GG611 2015 Lecture 3 v3 · 2015. 11. 4. · 11/3/15 1 GG612 Lecture%3% Strain% 11/3/15 GG611 1 Outline% • Mathemacal%Operaons% • Strain% – General%concepts% – Homogeneous%strain%

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1

GG612 Lecture 3

Strain

11/3/15 GG611 1

Outline

•  Mathema8cal Opera8ons •  Strain – General concepts – Homogeneous strain – E (strain matrix) – ε (infinitesimal strain)

– Principal values and principal direc8ons

11/3/15 GG611 2

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Main Theme

•  Representa8on of strain at a point in a clear concise manner

11/3/15 GG611 3

Vector Conven8ons

•  X = ini8al posi8on •  X’ = final posi8on •  U = displacement X

X’

U

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Matrix Inverses

•  [AA]-‐1 = [A]-‐1[A] = [I] •  [AB]-‐1 = [B-‐1][A-‐1]

Proof

•  [AB]B-‐1A-‐1=A[I]A-‐1 =AA-‐1 =[I] •  [AB][AB]-‐1 = [I] •  The two leX sides of the equa8ons above are equal

•  [AB]B-‐1A-‐1=[AB][AB]-‐1 •  Dropping the [AB] terms on both sides yields

•  [B-‐1A-‐1]= [AB]-‐1

11/3/15 GG611 5

Matrix Transposes

•  a•b = [aT][b]

•  [AB]T = [BT][AT]

Anxq

=

!a1!a2"!an

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

; Bqxm

=!b1!b2 #

!bm⎡

⎣⎤⎦

AB =

!a1 •!b1!a1 •!b2 #

!a1 •!bm

!a2 •!b1!a2 •!b2 #

!a2 •!bm

" " " "!an •!b1!an •!b1 #

!an •!bm

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

AB[ ]T =

!a1 •!b1

!a2 •!b1 #

!an •!b1

!a1 •!b2

!a2 •!b2 #

!an •!b1

" " " "!a1 •!bm

!a2 •!bm #

!an •!bm

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

BT AT =

!b1!b2"!bm

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

!a1!a2 #

!an⎡⎣

⎤⎦ =

!b1 •!a1!b1 •!a2 #

!b1 •!an

!b2 •!a1!b2 •!a2 #

!b2 •!an

" " " "!bm •!a1!bm •!a1 #

!bm •!a1

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

AB[ ]T = BT AT

a1 a2 ! an⎡⎣

⎤⎦

b1b2"bn

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

= a1b1 + a2b2 +…+ anbn

Each of the n rows of [A] is row vector with q components. Each of the m columns of [B] is a column vector with q components

These match

11/3/15 GG611 6

Proof

A matrix [A] is symmetric if [A]T = [A]

1 22 3

⎡

⎣⎢

⎤

⎦⎥

1 32 3

⎡

⎣⎢

⎤

⎦⎥

Symmetric Not symmetric

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Rota8on Matrix [R]

•  Rota8ons change the orienta8ons of vectors but not their lengths (or the square of the lengths)

•  X•X = |X||X| •  X•X = X’•X’ •  X’ = RX •  X•X = [RX]•[RX] •  X•X = [RX]T[RX] •  X•X = [XTRT] [RX]

•  [XT] [X]= [XTRT] [RX] •  [XT][I][X]= [XT][RT] [R][X] •  [I] = [RT] [R] •  But [I] = [R-‐1] [R], so •  [RT] = [R-‐1]

X X’

11/3/15 GG611 7

Rota8on Matrix [R] 2D Example

X

X’ R = cosθ sinθ

− sinθ cosθ⎡

⎣⎢

⎤

⎦⎥ ; ′X[ ] = R[ ] X[ ]

′x′y

⎡

⎣⎢⎢

⎤

⎦⎥⎥= cosθ sinθ

− sinθ cosθ⎡

⎣⎢

⎤

⎦⎥

xy

⎡

⎣⎢⎢

⎤

⎦⎥⎥→

′x = cosθx + sinθy′y = − sinθx + cosθy

→ ′x 2 + ′y 2 = x2 + y2

RT = cosθ − sinθsinθ cosθ

⎡

⎣⎢

⎤

⎦⎥

RRT = cosθ sinθ− sinθ cosθ

⎡

⎣⎢

⎤

⎦⎥

cosθ − sinθsinθ cosθ

⎡

⎣⎢

⎤

⎦⎥ =

1 00 1

⎡

⎣⎢

⎤

⎦⎥

RT = R−1

θx

y

z

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General Concepts

Deforma8on = Rigid body mo8on + Strain Rigid body mo8on

Rigid body transla8on • Treated by matrix addi8on [X’] = [X] + [U]

Rigid body rota8on • Changes orienta8on of lines,

but not their length • Axis of rota8on does not rotate • Treated by matrix mul8plica8on

[X’] = [R] [X]

Transla8on

Transla8on + Rota8on

Transla8on + Rota8on + Strain

11/3/15 GG611 9

General Concepts •  Normal strains

Change in line length –  Extension (elonga8on) = Δs/s0 –  Stretch = S = s’/s0 –  Quadra8c elonga8on = Q = (s’/s0)2

•  Shear strains Change in right angles •  Dimensions: Dimensionless

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Homogeneous Strain

•  Parallel lines to parallel lines (2D and 3D)

•  Circle to ellipse (2D) •  Sphere to ellipsoid (3D)

11/3/15 GG611 11

′X[ ] = F[ ] X[ ]′x′y

⎡

⎣⎢⎢

⎤

⎦⎥⎥= a b

c d⎡

⎣⎢

⎤

⎦⎥

xy

⎡

⎣⎢⎢

⎤

⎦⎥⎥

X X’

Matrix of constants

x = cosθ y = sinθ

Parametric eqn. of an ellipse

Homogeneous strain Matrix Representa8on (2D)

′X[ ] = F[ ] X[ ]′x′y

⎡

⎣⎢⎢

⎤

⎦⎥⎥= a b

c d⎡

⎣⎢

⎤

⎦⎥

xy

⎡

⎣⎢⎢

⎤

⎦⎥⎥

x

y x’

y’ X[ ] = F[ ]−1 ′X[ ]xy

⎡

⎣⎢⎢

⎤

⎦⎥⎥= a b

c d⎡

⎣⎢

⎤

⎦⎥

−1′x′y

⎡

⎣⎢⎢

⎤

⎦⎥⎥

11/3/15 GG611 12

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Matrix Representa8ons: Posi8ons (2D)

d ′x =∂ ′x∂x

dx + ∂ ′x∂y

dy

d ′y =∂ ′y∂x

dx + ∂ ′y∂y

dy

d ′xd ′y

⎡

⎣⎢⎢

⎤

⎦⎥⎥=

∂ ′x∂x

∂ ′x∂y

∂ ′y∂x

∂ ′y∂y

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

dxdy

⎡

⎣⎢⎢

⎤

⎦⎥⎥

d ′X[ ] = F[ ] dX[ ]11/3/15 GG611 13

Matrix Representa8ons: Posi8ons (2D)

d ′xd ′y

⎡

⎣⎢⎢

⎤

⎦⎥⎥=

∂ ′x∂x

∂ ′x∂y

∂ ′y∂x

∂ ′y∂y

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

dxdy

⎡

⎣⎢⎢

⎤

⎦⎥⎥

′X[ ] = F[ ] X[ ]

′x′y

⎡

⎣⎢⎢

⎤

⎦⎥⎥= a b

c d⎡

⎣⎢

⎤

⎦⎥

xy

⎡

⎣⎢⎢

⎤

⎦⎥⎥

If deriva8ves are constant (e.g., at a point), then the equa8ons are linear in x and y, and dx, dy, dx’, and dy’ can be replaced by x, y, x’, and y’, respec8vely.

11/3/15 GG611 14

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Matrix Representa8ons Displacements (2D)

du = ∂u∂xdx + ∂u

∂ydy

dv = ∂v∂xdx + ∂v

∂ydy

dudv

⎡

⎣⎢

⎤

⎦⎥ =

∂u∂x

∂u∂y

∂v∂x

∂v∂y

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

dxdy

⎡

⎣⎢⎢

⎤

⎦⎥⎥

dU[ ] = Ju[ ] dX[ ]11/3/15 GG611 15

Matrix Representa8ons Displacements (2D)

u = ∂u∂xx + ∂u

∂yy

v = ∂v∂xx + ∂v

∂yy

uv

⎡

⎣⎢

⎤

⎦⎥ =

∂u∂x

∂u∂y

∂v∂x

∂v∂y

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

xy

⎡

⎣⎢⎢

⎤

⎦⎥⎥

U[ ] = Ju[ ] X[ ] If deriva8ves are constant (e.g., at a point), then the equa8ons are linear in x and y

11/3/15 GG611 16

If deriva8ves are constant, then du, dv, dx, and dy can be replaced by u, v, x, and y, respec8vely.

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9

Matrix Representa8ons Posi8ons and Displacements (2D) U = ′X − X

U = FX − X = FX − IX

Ju[ ] = a −1 bc d −1

⎡

⎣⎢

⎤

⎦⎥

U = F − I[ ]XF − I[ ] ≡ Ju

11/3/15 GG611 17

F[ ] = a bc d

⎡

⎣⎢

⎤

⎦⎥

ε (Infinitesimal Strain Matrix, 2D)

Ju =

∂u∂x

∂u∂y

∂v∂x

∂v∂y

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

ε = 12Ju + Ju

T⎡⎣ ⎤⎦ ω = 12Ju − Ju

T⎡⎣ ⎤⎦

∂u∂x

∂u∂y

∂v∂x

∂v∂y

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

= 12

∂u∂x

+ ∂u∂x

⎛⎝⎜

⎞⎠⎟

∂u∂y

+ ∂v∂x

⎛⎝⎜

⎞⎠⎟

∂v∂x

+ ∂u∂y

⎛⎝⎜

⎞⎠⎟

∂v∂y

+ ∂v∂y

⎛⎝⎜

⎞⎠⎟

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

+ 12

∂u∂x

− ∂u∂x

⎛⎝⎜

⎞⎠⎟

∂u∂y

− ∂v∂x

⎛⎝⎜

⎞⎠⎟

∂v∂x

− ∂u∂y

⎛⎝⎜

⎞⎠⎟

∂v∂y

− ∂v∂y

⎛⎝⎜

⎞⎠⎟

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

Ju = ε + ω

ε is symmetric ω is an8-‐symmetric Linear superposi8on

11/3/15 GG611 18

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ε (Infinitesimal Strain Matrix, 2D) Meaning of components

ε =

∂u∂x

⎛⎝⎜

⎞⎠⎟

12

∂u∂y

+∂v∂x

⎛⎝⎜

⎞⎠⎟

12

∂u∂y

+∂v∂x

⎛⎝⎜

⎞⎠⎟

∂v∂y

⎛⎝⎜

⎞⎠⎟

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

Pure strain without rota8on

dudsdvds

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

=ε xx ε xyε yx ε yy

⎡

⎣⎢⎢

⎤

⎦⎥⎥

10

⎡

⎣⎢

⎤

⎦⎥ =

ε xxε yx

⎡

⎣⎢⎢

⎤

⎦⎥⎥

ds

First column in ε: rela8ve displacement vector for unit element in x-‐direc8on εyx is displacement in the y-‐direc8on of right end of unit element in x-‐direc8on

x

y ∂v∂x

=∂u∂y

11/3/15 GG611 19


ε =

∂u∂x

⎛⎝⎜

⎞⎠⎟

12

∂u∂y

+∂v∂x

⎛⎝⎜

⎞⎠⎟

12

∂u∂y

+∂v∂x

⎛⎝⎜

⎞⎠⎟

∂v∂y

⎛⎝⎜

⎞⎠⎟

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

dudsdvds

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

=ε xx ε xyε yx ε yy

⎡

⎣⎢⎢

⎤

⎦⎥⎥

01

⎡

⎣⎢

⎤

⎦⎥ =

ε xyε yy

⎡

⎣⎢⎢

⎤

⎦⎥⎥

ds

x

y

Second column in ε: rela8ve displacement vector for unit element in y-‐direc8on εxy is displacement in the x-‐direc8on of upper end of unit element in y-‐direc8on


∂v∂x

=∂u∂y

11/3/15 GG611 20

11/3/15

11


ε =

∂u∂x

⎛⎝⎜

⎞⎠⎟

12

∂u∂y

+∂v∂x

⎛⎝⎜

⎞⎠⎟

12

∂u∂y

+∂v∂x

⎛⎝⎜

⎞⎠⎟

∂v∂y

⎛⎝⎜

⎞⎠⎟

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

ε11 = εxx = elonga8on of line parallel to x-‐axis ε12 = εxy ≈ (Δθ)/2 ε21 = εyx ≈ (Δθ)/2 ε22 = εyy = elonga8on of line parallel to y-‐axis

Δθ2

= (ψ 2 – ψ 1)2

= 12

∂v∂x

+∂u∂y

⎛⎝⎜

⎞⎠⎟

Shear strain > 0 if angle between +x and +y axes decreases 11/3/15 GG611 21

Posi8ve angles are counter-‐clockwise

∂v∂x

=∂u∂y


ω (Infinitesimal Strain Matrix, 2D) Meaning of components

dudsdvds

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

≈0 −ω z

ω z 0

⎡

⎣⎢⎢

⎤

⎦⎥⎥

10

⎡

⎣⎢

⎤

⎦⎥ =

0ω z

⎡

⎣⎢⎢

⎤

⎦⎥⎥

ds x

y

First column in ω: rela8ve displacement vector for unit element in x-‐direc8on ω12 is displacement in the y-‐direc8on of right end of unit element in x-‐direc8on

Pure rota8on without strain

∂v∂x

=−∂u∂y

ω =0 1

2∂u∂y

−∂v∂x

⎛⎝⎜

⎞⎠⎟

12

∂v∂x

−∂u∂y

⎛⎝⎜

⎞⎠⎟

0

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

ωz

ωz << 1 radian

11/3/15 GG611 22

11/3/15

12


dudsdvds

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

≈0 −ω z

ω z 0

⎡

⎣⎢⎢

⎤

⎦⎥⎥

01

⎡

⎣⎢

⎤

⎦⎥ =

−ω z

0

⎡

⎣⎢⎢

⎤

⎦⎥⎥

ds

x

y

Second column in ω: rela8ve displacement vector for unit element in y-‐direc8on ω21 is displacement in the nega%ve x-‐direc8on of upper end of unit element in y-‐direc8on


∂v∂x

=−∂u∂y

ω =0 1

2∂u∂y

−∂v∂x

⎛⎝⎜

⎞⎠⎟

12

∂v∂x

−∂u∂y

⎛⎝⎜

⎞⎠⎟

0

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

ωz

ωz << 1 radian

11/3/15 GG611 23


ω11 = 0 ω12 = -‐ωz ω21 = ωz

ω22 = 0

ω z = (ψ 2 + ψ 1)2

= 12

∂v∂x

− ∂u∂y

⎛⎝⎜

⎞⎠⎟

11/3/15 GG611 24

ω =0 1

2∂u∂y

−∂v∂x

⎛⎝⎜

⎞⎠⎟

12

∂v∂x

−∂u∂y

⎛⎝⎜

⎞⎠⎟

0

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥


∂v∂x

=−∂u∂y

11/3/15

13

Principal Values (eigenvalues) and Principal Direc8ons (eigenvectors)

Mathema(cs

1  [X]’ = [F][X], where [X] is any posi8on vector and [F] is a matrix of constants

Meaning

•  [F] converts !

– We seek the lengths of the semi-‐axes of the ellipse and their direc8ons

–  The lengths of the semi-‐axes of the ellipse give the principal stretches if a unit circle is transformed to the ellipse

11/3/15 GG611 25

//


11/3/15 GG611 26

2 d(X•X) = 0 !X•dX = 0

• Max/min lengths of X are where X and its tangent(s) dX are perpendicular

Mathema(cs

Meaning

11/3/15

14


Mathema(cs

2 d(X•X) = 0 !X•dX = 0

3  [F][X] = λ[X] (Eigenvector equa8on)

4 If [F] = [F]T, a)  Xi•Xj = 0, etc.

b)  X1 || Xmax, Xn || Xmin

Meaning •  Max/min lengths of X are where X

and its tangent(s) dX are perpendicular

•  For solu8ons of (3), [X] (eigenvector) is not rotated by [F] but can be stretched, where λ is the stretch (eigenvalue)

•  If [F] is symmetric, then –  its eigenvectors are

perpendicular –  its eigenvectors are in direc8ons

of maximum/minimum X (from 2 and 4)

11/3/15 GG611 27


Mathema(cs 5 If [F] ≠ [F]T, let [B] = [F] T[F]

a)  [B] = [[F] T[F]] = [[F] T[F]] T b)  [B][X] = λ2 [X] = Q [X] (Eigenvector equa8on) c)  [F] = [R][U], where [U] = [C] ½,

[C] = [F] T [F], and [R] = [F][U]-‐1

d)  [F] = [V][R], where [V] = [B] ½, [B] = [F] [FT], and [R] = [V]-‐1[F]

Meaning •  Even if [F] is not symmetric, then

–  [B] = [F] T[F] and [C] = [F] [F] T are symmetric

–  The principal values of [F] T[F] include the maximum and minimum quadra8c elonga8ons

–  The principal direc8ons are the eigenvectors of [C]

–  F can be decomposed into the product of a symmetric stretch matrix ([U] or [V]), and a rota8on matrix [R]

11/3/15 GG611 28

11/3/15

15

Example 1: [F] is not symmetric >> F = [1 2;0 1] F = 1 2 0 1 >> [fvec,fval] = eig(F) fvec = 1.0000 -‐1.0000 0 0.0000 fval = 1 0 0 1

>> C=F'*F C = 1 2 2 5

>> U=C^(1/2) U = 0.7071 0.7071 0.7071 2.1213

>> R = F*inv(U)

R =

0.7071 0.7071 -‐0.7071 0.7071

>> [uvec,uval] = eig(U) uvec = -‐0.9239 -‐0.3827 0.3827 -‐0.9239 uval = 0.4142 0 0 2.4142

>> [cvec,cval] = eig(C) cvec = -‐0.9239 0.3827 0.3827 0.9239 cval = 0.1716 0 0 5.8284

>> principal_direc8ons = R*cvec

principal_direc8ons =

-‐0.3827 0.9239 0.9239 0.3827

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1  The two principal stretches are given by the eigenvalues of [C].

2  The eigenvectors of [F] are the vectors that [F] does not rotate.

3  The eigenvectors of [C] must be rotated to give the principal direc8ons of the ellipse.

Example 2: [F] is not symmetric

>> F = [1 2;0 1] F = 1 2 0 1 >> [fvec,fval] = eig(F) fvec = 1.0000 -‐1.0000 0 0.0000 fval = 1 0 0 1

>> B=F*F’ B= 1 2 2 5

>> V=B^(1/2) U = 0.7071 0.7071 0.7071 2.1213

>> R = inv(V)*F

R =

0.7071 0.7071 -‐0.7071 0.7071

>> [vvec,vval] = eig(V) vvec = 0.9239 -‐0.3827 0.3827 0.9239 vval = 2.4142 0 0 0.4142

>> [bvec,bval] = eig(B) bvec = 0.3827 -‐0.9239 -‐0.9239 -‐0.3827 bval = 0.1716 0 0 5.8284

>> principal_direc8ons = bvec

principal_direc8ons =

0.3827 -‐0.9239 -‐0.9239 -‐0.3827

11/3/15 GG611 30

1  The two principal stretches are given by the eigenvalues of [B].

2  The eigenvectors of [F] are the vectors that [F] does not rotate.

3  The eigenvectors of [B] give the principal direc8ons of the ellipse directly.

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16

Summary of Strain

•  Quan88es for describing strains are dimensionless •  Strain describes changes in distance between points and

changes in right angles •  Strain at a point can be represented by the orienta8on and

magnitude of the principal stretches •  Symmetric strains and stress are symmetric: eigenvalues

are orthogonal and do not rotate •  Asymmetric strain matrices involve rota8on •  Infinitesimal strains can be superposed linearly •  Finite strains involve matrix mul8plica8on •  The same final deforma8on can be achieved by different

paths

11/3/15 GG611 31

Documents

GG611 2015 Lecture 3 v3 · 2015. 11. 4. · 11/3/15 1 GG612 Lecture%3% Strain% 11/3/15 GG611 1 Outline% • Mathemacal%Operaons% • Strain% – General%concepts% – Homogeneous%strain%