Post on 09-Jun-2019
Stress, Strain, & Elasticity
Mostly from Dieter
Stress
• Nine quantities are required to define the state of stress at a point.
• Moment balance shows; tij = tji
• Six independent quantities
zxyzxyzyx ,,,,,
zyzzx
yzyxy
zxxyx
Plane Stress
• 2-D state of stress
• Approached when one dimension of the body is relatively small (example: thin plates loaded in the plane of the plate)plates loaded in the plane of the plate)
• Plane stress when s3 = 0
• Only three stresses are required
xyyx and ,,
Principal Stresses• For any state of stress, we can find a set of
planes on which only normal stresses act and the shearing stresses are zero.
• Called Principal Planes and the normal stresses acting on these planes are Principal stresses acting on these planes are Principal Stresses denoted as s1, s2 and s3
• Convention, s1> s2 > s3
• The principal directions are orthogonal to each other
2/122
2/122
4
2/2cos
42sin
xyyx
yx
xyyx
xy
Orientation
Magnitude2/1
22
2min
1max
22
xyyxyx
Magnitude
221
max
State of stress in 3-D
• If , triaxial state of stress
• If , cylindrical state of stress
0321
321 • If , cylindrical state of stress
• If , hydrostatic state of stress
321
321
Invariants
2223
2222
1
2 xyzzxyyzxzxyzxyzyx
zxyzxyxzzyyx
zyx
I
I
I
The sum of normal stresses for any orientation in the coordinate system is equal to the sum of thenormal stresses for any other orientation
Stress Tensor• Stress is a second-rank tensor quantity• Vector is a first-rank tensor quantity
jiji SaS '
1131211
'1 SaaaS
• A scalar, which remains unchanged with transformation of axes, is a zero-rank tensor
• No. components =3n n –tensor-rank
3
2
1
333231
232221
131211
'3
'2
1
S
S
S
aaa
aaa
aaa
S
S
S
Kronecker delta, dij
• A second-rank unit isotropic tensor
ji
1
010
001
• If we multiply a tensor of nth rank with dij, the product tensor will have (n-2)th rank
ji
jiij
0
1
100
010
• Components of a stress tensor, sij
• Stress is a symmetric tensor
• First invariant of the stress tensor, I1
I1 is a scalar
• Second Invariant, I2, is the sum of principal minors
• Third Invariant, I3, is the determinant of the matrix.
• The three invariants are given by the roots of the following equation.
2
2
20
02
yxxy
xyyx
yx
yx
yxy
xyx
• Hydrostatic stress is given by
• Decomposition of the stress tensor:
• J1, J2, and J3 are the principal values of the deviatoric stress tensor.
• J1 is the sum of the diagonal terms:
• J2 is the sum of the principal minors:
0)()()(1 mzmymxJ
)()()(1222
yxxzzy
• J3 is the determinant of the deviatoric stress matrix.
)(6
)()()(
61
2222
xzyzxy
yxxzzyJ
Strain
• A point can be displaced by translation, rotation, and deformation.
• Deformation can be made up of • Deformation can be made up of – dilatation---change in volume
– distortion-change in shape
Displacement, ui
Q(x,y,z) to Q’(x+u, y+v, z+w)
u =f(u, v, w)
1-D Strain
uAB
ABBA
L
Lex
''
x
udx
dxdxxu
dx
u = exx
Generalization to 3-D
• For 1-D, u = exx
• Generalizexzxyxx zeyexeu
jiji
zzzyxz
yzyyyx
xzxyxx
xeu
zeyexew
zeyexev
zeyexeu
eij= ui/ xj
Displacement Tensor
Produces both shear strain and rigid body rotation
• Is eij a satisfactory measure of the strain?
• If yes, eij=0 when there is no distortion
• Consider a rigid body rotation
0
0
ije
Decomposition• eij needs to be decomposed into shear strain and
rigid-body rotation
• Any second-rank tensor can be decomposed into a symmetric and an anti-symmetric tensor.
e
jiijijjiijij
ijijij
eeandee
e
2
1
2
1
Strain Tensor Rotation Tensor
Generalized Displacement Relation: jijjiji xxu
Principal Strains• Similar in concept to principal stresses• Can identify, principal axes along which there
are no shear strains or rotations, only pure extension or contraction.
• For isotropic solids, principal strain axes • For isotropic solids, principal strain axes coincide with the principal stress axes
• Definition of principal strain axes: Three mutually perpendicular directions in the body which remain mutually perpendicular during deformation.
• Remain unchanged if and only if vij =0
Dilatation, D• Volume change or dilatation
• Note D is the first invariant of the strain tensor
• Mean Strain, e = D/3
1'
1)1)(1)(1(
321
321
sfor
• Mean Strain, em = D/3
• Strain deviator, eij, is the part of the strain tensor that represents shape change at constant volume
ijijmijíj 3
'
Engineering Shear Strains
h
Shear Strain, g = a/h = tanq ~ q
h
Simple Shear
+Rotation
=
Pure Shear
xy
yxxy
yxxyxy ee
2
Tensor shear strains
Elasticity (for isotropic solids)• Equations that relate stresses to strains are
known as “Constitutive Equations”
• Hooke’s law: sx=Eex
• Poisson’s Relation:E
xxzy
zxzxyzzyxyxy GGG ;;
Need only two elastic constants, E and n
)1(2
EGShear Modulus,
ijkkijij EE
1
Other Elastic Constants
• Bulk Modulus,
p
K m
E
K
• Compressibility, b = 1/K
• Lamé’s Constant,
)21(3
EK
)21(
2
G
Inversion
ijijE
ijkkijij EE
1
E
ijijmijíj 3
'
ijij 1
ijij G 2
kkii K 3
Distortion:
Dilatation:
)1(2
EG
p
K m
• Plane Stress (s3=0):
• Plane Strain (e =0):
1222
2121
1
1
E
E
2133 01 E• Plane Strain (e3=0):
213
2133 0
E
12
22
212
1
)1(11
)1(11
E
E
Strain Energy• Elastic strain energy, U = energy spent by the
external forces in deforming an elastic body
• dU=0.5P du = 0.5(sxA)(exdx) = 0.5(sxex)Adx
• Strain Energy/vol., 1 22
xx EU
• Strain Energy/vol., 222
10
xxxx
E
EU
ijijzxzxyzyzxyxy
zzyyxxU
2
1
2
10
1
zxzx
yzzy
xyxy
G
G
G
222
2220
2
12
1
zxyzxy
xzzyyxzyx
G
EEU
ijij
U
0
Atomistic Aspects(from Ashby and Jones)
• E is influenced by two factors(a) the interatomic bonds spring constant(b) the packing of atoms no. of springs
• Different types of Bonds– Primary: Metallic, ionic, and covalent– Primary: Metallic, ionic, and covalent
Strong– Secondary: van der Waals and Hydrogen
Weak
Ionic Bondni
r
B
r
qUrU
0
2
4)(
Attractive
part
Repulsivepart
q-chargee -permitivity e0-permitivity of vacuumn~12
Lacks Directionality
Covalent Bond )()( nmr
B
r
ArU
nm
Highly Highly directional
Metallic bond issimilar
InteratomicForces
dr
dUF ,Force
0rrF for small displacements
2
2
dr
Ud
dr
dFS Stiffness,
When stretching is small, S is a constant
2
0
2
2
0rrdr
UdS
Spring Constant
of the Bond
00 rrSF
00 rrNS
0
0
r
rrn
No. of bonds/area, 20/1 rN
0r
0
0
r
SE
n
Generalized Hooke’s Law
• Fourth-rank tensors (81 components)
• Symmetry: s = s and e = e
klijklij
klijklij
C
S
Sijkl – Compliance Tensor
Cijkl – Stiffness Tensor
(from Nye: Physical Properties of Crystals)
• Symmetry: sij = sji and eij = eji
ijlkijkllkijlkklijkl
lkijlkijklijklij
SSSS
SS
Reduces the no. independent components from 81 to 36
Contracted Notation
345
426
561
332313
232212
131211
56111
345
426
561
332313
232212
131211
2
1
2
12
1
2
122
666564636261
565554535251
464544434241
363534333231
262524232221
161514131211
CCCCCC
CCCCCC
CCCCCC
CCCCCC
CCCCCC
CCCCCC
666564636261 CCCCCC
Energy consideration Cmn=Cnm
Reduces the no. of independent constants to 21Possible to reduce no. of independent constants further by considering crystal symmetry
nmnm S Tensor notation 11 22 33 23, 32 31, 13 12, 21
Matrix notation 1 2 3 4 5 6
Sijkl = Smn when m and n are 1, 2, or 3 Sijkl = Smn when m and n are 1, 2, or 3 2Sijkl = Smn when either m or n is 4, 5, or 64Sijkl = Smn when both m and n are 4, 5, or 6
S1111 = S11
2S1123 = S14
4S2323 = S44
Example
• Measures extension in Ox3 direction when the crystal is sheared about the Ox direction
S34 in orthorhombic crystal
Possible to reduce no. of independent constants further by considering crystal symmetry
crystal is sheared about the Ox1 direction
• Operate a diad axis parallel to Ox2 direction
• Crystal remains unaltered because of symmetry
• So does extension parallel to Ox3, now under reverse forces
• Implies that S34 has to be necessarily equal to zero
33
Cubic Crystals• Let Ox, Oy, and Oz be parallel to [100],
[010] and [001], respectively.
• Rotate by 90º. The crystal will look the same.
332211 CCC
665544
312312
332211
CCC
CCC
CCC
Remaining constants vanish
12111211
1212
12111211
121111
2
2
SSSS
SC
SSSS
SSC
4444
12111211
1
SC
Isotropic Solids
Form of the matrix can be obtained from thecubic matrix by requiring that the componentsshould be unaltered by a 45º rotation
GS
ES
ES
11441211
12112 SSX
Composites—Isostrain AnalysisCompatibility:
P
EEE m
m
f
f
c
c
mfc
fAAfAA
PPP
EA
P
EA
P
EA
P
cmcf
mfc
mm
m
ff
f
cc
c
1/;/
mfc EffEE )1( Rule-of-Mixtures
Isostress Ananlysis
mfc
mfc
mfc
E
f
E
f
E
ff
)1(
)1(
mfc EEE
fm
mfc EffE
EEE
)1(
Upper and Lower Bounds