gg611 2015 lecture 3 v3 · 2015. 11. 4. · 11/3/15 1 gg612 lecture%3% strain% 11/3/15 gg611 1...
TRANSCRIPT
11/3/15
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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
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Main Theme
• Representa8on of strain at a point in a clear concise manner
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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
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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
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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’
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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
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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)
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′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
⎡
⎣⎢⎢
⎤
⎦⎥⎥
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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.
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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
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If deriva8ves are constant, then du, dv, dx, and dy can be replaced by u, v, x, and y, respec8vely.
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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
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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
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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
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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
⎛⎝⎜
⎞⎠⎟
⎡
⎣
⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥
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
Pure strain without rota8on
∂v∂x
=∂u∂y
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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
⎛⎝⎜
⎞⎠⎟
⎡
⎣
⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥
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
Pure strain without rota8on
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
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ω (Infinitesimal Strain Matrix, 2D) Meaning of components
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
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 23
ω (Infinitesimal Strain Matrix, 2D) Meaning of components
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
⎡
⎣
⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥
Pure rota8on without strain
∂v∂x
=−∂u∂y
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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
//
Principal Values (eigenvalues) and Principal Direc8ons (eigenvectors)
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
Principal Values (eigenvalues) and Principal Direc8ons (eigenvectors)
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
Principal Values (eigenvalues) and Principal Direc8ons (eigenvectors)
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
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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
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