain 35
[0/90] s Laminate 3.33 m -1 Stress Resultants and Defrmatin 36
Aluminum 3.33 m -1 37