1 10 - hyperelastic materials
10 - hyperelastic materials
holzapfel �nonlinear solid mechanics� [2000], chapter 6, pages 205-222
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me338 - syllabus
10 - hyperelastic materials
10 - hyperelastic materials
entropy inequality
local entropy inequality
3 10 - hyperelastic materials
objectivity change of observer
material objectivity / frame indifference
... are objective strain measures
... are objective stress measures
... are objective 2nd, 1st, 0th order tensors
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10 - hyperelastic materials
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
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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
7 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
8 10 - hyperelastic materials
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
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chadwick �continuum mechanics� [1976]
10 - hyperelastic materials
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.
constitutive equations
10 - hyperelastic materials
constitutive equations
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))
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10 - hyperelastic materials
coleman noll evaluation
assume
with and
first PK
no entropy
11 10 - hyperelastic materials
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)
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13 10 - hyperelastic materials
stress tensors
gustav robert kirchhoff [1824-1887]
augustin louis caucy
[1789-1857]
first piola kirchhoff
second piola kirchhoff cauchy
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stress tensors
gustav robert kirchhoff [1824-1887]
augustin louis caucy
[1789-1857]
second piola kirchhoff
first piola kirchhoff
cauchy
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15 10 - hyperelastic materials
• 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]
!
16 10 - hyperelastic materials
• 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
10 - hyperelastic materials
helmholtz free energy
Isotropic hyperelasticity
(FQ) = (F ) = (C) = (IC , IIC , IIIC)
17 18 10 - hyperelastic materials
tensor analysis - basic derivatives
19 10 - hyperelastic materials
• (principal) invariants of second order tensor
• derivatives of invariants wrt second order tensor
tensor algebra - basic derivatives
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tensor analysis - basic derivatives
21 10 - hyperelastic materials
• 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)
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• stress
• free energy
• elasticity tesnor
example: st venant-kirchhoff material
(E) =1
2�(trE)2 + µE:E
S = �(trE)I + 2µE
C = �I ⌦ I + 2µI⌦I (E) =
1
2�(trE)2 + µE:E
S = �(trE)I + 2µE
C = �I ⌦ I + 2µI⌦I (E) =1
2�(trE)2 + µE:E
S = �(trE)I + 2µE
C = �I ⌦ I + 2µI⌦I
23 10 - hyperelastic materials
• stress
• free energy
• elasticity tesnor
example: st venant-kirchhoff material
=1
2�(EKK)2 + µEIJEIJ
SIJ = �(EKK)�IJ + 2µEIJ
CIJKL = ��IJ�KL + 2µ�IK�JL =1
2�(EKK)2 + µEIJEIJ
SIJ = �(EKK)�IJ + 2µEIJ
CIJKL = ��IJ�KL + 2µ�IK�JL =1
2�(EKK)2 + µEIJEIJ
SIJ = �(EKK)�IJ + 2µEIJ
CIJKL = ��IJ�KL + 2µ�IK�JL
24 10 - hyperelastic materials
• uniaxial stretch
example: st venant-kirchhoff material
F =
0
@lx 0 00 1 00 0 1/lx
1
A� = 0.0.7141
µ = 0.1785
• parameters
25 10 - hyperelastic materials
• simple shear
example: st venant-kirchhoff material
� = 0.0.7141
µ = 0.1785
• parameters
F =
0
@1 �
xy
00 1 00 0 1
1
A
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• free energy
undeformed potato
deformed potato
example: neo hooke‘ian elasticity
27 10 - hyperelastic materials
• definition of stress
• free energy
• definition of tangent operator
example: neo hooke‘ian elasticity
28 10 - hyperelastic materials
example: neo hooke‘ian elasticity
• definition of stress
• free energy
• definition of tangent operator
29 10 - hyperelastic materials
• 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
30 10 - hyperelastic materials
• 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