Continuum mechanics V. Constitutive equations Aleˇ s Janka office Math 0.107 [email protected]http://perso.unifr.ch/ales.janka/mechanics Mars 16, 2011, Universit´ e de Fribourg Aleˇ s 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, {e k }, {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 Aleˇ s Janka V. Constitutive equations
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1. Constitutive equation: definition and basic axioms
Constitutive equation: relation between two physical quantitiesspecific 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
Ales 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, measureof the ”useful” work obtainable from a closed thermodynamicsystem)
Strain energy density Ψ
Stress tensor τ ij
Heat flux qi
Internal energy ε
Entropy S
Ales Janka V. Constitutive equations
1. Basic axioms: determinism and equipresence
Axiom of determinisimResponses of the constitutive functions at a material point x attime t are determined by the history of the motion and historyof the temperature of all points of the body.
Axiom of equipresenceIf an independent variable enters in one function of response, itshould be present in all constitutive laws (until the proof of thecontrary)
Ales Janka V. Constitutive equations
1. Basic axioms: neighbourhood
Axiom of neighbourhoodResponses at a point x are not much influenced by values ofindependent variables (temperature and displacement) at adistant point x.
Hypothesis: functions y(x, t) and T (x, t) are sufficiently smoothto be expanded into a Taylor series:
y i (x, t) = y i (x, t)+∂y i
∂x j
∣∣∣∣x,t
(x j−x j)+1
2
∂2y i
∂x j∂xk
∣∣∣∣x,t
(x j−x j)(xk−xk)+. . .
with negligible higher-order terms.Simple thermomechanical material: Taylor expansion termswith the second+higher derivatives are negligible:
τ(x, t) = T [y(x, t ′), y,x(x, t ′),T (x, t ′),T,x(x, t ′); x, t ′ ≤ t]
This class of material also called gradient continua.Ales Janka V. Constitutive equations
1. Basic axioms: memory
Axiom of memoryValues of constitutive variables from a distant past do not affectappreciably the values of constitutive laws now.
Smooth memory material: constitutive variables can beexpanded to Taylor series in time with negligible higher order terms
Fading memory: response functionals must smooth possiblediscontinuities in memory
Ales Janka V. Constitutive equations
1. Basic axioms: objectivity
Axiom of objectivity
Invariance of constitutive laws with respect to rigid body motionof the spatial frame of reference (spatial coordinates).
Simple consequence: constitutive laws depend of thedeformation gradient (or strain tensor) rather than y(x).
Ales Janka V. Constitutive equations
1. Basic axioms: material invariance
Axiom of material invariance
Invariance of constitutive laws with respect to certainsymmetries/transformations of the material frame of reference(material coordinates).
Symmetries in material properties due to crystallographicorientation.
Hemitropic continuum: invariant w.r.t. all rotationsIsotropic continuum: hemitropic + invariant to reflectionAnisotropic continuum: otherwise (can have some invarianceproperties, but not all)Homogeneous continuum: invariance w.r.t. shift of coord system
Ales 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 someconstitutive variables.
Ales Janka V. Constitutive equations
2. Constitutive laws for simple thermo-mechanical continua
Thermo-elastic continua: simple thermo-mechanical continuawith no memory.
By applying the basic axioms, all material properties depend onlyon the current values of deformation and temperature: hence alsofor Helmholtz free energy:
ϕ = ϕ({eij)},T )
For simplicity, consider only small deformations (Cauchy straintensor eij).
Are we able to say more about the form of the constitutivelaws in this case?
Ales Janka V. Constitutive equations
2. Constitutive laws for simple thermo-mechanical continua
Axiom of admissibility: we need to be consistent withthermodynamical laws and basic equilibria.
Let us derive ϕ with respect to time
ϕ =∂ϕ
∂eijeij +
∂ϕ
∂TT
and substitute it into the dissipation inequality
ρ ϕ+ ρ T η − τ ij ∇jvi +qi
T∇iT ≤ 0.
We get
ρ∂ϕ
∂eijeij + ρ
∂ϕ
∂TT + ρ T η − τ ij ∇jvi +
qi
T∇iT ≤ 0.
NB. Due to the symmetry of τ ij = τ ji , we have
τ ij ∇jvi =1
2
(τ ij ∇jvi + τ ij ∇ivj
)= τ ij eij
Ales Janka V. Constitutive equations
2. Constitutive laws for simple thermo-mechanical continua
Hence, the dissipation inequality looks now like:
eij
(ρ∂ϕ
∂eij− τ ij
)+ T ρ
(η +
∂ϕ
∂T
)+
qi
T∇iT ≤ 0.
This inequality must hold for any time-dependent process, ie.for any eij and T !
Hence there must be:
ρ∂ϕ
∂eij− τ ij = 0 ⇒ τ ij = ρ
∂ϕ
∂eij,
η +∂ϕ
∂T⇒ η = − ∂ϕ
∂T,
qi
T∇iT ≤ 0.
Ales Janka V. Constitutive equations
3. Simple thermo-mechanical continuum: large deformation
Small deformations: (Cauchy stress tensor)
τ ij = ρ∂ϕ
∂eij, η = − ∂ϕ
∂T,
qi
T∇iT ≤ 0
Large deformations: (2nd Piola-Kirchhoff)
T ij = ρ0∂ϕ
∂εij, η = − ∂ϕ
∂T,
qi
T∇iT ≤ 0
Define strain (or stored) energy density Ψ = ρ0 ϕ, then:
T ij =∂Ψ
∂εij, η = − 1
ρ0
∂Ψ
∂T,
qi
T∇iT ≤ 0
NB: T = ∂Ψ∂ε is a derivative of a scalar function with respect to a
tensor, see M2, Section 2.
Hyperelastic material: material for which T ij =∂Ψ
∂εijAles Janka V. Constitutive equations
4. Hooke’s law (Robert Hooke 1635–1703)
Neglect temperature: Taylor expansion of strain energy density:
Ψ(e) = Ψ0 + E ij eij +1
2E ijk` eij ek` + . . .
with material coefficients E ij , E ijk` called elastic tensors.
Suppose small deformations: take only the 3 first terms of theexpansion: then from τ ij = ∂Ψ
∂eijwe obtain the Hooke’s law (1660):
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`
Ales Janka V. Constitutive equations
4.1. Hooke’s law: elastic tensor E ijk`
Elastic tensor E ijk`: 34 = 81 components depending only onmaterial 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.
Ales Janka V. Constitutive equations
4.2. Hooke’s law in deviator-form
Consider only small deformations. Physical meaning of Cauchy strain e: