1 第二讲 固体力学基础 张俊乾 [email protected] 近代力学基础. 2 contents 1.stress...
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Contents
1. Stress and Kinetics
2. Strain and Kinematics
3. Constitutive Models for Materials
4. Material Failure
5. Boundary and Initial Value Problems
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Stress vector at a point
Stress and Kinetics
SSn
nF
0
lim :stress Normal
Uniformly distributed Stress
Normal stress: F
A
Dimension of stress: [force] / [length]2 = N / m2 (Pa) = 10-6 MPa
Non-uniformly distributed Stress
force ：F
Outward normalOutward normal
SS
F
0
lim :stressShear
SS
Ft n
0
lim : vector)(StressTraction )(
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Stress tensor at a point
Stress and Kinetics
zxzyxyxxxx iiit )(
zyzyyyxyxy iiit )(
zzzyzyxzxz iiit )(
Stress vectors on the plane perpendicular to x-axis, to y-axis, to y-axis, respectively :
xx xy xz
yx yy yz
zx zy zz
σStress tensor:
1. The first subscript indicates the direction of the plane normal upon the stress acts, the second subscript the direction of stress component.
2. Positive stress rule: The directions of stress component and of the plane are both positive, or both negative.
1. The first subscript indicates the direction of the plane normal upon the stress acts, the second subscript the direction of stress component.
2. Positive stress rule: The directions of stress component and of the plane are both positive, or both negative.
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Cauchy’s formulaStress vectors on the plane with normal vector n
Remark: Cauchy’s formula assures us that the nine components of stress are necessary and sufficient to define the traction on any surface element across a body. Hence the stress state in a body is characterized completely by the set of stress tensor σ
Remark: Cauchy’s formula assures us that the nine components of stress are necessary and sufficient to define the traction on any surface element across a body. Hence the stress state in a body is characterized completely by the set of stress tensor σ
( )
( )
( )
x xx yx zx
y xy yy zy
z xz yz zz
t l m n
t l m n
t l m n
n
n
n
zzyyxx ttt iiit nnnn) )()()((
( ) n Tt σ n
Relationship: stress vector - stress tensor
),cos( xl in ),cos( ym in ),cos( zn in Stress tensor σ is symmetric
Stress and Kinetics
6
Index notation and transformation of coordinates
Coordinates: ),,(),,( 321 iiiiii zyx
Subscript: (x, y, z) Subscript: (x, y, z) index (1, 2, 3)
Stress vector: (tx, ty, tz)
11 12 13
21 22 23
31 32 33
xx xy xz
yx yy yz
zx zy zz
Stress tensor:
(t1, t2, t3)
Summation convention: Summation convention:
jiii jiii
ii ttt
3
1
The summation is implied by the repeated index, called dummy index. Use of any other index instead of i does not change the meaning.
Stress and Kinetics
7
Index notation and transformation of coordinates
Transformation of stress vector: Transformation of stress vector:
lkkllkll
lklk
lklll
klk
QQQ
xQxQx
iiiii
'3
1
'
3
1
'
,
pqjqipij QQ '
Transformation of coordinates: Transformation of coordinates:
1 11 1 12 2 13 3
2 21 1 22 2 23 3
3 31 1 32 2 33 3
i ij j
t Q t Q t Q t
t Q t t Q t Q t Q t
t Q t Q t Q t
Transformation of stress tensor: Transformation of stress tensor:
Stress and Kinetics
8
Equations of motion (equilibrium)
3-dimensional3-dimensional
3111 211 1
1 2 3
3212 222 2
1 2 3
13 23 333 3
1 2 3
(=0)
(=0)
(=0)
f vx x x
f vx x x
f vx x x
( 0)jii i
j
f vx
2-dimensional 2-dimensional
11 211 1
1 2
12 222 2
1 2
(=0)
(=0)
f vx x
f vx x
1211
2122
x1
x2
O
f1
f2
2121 2
2
dxx
1212 1
1
dxx
1111 1
1
dxx
2222 2
2
dxx
ij ji Symmetry:Symmetry:
--- derived from the linear momentum balance
or Newton’s second law of motion
--- derived from the linear momentum balance
or Newton’s second law of motion
--- derived from the angular momentum balance --- derived from the angular momentum balance
Stress and Kinetics
9
Strain tensor (deformation measure)
S
yx
z
P
'S
'Pu
rr
Strain and Kinematics
Displacement vector: u r r
'
'1 1 1 2 2 2 3 3 3
or
u x x v y y w z z
u x x u x x u x x
3 components:
x
ux
y
vy
z
wz
x
v
y
uxy
z
v
y
wyz
z
u
x
wxz
Strain-displacement relationship:
i
j
j
iij x
u
x
u
2
1
11 22 33 23 13 12, , , 2 , 2 , 2 , , , , ,x y z yz xz xy
,,
ororNormal strainNormal strain
Shear strainShear strain
The change of volume (volumetric strain): The change of volume (volumetric strain): x y z kk
x
y
Ou
u dxx
P
A
B
P A
dx
B
dy
u
v vv dx
x
uu dy
y
vv dy
y
dy
vdyyvv
yvy
dx
udxxuu
xux
y
u
x
vxy
yu
dy
udyyuu
tan
tan
xv
dx
vdxxvv
xy
Distortion of the right angle between two lines (Shear strain)：
Elongation of PA (Normal strain):
Elongation of PB (Normal strain):
Properties of Strain tensor
Strain and Kinematics
11
Transformation of coordinates
lkkllkll
lklk
lklll
klk
QQQ
xQxQx
iiiii
'3
1
'
3
1
'
,
'ij ip jq pqQ Q
Transformation of coordinates: Transformation of coordinates:
Transformation of strain tensor: Transformation of strain tensor:
Strain and Kinematics
Constitutive Model : Isotropic, linear elastic materials
Uniaxial tension: Uniaxial tension:
E
e1
e2
e3
Thermoelastic constitutive equations in multiaxial-stress state: Thermoelastic constitutive equations in multiaxial-stress state:
Typical materials: polycrystalline metal , polymers and concrete etc.Typical materials: polycrystalline metal , polymers and concrete etc.
Young’s modulusYoung’s modulus
Poisson’s ratioPoisson’s ratio
Coefficient of thermal expansionCoefficient of thermal expansion
Constitutive Model : Anisotropic linear elastic materials
Stiffness matrixStiffness matrix
compliance matrix compliance matrix
Coefficient of thermal expansionCoefficient of thermal expansion
Strain energy density:Strain energy density:
ijij
U
Constitutive Model : Linear elastic orthotropic materials
T σ C ε α
T ε S σ α
1
2
3
0
0
0
α
9 independent elastic constants;
3 CTE constants
9 independent elastic constants;
3 CTE constants
Constitutive Model : Transversely isotropic materials
T ε S σ α
T σ C ε α
1
1
3
0
0
0
α
5 independent elastic constants;
2 CTE constants
5 independent elastic constants;
2 CTE constants
Constitutive Model : Rate independent plasticity
Unloading
Stress
Strain
LinearElastic
PermanentStrain
Hold atconstant strain
Hold atconstant stress
Features of the inelastic response of metals Features of the inelastic response of metals
Decomposition of strain into elastic and plastic parts:Decomposition of strain into elastic and plastic parts:
E
p e
Uniaxial loading: e p
Yield: If the stress exceeds a critical magnitude, the stress-strain curve ceases to be linear.
Bauschinger effect: If the specimen is first deformed in compression, then loaded in tension, it will generally start to deform plastically at a lower tensile stress than an annealed specimen.
Yield: If the stress exceeds a critical magnitude, the stress-strain curve ceases to be linear.
Bauschinger effect: If the specimen is first deformed in compression, then loaded in tension, it will generally start to deform plastically at a lower tensile stress than an annealed specimen.
Multiaxial loading: e pij ij ij
Constitutive Model : Rate independent plasticity
Yield Criteria Yield Criteria
are the components of the `von Mises effective stress’ and `deviatoric stress tensor’ respectively.are the components of the `von Mises effective stress’ and `deviatoric stress tensor’ respectively.
1. A hydrostatic stress (all principal stresses equal) will never cause yield, no matter how large the stress;
2. Most polycrystalline metals are isotropic.
1. A hydrostatic stress (all principal stresses equal) will never cause yield, no matter how large the stress;
2. Most polycrystalline metals are isotropic.
( , ) ( ) 0p pij ef Y
ij
ijd
Y
Constitutive Model : Rate independent plasticity
Y
p
Y0h
Y
pY0
Y
p
Isotropic hardening model Isotropic hardening model
Perfectly plastic solid: Perfectly plastic solid: Linear strain hardening solid Linear strain hardening solid Power-law hardening material Power-law hardening material
Constitutive Model : Rate independent plasticity
Plastic flow law
Plastic flow law
is the slope of the plastic stress-strain curve. is the slope of the plastic stress-strain curve.
Complete incremental stress-strain relations Complete incremental stress-strain relations
Constitutive Model : Rate independent plasticity
Material Yield Stress (MPa)
Material Yield Stress (MPa)
Tungsten Carbide 6000 Mild steel 220
Silicon Carbide 10 000 Copper 60
Tungsten 2000 Titanium 180 - 1320
Alumina 5000 Silica glass 7200
Titanium Carbide
4000 Aluminum & alloys
40-200
Silicon Nitride 8000 Polyimides 52 - 90
Nickel 70 Nylon 49 - 87
Iron 50 PMMA 60 - 110
Low alloy steels 500-1980 Polycarbonate 55
Stainless steel 286-500 PVC 45-48
Constitutive Model : Rate independent plasticity
Typical values for yield stress of some materials Typical values for yield stress of some materials
Constitutive Model : Viscoplasticity
Primarycreep
Secondarycreep
Tertiarycreep
Time
Increasingstress
Features of creep behavior (constant stress)Features of creep behavior (constant stress)
Features of high-strain rate behavior Features of high-strain rate behavior10000
8000
6000
4000
2000Shea
r St
ress
(M
Pa)
Shear strain rate (s-1)10 -2 10 0
10 2 10 4 10 6
1. If a tensile specimen of a solid is subjected to a time independent stress, it will progressively increase in length.
2. The length-time plot has three stages
3. The rate of extension increases with stress
4. The rate of extension increases with temperature
1. If a tensile specimen of a solid is subjected to a time independent stress, it will progressively increase in length.
2. The length-time plot has three stages
3. The rate of extension increases with stress
4. The rate of extension increases with temperature
1. The flow stress increases with strain rate
2. The flow stress rises slowly with strain rate up to a strain rate of about 106 , and then begins to rise rapidly.
1. The flow stress increases with strain rate
2. The flow stress rises slowly with strain rate up to a strain rate of about 106 , and then begins to rise rapidly.
Constitutive Model : Viscoplasticity
Flow potential for creep: Flow potential for creep:
Flow potential for High strain rate:Flow potential for High strain rate:
Strain rate decomposition: Strain rate decomposition:
Plastic flow rule: Plastic flow rule:
Material Failure : Introduction
The mechanisms involved in fracture or fatigue failure are complex, and are influenced by material and structural features that span 12 orders of magnitude in length scale, as illustrated in the picture below
The mechanisms involved in fracture or fatigue failure are complex, and are influenced by material and structural features that span 12 orders of magnitude in length scale, as illustrated in the picture below
10-10 10-3 10-1 102
Atoms Microstructure Defects Testing
10-6
Applications
Continuum Mechanics
m m m m m
Material Failure : Mechanisms
Brittle Ductile
Failure under monotonic loadingFailure under monotonic loading
Brittle1. Very little plastic flow occurs in the specimen prior to failure
2. The two sides of the fracture surface fit together very well after failure
3. In many materials, fracture occurs along certain crystallographic planes. In other materials, fracture occurs along grain boundaries
Brittle1. Very little plastic flow occurs in the specimen prior to failure
2. The two sides of the fracture surface fit together very well after failure
3. In many materials, fracture occurs along certain crystallographic planes. In other materials, fracture occurs along grain boundaries
Ductile1. Extensive plastic flow occurs in the material prior to fracture
2. There is usually evidence of considerable necking in the specimen
3. Fracture surfaces don’t fit together
4. The fracture surface has a dimpled appearance, you can see little holes, often with second phase particles inside them.
Ductile1. Extensive plastic flow occurs in the material prior to fracture
2. There is usually evidence of considerable necking in the specimen
3. Fracture surfaces don’t fit together
4. The fracture surface has a dimpled appearance, you can see little holes, often with second phase particles inside them.
Material Failure : Mechanisms
Failure under cyclic loadingFailure under cyclic loading
max
min
m
t
a
Endurance limitFatigue limit
High cycle fatigue
Low cycle fatigue
1. S-N curve normally shows two different regimes of behavior, depending on stress amplitude
2. At high stress levels, the material deforms plastically and fails rapidly. In this regime the life of the specimen depends primarily on the plastic strain amplitude, rather than the stress amplitude. This is referred to as `low cycle fatigue’ behavior
3. At lower stress levels life has a power law dependence on stress, this is referred to as `high cycle’ fatigue behavior
4. In some materials, there is a clear fatigue limit, if the stress amplitude lies below a certain limit, the specimen remains intact forever. In other materials there is no clear fatigue threshold. In this case, the stress amplitude at which the material survives 108 cycles is taken as the endurance limit of the material. (The term `endurance’ appears to refer to the engineer doing the testing, rather than the material)
1. S-N curve normally shows two different regimes of behavior, depending on stress amplitude
2. At high stress levels, the material deforms plastically and fails rapidly. In this regime the life of the specimen depends primarily on the plastic strain amplitude, rather than the stress amplitude. This is referred to as `low cycle fatigue’ behavior
3. At lower stress levels life has a power law dependence on stress, this is referred to as `high cycle’ fatigue behavior
4. In some materials, there is a clear fatigue limit, if the stress amplitude lies below a certain limit, the specimen remains intact forever. In other materials there is no clear fatigue threshold. In this case, the stress amplitude at which the material survives 108 cycles is taken as the endurance limit of the material. (The term `endurance’ appears to refer to the engineer doing the testing, rather than the material)
Material Failure : Stress and strain based failure criteria
Failure criteria for isotropic materials: Failure criteria for isotropic materials:
Tsai-Hill criterion for brittle fiber-reinforced composites and wood: Tsai-Hill criterion for brittle fiber-reinforced composites and wood:
..
e1e2
Ductile Fracture Criteria: Ductile Fracture Criteria:
Criteria for failure by low cycle fatigue :Criteria for failure by low cycle fatigue :
Criteria for failure by high cycle fatigue: Criteria for failure by high cycle fatigue: