Learning Units of Module 3
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1 Module
2 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
3 Action of force (F) on a body Figure: Action of force (F) on a body
4 Action of force (F) on a body Figure: tress at a Point 4
5 Action of force (F) on a body Figure: lements -dimensional stress. All stresses have positive sense. 5
6 Global D train: Figure: Global -dimensional strain. 6
7 Global D train: Figure: Infinitesimal -dimensional strain. 7
8 Global D train: Figure M..4: General definition of D-strain 8
9 Global D train: Figure M..4: General definition of D-strain 9
10 tress-train urve of omposites omposite behavior: Matrix yields, fibers take load Failure at, f but not catastrophic
11 Lamina tress-train Relationships
12 tress omponents 9 components of stress: ij i, j,,
13 Types of tress. Normal tress i j. hear tress i j
14 ubscripts First subscript refers to direction of outer normal econd subscript refers to the direction in which the stress acts 4
15 tress ube 5
16 tress ube τ τ τ τ τ τ 6
17 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
18 TR TRAIN Tensor Notation ontracted Notation Tensor Notation ontracted Notation ( ) ( ) ( ) ( ) ( ) ( ) τ ( ) 4 γ 4 τ ( ) 5 γ 5 τ ( ) 6 γ 6 8
19 General Formula tresses and strains are related to each other. The most general form of this relationship is: (,,,,,,, ) ij f ij, 9
20 Linear lastic Material
21 9 tresses x 9 trains 8 omponents in relationship
22 ymmetry ij ji i j and ij ji i j
23 ymmetry ijkl ijkl and jikl ijlk 6 tresses x 6 trains 6 omponents in relationship
24 ontracted Notation: tresses
25 ontracted Notation : trains γ γ 4 γ γ 5 γ γ 6 5
26 Hooke s Law (tiffness) is the inverse of. These equations encompass all anisotropic crystalline behavior. 6
27 Hooke s Law (tiffness) i ij j i, j,, Κ,6 or { } [ ]{} 7
28 Hooke s Law
29 Hooke s Law (ompliance) i ij j i, j,, Κ,6 or {} [ ]{ } 9
30 Inverse Relationship [] and [] are symmetric matrices!
31 Heterogeneous omposite quivalent Homogeneous omposite
32 effective modulus. stress strain
33 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
34 Average Values {} [ ]{} and {} []{} We use the effective (or average ) values of stress, strain and moduli when referring to lamina behavior. 4
35 train nergy Density W ij i j and W i i ij j 5
36 econd Derivatives W i j ij and W j i ji 6
37 ymmetry ij ji and ij ji 7
38 YM Μ Μ Μ Μ Μ ymmetry: omponents
39 9 Monoclinic Material; omponents plane of symmetry YM Μ Μ Μ Μ Μ
40 Orthotropic Material: 9 onstants Three planes of symmetry θ y x 4
41 pecially Orthotropic Material Three planes of symmetry -- Directions are principal coordinate directions corresponding to symmetry planes, as shown on previous slide. 4
42 4 Orthotropic tiffness Matrix YM Μ Μ Μ Μ Μ 9 omponents
43 Transversely Isotropic Material Three planes of symmetry and directions the same. θ y x 4
44 44 tiffness Matrix ( ) YM Μ Μ Μ Μ Μ 5 omponents
45 Isotropic Material Three planes of symmetry, and directions the same. 45
46 46 tiffness Matrix ( ) ( ) ( ) YM Μ Μ Μ Μ Μ omponents
47 47 Anisotropic Material YM Μ Μ Μ Μ Μ
48 48 Anisotropic Material xtension xtension YM Μ Μ Μ Μ Μ
49 49 Anisotropic Material xtension xtension YM Μ Μ Μ Μ Μ xtension xtension-xtension oupling xtension oupling
50 5 xtension xtension YM Μ Μ Μ Μ Μ xtension xtension-xtension xtension oupling oupling hear-xtension oupling
51 5 xtension xtension YM Μ Μ Μ Μ Μ xtension xtension-xtension xtension oupling oupling hear-xtension oupling hear
52 5 xtension xtension YM Μ Μ Μ Μ Μ xtension xtension-xtension xtension oupling oupling hear-xtension oupling hear-hear oupling hear
53 Material D Anisotropic Generally Orthotropic pecially Orthotropic Transversely Isotropic Isotropic Nonzero Terms 6 6 Independent terms D Anisotropic Generally Orthotropic pecially Orthotropic Balanced Orthotropic Isotropic
54 Uniaxial Load in Fiber Direction 54
55 Resulting trains ν ν ν ν γ γ γ or
56 Transverse Load 56
57 Resulting trains ν ν ν ν or γ γ γ 57
58 hear Load
59 Resulting trains γ 6 G 6 γ γ or
60 6 τ τ τ ν ν ν ν ν ν γ γ γ G G G ngineering Material Properties
61 ngineering Material Properties ν ν ν ν ν ν 4 4 G G G 6
62 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
63 ymmetry [] [ ] Due to ymmetry ν ij i ν ji j : 6
64 L L 64
65 L L 65
66 L ν ν L ν ν L L 66
67 ν L ν ν ν L 67
68 Inverse Relationship [ ] [ ] [] and [] are symmetric matrices! 68
69 Inverse Relationship
70 Inverse Relationship ν ν ν ν ν ν G G G 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
71 Isotropic Material G ( +ν) <ν<.5 7
72 Orthotropic Material onstraints,,,,, > ,,,G,G,G > 7
73 Orthotropic Material onstraints,,,,, > ( ) ( ) ( ) ν ν, ν ν, ν ν > ν ν ν ν ν ν ν ν ν > 7
74 Orthotropic Material onstraints < < < 74
75 Orthotropic Material onstraints ( ) ( ) ( ) ν ν, ν ν, ν ν > ν ij i νji i,j,, j ν < ν < ν < ν < ν < ν < 75
76 Plane tress Orthotropic Material 4 5 or τ τ γ 66 76
77 Plane tress Orthotropic Material γ τ
78 Plane tress Orthotropic Material + 4 γ γ 5 78
79 ompliances : ompliance ν ν 66 G 79
80 ompliance ν 6 6 G ν 8
81 ompliance { } [ ]{ } ompliances: * 44 G * 55 G 66 ν G 8
82 tiffness { } [ Q]{ } 8
83 Lamina tiffness Matrix Q Q Q Q Q γ 66 8
84 Q Q Q Q ( ) ( ) ν ν ( ) ( ) ν ν Q G tiffness Terms ( ) ( ) ν ν ν 84
85 ome Typical Properties 85
86 Material (Msi) (Msi) G (Msi) n T/94 Graphite /poxy A/5 Graphite /poxy.... p-/rl 96 Pitch Graphite /poxy Kevlar 49 /94 Aramid/poxy cotchply -glass/poxy Boron/555 Boron/poxy pectra 9/86 Polyethylene/poxy glass/47-6 -glass/vinylester
87 Generally Orthotropic Lamina y + θ x Positive Angle 87
88 Generally Orthotropic Lamina y θ x Negative Angle 88
89 y tress lement da sinθ da sin θ θ x dacosθ θ xy da x da da cos θ 89
90 quilibrium F x + x da dacos dasinθcosθ θ dasin θ + F y da xy da ( sin θ cos θ) dasinθcosθ + dasinθcosθ 9
91 tress Transformation cos θ+ sin θ sin θcos θ x xy ( ) sin θcos θ sin θcosθ+ cos θsin θ imilar derivation for y 9
92 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
93 ondensed Matrix Form c s cs x s c cs y cs cs c s xy 6 c cosθ and s sinθ 9
94 94 Transformation Matrix: [T] [ ] [ ] xy y x xy y x T T or
95 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 96 train [ ] γ γ γ γ T or s c cs cs cs c s cs s c xy y x xy y x
97 tress and train x [ ] y T τ and τxy x [ ] y T γ xy γ 97
98 tress and train γ γ x x y y γ γ Reuter's Matrix: xy xy R [ ] 98
99 Lamina tiffness Matrix Q Q Q Q Q γ 66 99
100 tiffness Terms Q Q Q Q ( ) ( νν ) ( ) ( νν ) Q G ( ) ( νν ) ν
101 x [ ] y T xy x [ ] [ ] y T Q xy [ ] Q γ γ
102 [ R ] γ γ [ ] R γ γ
103 x [ ] [ ] y T Q γ xy [ ] R γ γ x [ ] [ ][ ] y T Q R γ xy
104 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
105 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
106 x x [ T] [ Q][ R][ T][ R] y y γ xy xy T [ R][ T][ R] [ T] -T inverse transpose 6
107 General tress-train Behavior x x T [ T] [ Q][ T] y y γ xy xy [ ] [ ] Q T [ Q][ T] 7
108 8 tress-train Behavior γ xy y x xy y x Q Q Q Q Q Q Q Q Q
109 xplicit Relationships ( ) Q Q cos θ+ Q sin θ+ Q + Q sin θcos θ ( ) ( 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
110 xplicit Relationships ( ) ( ) ( ) ( ) ( ) ( ) θ θ + + θ θ + θ θ + θ θ θ θ + θ θ 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
111 tress-train Behavior γ xy y x xy y x
112 tress-train Behavior T [ ] [ ][ ] T T x x y y γ xy xy
113 Inverse Relationship [ Q] [ ] [ ] [ ] Q ymmetric matrices!
114 xplicit Relationships θ + ( ) ( ) ( cos θsin θ + sin θ + cos θ) cos sin 4 4 θ + 66 sin cos 4 4 θ + θ + + θcos θ ( ) + sin θcos θ sin 4
115 5 xplicit Relationships ( ) ( ) ( ) ( ) ( ) ( ) θ θ + + θ θ + θ θ + θ θ θ θ + θ θ sin cos sin cos 4 sin cos sin cos sin cos sin cos
116 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
117 ngineering onstants 6 η xy, x 6 66 η y, xy 7
118 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
119 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
120 ngineering onstants c c s G s s c s G c y x + ν + + ν +
121 ngineering onstants ( ) ( ) s c G c s G G c s G s c xy x xy + + ν ν ν
122 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
123 T/94 Graphite/poxy x y Gxy
124 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
125 Alternate Form for tiffness Q Q Q Q Q Q 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
126 6 Invariants ( ) ( ) ( ) ( ) Q 6Q Q Q 8 U 4Q Q Q Q 8 U Q Q U 4Q Q Q Q 8 U
127 Alternate Form for ompliances 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
128 8 Invariants ( ) ( ) ( ) ( ) V 8 V V 8 V
129 Q U 4 θ p 4θ p U U U Q 9
130 Transversely Isotropic Material G ν ν G G ν ν ( + ν )
131 Balanced Orthotropic Material
132 Balanced Orthotropic Lamina. o and 9 o cross-plies. Woven Materials. omponents Q Q
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