Module
Learning Units of Module M. tress-train oncepts in Three- Dimension M. Introduction to Anisotropic lasticity M. Tensorial oncept and Indicial Notations M.4 Plane tress oncept
Action of force (F) on a body Figure: Action of force (F) on a body
Action of force (F) on a body Figure: tress at a Point 4
Action of force (F) on a body Figure: lements -dimensional stress. All stresses have positive sense. 5
Global D train: Figure: Global -dimensional strain. 6
Global D train: Figure: Infinitesimal -dimensional strain. 7
Global D train: Figure M..4: General definition of D-strain 8
Global D train: Figure M..4: General definition of D-strain 9
tress-train urve of omposites omposite behavior: Matrix yields, fibers take load Failure at, f but not catastrophic
Lamina tress-train Relationships
tress omponents 9 components of stress: ij i, j,,
Types of tress. Normal tress i j. hear tress i j
ubscripts First subscript refers to direction of outer normal econd subscript refers to the direction in which the stress acts 4
tress ube 5
tress ube τ τ τ τ τ τ 6
train orresponding to each stress component, there is a strain component, ij describing the deformation at a point. Normal strains describe the extension per unit length. hear strains describe distortional deformation. Tensor and ngineering hear trains 7
TR TRAIN Tensor Notation ontracted Notation Tensor Notation ontracted Notation ( ) ( ) ( ) ( ) ( ) ( ) τ ( ) 4 γ 4 τ ( ) 5 γ 5 τ ( ) 6 γ 6 8
General Formula tresses and strains are related to each other. The most general form of this relationship is: (,,,,,,, ) ij f ij, 9
Linear lastic Material
9 tresses x 9 trains 8 omponents in relationship
ymmetry ij ji i j and ij ji i j
ymmetry ijkl ijkl and jikl ijlk 6 tresses x 6 trains 6 omponents in relationship
ontracted Notation: tresses 4 5 6 4
ontracted Notation : trains γ γ 4 γ γ 5 γ γ 6 5
Hooke s Law (tiffness) is the inverse of. These equations encompass all anisotropic crystalline behavior. 6
Hooke s Law (tiffness) i ij j i, j,, Κ,6 or { } [ ]{} 7
8 6 5 4 66 65 64 6 6 6 56 55 54 5 5 5 46 45 44 4 4 4 6 5 4 6 5 4 6 5 4 6 5 4 Hooke s Law
Hooke s Law (ompliance) i ij j i, j,, Κ,6 or {} [ ]{ } 9
Inverse Relationship [] and [] are symmetric matrices!
Heterogeneous omposite quivalent Homogeneous omposite
effective modulus. stress strain
Average tresses and trains i V V i dv dv V i V dv i,, Κ,6 i V V i dv dv V i V dv
Average Values {} [ ]{} and {} []{} We use the effective (or average ) values of stress, strain and moduli when referring to lamina behavior. 4
train nergy Density W ij i j and W i i ij j 5
econd Derivatives W i j ij and W j i ji 6
ymmetry ij ji and ij ji 7
8 6 5 4 66 56 55 46 45 44 6 5 4 6 5 4 6 5 4 6 5 4 YM Μ Μ Μ Μ Μ ymmetry: omponents
9 Monoclinic Material; omponents plane of symmetry 6 5 4 66 55 45 44 6 6 6 6 5 4 YM Μ Μ Μ Μ Μ
Orthotropic Material: 9 onstants Three planes of symmetry θ y x 4
pecially Orthotropic Material Three planes of symmetry -- Directions are principal coordinate directions corresponding to symmetry planes, as shown on previous slide. 4
4 Orthotropic tiffness Matrix 6 5 4 66 55 44 6 5 4 YM Μ Μ Μ Μ Μ 9 omponents
Transversely Isotropic Material Three planes of symmetry and directions the same. θ y x 4
44 tiffness Matrix ( ) 6 5 4 66 66 6 5 4 YM Μ Μ Μ Μ Μ 5 omponents
Isotropic Material Three planes of symmetry, and directions the same. 45
46 tiffness Matrix ( ) ( ) ( ) 6 5 4 6 5 4 YM Μ Μ Μ Μ Μ omponents
47 Anisotropic Material 6 5 4 66 56 55 46 45 44 6 5 4 6 5 4 6 5 4 6 5 4 YM Μ Μ Μ Μ Μ
48 Anisotropic Material xtension xtension 6 5 4 66 56 55 46 45 44 6 5 4 6 5 4 6 5 4 6 5 4 YM Μ Μ Μ Μ Μ
49 Anisotropic Material xtension xtension 6 5 4 66 56 55 46 45 44 6 5 4 6 5 4 6 5 4 6 5 4 YM Μ Μ Μ Μ Μ xtension xtension-xtension oupling xtension oupling
5 xtension xtension 6 5 4 66 56 55 46 45 44 6 5 4 6 5 4 6 5 4 6 5 4 YM Μ Μ Μ Μ Μ xtension xtension-xtension xtension oupling oupling hear-xtension oupling
5 xtension xtension 6 5 4 66 56 55 46 45 44 6 5 4 6 5 4 6 5 4 6 5 4 YM Μ Μ Μ Μ Μ xtension xtension-xtension xtension oupling oupling hear-xtension oupling hear
5 xtension xtension 6 5 4 66 56 55 46 45 44 6 5 4 6 5 4 6 5 4 6 5 4 YM Μ Μ Μ Μ Μ xtension xtension-xtension xtension oupling oupling hear-xtension oupling hear-hear oupling hear
Material D Anisotropic Generally Orthotropic pecially Orthotropic Transversely Isotropic Isotropic Nonzero Terms 6 6 Independent terms 9 9 5 D Anisotropic Generally Orthotropic pecially Orthotropic Balanced Orthotropic Isotropic 9 9 5 5 5 6 4 4 5
Uniaxial Load in Fiber Direction 54
Resulting trains ν ν ν ν γ γ γ or 6 4 5 55
Transverse Load 56
Resulting trains ν ν ν ν or 6 4 5 γ γ γ 57
hear Load 6 6 58
Resulting trains γ 6 G 6 γ γ or 4 5 59
6 τ τ τ ν ν ν ν ν ν γ γ γ G G G ngineering Material Properties
ngineering Material Properties ν ν ν ν ν ν 4 4 G 5 5 6 6 G G 6
ngineering Material Properties,, Young's moduli in -, - and - directions ν, ν, ν Poisson's ratios (extension-extension coupling) G,G,G hear moduli in -,-,and -directions 6
ymmetry [] [ ] Due to ymmetry ν ij i ν ji j : 6
L L 64
L L 65
L ν ν L ν ν L L 66
ν L ν ν ν L 67
Inverse Relationship [ ] [ ] [] and [] are symmetric matrices! 68
Inverse Relationship 44 55 66 44 55 66 + 69
Inverse Relationship ν ν ν ν ν ν G G G 44 55 66 ν +ν ν ν +ν ν ν +ν ν ν +ν ν ν +ν ν ν +ν ν ν ν ν ν ν ν ν ν ν 7
Isotropic Material G ( +ν) <ν<.5 7
Orthotropic Material onstraints,,,,, > 44 55 66,,,G,G,G > 7
Orthotropic Material onstraints,,,,, > 44 55 66 ( ) ( ) ( ) ν ν, ν ν, ν ν > ν ν ν ν ν ν ν ν ν > 7
Orthotropic Material onstraints < < < 74
Orthotropic Material onstraints ( ) ( ) ( ) ν ν, ν ν, ν ν > ν ij i νji i,j,, j ν < ν < ν < ν < ν < ν < 75
Plane tress Orthotropic Material 4 5 or τ τ γ 66 76
Plane tress Orthotropic Material γ τ 6 66 77
Plane tress Orthotropic Material + 4 γ γ 5 78
ompliances : ompliance ν ν 66 G 79
ompliance ν 6 6 G ν 8
ompliance { } [ ]{ } ompliances: * 44 G * 55 G 66 ν G 8
tiffness { } [ Q]{ } 8
Lamina tiffness Matrix Q Q Q Q Q γ 66 8
Q Q Q Q ( ) ( ) ν ν ( ) ( ) ν ν 66 66 Q G tiffness Terms ( ) ( ) ν ν ν 84
ome Typical Properties 85
Material (Msi) (Msi) G (Msi) n T/94 Graphite /poxy 9..5.. A/5 Graphite /poxy.... p-/rl 96 Pitch Graphite /poxy 68..9.8. Kevlar 49 /94 Aramid/poxy..8..4 cotchply -glass/poxy 5.6..6.6 Boron/555 Boron/poxy 9.6.678.8. pectra 9/86 Polyethylene/poxy 4.45.5.. -glass/47-6 -glass/vinylester.54..4. 86
Generally Orthotropic Lamina y + θ x Positive Angle 87
Generally Orthotropic Lamina y θ x Negative Angle 88
y tress lement da sinθ da sin θ θ x dacosθ θ xy da x da da cos θ 89
quilibrium F x + x da dacos dasinθcosθ θ dasin θ + F y da xy da ( sin θ cos θ) dasinθcosθ + dasinθcosθ 9
tress Transformation cos θ+ sin θ sin θcos θ x xy ( ) sin θcos θ sin θcosθ+ cos θsin θ imilar derivation for y 9
9 Transformation in Matrix Form θ θ θ θ θ θ θ θ θ θ θ θ θ θ xy y x sin cos sin cos sin cos sin cos cos sin sin cos sin cos
ondensed Matrix Form c s cs x s c cs y cs cs c s xy 6 c cosθ and s sinθ 9
94 Transformation Matrix: [T] [ ] [ ] xy y x xy y x T T or
95 Matrices [ ] [ ] s c cs cs cs c s cs s c T s c cs cs cs c s cs s c T
96 train [ ] γ γ γ γ T or s c cs cs cs c s cs s c xy y x xy y x
tress and train x [ ] y T τ and τxy x [ ] y T γ xy γ 97
tress and train γ γ x x y y γ γ Reuter's Matrix: xy xy R [ ] 98
Lamina tiffness Matrix Q Q Q Q Q γ 66 99
tiffness Terms Q Q Q Q ( ) ( νν ) ( ) ( νν ) 66 66 Q G ( ) ( νν ) ν
x [ ] y T xy x [ ] [ ] y T Q xy [ ] Q γ γ
[ R ] γ γ [ ] R γ γ
x [ ] [ ] y T Q γ xy [ ] R γ γ x [ ] [ ][ ] y T Q R γ xy
x [ ] [ ][ ] y T Q R γ xy x [ ] T y γ γ xy x x [ T] [ Q][ R][ T] y y γ xy xy 4
x x [ R ] y y γ γ xy xy x x [ T] [ Q][ R][ T] y y γ xy xy x x [ T] [ Q][ R][ T][ R] y y γ xy xy 5
x x [ T] [ Q][ R][ T][ R] y y γ xy xy T [ R][ T][ R] [ T] -T inverse transpose 6
General tress-train Behavior x x T [ T] [ Q][ T] y y γ xy xy [ ] [ ] Q T [ Q][ T] 7
8 tress-train Behavior γ xy y x 66 6 6 6 6 xy y x Q Q Q Q Q Q Q Q Q
xplicit Relationships ( ) Q Q cos θ+ Q sin θ+ Q + Q sin θcos θ 4 4 66 ( ) ( 4 4 ) Q Q + Q 4Q cos θsin θ+ Q sin θ+ cos θ 66 ( ) Q Q sin θ+ Q cos θ+ Q + Q sin θcos θ 4 4 66 9
xplicit Relationships ( ) ( ) ( ) ( ) ( ) ( ) θ θ + + θ θ + θ θ + θ θ θ θ + θ θ 4 4 66 66 66 66 66 6 66 66 6 sin cos sin cos sin cos sin cos sin cos sin cos Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q
tress-train Behavior γ xy y x 66 6 6 6 6 xy y x
tress-train Behavior T [ ] [ ][ ] T T x x y y γ xy xy
Inverse Relationship [ Q] [ ] [ ] [ ] Q ymmetric matrices!
xplicit Relationships θ + ( ) ( ) ( 4 4 + cos θsin θ + sin θ + cos θ) cos sin 4 4 θ + 66 sin cos 4 4 θ + θ + + θcos θ ( ) + sin θcos θ 66 66 sin 4
5 xplicit Relationships ( ) ( ) ( ) ( ) ( ) ( ) θ θ + + θ θ + θ θ + θ θ θ θ + θ θ 4 4 66 66 66 66 66 6 66 66 6 sin cos sin cos 4 sin cos sin cos sin cos sin cos
6 ngineering onstants xy xy y y y xy xy xy x x x xy y yx x xy xy y x G G G,, 6,, 6 66 η η η η ν ν
ngineering onstants 6 η xy, x 6 66 η y, xy 7
oefficients of Mutual Influence η η x,xy x,xy and η γ x xy y,xy oefficients of mutual influence of the first kind haracterize stretching in the x or y direction caused by shear stress in the xy plane. for xy τ all other stresses 8
oefficients of Mutual Influence η η xy,x xy,x and γ xy x η xy,y oefficients of mutual influence of the second kind. haracterizes shearing in the xy plane caused by normal stress in the xy for x plane. all other stresses 9
ngineering onstants 4 4 4 4 c c s G s s c s G c y x + ν + + ν +
ngineering onstants ( ) ( ) 4 4 4 4 4 s c G c s G G c s G s c xy x xy + + ν + + + + ν ν
ngineering onstants xy,y y xy y,xy xy,x x xy x,xy y xy,y x xy,x G G sc G c s G c s G sc G η η η η ν + ν + η ν + ν + η
T/94 Graphite/poxy 4 8 6 x y Gxy 4 4 5 6 7 8 9
cos sin cos cosθsin cos 4 4 Trig Identities θ θ 8 8 θsinθ θsin ( + 4cosθ + cos4θ) ( 4cosθ + cos4θ) θ 8 8 θ ( sinθ + sin4θ) ( sinθ sin4θ) 8 ( cos4θ) 4
Alternate Form for tiffness Q Q Q Q Q Q 6 6 66 U U U U 4 U + U U U cosθ + U cos4θ sinθ + U cosθ + U sinθ U sin4θ sin4θ cos4θ cos4θ ( U U ) U cos4θ 4 5
6 Invariants ( ) ( ) ( ) ( ) 66 4 66 66 4Q 6Q Q Q 8 U 4Q Q Q Q 8 U Q Q U 4Q Q Q Q 8 U + + + + + +
Alternate Form for ompliances 6 6 66 V V V V V 4 + V V V sinθ + sinθ cosθ + cos4θ cosθ + V V V V sin4θ sin4θ cos4θ cos4θ ( V V ) 4V cos4θ 4 7
8 Invariants ( ) ( ) ( ) ( ) 66 4 66 66 6 8 V 8 V V 8 V + + + + + +
Q U 4 θ p 4θ p U U U Q 9
Transversely Isotropic Material G ν ν G G ν ν ( + ν )
Balanced Orthotropic Material
Balanced Orthotropic Lamina. o and 9 o cross-plies. Woven Materials. omponents Q Q