Constitutive models

Part 1

Background and terminology

Elasticity

An initial excuse

• One of the difficulties of the presentation of a difficult theory is that the subject will not be fully appreciated until it is studied in detail

• Difficulties of this sort are encountered frequently in teaching a mature subject

• The wholeness of the subject can rarely be communicated quickly

Material behaviour – models

• Linear elastic

• Nonlinear elastic

• Hyperelastic

• Hypoelastic

• Elastoplastic

• Creep

• Viscoplasticity

Preliminaries

• Invariants

• Strain energy

• Continuum mechanics background

Sometimes hydrostaticor spherical part of stress tensor

Interpretation of 1D tensile test

x

L

Lx

x

x

xxx

x

LLLdL

LL

AT

LAT

LLLL

0

ln)/ln(/ strain true

/ddincrement strain true

/

1/,/

,/ stretch

0true

true

true

0eng

0eng

00

force

elongation

L … current lengthA … current area

Rate dependent properties of material

stretching of rate the toequivalent is rate stress gengineerin theSo

// and since

/ is rate stress gengineerin

eng

00

0eng

xx

x

x

LLLL

L

Loading, unloading – different models of material behaviour

Non-linear elastic

Linear elastic

Elastic plastic

Brittle material which is damaged due to formation of microcracks during loading,microcracks are closed upon removal of the load

0.2% for steel

Theory of elasticity

• There is a unique relationship between stress and strain • Elastic actually means no hysteresis, it does not mean

‘linear’ or ‘force proportional to displacement’.• Could be linear or nonlinear.• Strains are said to be reversible.• Elastic material is rate independent• It is purely mechanical theory – no thermodynamic

effects – are considered.

Nonlinear elasticity, 1D, small strains 1/2

x

xx

xxx

xxx

W

W

Wx

d

)(d

potentiala is...d)(density energy strain

d)(ddensity energy strain ofIncrement

0

unstable thenis response its and

softening strainexhibit tosaid is material not, if

hardening) strain (i.e.

increasinglly monotonica)(...0d/d if

iprelationsh unique)(

xx

xx

ss

s

Convex function

Linear elasticity

221

x

xx

EW

E

Nonlinear elasticity, 1D, small strains 2/2

Non-convex function

unstable thenis response its and

softening strainexhibit tosaid is material

convex -non is functionenergy strain

increasingy motonicallnot is)( then(locally)0d/d if xx ss

1D behaviour of elastic material is characterized by• path independence• reversibility• non-dissipativeness

Nonlinear elasticity, 1D, large strains 1/2

)(

strain Lagrange-Green stress, Kirchhoff- Piola2ndd

d

x

xx

EWW

E

WS

Elastic stress-strain relationships in which the stress can be obtainedfrom a potential function of the strains are called hyperelastic

• path independence

• reversibility

• non-dissipativeness

1D element – small strains, small rotations, no hysteresis

strain

stre

ss

L

A

F F

)()( 0 gEf

0tan E

)()( 00 LgAEgAEF

LgAE

gAE

gAE

FkT

1)(

d

d

d

)(d

d

)(d

d

d000

11

11)(0 g

L

AEk

stress

y=K

(q)

q

q2 q1 q0

Equilibrium reached

Q

alpha1

alpha0

Secant stiffness method

QqKkQqQyqky

qKq

qqKk

QqKkQqQyqky

qKq

qqKk

qQ

11121

11

1111

10010

00

0000

0

)]([/ gives and

)()(

tan

step second

)]([/ gives and

)()(

tan

stepfirst

... quess initial and given

Linear elastic model

• Good for many materials under reasonable temperatures provided that the stresses and strains are small

• Examples– Steel– Cast iron– Glass– Rock– Wood

Linear elastic model, Hooke’s law• Linear

– Strain is proportional to stress

• Homogeneous– Material properties are independent of the size of specimen

– corpuscular structure of matter is disregarded

• Anisotropic– Stress-strain coefficients in C depend on direction

• Fourth tensor C is constant and has 81 components– Generally, however, there are 21 independent elastic

constants

– Orthotropic material … 9

– Cubic anisotropy … 3

– Isotropic material … 2 (Young modulus, Poisson ratio)

VVm limlimlim

klijklij C

Hooke’s law – Voigt’s notation

zx

yz

xy

z

y

x

zx

yz

xy

z

y

x

E

2

2100000

02

210000

002

21000

0001

0001

0001

211

E

E

1200000

0120000

0012000

0001

0001

0001

1

E1ED

Nonlinear elastic

)( klklijklij fC

Tensor C is a function of strainThere is no hysteresis

strain

stre

ss

For large displacements (but small strains) engineering strain (Cauchy) is replaced by Green-Lagrange strain and engineering stress is replaced by 2PK stressand TL or UL formulations should be used

Hyperelastic• Stress is calculated from strain energy

functional by

• Good for rubberlike materials• W is assumed by

– Mooney-Rivlin model,– Ogden model,– Etc.

• Path-independent and fully reversible

ijij E

WS

Hypoelastic

• Path-dependent• Stress increments are calculated from strain

increments

• Models for concrete

,...),,,,( unloadingloadingcriteriafracturestrainstressfC

C

ijkl

klijklij