Mathematics for Plasticity Models in Geomechanics Prof. Guy … · 2018-10-11 · Mathematics for...

Oxford University Civil EngineeringOxford University Civil Engineering

MathMathematics for Plasticity Modelsematics for Plasticity Modelsin in GeoGeomechanicsmechanics

Prof. Guy HoulsbyProf. Guy Houlsby

Oxford UniversityOxford University


AcknowledgementsAcknowledgements

Assoc. Prof. Alexander PuzrinGeorgia Tech

Prof. Ian CollinsAuckland University


OutlineOutline

• Potentials and the Legendre transform

• Convex analysis, subgradients and the

Fenchel dual

• Functionals and Frechet differentiation


PotentialsPotentials

( )ijEE ε=

ijij

Eε∂

∂=σ

E

σ

ε

dEdσ


Why use a potential?Why use a potential?

• Hyper-elastic

• Elastic

• Hypo-elastic

( )ijEE ε=

( )ijijij f ε=σ

( ) klijijklij dDd εε=σ

ijij

Eε∂

∂=σ

klklij

ij dE

d εε∂ε∂

∂=σ

DifferentiateScalar

2nd order tensor

4th order tensor


The Legendre transformThe Legendre transform

ii x

Xy

∂∂

=

ii yxYX =+

( )ixXX =

( )iyYY =

ii

jj

ii

j

jx

y

xy

yY

y

x

xX

+∂

∂=

∂∂

+∂

∂

∂∂

ii y

Yx

∂∂

=

If there is a function:

such that

Then define:

where

so that:

Differentiating w.r.t. yi :


Legendre transform: summaryLegendre transform: summary

ii x

Xy

∂∂

=

ii yxYX =+

( )ixXX = ( )iyYY =

ii y

Yx

∂∂

=


Complementary energyComplementary energy

C

E

σ

ε

dEdσ

dCdε

ijij

Eε∂

∂=σ

ijijCE εσ=+

( )ijEE ε= ( )ijCC σ=

ijij

Cσ∂∂

=ε


Thermodynamic terminologyThermodynamic terminology

fuE ==

ghC −=−=

Internal energy or Helmholtz free energy

Enthalpy or Gibbs free energy

ijij

fε∂∂

=σ

ijijgf εσ=−

( )ijff ε= ( )ijgg σ=

ijij

gσ∂∂

−=ε


Passive variablesPassive variables

ii x

Xy

∂∂

=

ii yxYX =+

( )ii axXX ,= ( )ii ayYY ,=

ii y

Yx

∂∂

=

ii a

Xb

∂∂

=i

i aY

b∂∂

−=


Chain of Legendre transformsChain of Legendre transforms

ii

ii

ii

aX

bxX

y

axX

∂∂

=∂∂

= ,

),(

ii

ii

ii

a

Yb

y

Yx

ayY

∂∂

−=∂∂

= ,

),(

ii

ii

ii

b

Wa

x

Wx

byW

∂∂

−=∂∂

−= ,

),(

ii

ii

ii

b

Va

x

Vy

bxV

∂∂

=∂∂

−= ,

),(

ii yxYX =+

iibaWY −=+ ii yxVW −=+

iibaXV =+

0=+++ WVYX


The thermodynamic chainThe thermodynamic chain

suu

su

ijij

ij

∂∂

=θ∂ε∂

=σ

ε

,

),(

shh

sh

ijij

ij

∂∂

=θ∂σ∂

−=ε

σ

,

),(

∂θ∂

−=∂σ∂

−=ε

θσ

gs

g

g

iji

ij

,

),(

∂θ∂

−=∂ε∂

=σ

θε

fs

f

f

ijij

ij

,

),(

ijijhu εσ=−

sgh θ=− ijijgf εσ=−

sfu θ=−

0=−+− fghu

Helmholtz free energy

Gibbs free energy

Enthalpy

Internal energy


Example: stiffness proportional to pressureExample: stiffness proportional to pressure

κε

+κ

κ=2

3exp

2gvpf o

gpq

pp

pgo 6

1log2

−

−

κ−=

ε

κ+

κ

κκ=

&

&

&

& v

pq

gpq

qp

q

p2

3

ε

=

&

&

&

& v

G

K

q

p

30

0

23

2

22 ε+=

GKvf

Gq

Kp

g62

22−−=

Linear Non-linear

Triaxial variables, stresses: strains( )qp, ( )ε,v


A thermodynamic approach to plasticity theoryA thermodynamic approach to plasticity theory

( )ijff ε=

ijij

fε∂∂

=σ

( )ijijff αε= ,

( )ijijijdd ααε= &,,

ijij

fε∂∂

=σ

ijij

dfα∂∂

+α∂∂

=&

0

ijij

ijij

ijij

d

f

χ=χ

α∂∂

=χ

α∂∂

−=χ

&

Hyper-elasticity Hyper-plasticity


What is the generalised stress?What is the generalised stress?

( )ijijg ασ ,

ijij

gσ∂∂

−=ε

ijij

g

α∂∂

−=χ

( ) ( )ijijijij ggg α+ασ−σ= 21

Example:

pij

eijij

ijij

gε+ε=α+

σ∂∂

−=ε 1

ijijij

gα∂

∂−σ=χ 2


Eg

2

2σ−=

Eg σ

=σ∂

∂−=ε

σα−α

+σ

−=22

22 HE

g

α+σ

=σ∂

∂−=ε

Eg

σ+α−=α∂

∂−=χ H

g

( )α=α∂

∂−=χ &

&sgnk

d

α= &kd

( ) α+α=σ Hk &sgn

H

k

E

α

εεe

σE

ε

σ

ε

σ

ε

σ


The dissipation functionThe dissipation function

The dissipation is homogeneous first-order in

( )ijijijdd ααε= &,,

ijα&

dd

ijijijij

=αχ=αα∂∂

&&&

(Euler’s theorem)

ijijd αχ=+ &?

Legendre transform?


Legendre transformation: a special caseLegendre transformation: a special case

ii x

Xy

∂∂

=

ii yxYX =λ+( )ixXX =

( ) 0=λ=λ iyYY

ii y

Yx

∂∂

λ=

A first order function:

such that

wherei

ix

xX

X∂∂

=

then:

Define:

0=λ ( ) 0=iyYor


Dissipation or yieldDissipation or yield

( )ijijg ασ ,

( )ijijijd αασ &,,

ijij

gσ∂∂

−=ε

ijij

g

α∂∂

−=χ

ijij

dα∂∂=χ&

( )ijijg ασ ,

( ) 0,, =χασ ijijijy

ijij

gσ∂∂

−=ε

ijij

g

α∂∂

−=χ

ijij

y

χ∂∂

λ=α&


Example: von Mises plasticityExample: von Mises plasticity

ijijkd αα= &&2

klkl

ijij k

α′α′

α′=χ′

&&

&2

02 2 =−χ′χ′= ky ijij

0=α= kkc &

cdd Λ+=′

ijklkl

ij

ijij k

dδΛ+

αα

α=

α∂′∂

=χ&&

&

&2

ijij

ijy

χ′λ=χ∂∂

λ=α 2&


Convex analysisConvex analysis

X

Y

Non-convex

X

Y

Convex

λ1 - λ

( ) 10,,,1 <λ<∀∈∀∈λ+λ− CyxCyx

A set C is convex if and only if:


Convex functionConvex function

x

w

Q

P

YNX

w = f(x)

(1-λ)x + λy

λ 1 - λ

( )( ) ( ) ( ) ( ) 10,,,11 <λ<∀∈∀λ+λ−≤λ+λ− Cyxyfxfyxf


Subdifferential Subdifferential and and subgradientsubgradient

x

w

w = f(x)

P

( ) ( ) ( ) ( ){ }zxyxfyfzxf ,−≥−=∂


Example: Example: subgradientsubgradient of abs(of abs(xx))

x

x

1

-1

( )xw∂

( ) [ ]

>+

=+−

<−

=∂

0,1

0,1,1

0,1

x

x

x

xw

xw =( )xw


Fenchel dualFenchel dual

( ) ( ){ }xfzxzfVx

−=∈

,sup*

( )xfz ∂∈

( )zfx *∂∈

( )xxf

z∂

∂=

( )z

zfx

∂∂

=*

( ) ( )xfzxzf −= ,*

Fenchel dualLegendre transform


Some functions of setsSome functions of sets

• Indicator function

• Gauge function (Minkowski function)

• Support function


Indicator functionIndicator function

( )

∉∞+

∈=δ

Cx

CxCx

,

,0

x

z = δ(x | C)

x=a x=b

[ ] { }bxaxbaC ≤≤== ,


Some useful indicator functionsSome useful indicator functions

( ) [ ]( )0,0 ∞−δ=δ − xx

( ) { }( )00 xx δ=δ

C is set of non-positive real numbers

C is set containing zero only


x

x

x

x

( )x−δ∂ 0 ( )x0δ∂

( )x−δ0 ( )x0δ

( )xy iz=0=x


Subgradients Subgradients and duals of indicator functionsand duals of indicator functions

( )xf ( )xfx ∂∈* ( )** xf ( )** xfx ∈

0 0( )x0δ ( )xiz

( )x−δ0 ( )x−δ0 ( )*0 x−δ− −( )*0 x−δ −


Indicator functions for elasticity (1): rigid materialIndicator functions for elasticity (1): rigid material

ε

f( ε )

ε

f(ε )

ε

f( ε)

2

2ε=

Ef

ε=ε∂

∂=σ E

f( )εδ= 0f

( ) ( )ε=ε∂∈σ izf

⇒ Increasing E ⇒ Infinite E

0=ε


ε

σ

ε

f(ε)

σ

ε

σ

-g(σ)

Indicator functions for elasticity (2): cracking materialIndicator functions for elasticity (2): cracking material

2

2ε−=

Ef

( ) ε−−=ε∂∈σ Ef

( )σδ+σ

=− −0

2

2Eg

( ) ( )σδ+σ

=σ−∂∈ε −0Eg


The gauge functionThe gauge function

( ) { }CxCx λ∈≥λ=γ 0inf

C

γ > 1

γ < 1

γ = 1

γ = 0

( ) ( ) 1−χγ=χ Cy

Yield function


Indicator function revisited…Indicator function revisited…

For a convex set C which contains the origin:

( ) 01<−γ Cx for any point within the set, so that:

( ) ( )( )10 −γδ=δ − CxCx


The support functionThe support function

( ) { }CxzxCz ∈=δ ,sup*

• The support function is a function of the variables z conjugate to x

under the inner product zx,

• The support function is the Fenchel dual of the indicator function

(hence the δ* notation)

• The support function is a homogeneous 1st-order function of z

• Corollary: any homogeneous 1st-order function of z can be interpreted

as a support function and therefore defines a set C in its dual x-space


Dissipation and yield revisited…Dissipation and yield revisited…

• The dissipation function is 1st-order and so it is a support function

• It defines a set in (generalised) stress space: the set of accessible

(generalised) stresses

( ) ( )Cd αδ=α && *...,

( ) ( )( ) ( )( ) ( )χ=χδ=−χγδ=χδ −− ...,1 00 wyCC

with Fenchel dual:


Plasticity in convex analysis terminologyPlasticity in convex analysis terminology

( )α&...,d

( )α∂∈χ &...,d

( )χ...,w

( )χ∂∈α ...,w&

( ) ( )( )χδ=χ − yw 0...,

( )( ) ( )χ∂χδ∈α − yy0&

α∂∂

=χ&

d ⇒χ∂

∂λ=α

y& ⇒


Plasticity example (1): dissipation and yieldPlasticity example (1): dissipation and yield

α

d

c

1

χ

w

χ = -c χ = c

α= &cd [ ]( )( )c

w cc

−χδ=

χδ=

−

−

0

,

Dissipation Yield

Indicator functionSupport function

Indicator(yield function)

Indicator(gauge function - 1)

.


Plasticity example (2): avoiding constraintsPlasticity example (2): avoiding constraints

ijijkd αα= &&2 0=α= kkc &

( )kkijijkd αδ+αα= &&& 02

( ) ( )kkklkl

ijijij kd α+

αα

α=α∂∈χ &

&&

&& iz2

( )20 kw ijij −χ′χ′δ= −

( ) ( )klkl

ijijijijij kw

χ′χ′

χ′−χ′χ′δ=χ∂∈α − 20&


SummarySummary

• Potentials and Legendre transforms– Examples in elasticity

– Chains of transforms

– Special case: first order functions

– Dissipation and yield

• Convex analysis, subdifferentials and Fenchel duals– Special elastic models (rigid, cracking etc.)

– [Flow constraints in plasticity]

– Formulation of plasticity theory

– Relationship between yield and dissipation


What is the point? ……..What is the point? ……..

• Elasticity• Plasticity

• Friction• Dilation

• Non-associated flow• Small-strain non-linearity• History effects

• Pressure-dependent stiffness• Critical state• Consolidation• Thermal expansion

• Heat conduction

• Heat capacity• Solid/liquid phase change• Latent heat

• Effective stress• Fluid flow

• Rate effects/ viscosity• Creep


Piecewise linear to nonPiecewise linear to non--linearlinear

ε

σ

ε

σ

ε

σ


∑∑==

ασ−α+σ−=N

nn

N

nnnH

Eg

11

22

21

21

∑=

α=N

nnnkd

1

&

H1

k1

E

α1

εεe

σ

HNH2

kNk2

αNα2

ε

σ

E

2k1

k1

α1εe

E

E

2kN

2k2

k2

kN

αNα2


History of loadingHistory of loading

N

α

1 2 3 4

ε

σ

E E1

k1

α1εe

Ek2

kN

E1

E2

EN

αN

α2


Continuous plasticityContinuous plasticity

N

α

1 2 3 4

α̂

ε

σ

ε

σ

η0 1


Functions and Functions and functionfunctionalalss

function(variables)

variable = d(function)

d(variable)

functional(variables,functions)

linear operator = d(functional)

d(function)


Frechet differentialFrechet differential

[ ]uf ˆ

[ ] [ ] [ ]0

ˆ

ˆˆˆˆˆlim

0ˆ=

δδ′−−δ+

→δ u

uufufuuf

u

[ ] ( )( )∫Υ

ηηη= dufuf ,ˆˆˆ

[ ] ( )( )( ) ( )∫

Υ

ηηη∂

ηη∂=′ dud

uuf

uduf ˆˆ

,ˆˆˆˆ

Particular case of interest:


MultilinearMultilinear to continuous hyperplasticityto continuous hyperplasticity

∑∑==

ασ−α

+σ

−=N

nn

N

n

nnHE

g11

22

22

∑=

α=N

nnnkd

1

&

[ ] ( ) ( )( ) ( ) ηηασ−ηηαη

+σ

−=ασ ∫∫ ddH

Eg

1

0

1

0

22

ˆˆ22

ˆ,

[ ] ( ) ( ) ηηαη=α ∫ dkd1

0

ˆˆ &&

[ ] ( ) ηηαη=α ∫ dkd1

0

ˆˆ &&


How do we determine H(How do we determine H(ηη)?)?

2

2

1

σ

ε=

σ

d

dk

kH

( )σε=ε σ−σ

=εkE

k (hyperbolic)

( )2

31

2

2

2

11

k

kEkd

dkk

Hσ−

=

σ

ε=

σ

−

( ) ( )2

1 3η−=η

EH

ε

σ

k

E


Block models to material modelsBlock models to material models

Eg

2

2σ−=

σα−α

+σ

−=22

22 H

Eg

α= &kd

ijijijijijijjjiih

GKg ασ−α′α′+σ′σ′−σσ−=

24

1

18

1

ijijjjii GKg σ′σ′−σσ−=

4

1

18

1

ijijkd α′α′= &&2

∑∑==

ασ−α

+σ

−=N

nn

N

n

nnHE

g11

22

12

∑=

α=N

nnnkd

1

&

( ) ( ) ( ) ( )∑∑==

ασ−α′α′+σ′σ′−σσ−=N

n

nijij

N

n

nij

nij

n

ijijjjiih

GKg

11 24

1

18

1

( ) ( ) ( )∑=

α′α′=N

n

nij

nij

nkd1

2 &&

[ ] ( ) ( ) ( ) ( ) ηηασ−ηηα′ηα′η+

σ′σ′−

σσ−= ∫∫ dd

h

GKg ijijijij

ijijjjii1

0

1

0

ˆˆˆ2418

[ ] ( ) ( ) ηηα′ηα′η= ∫ dkd ijij

1

0

ˆˆ2 &&

[ ] ( ) ( )( ) ( ) ηηασ−ηηαη

+σ

−= ∫∫ ddH

Eg

1

0

1

0

22

ˆˆ22

[ ] ( ) ηηαη= ∫ dkd1

0

&̂


Example: nonExample: non--linearity at small strainlinearity at small strain

Puzrin, Houlsby and Burland (2001) “Formulation of a Small Strain Model for Overconsolidated Clays"

Proc. Roy. Soc., Series A, Vol. 457, No. 2006, pp 425-440

Gs

ln(γ)

Go


Unloading: “Unloading: “MasingMasing rules”rules”

ε

σ

kE

Reverse curve at double scale


SummarySummary

• Functionals and Frechet derivatives– Infinite number of internal variables ⇒ internal functions– Model small-strain nonlinearity

• History effects

• Damping


ReferencesReferencesHoulsby, G.T. (1992) "Interpretation of Dilation as a Kinematic Constraint", Proceedings of the Workshop

on Modern Approaches to Plasticity, Horton, Greece, June 12-16, pp 19-38Houlsby, G.T. (1996) "Derivation of Incremental Stress-Strain Response for Plasticity Models Based on

Thermodynamic Functions", Proc. IUTAM Symp. on Mech. of Granular and Porous Materials, Cambridge, 15-17 July, Kluwer Academic Publishers, pp 161-172

Collins, I.F. and Houlsby, G.T. (1997) "Application of Thermomechanical Principles to the Modelling of Geotechnical Materials", Proc. Royal Society of London, Series A, Vol. 453, pp 1975-2001

Houlsby, G.T. and Puzrin, A.M. (1999) "An Approach to Plasticity Based on Generalised Thermodynamics", Proc. of the Int. Symposium on Hypoplasticity, Horton, Greece, pp 233-245

Houlsby, G.T. (2000) “Critical State Models and Small-Strain Stiffness”, Developments in Theoretical Geomechanics, Proc. Booker Memorial Symp., Sydney, 16-17 November, Balkema, pp 295-312

Houlsby, G.T. and Puzrin, A.M. (2000) "A Thermomechanical Framework for Constitutive Models for Rate-Independent Dissipative Materials", Int. Journal of Plasticity, Vol. 16 No. 9, pp 1017-1047

Puzrin, A.M., Houlsby, G.T. and Burland, J.B. (2001) "Thermomechanical Formulation of a Small Strain Model for Overconsolidated Clays", Proc. Roy. Soc., Series A, Vol. 457, No. 2006, pp 425-440

Puzrin, A.M. and Houlsby, G.T. (2001) “Strain-based plasticity models for soils and the BRICK model as an example of the hyperplasticity approach”, Géotechnique, Vol. 51, No. 2, pp 169-172

Puzrin, A.M. and Houlsby, G.T. (2001) "Fundamentals of Kinematic Hardening Hyperplasticity", International Journal of Solids and Structures, Vol. 38, No. 21, pp 3771-3794

Puzrin, A.M. and Houlsby, G.T. (2001) "On the Non-Intersection Dilemma in Multi-Surface Plasticity", Géotechnique, Vol. 51, No. 4, pp 369-372

Puzrin, A.M. and Houlsby, G.T. (2001) "A Thermomechanical Framework for Rate-Independent Dissipative Materials with Internal Functions", Int. Journal of Plasticity, Vol. 17, pp 1147-1165

Houlsby, G.T. and Puzrin, A.M. (2002) “Rate-Dependent Plasticity Models Derived from Potential Functions”, Journal of Rheology, Vol. 46, No. 1, pp 113-126

Mathematics for Plasticity Models in Geomechanics Prof. Guy … · 2018-10-11 · Mathematics for...

Documents

Transcript of Mathematics for Plasticity Models in Geomechanics Prof. Guy … · 2018-10-11 · Mathematics for...