Continuum mechanics V. Constitutive equations

AleˇsJanka

office Math 0.107 [email protected] http://perso.unifr.ch/ales.janka/mechanics

Mars 16, 2011, Universit´ede Fribourg

AleˇsJanka 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 , F k , T ) { k`} { ` } 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

AleˇsJanka 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 qi Internal energy  Entropy S

AleˇsJanka 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)

AleˇsJanka 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 x¯.

Hypothesis: functions y(x, t) and T (x, t) are sufficiently smooth to be expanded into a Taylor series:

∂y i 1 ∂2y i y i (x¯, t) = y i (x, t)+ (¯xj xj )+ (¯xj xj )(¯xk xk )+... ∂xj − 2 ∂xj ∂xk − − x,t x,t

with negligible higher-order terms. Simple thermomechanical material: Taylor expansion terms with the second+higher derivatives are negligible:

τ(x, t) = y(x, t0), y (x, t0), T (x, t0), T (x, t0); x, t0 t T ,x ,x ≤ This class of material also called gradient continua.  AleˇsJanka 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

AleˇsJanka 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).

AleˇsJanka 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

AleˇsJanka 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.

AleˇsJanka 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( e ) , T ) { ij } 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?

AleˇsJanka 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 ϕ˙ = e˙ij + T˙ ∂eij ∂T and substitute it into the dissipation inequality qi ρ ϕ˙ + ρ T˙ η τ ij v + T 0. − ∇j i T ∇i ≤ We get i ∂ϕ ∂ϕ ij q ρ e˙ij + ρ T˙ + ρ T˙ η τ j vi + i T 0. ∂eij ∂T − ∇ T ∇ ≤ NB. Due to the symmetry of τ ij = τ ji , we have 1 τ ij v = τ ij v + τ ij v = τ ij e˙ ∇j i 2 ∇j i ∇i j ij

AleˇsJanka V. Constitutive
equations 2. Constitutive laws for simple thermo-mechanical continua

Hence, the dissipation inequality looks now like:

∂ϕ ∂ϕ qi e˙ ρ τ ij + T˙ ρ η + + T 0. ij ∂e − ∂T T ∇i ≤  ij    This inequality must hold for any time-dependent process, ie. for any e˙ij and T˙ !

Hence there must be: ∂ϕ ∂ϕ ρ τ ij = 0 τ ij = ρ , ∂eij − ⇒ ∂eij

∂ϕ ∂ϕ η + η = , ∂T ⇒ −∂T qi T 0. T ∇i ≤

AleˇsJanka V. Constitutive equations 3. Simple thermo-mechanical continuum: large deformation

Small deformations: (Cauchy stress tensor) i ij ∂ϕ ∂ϕ q τ = ρ , η = , i T 0 ∂eij −∂T T ∇ ≤ Large deformations: (2nd Piola-Kirchhoff) i ij ∂ϕ ∂ϕ q T = ρ0 , η = , i T 0 ∂εij −∂T T ∇ ≤

Define strain (or stored) energy density Ψ = ρ0 ϕ, then:

i ij ∂Ψ 1 ∂Ψ q T = , η = , i T 0 ∂εij −ρ0 ∂T T ∇ ≤

∂Ψ NB: T = ∂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 AleˇsJanka V. Constitutive equations 4. Hooke’s law (Robert Hooke 1635–1703)

Neglect temperature: Taylor expansion of strain energy density: 1 Ψ(e) = Ψ + E ij e + E ijk` e e + ... 0 ij 2 ij k` with material coefficients E ij , E ijk` called elastic tensors.

Suppose small deformations: take only the 3 first terms of the expansion: then from τ ij = ∂Ψ we obtain the Hooke’s law (1660): ∂eij

ij ij ijk` τ = E + E ek`

with the pre-stress E ij at initial configuration. If no pre-stress: ij ijk` τ = E ek`

AleˇsJanka 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 e symmetry of E ijk` within ij and k`: 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.

AleˇsJanka V. Constitutive equations 4.2. Hooke’s law in deviator-form

Consider only small deformations. Physical meaning of Cauchy strain e: dV dV0 1 2 3 ` Relative volume change: − = e1 + e2 + e3 = 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 e˜: volume-preserving deformation component e˜ = e e g e˜i = ei e δi ij ij − ij j j − j changes only shape, not the volume, tr(˜e) = 0. Hydrostatic tension s: forces opposed to volume change 1 1 s = tr(τ) = τ j 3 3 j Stress deviator τ˜: forces opposed to shape change: τ˜ij = τ ij s g ij τ˜i = τ i s δi − j j − j AleˇsJanka V. Constitutive equations 4.2. Hooke’s law in deviator-form, shear and bulk moduli

Hooke’s law for isotropic materials (in deviator form):

i i τ˜j = 2 µ e˜j volume-preserving deformations s = 3 K 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:

i i i i i τj =τ ˜j + s δj = 2 µe˜j + 3 K e δj 1 = 2 µ ei e` δi + K e` δi j − 3 ` j ` j   2 µ = 2 µ ei + K e` δi j − 3 ` j  

AleˇsJanka V. Constitutive equations 4.3. Hooke’s law for isotropic materials: elastic tensor E ijk`

Total strain tensor: 2 µ τ ij = 2 µ eij + K e` g ij − 3 `   2 µ = 2 µ g ik g j` e + K g ij g k` e k` − 3 k`   =λ ik j` i` jk ij k` = µ g g + g g| {zek` + }λ g g ek`   2 µ + 3 λ Here, λ and µ are the so called Lam´ecoefficients, K = . 3

ijk` ij ijk` The corresponding elastic tensor E is thus (τ = E ek`):

E ijk` = µ g ik g j` + g i` g jk + λ g ij g k`  

AleˇsJanka V. Constitutive equations 4.4. Hooke’s law for isotropic materials: compliance Cijk` The tensor-inverse of E ijk` is called compliance C:

ij ijk` k` τ = E ekl eij = Cijk` τ . 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 λ τ = 2 µe + tr(τ) Id 2 µ + 3 λ Hence, 1 λ tr(τ) 1 λ e = τ Id or e = g g g g τ k` 2µ − 2µ+3λ ij 2µ ik j` − 2µ+3λ k` ij     Cijk` AleˇsJanka V. Constitutive| equations {z } 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) τ yy(u) force F In 1D, spring stiffness E = relative elongation ε ε (u) u(x,y) yy F τ (u) yy = du Compare with a special case in 3D: dy

y F

Mono-axial loading x Suppose τ ij = 0 for all i, j, except τ 11 = 0. 6 Compliance-form of a 3D Hooke’s law gives:

1 λ µ + λ e = 1 τ = τ 11 2µ − 2µ + 3λ 11 µ (2µ + 3λ) 11   λ e = e = τ 22 33 −2µ (2µ + 3λ) 11

AleˇsJanka V. Constitutive equations 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: τ 11 µ + λ E = = e11 µ (2µ + 3λ)

11 with τ and e11 from the mono-axial loading in cartesian coordinates.

Poisson’s ratio measures transversal vs. axial elongation e λ ν = 22 = −e11 2 (µ + λ) Relative volume change:

dV dV0 − = e11 + e22 + e33 = (1 2 ν) e11 dV0 − ie. ν = 0.5 for incompressible materials AleˇsJanka 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, E 1 1 E µ = K = (2µ + 3λ) = 2 (1 + ν) 3 3 1 2ν − Lam´e’scoefficients µ and λ, E E ν 2 µ µ = λ = = K 2 (1 + ν) (1 + ν)(1 2ν) − 3 − Young’s modulus E and Poisson’s ratio ν,

µ (2µ + 3λ) λ E = ν = µ + λ 2(µ + λ)

AleˇsJanka 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: i i τ˜j = 2 µ e˜j , s = 3 K e using Lam´e’scoefficients µ and λ: ij ik j` i` jk ij k` τ = µ g g + g g ek` + λ g g ek` Or in global form  λ τ = 2 µe + tr(τ) Id 2 µ + 3 λ using Young’s modulus E and Poisson’s ratio ν: E 2ν τ ij = g ij g k` + g ik g j` + g i` g jk e 2(1 + ν) 1 2ν k`  −  large deformations: Saint Venant-Kirchhoff material λ ∂Ψ Ψ(ε) = µ tr(ε2) + tr(ε) 2 , T ij = 2 ∂εij AleˇsJanka V. Constitutive equations 4.7. Hooke’s law: measuring stress-strain curve for steel

3 B stress A 1 4 5 3 2

elasticuniform plastic necking 0 strain e11 1 Ultimate Strength 5 Necking region 2 Yield Strength (elastic limit) F 3 Rupture A 1st Piola-Kirchhoff stress σ = A0 4 Strain hardening region F B Euler stress τ = A

AleˇsJanka V. Constitutive equations 5. Linear thermo-elasticity: Duhamel-Neumann’s law Consider also temperature: Taylor expansion of strain energy: 1 Ψ(e, T ) = Ψ (T ) + E ij (T ) e + E ijk`(T ) e e + ... 0 ij 2 ij k` ∂E ij = Ψ (T ) + E ij (T ) + (T T ) + ... e 0 0 ∂T − 0 ij   1 ∂E ijk` + E ijk`(T ) + (T T ) + ... e e + ... 2 0 ∂T − 0 ij k`   Suppose T T << T and small deformations and neglect all | − 0| 0 (mixed) 3rd order terms and higher.

Duhamel-Neumann’s law: from τ ij = ∂Ψ we obtain: ∂eij

ij ij ijk` ij τ = E + E ek` β (T T0) T0 − −

ij with βij = ∂E . For isotropic materials βij = β g ij . − ∂T Usually, we take T with no pre-strain, E ij = 0. 0 T0 AleˇsJanka V. Constitutive equations 5. Linear thermo-elasticity: Duhamel-Neumann’s law

From the Hooke’s law, we can write:

τ i = 2 µ ei + λ δi e` βi (T T 0) j j j ` − j − Let us derive the compliance-form e = e(τ, T ): Index-contraction of the above gives 1 τ i = (2µ+3λ) e` βk (T T ) ie. e` = τ i + βk (T T ) i ` − k − 0 ` 2µ + 3λ i k − 0 h i Substitute it back to the Duhamel-Neumann’s law to obtain 1 λ 1 λ ei = δi δ` δi δ` τ k δi βm βi (T T ) j 2µ k j − 2µ+3λ j k ` −2µ 2µ+3λ j m − j − 0     i αj ...thermal dilatation coeff | {z }

AleˇsJanka 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: qi = ki + ki (T T ) + kij T + kijk e 0 1 − 0 2 ∇j 3 jk i i ijk with some coefficients k0, k1 and k3 . This law must not contradict the 2nd law of thermodynamics in particular there must be (cf. Section 2 and 3 above): qi 1 T = ki + ki (T T ) + kij T + kijk e T 0 T ∇i T 0 1 − 0 2 ∇j 3 jk ∇i ≤   for any state of the continuum, ie. T > 0 and e. This is ∀ ∀ satisfied only if ki = ki 0 , kijk 0 and [kij ] is sym.positive definite 0 1 ≡ 3 ≡ − 2

AleˇsJanka V. Constitutive equations 6. Constitutive law for heat flux q: Fourier’s law

Hence, we have derived the Fourier’s law for heat flux:

qi = kij T , q = k T − ∇j · ∇

with the heat conductivity tensor k, kij = kij ,[kij ] is a − 2 symmetric positive definite matrix.

For isotropic materials: k = k Id.

AleˇsJanka V. Constitutive equations