2 Modeling of Solid Mechanics
2.2 Some Results from Convex Analysis
Equivalently to the comment above, any constitutive model which fulfills inequality (2.12) fulfills all formal requirements enforced by thermodynamics, see [Ottosen and Ristinmaa, 2005, page 552]. Note that the first law of thermodynamics provides no additional restrictions on the constitutive model; we will see in Section 2.3 that it determines the heat equation.
Helmholtz free energy. Due to the presence of the specific internal energy z it is more convenient to express the dissipation inequality (2.12) in terms of the Helmholtz free energy ψ which is defined as
ψ := z − s θ.
With the Helmholtz free energy ψ we obtain from (2.12) with ˙ψ = ˙
z − ˙s θ − s ˙θ the following alternative formulation of the dissipation inequality,
−% ( ˙ψ + s ˙θ) + ε( ˙u) : σ +κ ∇θ · ∇θ
θ ≥ 0. (2.13)
2.2
Some Results from Convex Analysis
In this section we discuss some results from convex analysis, especially the calculus of convex conjugates. We will exploit the results of this section in Section 2.3 in order to obtain a reformulation of the maximum plastic work inequality (2.9) which describes the development of the plastic deformation.
We start with the definition of the (convex) subdifferential, see [Ekeland and Temam, 1999, Chapter I, Definition 5.1]. We remark that for a Banach space X and its dual space X0, we denote the duality product as h·, ·iX or simply h·, ·i if no ambiguity arises.
Definition 2.2.1 (Convex Subdifferential). Let X be a normed vector space. The (convex) subdifferential ∂f (x) ⊆ X0 of a convex function f : X → (−∞, +∞] is defined by
∂f (x) :=x∗∈ X0: hx∗, y − xi + f (x) ≤ f (y) for all y ∈ X .
An element of ∂f (x) is called a subgradient of f at x.
Next, the convex conjugate of a function is defined as follows, see [Ekeland and Temam, 1999, Chapter I, Definition 4.1].
Definition 2.2.2 (Convex Conjugate). Let X be a normed vector space. The convex conjugate f∗ : X0 → (−∞, +∞] of a function f : X → (−∞, +∞] is defined by
f∗(x∗) := sup
x∈X
hx∗, xi − f (x) .
Remark 2.2.3. The convex conjugate f∗is always convex and lower semicontinuous, see [Ekeland and Temam, 1999, Remark after Chapter I, Definition 4.1].
We will see in Section 2.3 that we can re-interpret the maximum plastic work inequality (2.9) as
˙p ∈ ∂f [σ + χ]D
for a suitable function f . During the reformulation of this equation, also called the plastic flow rule, we will benefit from a result, which describes the relation between the convex subdifferential of a convex, lower semicontinuous and proper function and its subdifferential. Lemma 2.2.4. Let X be a normed vector space and f : X → (−∞, ∞] convex, lower semicontinuous and proper. Then
x∗∈ ∂f (x) ⇔ x ∈ ∂f∗(x∗).
Proof. See [Ekeland and Temam, 1999, Chapter I, Corollary 5.2]. Finally, we will apply the following projection theorem to obtain a reformulation of the (visco)plastic flow rule as a Banach space-valued ODE, see Section 4.3.
Lemma 2.2.5 (Projection Theorem). Suppose C is a nonempty, closed, convex subset of a Hilbert space X with inner product (·, ·) and y ∈ X. Then x∗= projC(y) := argmin
x∈C
ky − xkX if and only if
x∗∈ C and (x∗− y, x − x∗) ≥ 0 for all x ∈ C. (2.14) Proof. See [Alt, 2011, 2.3 Projektionssatz].
Especially, we will use Lemma 2.2.5 in the situation where C is a closed ball in R3×3dev. We recall that we denote the Frobenius norm of a matrix
2.3 Modeling of Thermoplasticity 23
Corollary 2.2.6. Let C :=v ∈ R3×3dev : |v| ≤ M ⊂ R3×3dev with M > 0 and y ∈ R3×3dev. Then the projection x
∗= proj C(y) = argmin x∈C |y − x| is given by x∗= projC(y) := ( y for |y| ≤ M, M|y|y for |y| > M,
)
= min(M, |y|) y |y|.
Proof. Obviously, the set C is convex and closed in R3×3dev. We apply
Lemma 2.2.5 and show that projC(y) ∈ R3×3dev satisfies inequality (2.14). We distinguish among the following two cases.
Case 1. |y| ≤ M . We insert projC(y) = y in the left-hand side of
(2.14) and see
(y − y) : (v − y) = 0 for all v ∈ C. Case 2. |y| > M . Now we choose projC(y) = M
y
|y| in the left-hand
side of (2.14) and calculate M y |y|− y :v − M y |y| =M − |y|y |y|: v − M y |y| ≥M − |y| y |y| |v| − M − |y|M ≥M − |y|M −M − |y|M = 0 for all v ∈ C.
2.3
Modeling of Thermoplasticity
In this section we will derive the constitutive equations for a rate- independent thermoplastic model with linear kinematic hardening and the von Mises yield criterion. We assume that all material parameters apart from the uni-axial yield stress are independent of the temperature. In order to keep the overview we will indicate all the dependencies of the temperature.
We establish the Helmholtz free energy ψ in the format, compare [Ottosen and Ristinmaa, 2005, eq. (23.4)],
% ψ(θ, ε(u) − p, ξ) = % cp(θ − θ ln(θ)) +
1
− t(θ) : C(ε(u) − p) +1
2ξ : H ξ. (2.15) Here, % denotes the mass density, cp the specific heat capacity, t(θ)
the thermally induced strains and C and H the elastic and hardening moduli. We recall that the von Mises yield criterion with linear kinematic hardening reads ˆ φ(σ, χ; θ) = φ(σ + χ; θ) = [σ + χ]D − r 2 3σ0(θ) = 0
and that the internal force equals the plastic strain, ξ = p. Notice that we denote in the following the partial derivatives of the function ψ by ψ0:=∂ψ∂θ,∂ψ∂ε,∂ψ∂ξ.
Stress-strain relation and conjugate forces. We obtain from [Ottosen and Ristinmaa, 2005, eq. (21.37)] that the stress-strain relation is given by Hooke’s law (as assumed on page 13) and the conjugate thermodynamic forces by a linear equation;
σ = %∂ψ
∂ε = C(ε(u) − p − t(θ)) and χ = −% ∂ψ
∂ξ = −H ξ. (2.16) We remark that the hardening modulus H : Ω → Lin(R3×3
sym) satisfies
the same symmetry properties as the elastic modulus C, see page 13. Moreover, we will assume that the tensor H is invertible in order to guarantee that there is a one-to-one relation between the conjugate forces and the internal variables.
Plastic flow rule. We develop the plastic flow rule starting from the maximum plastic work inequality exploiting the results from convex analysis. Since the aim of this chapter is to motivate the constitutive equations, we will only sketch the idea. However, similar techniques will be applied in Section 4.3 in order to obtain a reformulation for the viscoplastic flow rule. More details can be found in [Han and Reddy, 1999, Chapter 4].
First, we insert ξ = p into the maximum plastic work inequality (2.9) and re-interpret the inequality as the plastic flow rule
˙p ∈ ∂IB(θ) [σ + χ]D
(2.17) where the function IB(θ): R3×3dev → (−∞, +∞] is given by the indicator
function according to the nonempty, convex and closed set B(θ) := {v ∈ R3×3dev : φ(v; θ) ≤ 0},
IB(θ)(v) :=
(
0 for v ∈ B(θ), ∞ for v /∈ B(θ).
2.3 Modeling of Thermoplasticity 25 Note that we used in (2.17) the fact that f : q = [f ]D
: q for all f ∈ R3×3
and q ∈ R3×3dev. Moreover, equation (2.17) is sometimes called the dual
formulation of the plastic flow rule. With Lemma 2.2.4 at hand we are now able to derive an equivalent formulation of the plastic flow rule (2.17), the so called primal formulation.
Proposition 2.3.1 (Plastic Flow Rule). The plastic flow rule (2.17) can be equivalently reformulated as
[σ + χ]D∈ ∂p˙D( ˙p, θ) or short: σ + χ ∈ ∂p˙D( ˙p, θ) (2.18)
where the function D : R3×3dev × R → R is given by
D( ˙p, θ) := r
2
3σ0(θ) | ˙p|. (2.19) Proof. Since IB(θ)is a convex, lower semicontinuous and proper function,
applying Lemma 2.2.4 we find the equivalence ˙p ∈ ∂IB(θ) [σ + χ]D
⇔ [σ + χ]D∈ ∂IB(θ)∗ ( ˙p),
where IB(θ)∗ : (R3×3dev)0= R3×3dev → (−∞, ∞] denotes the convex conjugate of IB(θ), see Definition 2.2.2. We calculate
IB(θ)∗ (q∗) = sup q∈R3×3dev q∗: q − I B(θ)(q) = sup q∈B(θ) {q∗: q} = sup α∈R α sup r∈R3×3 dev n q∗: r | 0 ≤ α ≤p2/3 σ0(θ), |r| = 1 o = r 2 3σ0(θ) |q ∗| =: D(q∗, θ).
This shows the assertion.
Note that the abbreviated form in (2.18) is justified since the solution of the (visco)plastic flow rule will depend only on the deviatoric part of the left-hand side in (2.18). We further remark that the function D is also called dissipation function in the literature since D( ˙p, θ) = supσ+χ∈B(θ){ ˙p : (σ + χ)} according to the definition of the convex conjugate.
Heat equation. Finally, we will derive the heat equation from the first law of thermodynamics (2.10). Starting with equation (2.10) we obtain together with the time derivative of the Helmholtz free energy,
˙
ψ = ˙z − ˙s θ − s ˙θ, the equation
Next, we determine in a formal way the time derivative of the Helmholtz free energy ψ, (2.15), and the specific entropy s which is according to [Ottosen and Ristinmaa, 2005, eq. (21.35)] given by s = −∂ψ∂θ. We calculate ˙ ψ = −s ˙θ +∂ψ ∂ε : ε( ˙u) − ∂ψ ∂ε : ˙p + ∂ψ ∂ξ : ˙ξ, (2.21) ˙s = −∂ 2ψ ∂θ2 θ −˙ ∂2ψ ∂ε ∂θ : ε( ˙u) + ∂2ψ ∂ε ∂θ : ˙p − ∂2ψ ∂ξ ∂θ : ˙ξ. (2.22) Now, we insert the terms (2.21) and (2.22) into the first law of thermo- dynamics, (2.20), and discover the abstract heat equation
− % θ∂ 2ψ ∂θ2 θ − % θ˙ ∂2ψ ∂ε ∂θ : ε( ˙u) + % θ ∂ 2ψ ∂ε ∂θ − ∂ψ ∂ε : ˙p (2.23) +%∂ψ ∂ε − σ : ε( ˙u) + %∂ψ ∂ξ − θ ∂2ψ ∂ξ ∂θ : ˙ξ = r + div(κ ∇θ). We calculate all terms explicitly and find as in [Ottosen and Ristinmaa, 2005, eq. (23.23)] that the heat equation related to linear kinematic hardening and the von Mises yield condition is given by
(% cp− θ t00(θ) : C(ε(u) − p)) ˙θ − div(κ ∇θ) (2.24)
= r + σ : ˙p + χ : ˙ξ − θ t0(θ) : C(ε( ˙u) − ˙p)
with a given temperature load r. We recall that in the setting of linear kinematic hardening the internal variable equals the plastic strain, i.e. ξ = p; see page 18.
Simplified heat equation. It has been observed in applications that the term θ t00(θ) : C(ε(u) − p) ˙θ is often small, see [Ottosen and Ristinmaa, 2005, Subsection 23.2.1]. Therefore we neglect this term in the heat equation (2.24) and study the simplified heat equation
% cpθ − div(κ ∇θ) = r + (σ + χ) : ˙p − θ t˙ 0(θ) : C(ε( ˙u) − ˙p). (2.25)
In Remark 2.3.2 we will discuss the effect of this simplification on the Clausius–Duhem inequality (2.11).
Equations for thermoplasticity. We summarize the equations for a rate-independent thermoplastic model with linear kinematic hardening and the von Mises yield criterion: the stress-strain relation (2.16), the conjugate forces (2.16), the quasistatic balance of momentum (2.8), the plastic flow rule in primal formulation (2.18) and the simplified heat equation (2.25). The given mechanical and thermal loads are denoted by ` and r, respectively.
2.3 Modeling of Thermoplasticity 27 We obtain the thermoplastic forward equations
stress-strain relation: σ = C(ε(u) − p − t(θ)), conjugate forces: χ = −H p,
plastic flow rule: ∂p˙D( ˙p, θ) 3 σ + χ,
balance of momentum: − div σ = `,
simplified heat equation: % cpθ − div(κ ∇θ) = r + (σ + χ) : ˙p˙
− θ t0(θ) : C(ε( ˙u) − ˙p). When enriched with viscous material effects in the following section these equations will lead to the forward system investigated in this work.
Remark 2.3.2 (Thermodynamical Consistency of the Model, revised version of [Herzog et al., 2017, Remark 9]). To check the thermody- namical consistency of the model with simplified heat equation (2.25), we calculate the specific entropy s associated with our thermoplastic system related to the Helmholtz free energy ψ presented in (2.15) by
% s = −%∂ψ
∂θ = % cp ln θ + t
0
(θ) : C(ε(u) − p).
Then our simplified heat equation (2.25) can be formulated in terms of the entropy equation as
% θ ˙s − div(κ ∇θ) = % cpθ + θ t˙ 00(θ) : C(ε(u) − p) ˙θ
+ θ t0(θ) : C(ε( ˙u) − ˙p) − div(κ ∇θ)
= r + (σ + χ) : ˙p + θ t00(θ) : C(ε(u) − p) ˙θ. (2.26) Therefore, we obtain for the left-hand side of the Clausius–Duhem inequality (2.11) applying (2.26) the equality
% ˙s −div(κ ∇θ) θ − r θ + κ ∇θ · ∇θ θ2 (2.27) = (σ + χ) : ˙p θ + t 00 (θ) : C(ε(u) − p) ˙θ +κ ∇θ · ∇θθ2 .
Using the definition of the convex subdifferential and the positive homo- geneity of D it follows that D( ˙p, θ) = (σ + χ) : ˙p, cf. also [Bartels and Roub´ıˇcek, 2008, eq. (2.4)]. Consequently, assuming θ > 0 (measured in Kelvin, θ ≥ 0 proven in Proposition 4.2.18) the Clausius–Duhem inequality (2.11) would follow, provided that the second-to-last term in (2.27) is non-negative. Note that this term does not appear in case of the
heat equation (2.24), which implies the thermodynamical consistency of the latter.
To summarize, the thermoplastic model with simplified heat equation is thermodynamically consistent, if the second-to-last term in (2.27) is non-negative. This is the case, in particular, if the thermal strain t is an affine-linear function of the temperature. However, an affine- linear thermal strain does not fulfill the global boundedness condition in Assumption 3.2.1, unless it is constant, which substantially simplifies the entire model.
We wish to point out that this lack of physical rigor in the simplified model can be compensated in an optimization framework. In many application problems, it makes sense to impose pointwise bounds on the temperature in order to avoid destruction of the material. Then one can choose the thermal strain t to be an affine-linear function within the temperature range associated with these bounds and impose a bound on t for temperatures outside this range in order to fulfill Assumption 3.2.1. In this way we obtain both a thermodynamically consistent model, and the bound for t needed in our analysis implicitly through the restrictions on the temperature. The simplified model is therefore especially well suited to optimization problems involving pointwise constraints on the temperature as in Problem 6.0.1.