Hyperplasticity: from micro to macro

Prof. Guy HoulsbyDepartment of Engineering Science, University of Oxford

micro to MACRO mathematical modelling in soil mechanicsReggio Calabria, 31 May 2018

Hyperplasticity: from micro to macro 2

• Hyperplasticity

• Micromechanical models

– Elasticity

– Plasticity

– Towards a model forgranular material

• Shortcomings and areas for future work

Hyperplasticity: from micro to macro

For each model:

• Underlying physics• Derivation from hyperplasticity• Casting the model in matrix form for

numerical solution• Some results

Hyperplasticity: from micro to macro 3

• An approach to plasticity theory based onthermodynamic principles

• Encompasses many (not all) versions of plasticitycurrently in use

• Advantages:– guarantees thermodynamic acceptability (in framework of

Thermodynamics with Internal Variables: TIV)

– compact mathematically - the entire model is specified byjust two scalar functions:• stored energy

• dissipated energy

– much use of Legendre Transforms to express the model inways convenient for computation

Hyperplasticity

Hyperplasticity: from micro to macro 4

• Strain energy

• Stress

• Alternative form: complementary energy

Hyperelasticity

E E ε

E

σε

C C σ

C

εσ

Legendre Transform

Thermodynamics terminology:

E f Helmholtz free energy

C -g (negative) Gibbs free energy

:E C σ ε

Hyperplasticity: from micro to macro 5

• Strain energy

• Stress

• Alternative form: complementary energy

Hyperelasticity

f f ε

f

σε

g g σ

g

ε

σLegendre Transform

Thermodynamics terminology:

E f Helmholtz free energy

C -g (negative) Gibbs free energy

:f g σ ε

Hyperplasticity: from micro to macro 6

• Free energy

• Dissipation

• Stress

Hyperplasticity

,f f ε α

f

σε

"internal variable"α

, ,d d ε α α

0f d

α α

f

χ

α

d

χα

χ χ

“Generalised stress” – in thispaper always equal to real stress

(often = plastic strain)

0f d

αα α

Thermodynamics

Hyperplasticity: from micro to macro 7

• Free energy

• Yield

Reform with Legendre Transform (special case)

,f f ε α

f

σε

, ,y y ε α χ

y

α

χ

χ χ

“Karush Kuhn Tucker”conditions

0, 0, 0y y

f

χ

α

Hyperplasticity: from micro to macro 8

Elastic bar assembly

Periodic boundaries

Hyperplasticity: from micro to macro 9

Bar assembly in hyperplasticity

Free energy in RVE of size (a x a) is F = fa2, so:

Free energy = sum of elastic strain energy in bars:

2

1 F

a

σ

ε

2

12

n

N

n

n ek

F

ijijijijij vvuue sincos

02

2m x m m mu u x y p

mmymm qyxvv

2

20

Const. PerturbationsAffine

qij

Stiffness kij

Extension eij

i

juj

vj

x

y

Hyperplasticity: from micro to macro 10

... apply standard results

strains,perturbationsF F

2

2

2

1

1

1

xx

yy

F

a

F

a

F

a

2

2

2

2

11

1 1

0

0

mm

m m

F

pa

F

qa

D

pa

D

qa

Stresses

Equilibrium of each node

Treat as “internal variables”

Hyperplasticity: from micro to macro 11

Stress

Strain

Perturbations

Extensions

Matrix formulation

Tyx σ

Tyx ε

TMM qpqp 11p

TNee 1e

BpAεe

σ Fe

1

0 Ge p GB GAε

1

σ εF A B GB G ε DA

If p = 0, σ FAε“Correction” for

perturbations

Hyperplasticity: from micro to macro 12

Results

33.7 19.8 2.3

19.8 42.7 5.9

2.3 5.9 10.1

x x

y y

129.4 44.7 1.1

44.7 83.6 0.7

1.1 0.7 44.7

x x

y y

If p = 0

Note: symmetric matrix butnot isotropic elasticity

Hyperplasticity: from micro to macro 13

Plasticity

21

2nn

N

n

n ek

F

n

N

nncD

1

2 2 0n n ny c e Aε Bp

* σ F e α F e P

* 0 G e α G e Q

* * * * * *

1 1

Dε Rσ F A B G B G A ε F B G B Q P

Hyperplasticity: from micro to macro 14

Response of assembly of elastic-plastic bars

Elastic stiffness

First bar yields

Hyperplasticity: from micro to macro 15

Yield surfaces

First bar yields

Ultimate failuresurfaceFirst yield if

kcomp = ktens/10

Hyperplasticity: from micro to macro 16

Connect bars rigidly at joints and introduce “radial” and“tangential” movement for each bar

22

22tn

tn

tnr

nrn

rn e

ke

kf

Additional “rotational”perturbation at each node, bm

Hyperplasticity: from micro to macro 17

Stress-strain curve with radial and tangential movement

2 x initial loading curve“Masing rules”

Hyperplasticity: from micro to macro 18

Towards a model for granular material

Contact

Hyperplasticity: from micro to macro 19

Towards a model for granular material

22

22tn

tn

tnr

n

rn e

ke

kf

r tn n n nd k e

No tension

Friction

22

0t rn n n ny k e

Hyperplasticity: from micro to macro 20

Stress-strainresponse and

volume changes

Hyperplasticity: from micro to macro 21

• Contact rule is simple

– easy to introduce more sophisticated contact laws

• Contact is at midpoint between nodes and normal tothe line joining them

– easy to fix at expense of more complicated equationsdefining the geometry

• Only originally defined contacts

– as particles move, a detection algorithm is required to makenew contacts and break old ones

– necessary to develop anisotropic structure that governsdilative behaviour

Limitations

Hyperplasticity: from micro to macro 22

• Hyperplasticity is a mathematically compact formulationof plasticity theory that obeys thermodynamics

• Uses just two scalar functions to define material

(+ reformulation using Legendre Transforms)

• Well established for MACRO behaviour, but potential formicroMACRO

– Define energy functions for RVE

– Express in matrix form

– Elastic bars elastic-plastic radial/tangential frictional

– Shows promise but still has some limitations

Summary