11 Elements of Solid Mechanics Zhang Junqian jqzhang2@shu.edu.cn

Elements of Solid Mechanics

Zhang Junqian

jqzhang2@shu.edu.cn

Contents

1. Stress and Kinetics

2. Strain and Kinematics

3. Constitutive Models for Materials

4. Material Failure

5. Boundary and Initial Value Problems

Stress vector at a point

Stress and Kinetics

lim :stress Normal

Uniformly distributed Stress

Normal stress: F

Dimension of stress: [force] / [length]2 = N / m2 (Pa) = 10-6 MPa

Non-uniformly distributed Stress

force ：F

Outward normalOutward normal

lim :stressShear

lim : vector)(StressTraction )(

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.

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

x xx yx zx

y xy yy zy

z xz yz zz

t l m 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

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

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

Index notation and transformation of coordinates

Transformation of stress vector: Transformation of stress vector:

lkkllkll

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

Equations of motion (equilibrium)

3-dimensional3-dimensional

3111 211 1

3212 222 2

13 23 333 3

f vx x x

( 0)jii i

2-dimensional 2-dimensional

11 211 1

12 222 2

f vx x

2121 2

1212 1

1111 1

2222 2

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

Strain tensor (deformation measure)

Strain and Kinematics

Displacement vector: u r r

'1 1 1 2 2 2 3 3 3

u x x v y y w z z

u x x u x x u x x

3 components:

Strain-displacement relationship:

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

v vv dx

vdyyvv

udxxuu

udyyuu

vdxxvv

Distortion of the right angle between two lines (Shear strain)：

Elongation of PA (Normal strain):

Elongation of PB (Normal strain):

Properties of Strain tensor

Transformation of coordinates

lkkllkll

'ij ip jq pqQ Q

Transformation of coordinates: Transformation of coordinates:

Transformation of strain tensor: Transformation of strain tensor:

Constitutive Model : Isotropic, linear elastic materials

Uniaxial tension: Uniaxial tension:

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:

Constitutive Model : Linear elastic orthotropic materials

T σ C ε α

T ε S σ α

9 independent elastic constants;

3 CTE constants

Constitutive Model : Transversely isotropic materials

T ε S σ α

T σ C ε α

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:

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

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

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

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

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

Typical values for yield stress of some materials Typical values for yield stress of some materials

Constitutive Model : Viscoplasticity

Primarycreep

Secondarycreep

Tertiarycreep

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

2000Shea

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

10-10 10-3 10-1 102

Atoms Microstructure Defects Testing

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

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:

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:

6 components of stress:

, , , , , ,ij xx yy zz xy yz zx

3 components of displacements: , , ,iu u v w

6 components of strain:

, , , , , ,ij x y z xy yz zx

15 unknown mechanical variables 15 unknown mechanical variables

Boundary Value Problems: Basic equations

Equations of equilibrium

Boundary Value Problems: Basic equations

Strain-displacement relations

Constitutive relations

or ij ijkl kl ij ij ijkl kl klS T C T

Thermoelastic:Thermoelastic:

Plastic:Plastic:

on i i uu u S

Boundary conditions

on ij j in t S

15 field equations 15 field equations

Boundary Value Problems: Boundary conditions

on i i uu u SBoundary conditions on ij j in t S

1. The displacement boundary condition and traction boundary condition are mutually exclusive. Either displacement or traction is specified on the boundary. They can not specified simultaneously.

2. A boundary may be subjected to a combination of displacement and traction (“mixed”) boundary conditions, in other words, displacement boundary conditions in some directions may be given whereas the traction boundary conditions in remaining directions are specified.

3. If you are solving a static problem with only tractions prescribed on the boundary, you must ensure that the total external force acting on the solid sums to zero (otherwise a static equilibrium solution cannot exist).

1. The displacement boundary condition and traction boundary condition are mutually exclusive. Either displacement or traction is specified on the boundary. They can not specified simultaneously.

2. A boundary may be subjected to a combination of displacement and traction (“mixed”) boundary conditions, in other words, displacement boundary conditions in some directions may be given whereas the traction boundary conditions in remaining directions are specified.

3. If you are solving a static problem with only tractions prescribed on the boundary, you must ensure that the total external force acting on the solid sums to zero (otherwise a static equilibrium solution cannot exist).

Examples:Examples:

Saint-Venant Principle

若把物体的一小部分边界上的面力，变换为分布不同但静力等效的面力，则近处的应力分布将有显著改变，而远处所受的影响可忽略不计。

P/2 P/2

Boundary Value Problems: Boundary conditions

Boundary Value Problems: Interfacial conditions

Perfect interface:Perfect interface:

Two materials jointed together Two materials jointed together

1 2 1 0ij ij jn

Interface crack (debonding):Interface crack (debonding): 1 1 2 20, 0ij j ij jn n

jiji uKt Spring-like interface:Spring-like interface:

Navier’s equations

Boundary Value Problems: in terms of displacements

3 field equations 3 field equations

0 for anisotropic materialskijkl i

( ) 0 for isotropic materialsk ii

i k j j

u uG G f

x x x x

Papkovich–Neuber’s solution (without body force)

4 harmonic functions4 harmonic functions

k kx x

4(1 ) 4(1 )

i i k ki

0 k kx x

Boussinesq problem

Ψ log( )B r z

(1 )(1 2 )

P xz xu

E r r r z

(1 )(1 2 )

P yz yv

E r r r y

(1 ) 12(1 )

2 2 2r x y z

Boundary conditions:Boundary conditions: 0, 0, 0, for 0 but 0, 0zz xz yz z x y

, for constantzzdxdy P z

5 3 2 2 3 2

3 (2 ) 2 2(1 2 )

2 ( ) ( ) ( )x

P x x xz r z x x

r r r z r r z r r z

5 3 2 2 3

3 (2 ) 2(1 2 )

2 ( ) ( )y

P xy xy xz r z xy

r r r z r r z

5 3 2 2 3 2

3 2(1 2 )

2 ( ) ( ) ( )xy

P x y x y x y y

r r r z r r z r r z

Cerruti’s problem

2 2 2r x y z

Boundary conditions:Boundary conditions: 0, 0, 0, for 0 but 0, 0zz xz yz z x y

, for constantzxdxdy P z

( ) ( ) ( )

y Bxy xA C

r z r r z r r z r

(1 )1 (1 2 )

P x r xu

Er r r z r z

(1 ) 1 1(1 2 )

Er r r z

(1 ) 1(1 2 )

Er r r z

5 3 2 2 3 2

3 (2 ) 2 2(1 2 )

2 ( ) ( ) ( )x

P x x xz r z x x

r r r z r r z r r z

5 3 2 2 3

3 (2 ) 2(1 2 )

2 ( ) ( )y

P xy xy xz r z xy

r r r z r r z

5 3 2 2 3 2

3 2(1 2 )

2 ( ) ( ) ( )xy

P x y x y x y y

r r r z r r z r r z

Flat punch indenting a half-space

Boundary conditions:Boundary conditions:

1 ( , )

( ) ( )

p x yh dx dy

E x x y y

2 2, for and 0w h x y a z

2 20, for and 0zz x y a z

0, 0, for 0xz yz z

( , ) ( , ,0)zp x y x ydistributed pressure:distributed pressure:

Governing eqaution:Governing eqaution:22

2 20 0

1 ( ) for

( 2 cos

a ph d d r a

2 2 2( )

Solution:Solution:

Plane strain

Boundary Value Problems: 2-dimensional

btat ,

0, 0, 0z zy yz zx xz

Plane stress

0, 0, 0z zx zy

8 field variables: , , , ( and 1,2)u

Plane strain

Plane stress

Constitutive equations for isotropic elasticityConstitutive equations for isotropic elasticityConstitutive equations for isotropic elasticityConstitutive equations for isotropic elasticity

Rectangular coordinates

Polar coordinates

xy x y

r rr r

r r r r r

Airy Function

Polynomial of degree Polynomial of degree 22 Y

2A22 2

0 1 2A x A xy A y

22Axx 02Ayy

Chapter 5.424

Airy Function: Polynomials

Polynomial of degree Polynomial of degree 33

3 2 2 30 1 2 3A x A x y A xy A y

yAxAyy 10 26 yAxAxy 21 22 yAxAxx 32 62

0,0 3210 AAAA

Pure bending Pure bending

yAxx 36

0 yyxy

Chapter 5.4

Lateral Bending of a Slender Rectangle

BCs :BCs :

0xxFdy

0yy 0xy

Chapter 5.426

Fyx y b

A Hole Under Remote Shear

Chapter 5.548

BCs : BCs :

arrrr at 0

2 222 24 sin 2

2r A A r

2sin64

Arr 2cos

Ar 2sin

Chapter 5.5

StressesStresses

2sin34

2cos32

ar 2sin

Along the rim of the holeAlong the rim of the hole 2sin4ar

The maximum hoop stressThe maximum hoop stress 4

A Hole Under Remote Shear

A Circular Hole Under Tension

Chapter 5.554

BCs : BCs :

arrrr at 0

2 21 cos 2 ln cos 2 cos 24

r A r B Cr

Pure Bending of Curved Beams

bbaabN 222222 ln43

2 2 ln lnAr Br r C r D

0, for and rr r r a b Mrrb

1ln22r

CrBArr 0r

3ln22r

Boundary conditions:Boundary conditions: Weak formWeak form

aabbabN

MA ln2ln2 2222

A curved beam loaded by a transverse force

3 1 ln sin cosAr Br Cr r Dr F

bar ,on 0 rrrboundary conditions

r d0d b

rr 0on

sin222 113 DrCrBrArrr cos22 13 CrBrArr

sin26 13 CrBrAr

bbabaN ln2222

1 2 3 4sin cos ln cos ln sinC r C r C r r C r r

sincossin2cos2 43211 CCCCrrr

cossin 431 CCrr

sincos 431 CCr

, ,0 r 043 CC

Stresses:Stresses:

BCs:BCs:

0dcossincos2 21

0dsinsincos2 21

sin2 P

sinsin

coscos2

Chapter 6.322

Prr 0 r oror

11 Elements of Solid Mechanics Zhang Junqian jqzhang2@shu.edu.cn

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