is locally Lipschitz continuous. Using that {θn ≥ 0 a.e. in (0, T ) ×
Ω}n≥0 is a closed subset in C(0, T ; L2(Ω)) and the continuity of the
solution operator mentioned above we see that the temperature θ := limn→∞θn≥ 0 a.e. in (0, T ) × Ω.
4.3
Reformulation of the Viscoplastic Flow Rule
The notion of the viscoplastic flow rule (1.3) indicates that it is a set- valued equation. But as we have seen in equation (4.12) in the proof of Proposition 4.2.6 we are able to reformulate (for a given temperature field) the mechanical system (1.1)–(1.4) as a Banach space-valued ODE system, ˙u ˙p = Φu(F (`, u, p, θ)) Φp(θ, Φσ(u, p, θ) + Φχ(u, p, θ)) . (4.12)
This section is devoted to calculating the operator Φp explicitly and to
showing that the viscoplastic flow rule is indeed a Banach space-valued ODE with non-differentiable nonlinear right-hand side. We will exploit this kind of formulation for the viscoplastic flow rule for example in the proof of the directional differentiability of the control-to-state mapping in Theorem 5.4.11 or during the implementation, see Section 7.3. The reformulation of the viscoplastic flow rule (1.3) is based on a result from the calculus of convex conjugates and the projection theorem, compare Section 2.2. With Lemma 2.2.4 at hand we are able to derive the reformulation of the viscoplastic flow rule (1.3).
Proposition 4.3.1 (Reformulation Viscoplastic Flow Rule; [Herzog and St¨otzner, 2018, Proposition 2.5]). The viscoplastic flow rule (1.3) can be equivalently reformulated as
˙p = −−1min
σ(θ)˜
|τ (u, p, θ)|− 1, 0
τ (u, p, θ) (4.19) a.e. in (0, T ) × Ω where τ (u, p, θ) := [Φσ(u, p, θ) + Φχ(u, p, θ)]D =
[C ε(u) − p − t(θ) − H p]D.
The right-hand side of (4.19) is understood to be zero by continuous extension when τ (u, p, θ) = 0.
Proof. We can understand the viscoplastic flow rule (1.3) in a pointwise sense, see Remark 4.2.5. Therefore we fix an arbitrary (t, x) ∈ (0, T ) × Ω
and prove the equivalence of (1.3) and (4.19) pointwise. For brevity we omit in the following the argument (t, x) for all functions.
Since the mapping q 7→ D(q, θ) = ˜σ(θ)|q| is proper, convex and lower semicontinous, we apply Lemma 2.2.4 and obtain
viscoplastic flow rule (1.3) ⇔ − ˙p + τ (u, p, θ) ∈ ∂D( ˙p, θ) ⇔ ˙p ∈ ∂D∗(− ˙p + τ (u, p, θ), θ), where τ (u, p, θ) := [Φσ(u, p, θ) + Φχ(u, p, θ)]D. It remains to show that
the subdifferential of D∗(· , θ) is a singleton and can be characterized as in the assertion.
We start by calculating D∗(· , θ) : (R3×3dev)0= R3×3dev → (−∞, ∞] explicitly using Definition 2.2.2, D∗(q∗, θ) = sup q∈R3×3 dev {q∗: q − D(q, θ)} = sup α≥0,|r|=1, r∈R3×3 dev 1 ˜ σ(θ)q ∗: α r − |α r| set r=|q∗ |q∗ = sup α≥0 α 1 ˜ σ(θ)|q ∗| − 1 = ( 0 if |q∗| ≤ ˜σ(θ), ∞ if |q∗| > ˜σ(θ), ) = IB(θ)(q∗),
where IB(θ) is the indicator function of the set B(θ) = {v ∈ R3×3dev :
|v| ≤ ˜σ(θ)}. Therefore, we have to determine the subdifferential of the indicator function IB(θ) which is nonempty only for q∗∈ B(θ):
q ∈ ∂IB(θ)(q∗) = ∂D∗(q∗, θ)
⇔ IB(θ)(v) ≥ IB(θ)(q∗) + q : (v − q∗) ∀v ∈ B(θ)
⇔ 0 ≥ q : (v − q∗) ∀v ∈ B(θ). (4.20)
We multiply (4.20) with β > 0 and add a zero term in order to exploit the projection theorem, see Lemma 2.2.5.
⇔ (q∗− (q∗+ β q)) : (v − q∗) ≥ 0 ∀v ∈ B(θ), β > 0
⇔ q∗= projB(θ)(q∗+ β q). (4.21)
Using the fact that the orthogonal projection w.r.t. the Frobenius norm onto the ball B(θ) in (4.21) can be calculated explicitly by
q∗= projB(θ)(q∗+ β q) = ( q∗+ β q for |q∗+ β q| ≤ ˜σ(θ), ˜ σ(θ)|qq∗∗+β q+β q| else, = min (˜σ(θ), |q∗+ β q|) q ∗+ β q |q∗+ β q|,
4.3 Reformulation of the Viscoplastic Flow Rule 65
see Corollary 2.2.6, we obtain altogether
q ∈ ∂D∗(q∗, θ) ⇔ q∗= min (˜σ(θ), |q∗+ β q|) q
∗+ β q
|q∗+ β q|. (4.22)
Note that (4.22) implies |q∗| = |min (˜σ(θ), |q∗+ β q|)| ≤ ˜σ(θ). There- fore, the equivalence (4.22) extends to the case q∗ ∈ B(θ), when/ ∂D∗(q∗, θ) = ∅.
Finally we insert q ≡ ˙p ∈ R3×3dev and q∗≡ − ˙p + τ (u, p, θ) ∈ R 3×3 dev into
(4.22), choose β = and obtain the assertion via viscoplastic flow rule (1.3)
⇔ ˙p ∈ ∂D∗(− ˙p + τ (u, p, θ), θ) ⇔ ˙p = −−1min ˜ σ(θ) |τ (u, p, θ)|− 1, 0 τ (u, p, θ).
Remark 4.3.2 (Physical Interpretation of Viscoplasticity). The refor- mulation (4.19) of the viscoplastic flow rule (1.3) illustrates for the first time the effect of the viscosity term ˙p on the behavior of the plastic strain. Obviously, the plastic strain only develops if the right-hand side of (4.19) is non-zero which means that the Frobenius norm of the deviatoric part of the stress plus the back-stress has to exceed the yield stress,
[Φσ(u, p, θ) + Φχ(u, p, θ)]D =
[σ + χ]D > ˜σ(θ).
This is in contrast to the non-viscous case where plasticity only arises if the Frobenius norm of the deviatoric part of the stress and the back-stress is equal to the yield stress, see Section 2.3 and Section 2.4. We reduce the ODE (4.19) to a simpler one-dimensional equation with similar structure in order to motivate the influence of the plastic Perzyna viscosity. We study for given constants , σ ∈ R the ODE
˙p(t) = σ − p(t)
whose solution is given by p(t) = σ + B e−1t with B ∈ R. A creep test started with a plastic strain p(0) = σ1+ p1 where we generate
a certain constant stress σ = σ1< p1 during the test shows that the
plastic strain develops with the formula p(t) = σ1+ p1e−
1
t. We see that the viscoplastic response relaxes, i.e. p(t) → σ1for t → ∞, in order
to satisfy the yield condition for the purely plastic case.
Finally, we observe as in the viscoelastic setting, see Remark 2.4.1, that the plastic viscosity parameter influences the degree of the viscosity;
we reach half of the gap p1− σ1at time t = lnp12 p−σ11. Therefore, the
viscous effects decrease if → 0; compare the right picture in Figure 4.2.
t σ σ1 t p σ1 σ1+ p1
Figure 4.2. Creep test for a Perzyna viscoplastic material; time- stress diagram (left) and time-plastic strain diagram (right) for (solid, orange), 0.5 (dotted, dark-green), 0.1 (dashed, blue).