Continuum mechanics
V. Constitutive equations
Aleš Janka
office Math 0.107
ales.janka@unifr.ch
http://perso.unifr.ch/ales.janka/mechanics
Mars 16, 2011, Université de Fribourg
Aleš Janka
V. Constitutive equations
1. Constitutive equation: definition and basic axioms
Constitutive equation: relation between two physical quantities
specific to a material, e.g.:
τ ij = τ ij (u, {ek` }, {F`k }, T )
Basic axioms
Axiom of causality
Axiom of determinism
Axiom of equipresence
Axiom of neighbourhood
Axiom of memory
Axiom of objectivity
Axiom of material invariance
Axiom of admissibility
Aleš Janka
V. Constitutive equations
1. Basic axioms: causality
Axiom of causality:
Independent variables in the constitutive laws are:
Continuum position y i (x, t)
Temperature T
Dependent variables (responses) are e.g.:
Helmholtz free energy ϕ (thermodynamic potential, measure
of the ”useful” work obtainable from a closed thermodynamic
system)
Strain energy density Ψ
Stress tensor τ ij
Heat flux q i
Internal energy Entropy S
Aleš Janka
V. Constitutive equations
1. Basic axioms: determinism and equipresence
Axiom of determinisim
Responses of the constitutive functions at a material point x at
time t are determined by the history of the motion and history
of the temperature of all points of the body.
Axiom of equipresence
If an independent variable enters in one function of response, it
should be present in all constitutive laws (until the proof of the
contrary)
Aleš Janka
V. Constitutive equations
1. Basic axioms: neighbourhood
Axiom of neighbourhood
Responses at a point x are not much influenced by values of
independent variables (temperature and displacement) at a
distant point x̄.
Hypothesis: functions y(x, t) and T (x, t) are sufficiently smooth
to be expanded into a Taylor series:
2y i i
1
∂
∂y
(x̄ j −x j )(x̄ k −x k )+. . .
y i (x̄, t) = y i (x, t)+ j (x̄ j −x j )+
j
k
∂x x,t
2 ∂x ∂x x,t
with negligible higher-order terms.
Simple thermomechanical material: Taylor expansion terms
with the second+higher derivatives are negligible:
τ (x, t) = T y(x, t 0 ), y,x (x, t 0 ), T (x, t 0 ), T,x (x, t 0 ); x, t 0 ≤ t
This class of material also called gradient continua.
Aleš Janka
V. Constitutive equations
1. Basic axioms: memory
Axiom of memory
Values of constitutive variables from a distant past do not affect
appreciably the values of constitutive laws now.
Smooth memory material: constitutive variables can be
expanded to Taylor series in time with negligible higher order terms
Fading memory: response functionals must smooth possible
discontinuities in memory
Aleš Janka
V. Constitutive equations
1. Basic axioms: objectivity
Axiom of objectivity
Invariance of constitutive laws with respect to rigid body motion
of the spatial frame of reference (spatial coordinates).
Simple consequence: constitutive laws depend of the
deformation gradient (or strain tensor) rather than y(x).
Aleš Janka
V. Constitutive equations
1. Basic axioms: material invariance
Axiom of material invariance
Invariance of constitutive laws with respect to certain
symmetries/transformations of the material frame of reference
(material coordinates).
Symmetries in material properties due to crystallographic
orientation.
Hemitropic continuum: invariant w.r.t. all rotations
Isotropic continuum: hemitropic + invariant to reflection
Anisotropic continuum: otherwise (can have some invariance
properties, but not all)
Homogeneous continuum: invariance w.r.t. shift of coord system
Aleš Janka
V. Constitutive equations
1. Basic axioms: admissibility
Axiom of admissibility
Consistence with respect to basic conservation laws (mass,
momentum, energy), and the 2nd law of thermodynamics
(entropy).
This axiom can also help to eliminate dependences on some
constitutive variables.
Aleš Janka
V. Constitutive equations
2. Constitutive laws for simple thermo-mechanical continua
Thermo-elastic continua: simple thermo-mechanical continua
with no memory.
By applying the basic axioms, all material properties depend only
on the current values of deformation and temperature: hence also
for Helmholtz free energy:
ϕ = ϕ({eij )}, T )
For simplicity, consider only small deformations (Cauchy strain
tensor eij ).
Are we able to say more about the form of the constitutive
laws in this case?
Aleš Janka
V. Constitutive equations
2. Constitutive laws for simple thermo-mechanical continua
Axiom of admissibility: we need to be consistent with
thermodynamical laws and basic equilibria.
Let us derive ϕ with respect to time
∂ϕ
∂ϕ
ėij +
Ṫ
ϕ̇ =
∂eij
∂T
and substitute it into the dissipation inequality
qi
ρ ϕ̇ + ρ Ṫ η − τ ∇j vi + ∇i T ≤ 0.
T
ij
We get
∂ϕ
∂ϕ
qi
ij
ρ
ėij + ρ
Ṫ + ρ Ṫ η − τ ∇j vi + ∇i T ≤ 0.
∂eij
∂T
T
NB. Due to the symmetry of τ ij = τ ji , we have
1 ij
τ ij ∇j vi =
τ ∇j vi + τ ij ∇i vj = τ ij ėij
2
Aleš Janka
V. Constitutive equations
2. Constitutive laws for simple thermo-mechanical continua
Hence, the dissipation inequality looks now like:
∂ϕ
qi
∂ϕ
ij
−τ
+ Ṫ ρ η +
+ ∇i T ≤ 0.
ėij ρ
∂eij
∂T
T
This inequality must hold for any time-dependent process, ie.
for any ėij and Ṫ !
Hence there must be:
ρ
∂ϕ
− τ ij = 0
∂eij
⇒
τ ij = ρ
∂ϕ
∂T
⇒
η=−
η+
∂ϕ
,
∂eij
∂ϕ
,
∂T
qi
∇i T ≤ 0.
T
Aleš Janka
V. Constitutive equations
3. Simple thermo-mechanical continuum: large deformation
Small deformations: (Cauchy stress tensor)
∂ϕ
τ =ρ
∂eij
ij
∂ϕ
η=−
∂T
,
qi
∇i T ≤ 0
T
,
Large deformations: (2nd Piola-Kirchhoff)
∂ϕ
T = ρ0
∂εij
ij
,
∂ϕ
η=−
∂T
,
qi
∇i T ≤ 0
T
Define strain (or stored) energy density Ψ = ρ0 ϕ, then:
∂Ψ
T =
∂εij
ij
,
1 ∂Ψ
η=−
ρ0 ∂T
,
qi
∇i T ≤ 0
T
NB: T = ∂Ψ
∂ε is a derivative of a scalar function with respect to a
tensor, see M2, Section 2.
∂Ψ
Hyperelastic material: material for which T ij =
∂εij
Aleš Janka
V. Constitutive equations
4. Hooke’s law (Robert Hooke 1635–1703)
Neglect temperature: Taylor expansion of strain energy density:
1
Ψ(e) = Ψ0 + E ij eij + E ijk` eij ek` + . . .
2
with material coefficients E ij , E ijk` called elastic tensors.
Suppose small deformations: take only the 3 first terms of the
∂Ψ
expansion: then from τ ij = ∂e
we obtain the Hooke’s law (1660):
ij
τ ij = E ij + E ijk` ek`
with the pre-stress E ij at initial configuration.
If no pre-stress:
τ ij = E ijk` ek`
Aleš Janka
V. Constitutive equations
4.1. Hooke’s law: elastic tensor E ijk`
Elastic tensor E ijk` : 34 = 81 components depending only on
material coordinates
Possible reduction of degrees of freedom:
Symmetry of τ ij and ek` ⇒ symmetry of E ijk` within ij and k`:
E ijk` = E jik` = E ij`k = E ji`k
no. of components reduced to 36.
Taylor expansion of Ψ(e) ⇒ symmetry in pairs ij and k`:
E ijk` = E k`ij
no. of components thus reduced to 21.
Aleš Janka
V. Constitutive equations
4.2. Hooke’s law in deviator-form
Consider only small deformations. Physical meaning of Cauchy strain e:
dV − dV0
Relative volume change:
= e11 + e22 + e33 = tr(e) = e``
dV0
(cf. “1. Kinematics”, section 4.)
Define:
Volumic dilatation e: volume-changing deformation component:
1
1
e = tr(e) = e``
3
3
Strain deviator ẽ: volume-preserving deformation component
ẽij = eij − e gij
ẽji = eji − e δji
changes only shape, not the volume, tr(ẽ) = 0.
Hydrostatic tension s: forces opposed to volume change
1
1
s = tr(τ ) = τjj
3
3
Stress deviator τ̃ : forces opposed to shape change:
τ̃ ij = τ ij − s g ij
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τ̃ji = τji − s δji
V. Constitutive equations
4.2. Hooke’s law in deviator-form, shear and bulk moduli
Hooke’s law for isotropic materials (in deviator form):
τ̃ji
= 2 µ ẽji
volume-preserving deformations
s = 3K e
volume-change
Material properties characterized only by 2 constants
shear modulus µ characterizes genuine shear
bulk modulus K characterizes (in)compressibility
(incompressible material for K → ∞)
Total strain tensor:
τji
= τ̃ji + s δji = 2 µẽji + 3 K e δji
1
= 2 µ eji − e`` δji + K e`` δji
3
2
µ
e`` δji
= 2 µ eji + K −
3
Aleš Janka
V. Constitutive equations
4.3. Hooke’s law for isotropic materials: elastic tensor E ijk`
Total strain tensor:
τ ij
2
µ
= 2 µ e ij + K −
e`` g ij
3
2
µ
= 2 µ g ik g j` ek` + K −
g ij g k` ek`
3
|
{z
}
=λ
= µ g ik g j` + g i` g jk ek` + λ g ij g k` ek`
Here, λ and µ are the so called Lamé coefficients, K =
2µ + 3λ
.
3
The corresponding elastic tensor E ijk` is thus (τ ij = E ijk` ek` ):
E
ijk`
ik
j`
i`
=µ g g +g g
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jk
+ λ g ij g k`
V. Constitutive equations
4.4. Hooke’s law for isotropic materials: compliance Cijk`
The tensor-inverse of E ijk` is called compliance C :
τ ij = E ijk` ekl
eij = Cijk` τ k` .
Let us inverse the Hooke’s law (ie. express e as a function of τ ):
τ = 2 µ e + λ tr(e) Id
Take a trace:
tr(τ ) = 2 µ tr(e) + 3 λ tr(e) = (2 µ + 3 λ) tr(e)
Plug back tr(e) into the Hooke’s law above to get
τ = 2 µe +
Hence,
1
e=
2µ
λ tr(τ )
τ−
Id
2µ+3λ
λ
tr(τ ) Id
2µ + 3λ
or
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1
eij =
2µ
|
gik gj` −
λ
gk` gij τ k`
2µ+3λ
{z
}
Cijk`
V. Constitutive equations
4.5. Hooke’s law: Young’s modulus E , Poisson’s ratio ν
Material constants in Hooke’s law by analogy with linear springs:
u(x,y)
E=
force F
relative elongation ε
Compare with a special case in 3D:
Mono-axial loading
Suppose τ ij = 0 for all i, j, except τ 11 6= 0.
Compliance-form of a 3D Hooke’s law gives:
λ
µ+λ
1
1−
τ11 =
τ11
e11 =
2µ
2µ + 3λ
µ (2µ + 3λ)
λ
e22 = e33 = −
τ11
2µ (2µ + 3λ)
Aleš Janka
V. Constitutive equations
τyy(u)
u(x,y)
F
F
y
x
εyy (u)
=
In 1D, spring stiffness
τyy(u)
du
dy
4.5. Hooke’s law: Young’s modulus E , Poisson’s ratio ν
Young’s modulus defined as apparent “1D spring stiffness” in the
case of mono-axial loading, ie:
µ+λ
τ 11
E=
=
e11
µ (2µ + 3λ)
with τ 11 and e11 from the mono-axial loading in cartesian
coordinates.
Poisson’s ratio measures transversal vs. axial elongation
ν=−
λ
e22
=
e11
2 (µ + λ)
Relative volume change:
dV − dV0
= e11 + e22 + e33 = (1 − 2 ν) e11
dV0
ie. ν = 0.5 for incompressible materials
Aleš Janka
V. Constitutive equations
4.6. Hooke’s law for isotropic materials: summary
Isotropic material characterized by two constants:
shear modulus µ and bulk modulus K ,
µ=
E
2 (1 + ν)
K=
1
1 E
(2µ + 3λ) =
3
3 1 − 2ν
Lamé’s coefficients µ and λ,
µ=
E
2 (1 + ν)
λ=
Eν
2µ
=K−
(1 + ν)(1 − 2ν)
3
Young’s modulus E and Poisson’s ratio ν,
E=
µ (2µ + 3λ)
µ+λ
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ν=
λ
2(µ + λ)
V. Constitutive equations
4.6. Hooke’s law for isotropic materials: summary
Corresponding form of Hooke’s law:
using shear modulus µ and bulk modulus K , in deviator form:
τ̃ji = 2 µ ẽji
,
s = 3K e
using Lamé’s coefficients µ and λ:
ij
ik j`
i` jk
τ =µ g g +g g
ek` + λ g ij g k` ek`
Or in global form
λ
tr(τ ) Id
2µ + 3λ
using Young’s modulus E and Poisson’s ratio ν:
2ν
E
g ij g k` + g ik g j` + g i` g jk ek`
τ ij =
2(1 + ν) 1 − 2ν
τ = 2 µe +
large deformations: Saint Venant-Kirchhoff material
2
λ
∂Ψ
Ψ(ε) = µ tr(ε2 ) +
tr(ε)
,
T ij =
2
∂εij
Aleš Janka
V. Constitutive equations
4.7. Hooke’s law: measuring stress-strain curve for steel
stress
3
B
1
A
5
4
2
elastic
uniform plastic
0
3
necking
strain e 11
1 Ultimate Strength
5 Necking region
2 Yield Strength (elastic limit)
A 1st Piola-Kirchhoff stress σ =
3 Rupture
4 Strain hardening region
B Euler stress τ =
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V. Constitutive equations
F
A
F
A0
5. Linear thermo-elasticity: Duhamel-Neumann’s law
Consider also temperature: Taylor expansion of strain energy:
Ψ(e, T ) =
=
1
Ψ0 (T ) + E ij (T ) eij + E ijk` (T ) eij ek` + . . .
2
ij
∂E
(T −T0 ) + . . . eij
Ψ0 (T ) + E ij (T0 ) +
∂T
ijk`
1
∂E
+ E ijk` (T0 ) +
(T −T0 ) + . . . eij ek` + . . .
2
∂T
Suppose |T − T0 | << T0 and small deformations and neglect all
(mixed) 3rd order terms and higher.
Duhamel-Neumann’s law: from τ ij =
∂Ψ
∂eij
we obtain:
τ ij = ETij0 + E ijk` ek` − β ij (T −T0 )
ij
ij
ij
with β ij = − ∂E
∂T . For isotropic materials β = β g .
Usually, we take T0 with no pre-strain, ETij0 = 0.
Aleš Janka
V. Constitutive equations
5. Linear thermo-elasticity: Duhamel-Neumann’s law
From the Hooke’s law, we can write:
τji = 2 µ eji + λ δji e`` − βji (T −T 0)
Let us derive the compliance-form e = e(τ , T ):
Index-contraction of the above gives
τii
=
(2µ+3λ) e`` −βkk
(T−T0 )
ie.
e``
h
i
1
i
k
τ + βk (T −T0 )
=
2µ + 3λ i
Substitute it back to the Duhamel-Neumann’s law to obtain
1
λ
λ
1
m
eji =
δji δk` τ`k −
δji βm
− βji (T−T0 )
δki δj` −
2µ
2µ+3λ
2µ 2µ+3λ
|
{z
}
αij ...thermal dilatation coeff
Aleš Janka
V. Constitutive equations
6. Constitutive law for heat flux q: Fourier’s law
Suppose simple thermo-mechanical continuum, small deformations:
q = q(T , ∇T , e)
Use first-order Taylor expansion around a deformation-free
configuration at T0 to approximate:
q i = k0i + k1i (T −T0 ) + k2ij ∇j T + k3ijk ejk
with some coefficients k0i , k1i and k3ijk .
This law must not contradict the 2nd law of thermodynamics
in particular there must be (cf. Section 2 and 3 above):
1 i
qi
ij
ijk
i
∇i T =
k + k1 (T −T0 ) + k2 ∇j T + k3 ejk ∇i T ≤ 0
T
T 0
for any state of the continuum, ie. ∀ T > 0 and ∀ e. This is
satisfied only if
k0i = k1i ≡ 0
,
k3ijk ≡ 0
and −[k2ij ] is sym.positive definite
Aleš Janka
V. Constitutive equations
6. Constitutive law for heat flux q: Fourier’s law
Hence, we have derived the Fourier’s law for heat flux:
q i = −k ij ∇j T
,
q = k · ∇T
with the heat conductivity tensor k, k ij = −k2ij , [k ij ] is a
symmetric positive definite matrix.
For isotropic materials: k = k Id.
Aleš Janka
V. Constitutive equations