10 - hyperelastic materials me338 - syllabus · 10 - hyperelastic materials 1 10 ... what s strain?...

1 10 - hyperelastic materials

10 - hyperelastic materials

holzapfel �nonlinear solid mechanics� [2000], chapter 6, pages 205-222

2

me338 - syllabus



entropy inequality

local entropy inequality


objectivity change of observer

material objectivity / frame indifference

... are objective strain measures

... are objective stress measures

... are objective 2nd, 1st, 0th order tensors

4


objectivity

principle of material objectivity principle of frame-indifference the constitutive laws governing

the internal conditions of a physical system and the interactions between its parts should not depend on whatever external frame of reference is used to describe them. “i was responsible for introducing the obsolete term in 1958 and now regret that I misled a lot of people.” walter noll

5 6 10 - hyperelastic materials

the potato equations

• kinematic equations - what�s strain?

• balance equations - what�s stress?

• constitutive equations - how are they related?

general equations that characterize the deformation of a physical body without studying its physical cause

general equations that characterize the cause of motion of any body

material specific equations that complement the set of governing equations


the potato equations

• kinematic equations - what�s strain?

• balance equations - what�s stress?

• constitutive equations - how are they related?

general equations that characterize the deformation of a physical body without studying its physical cause

general equations that characterize the cause of motion of any body

material specific equations that complement the set of governing equations


constitutive equations in structural analysis, constitutive rela-tions connect applied stresses or forces to strains or deformations. the constitutive relations for linear materials are linear. more generally, in physics, a constitutive equation is a relation between two physical quantities (often tensors) that is specific to a material, and does not follow directly from physical law. some constitutive equations are simply phenomenological; others are derived from first principles.

constitutive equations

9

chadwick �continuum mechanics� [1976]


constitutive equations or equations of state bring in the charac-terization of particular materials within continuum mechanics. mathematically, the purpose of these relations is to supply connections between kinematic, mechanical and thermal fields. physically, constitutive equations represent the various forms of idealized material response which serve as models of the behavior of actual sub- stances.




elastic materials: time independent stress

hyperelastic materials: path independent work P = P (F (X),X)

homogeneus materials: location independent stress

(F (X),X) =

Z t

t0

P (F (X),X):F dt

(F (X),X) = (F (X))

10


coleman noll evaluation

assume

with and

first PK

no entropy


helmholtz free energy

conditions on free energy (I) = 0

(F ) � 0

(F ) ! 1 as J ! 1 (F ) ! 1 as J ! 0

(F ⇤) = (QF ) = (U) = (C)

12


stress tensors

gustav robert kirchhoff [1824-1887]

augustin louis caucy

[1789-1857]

first piola kirchhoff

second piola kirchhoff cauchy


stress tensors

gustav robert kirchhoff [1824-1887]

augustin louis caucy

[1789-1857]

second piola kirchhoff

first piola kirchhoff

cauchy

14


• definition of stress: possibly nonlinear

• free energy: possibly nonlinear

• linearization: tangent operator

elasticity tensor

A =@P

@F=

@2

@F ⌦ @F

DP [u] =d

d✏

��✏=0

P (F ['(X) + ✏u])

=@P

@F

d

d✏

��✏=0

F ['(X) + ✏u])

=@P

@FDF [u]

!


• definition of stress: possibly nonlinear

• free energy: possibly nonlinear

• linearization: elasticity tensor

elasticity tensor (I) = 0

(F ) � 0

(F ) ! 1 as J ! 1 (F ) ! 1 as J ! 0

(F ⇤) = (QF ) = (U) = (C)

DS[u] =d

d✏

��✏=0

S(E['(X) + ✏u]) =@S

@EDE[u] !

C =@S

@E= 2

@S

@C

=@2

@E ⌦ @E= 4

@2

@C ⌦ @C


helmholtz free energy

Isotropic hyperelasticity

(FQ) = (F ) = (C) = (IC , IIC , IIIC)


tensor analysis - basic derivatives


• (principal) invariants of second order tensor

• derivatives of invariants wrt second order tensor

tensor algebra - basic derivatives


tensor analysis - basic derivatives


• free energy

undeformed potato

deformed potato

example: st venant-kirchhoff material

(E) =1

2�(trE)2 + µE:E

S = �(trE)I + 2µE

C = �I ⌦ I + 2µI⌦I(warning: used only for small strains)


• stress

• free energy

• elasticity tesnor


(E) =1

2�(trE)2 + µE:E


C = �I ⌦ I + 2µI⌦I (E) =

1

2�(trE)2 + µE:E


C = �I ⌦ I + 2µI⌦I (E) =1

2�(trE)2 + µE:E


C = �I ⌦ I + 2µI⌦I


• stress

• free energy

• elasticity tesnor


=1

2�(EKK)2 + µEIJEIJ

SIJ = �(EKK)�IJ + 2µEIJ

CIJKL = ��IJ�KL + 2µ�IK�JL =1



CIJKL = ��IJ�KL + 2µ�IK�JL =1



CIJKL = ��IJ�KL + 2µ�IK�JL


• uniaxial stretch


F =

0

@lx 0 00 1 00 0 1/lx

1

A� = 0.0.7141

µ = 0.1785

• parameters


• simple shear


� = 0.0.7141

µ = 0.1785

• parameters

F =

0

@1 �

xy

00 1 00 0 1

1

A


• free energy

undeformed potato

deformed potato

example: neo hooke‘ian elasticity


• definition of stress

• free energy

• definition of tangent operator




• definition of stress

• free energy

• definition of tangent operator


• uniaxial stretch

example: neo-hook‘ian elasticity

F =

0

@lx 0 00 1 00 0 1/lx

1

A� = 0.0.7141

µ = 0.1785

• parameters


• simple shear

example: neo-hook‘ian elasticity

� = 0.0.7141

µ = 0.1785

• parameters

F =

0

@1 �

xy

00 1 00 0 1

1

A

10 - hyperelastic materials me338 - syllabus · 10 - hyperelastic materials 1 10 ... what s strain?...

Documents

Transcript of 10 - hyperelastic materials me338 - syllabus · 10 - hyperelastic materials 1 10 ... what s strain?...