Stress Strain Elasticity

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II. Stress, Strain and Elasticity

Revision

Arcady Dyskin

A.V. Dyskin. Geomechanics Group, UWA

Plan

Elasticity

Boundary conditions

Wave propagation

A.V. Dyskin. Geomechanics Group, UWA Slide 2


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Learning objectivesUnderstanding stress and strain matrices, meaning

of components, equilibrium and compatibility

Change of components with coordinatetransformation, principal stress /strain, principalaxes, spherical and deviatoric parts

Assumptions of elasticity, generalised Hookes law

Isotropic and anisotropic rocks

Plane stress and lane strain


Elastic energy

Boundary conditions, the use of symmetry

Plane and Rayleigh waves

Stress and strain Stress: Definition and equations of equilibrium

tra n: e n t on an equat ons o compat ty

Co-ordinate rotations

Principal stress/strain and principal directions Spherical/deviatoric decomposition

Plane stress and lane strain


Mohrs circle

Cylindrical and spherical co-ordinates


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Stress: Definition and equations ofequilibrium

.

Surface forces. Stress

Stress matrix

Force equilibrium


omen equ r um

Body forces. Intensity of body forces Force is an abstract notion representing the action of other

bodies on the given one. Body forces

Surface forces

Body forces act on all points of the given bodyrepresenting the remote action of other bodies

Intensity



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Surface forces. Stress Surface forces represent the action of other bodies on the surface of

the given one by means of direct contact. Intensity of surface forcesis called stress .


Stress components are labelled by 2 indexes: the first refers to thedirection of the surface element; the second to the component offorce.

Internal stress state

Stress


matrix

zzyzx

yzyyx

xzxyx

zzzyzx

yzyyyx

xzxyxx


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Convenient index notations33

x3

22

11

31

21

2312

13

x2

333231

232221

131211


x1

9

Sign conventions


Rock Mechanics convention


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Stress on a surface element

z

y

x

F=(Fx, Fy, Fz)

n=(nx, ny, nz)

A

SF

An n n

SF

An n n

SF

An n n

xx

x x y yx z zx

yy

x xy y y z zy

zz

x xz y yz z z


Stress on a surface element in new notations

3232221212

2

3132121111

1

nnnF

S

nnnA

FS

F=(F1, F2, F3)

x3

1232221

3332321313

3

nnn

nnnA

FS

1, 2, 3A

x1

x2

Take advantage of new notations


13

1

2

3

1

i

i

i

iji

j

j

n

nA

FS

Einstein convention

1

ii

iji

j

j

nn

nA

FS

Summation over

repeated indices

Slide 12


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Force equilibrium


Force equilibrium in index notations

01312111

f

0

0

3

3

33

2

23

1

13

2

3

32

2

22

1

12

321

fxxx

fxxx

xxx

321


321

jxxxOr, in index notations

or 0

j

i

ijf

x

or 0, jiij f

Slide 14


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Moment equilibrium

332313

232212

131211


jiij

Example

Uniform stress field



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Strain: Definition and equations ofcompatibility

Displacement and strain

Meaning of strain components

Compatibility


Types of movement

Rigid body

displacement


Rigid body

rotationDeformation


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Displacement and strain

Small strains, ||


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CompatibilityEquations of compatibilityMeaning: the absence of

discontinuitiesCompatible


Incompatible

Co-ordinate rotations



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Principal directionsp pThree special orientations

of the volume element

p

axes which ensure the

absence of shear stresses

(strains)

I I0 0 0 0


p

II

III

II

III

0 00 0

0 00 0

,

Principal stress/strain and principal

directionsPrincipal stress

(no shear stresses)

Principal strain

(no distortion)

3


1


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Determination of principal stress andstrain

Stress


Eigenvalues and eigenvectors

Principal directions for stress and strain do not necessarilycoincide

Spherical/deviatoric decomposition

Stress


Strain


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Plane stress and plane strainPlane stress Plane strain


Mohrs circle

2D


3D


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Cylindrical co-ordinates


Cylindrical co-ordinates (cont.)

Equations of equilibrium Definition of strain



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Spherical co-ordinates


Spherical co-ordinates

e n on o s ra n


Equations of equilibrium


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Summary: stress and strain Stress Combination (Cartesian product) of 2 vectors: surface element and force

Represented by 3*3 matrix, symmetric due to moment equilibrium

Strain Combination of 2 vectors: elementary length and relative displacements

of its ends

Represented by symmetric 3*3 matrix

Equations of compatibility (represent continuity of rocks)

Rock Mechanics sign convention Normal vectors directed inwards hence signs of all stress and strain


components are reversed. Component change with coordinate rotation (tensorial property)

Principal stress (strain) and directions

Splitting into spherical (hydrostatic pressure or volumetricstrain) and deviatoric parts

Elasticity The need for constitutive equations

Equations of equilibrium and compatibility equations areinsufficient to uniquely determine stress and strain

Assumptions of elasticity

Generalised Hookes law

Isotropic rocks 3D Hookes law Plane stress and plane strain

Anisotropic rocks


Orthotropic (orthorombic) rocks

2D anisotropy

Elastic energy

Boundary conditions


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Assumptions of elasticity

Residual strain

-

curve for rocks


E

1. Reversibility (noresidual strains)

2. Linearity (Hookes law)3. Time independence

Generalised Hookes lawx3

131211131211

x1

x2

333231

232221

333231

232221 , klij

klijklijklijklij SC Summation over

repeated indexes


Tensor of elastic

moduli

Tensor of

compliances

81 components 21 independent


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Isotropy and anisotropy Isotropic rocks

Shear and normal stress and strain are not coupled

Principal directions for stress and strain coincide

Transversal isotropic and orthotropic rocks Moduli depend upon the direction of loading

Exist directions where shear and normal stress andstrain are not cou led


General case Moduli depend upon the direction of loading

Shear and normal stress and strain are coupled in alldirections

Isotropic material

Moduli are independent of the direction of loading

Hookes law two independent parameters

x x y z

y x y z

E E E

E E E

1

1

1

5.01


z x y z

xy xy yz yz zx zx

E E E

E E E

1 1 1, , ,


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Other forms of isotropic Hookes law


=0.5 orK= - incompressible material

Anisotropic rocks

Different elastic properties in different directions

Examples:

Transversal isotropic material

Foliated rock or layered rock mass

Rock or concrete with reinforcement in one direction Rock at later stages of uniaxial compression

Orthotropic material


Rock with two or three orthogonal sets of fractures

Blocky rock mass

Concrete with reinforcement in two or three directions


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Transversal isotropy z E2Hooks law

zyxz

zyxy

zyxx

EEE

EEE

22

2

2

11

1

2

2

1

1

1

1

1

1

x

y

zE1


xyxyzxzxyzyz

EGG

1

1

22

222

1,

2

1,

2

1

x

y

Orthotropy

x x 1 21 31

z E3

Hooks law

y x y z

z x y z

E E E

E E E

1

1

1 2 3

12

1 2

32

3

13

1

23

2 3

x

y

zE2E1


yz yz zx zx xy xyG G G

E E E E E E

1

2

1

2

1

223 13 12

1 21 2 12 2 32 3 23 3 13 1 31

, ,

, ,

x

y


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ExamplePressurised borehole in a layered rock.Stress concentration at the contour

90 90

q

2

E1

0

30

60120

150

180

210 330

0

30

60120

150

180

210 330

q q


E2/E1=0.2

G E2

2

12 1

Video Clip

240

270

300240

270

300

Isotropic Highly anisotropic

2D isotropic Hookes lawPlane stress

Plane strain



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2D orthotropic Hookes lawPlane stress


replacement

Elastic (strain) energy

1D case


General case


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Expression of elastic energy viamoduli and compliances

Isotro ic rock

or


General anisotropic rock

klijklijklijklij SCW 2121

Boundary conditions

At each point of the boundary 3 conditions (2 in

Tractions (stresses acting on the elements of theboundary surface)

Simulate the action of external bodies that are very soft ascompared to the rock

Displacements Simulate the action of external bodies that are very stiff as

compared to the rock


Combination

Symmetry

Saint Venant principle


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Example: Triaxial testing of a rock

sampleu0 0

,0 uuuu zr

p

z

y

r

0

zr

r p


Stiff loading frame

Full contact with looing platens, or

Full sliding

x

0,0 zr uuu

0 zzr instead of

Saint Venant Principle Complex load can be replaced with a statically equivalent

simpler one Total force and moment are preserved

This affects only a vicinity of the surface where the load isapplied

F, M


Affected area


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Summary: elasticity Constitutive relationship (equations) to close the system of equations

Linear elasticity Reversibility, Hookes law, time independence

Two elastic constants

Principal directions for stress and strain coincide

Transverse isotropic rock Five elastic constants

Plane of isotropy

Layered rock, one set of fractures

Orthotropic rock Nine elastic constants


Three planes of symmetry Blocky rock, two or three mutually orthogonal sets of fractures

Plane stress and plane strain the same equations ,adjusted moduli

Boundary conditions 3 at each point

Saint-Venant principle Complex load replaced with a statically equivalent one

Wave propagation

Plane waves

Rayleigh waves

Energy distribution



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Equations of motion

Adding inertial body forces

Equations

of motion


Plane waves. Isotropic infinite rock

Assumptions

Both and u are functions ofx only

Infinite rock (no boundaries)

Isotropic rock



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Wave equations. Planar waveDirection of wave

propagation01

2

2

22

2

t

u

cx

u xx

x

z

y

uxuz

uy

Wave front

P-waveS-waves

01

01

2

2

22

2

2

2

22

2

t

u

cx

u

t

u

cx

u

z

S

z

y

S

y


Non-planar wave front

Locally the front can be replaced with a

tangent plane

Plane waves

x

z

uxuz

Direction of wave

propagation


yuy

Wave front


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Rayleigh waves

Waves near the boundary of a semi-space

Direction of wave propagation

Direction of particle motion

s ntially

Velocity


Deca

y

expon

Energy distribution- Point source, energyE

Energy is concentrated in a surface layer of with

ct

or wave

h

Rayleigh wave

ct


2~

ct

EW hct

EW~

Rayleigh wave can carry energy farther than P and S waves


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Earthquakes

lHWr

11~

2

Rayleigh wave

H

l

1~


Seismic source

22 Hl

Summary: waves In isotropic rock, far from the boundaries, two

types of planar waves (the front is planar) exist-

propagation

S-wave: particles oscillate in the direction normal to

wave propagation Velocities of wave propagation are determined bymoduli and density. P-wave is faster than S-wave

Non-planar wave fronts locally treated as planar


Near boundary, the Rayleigh wave exists Backward circular particle motion

Slower than S-wave

Energy is distributed farther than by plane waves


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Conclusion Stress and strain

Basic notions of mechanics

Continuum

Three dimensional tensors

Equations of equilibrium and compatibility

Constitutive law Elasticity basic law

Isotropy and anisotropy

Boundary conditions


Dynamics Equations of motion

Plane and Rayleigh waves

Velocity of wave propagation

Energy

Literature

Mase, G.H. Schaum's outline of theory and

problems of continuum mechanics.

Timoshenko, S.P. & J.N. Goodier Theory

of Elasticity. New York : McGraw-Hill.

Brady, B.H.G. & E.T. Brown. Rock


.

George Allen & Unwin. London, Boston,

Sydney, 1985.

Stress Strain Elasticity

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