Chełmiński Krzysztof

On initial-boundary value problems for the inelastic material behaviour of metals

We prove existence of strong solutions for a subclass of noncoercive, monotone constitutive equations in the theory of inelastic material behaviour of metals.

1. Formulation of the initial-boundary value problem

Let Ω ⊆ IR^{3} be an open set with smooth boundary ∂Ω . This set represents the mass points of the body. Let u(x, t) ∈ IR^{3} be the displacement of the mass point x ∈ Ω at time t ≥ 0 , let ρ > 0 be the density, which we assume to be constant, and let T (x, t) ∈ S^{3} (the set of symmetric 3×3 matrices) be the Cauchy stress tensor. We study the problem

ρu_{tt}(x, t) = div_{x} T (x, t) , T (x, t) = D(ε(∇_{x}u(x, t)) − ε_{p}(x, t)) , z_{t}(x, t) = f(ε(∇_{x}u(x, t)), z(x, t)) . (P)

Here ε(∇_{x}u(x, t)) = \frac{1}{2}(∇_{x}u(x, t) + (∇_{x}u(x, t))^{T} ) ∈ S^{3} is the linear strain tensor, ε_{p}(x, t) ∈ S^{3} is the plastic strain tensor, and z(x, t) = (ε_{p}(x, t), z̃(x, t)) ∈ S^{3} × IR^{N−9} ⊆ IR^{N} is the vector of internal variables. D : S^{3} → S^{3} denotes a symmetric linear mapping, which is called the elasticity tensor. We assume that the mapping D is positive definite, and independent of x and t. Finally f : D(f) ⊆ S^{3} × IR^{N} → IR^{N} is a given function. We assume that one of the two homogeneous boundary conditions

u(x, t)_{|_{∂Ω}} = 0 or T (x, t)n(x)_{|_{∂Ω}} = 0 (BC)

is satisfied,(n is the exterior unit normal vector to ∂Ω . The initial conditions are

u(x, 0) = u^{(0)}(x) , u_{t}(x, 0) = u^{(1)}(x) , z(x, 0) = z^{(0)}(x) (IC)

with given functions u^{(0)} , u^{(1)} : Ω → IR^{3} , z^{(0)} : Ω → IR^{N} . We also require that there exists a function ψ : D(f) ⊆ S^{3} × IR^{N} → [0, ∞) with

ρ∇_{ε}ψ(ε, z) = T , ρ∇_{z}ψ(ε, z) · f(ε, z) ≤ 0

for all (ε, z) ∈ D(f) . The function ψ is called the free energy and the last inequality is called the dissipation inequality. From the constitutive equation (P2) it follows that ρψ(ε, z) = \frac{1}{2}[D(ε − ε_{p})] · (ε − ε_{p}) + ψ_{1}(z) with a suitable function ψ_{1} : D(ψ_{1}) ⊆ IR^{N} → [0, ∞) .

2. Existence theory for monotone and coercive constitutive equations

For mathematical reasons it is important to generalize equation (P2) and replace it by T (x, t) = D(ε(∇_{x}u(x, t)) − Bz(x, t)) , where B : IR^{N} → S^{3} is a given linear operator. For example we can choose for B the linear operator z → Bz = \frac{1}{2}(z_{p} + z_{p}^{T}) where z_{p} denotes the 3 × 3 matrix formed out of the first nine components of z. Moreover we assume that the free energy ψ has the form

ρψ(ε, z) = \frac{1}{2}[D(ε − Bz)] · (ε − Bz) + \frac{1}{2}(Lz) · z

with a symmetric, positive semi-definite N×N matrix L. Derivation of the above expression of ψ yields −ρ∇_{z}ψ(ε, z) = B^{T} D(ε − Bz) − Lz = \overline{L}ε − Mz with linear mapping \overline{L} = B^{T} D and M = B^{T} DB + L.

Definition . We say that a pair (f, B) of a function f : D(f) ⊆ S^{3} × IR^{N} → IR^{N} and a linear mapping B : IR^{N} → S^{3} is of monotone type, if there exists a symmetric, positive definite N×N matrix M with the property that the symmetric N×N matrix L = M − B^{T} DB is positive semi-definite, and if there exists a monotone function g : D(g) ⊆ IR^{N} → IR^{N} such that g(z) · z ≥ 0 for all z ∈ D(g) and

f(ε, z) = g(−ρ∇_{z}ψ(ε, z)) = g(B^{T} Dε − Mz)

for all (ε, z) ∈ D(f) , with the positive semi-definite quadratic form ρψ(ε, z) = \frac{1}{2}[D(ε − Bz)] · (ε − Bz) + \frac{1}{2}(Lz) · z .

We show that the following equivalent problem obtained from the initial–boundary value problem (P) by setting v = u_{t} has a unique solution:

v_{t} = div\frac{1}{ρ}D(ε − Bz) , ε_{t} = \frac{1}{2}(∇_{x}v + (∇_{x}v)^{T} ) , z_{t} = g( − ρ∇_{z}ψ(ε, z))

with the boundary condition (BC) and the initial conditions (IC). The solution is a mapping (v, ε, z) : Ω × [0, ∞) → IR^{3}×S^{3}×IR^{N} . We write the problem in the form w_{t}(t) = C(w(t)) , w(0) = w^{(0)} = (v^{(0)}, ε^{(0)}, z^{(0)}) of an initial value problem to an evolution equation with a maximal monotone operator −C, and with w(t) = (v(·, t), ε(·, t), z(·, t)) : Ω → IR^{3} × S^{3} × IR^{N} =: W .

T h e o r e m Let the quadratic form ρψ(ε, z) = \frac{1}{2}[D(ε − Bz)] · (ε − Bz) + \frac{1}{2}(Lz) · z be positive definite on S^{3} × IR^{N} and let g : IR^{N} → IR^{N} be maximal monotone with g(0) = 0 . Let Ω ⊆ IR^{3} be an open set for the Dirichlet boundary-value problem and let Ω be an open, bounded set with Lipschitz boundary for the Neumann boundaryvalue problem. Then the operator −C : L^{2}(Ω, W) → L^{2}(Ω, W) is maximal monotone, if the space L^{2}(Ω, W) is equiped with the scalar product

⟨(v, e, z), (\overline{v}, \overline{e}, \overline{z})⟩ = ∫_{Ω} ρv · \overline{v} + [D(e − Bz)] · (\overline{e} − B\overline{z}) + (Lz) · \overline{z} dx .

The theorem is proved in [1]. The theory of evolution equations to maximal monotone operators (see [2]) yields existence of global solutions to large initial data of our problem.

3. A noncoercive example - the monotone model of Bodner-Partom

In the model of Bodner-Partom the vector z of internal variables consists of the nine components of the plastic strain tensor ε_{p}, and of one positive variable y, which describes isotropic hardening. Thus N = 10 , and z(x, t) = (ε_{p}(x, t), y(x, t)) ∈ IR^{10} with y(x, t) > 0 . The constitutive equations are in the form

\frac{∂}{∂t} ε_{p} = F (\frac{|σ|}{y}) \frac{σ}{|σ|} , \frac{∂}{∂t} y = \frac{|σ|}{y} F (\frac{|σ|}{y})

where F : IR_{+} → IR_{+} , is a given function and σ = T − \frac{1}{3}(tr T) · I. It is proved, that the model is of monotone type only for F (s) = ds^{β} (a homogeneous power function) (β > 1). Thus the system of equations for the monotone model of Bodner-Partom has the form

\begin{matrix}ρv_{t} & = & div_{x} D(ε − ε_{p}) , & \frac{∂}{∂t} ε_{p} & = & d(\frac{|σ|}{y})^{β} \frac{σ}{|σ|} , \\ ε_{t} & = & \frac{1}{2}(∇_{x}v + (∇_{x}v)^{T} ) , & \frac{∂}{∂t} y & = & d(\frac{|σ|}{y})^{β+1} \end{matrix} (B-P)

(d > 0) with the free energy

ρψ(ε, ε_{p}, y) = \frac{1}{2}D(ε − ε_{p})(ε − ε_{p}) + \frac{β}{2(β + 1)}y^{2} .

The operator M(ε_{p}, y) = (B^{T} DB + L)(ε_{p}, y) = (Dε_{p}, \frac{β}{β+1}y) is positive definite, but L(ε_{p}, y) = (0, \frac{β}{β+1}y) is only semi-positive definite and consequently the model is noncoercive. We want now to approximate the noncoercive model of Bodner-Partom with a sequence of coercive models.

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ψ–positive semi-definite

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ψ–positive definite The rheological model corresponding to the coercive material shows that the stress deviator σ responsible for the nonelastic deformation is now “reduced” by the stress from the inserted spring. Thus the approximation satisfies

\begin{matrix}ρv_{t}^{k} & = & div_{x} D(ε^{k} − ε_{p}^{k}) & \frac{∂}{∂t} ε_{p}^{k} & = & d(\frac{|σ^{k} − \frac{1}{k} ε_{p}^{k}|}{y^{k}})^{β} \frac{σ^{k} − \frac{1}{k} ε_{p}^{k}}{|σ^{k} − \frac{1}{k} ε_{p}^{k}|} \\ ε_{t}^{k} & = & \frac{1}{2}(∇_{x}v^{k} + (∇_{x}v^{k})^{T} ) & \frac{∂}{∂t} y^{k} & = & d(\frac{|σ − \frac{1}{k} ε_{p}^{k}|}{y^{k}})^{β+1} \end{matrix} (C-B-P)

with the free energy

ρψ^{k}(ε^{k}, ε_{p}^{k}, y^{k}) = \frac{1}{2}D(ε^{k} − ε_{p}^{k})(ε^{k} − ε_{p}^{k}) + \frac{β}{2(β + 1)}(y^{k})^{2} + \frac{1}{2k} |ε_{p}^{k}|^{2} .

The operators M and L have the form

M(ε_{p}^{k}, y^{k}) = (B^{T} DB + L)(ε_{p}^{k}, y^{k}) = (Dε_{p}^{k} + \frac{1}{k} ε_{p}^{k}, \frac{β}{β + 1}y) L(ε_{p}^{k}, y^{k}) = (\frac{1}{k} ε_{p}^{k}, \frac{β}{β + 1}y)

and are positive definite. Hence by the existence theorem for coercive monotone problems we have for all k > 0 the unique solution

(v^{k}, ε^{k}, ε_{p}^{k}, y^{k}) ∈ W^{1,∞}([0, T), H)

to the problem (C-B-P) where H is the Hilbert space L^{2}(Ω, W) equiped with the scalar product defined by the usual L^{2}-scalar product for the first component v and by the free energy ψ^{k} for the other components and T > 0 is arbitrary. Thus for the approximate sequence we have

\begin{aligned}sup_{t∈[0,T)}(∥v^{k}(t)∥ + ∥ε^{k}(t) − ε_{p}^{k}(t)∥ + \frac{1}{\sqrt{k}} ∥ε_{p}^{k}(t)∥ + ∥y^{k}(t)∥) & < & C \\ sup_{t∈[0,T)}(∥∂_{t}v^{k}(t)∥ + ∥∂_{t}ε^{k}(t) − ∂_{t}ε_{p}^{k}(t)∥ + \frac{1}{\sqrt{k}} ∥∂_{t}ε_{p}^{k}(t)∥ + ∥∂_{t}y^{k}(t)∥) & < & C \end{aligned}

where the positive constant C does not depend on T . Moreover the special form of the constitutive equations for the Bodner-Partom model yields for almost every t ∈ [0, T)

∥∂_{t}ε_{p}^{k}(t)∥^{2} ≤ |Ω|^{\frac{1}{β+1}} ∥∂_{t}y^{k}(t)∥^{\frac{2β}{β+1}} .

and consequently

sup_{t∈[0,T)} ∥ε_{p}^{k}(t)∥ ≤ C(T) .

Next we prove L^{2} bounds for differences of two approximate solutions to the same initial data. Note that for a monotone approximation this kind of estimate is very naturall.

∂_{t} (\frac{1}{2}∥v^{k} − v^{m}∥^{2} + \frac{β}{2(β + 1)}∥y^{k} − y^{m}∥^{2} + \frac{1}{2} ∫_{Ω} D(ε^{k} − ε_{p}^{k} − ε^{m} + ε_{p}^{m})(ε^{k} − ε_{p}^{k} − ε^{m} + ε_{p}^{m}) dx)

= −(σ^{k} − \frac{1}{k} ε_{p}^{k} − σ^{m} + \frac{1}{m}ε_{p}^{m}, ∂_{t}ε_{p}^{k} − ∂_{t}ε_{p}^{m}) + \frac{β}{β + 1}(y^{k} − y^{m}, ∂_{t}y^{k} − ∂_{y}^{m}) − (\frac{1}{k} ε_{p}^{k} − \frac{1}{m}ε_{p}^{m}, ∂_{t}ε_{p}^{k} − ∂_{t}ε_{p}^{m})

≤ −(\frac{1}{k} ε_{p}^{k} − \frac{1}{m}ε_{p}^{m}, ∂_{t}ε_{p}^{k} − ∂_{t}ε_{p}^{m}) ≤ (\frac{1}{k} + \frac{1}{m})C(T)

This inequality yields for fixed T the strong convergence of the sequences v^{k}, ε^{k}−ε_{p}^{k}, y^{k} in the space L^{∞}([0, T), L^{2}(Ω)). Thus finally we can select a subsequence for which we have:

v^{k} → v , v_{t}^{k} ⇀^{∗} v_{t} , ε^{k} ⇀^{∗} ε , ε_{t}^{k} ⇀^{∗} ε_{t} , ε_{p}^{k} ⇀^{∗} ε_{p} , ∂_{t}ε_{p}^{k} ⇀^{∗} ∂_{t}ε_{p} , y^{k} → y , y_{t}^{k} ⇀ y_{t}^{∗} in the space L^{∞}([0, T), L^{2}(Ω)) .

Moreover we can assume that σ^{k} − \frac{1}{k} ε_{p}^{k} → σ = D(ε − ε_{p}) pointwise for almost every (t, x) ∈ [0, T) × Ω . This is enough to prove that the limit (v, ε, ε_{p}, y) is a strong solution of the problem (B-P).

4. Coercive limits for a subclass of monotone problems

We want now to generalize the phenomenon of the monotone Bodner-Partom model. Let us consider general constitutive equations of monotone type:

v_{t} = div\frac{1}{ρ}D(ε − Bz) , ε_{t} = \frac{1}{2}(∇_{x}v + (∇_{x}v)^{T} ) , z_{t} = g( − ρ∇_{z}ψ(ε, z))

with a continuous maximal monotone function g : IR^{N} → IR^{N} and with a suitable free energy function

ρψ(ε, z) = \frac{1}{2}[D(ε − Bz)] · (ε − Bz) + \frac{1}{2}(Lz) · z .

We assume that the operator M = B^{T} DB + L is positive definite and the operator L is positive semi-definite. We define now a coercive approximation of the problem: for all k > 0 let (v^{k}, ε^{k}, z^{k}) be the unique solution of the system

v_{t}^{k} = div\frac{1}{ρ}D(ε^{k} − Bz^{k}) , ε_{t}^{k} = \frac{1}{2}(∇_{x}v^{k} + (∇_{x}v^{k})^{T} ) , z_{t}^{k} = g( − ρ∇_{z}ψ^{k}(ε^{k}, z^{k}))

with the same function g and with the free energy in the form

ρψ^{k}(ε^{k}, z^{k}) = \frac{1}{2}[D(ε^{k} − Bz^{k})] · (ε^{k} − Bz^{k}) + \frac{1}{2}(Lz^{k}) · z^{k} + \frac{1}{2k} (Bz^{k} · Bz^{k}) .

We see that the approximate problem is coercive and monotone. Thus by the general existence theorem we have

(v^{k}, ε^{k}, ε_{p}^{k}, y^{k}) ∈ W^{1,∞}([0, T), H)

where H is the space L^{2}(Ω, W) with a suitable scalar product. Like the coercive approximation for the Bodner-Partom problem, our sequence satisfies

\begin{aligned}sup_{t∈[0,T)}(∥v^{k}(t)∥ + ∥ε^{k}(t) − Bz^{k}(t)∥ + \frac{1}{\sqrt{k}} ∥Bz^{k}(t)∥ + ∥L^{\frac{1}{2}} z^{k}(t)∥) & < & C \\ sup_{t∈[0,T)}(∥∂_{t}v^{k}(t)∥ + ∥∂_{t}ε^{k}(t) − ∂_{t}Bz^{k}(t)∥ + \frac{1}{\sqrt{k}} ∥∂_{t}Bz^{k}(t)∥ + ∥∂_{t}L^{\frac{1}{2}} z^{k}(t)∥) & < & C \end{aligned}

where the positive constant C does not depend on T . For the next step (estimation of differences of two approximate solutions for the same initial data) we need the following general assumption:

Assumption . There exists a continuous function F : IR_{+}^{2} → IR_{+} such that for all measurable functions y : Ω → IR^{N} with |y(x)| ≥ 1 for almost every x ∈ Ω it holds that

∥Bg(y)∥ ≤ F(∥L^{\frac{1}{2}} g(y)∥, ∥y∥) .

Note that for the Bodner-Partom’s model F(a, b) = da^{\frac{β}{β+1}} where d > 0 is a positive constant. Therefore as above we have

sup_{t∈[0,T)} (∥Bz^{k}(t)∥ + ∥Bz_{t}^{k}(t)∥) ≤ C(T) .

Consequently for differences of two approximate solutions we obtain

∂_{t} (\frac{1}{2}∥v^{k} − v^{m}∥^{2} + \frac{1}{2}∥L^{\frac{1}{2}} z^{k} − L^{\frac{1}{2}} z^{m}∥^{2} + \frac{1}{2} ∫_{Ω} D(ε^{k} − Bz^{k} − ε^{m} + Bz^{m})(ε^{k} − Bz^{k} − ε^{m} + Bz^{m}) dx)

≤ −(\frac{1}{k} Bz^{k} − \frac{1}{m}Bz^{m}, ∂_{t}Bz^{k} − ∂_{t}Bz^{m}) ≤ (\frac{1}{k} + \frac{1}{m})C(T)

Thus v^{k}, ε^{k} − Bz^{k}, L^{\frac{1}{2}} z^{k} are Cauchy sequences in the space L^{∞}([0, T), L^{2}(Ω)) . Finally we select a subsequence satisfying

(v^{k}, ε^{k}, z^{k}) ⇀^{∗} (v, ε, z) , (v_{t}^{k}, ε_{t}^{k}, z_{t}^{k}) ⇀^{∗} (v_{t}, ε_{t}, z_{t}) , (v^{k}, ε^{k} − Bz^{k}, L^{\frac{1}{2}} z^{k}) → (v, ε − Bz, L^{\frac{1}{2}} z)

in L^{∞}([0, T), L^{2}(Ω)) , and moreover

∇_{z}ψ^{k}(t, x) → ∇_{z}ψ(t, x)

pointwise for almost every (t, x) ∈ [0, T) × Ω . Thus the limit functions (v, ε, z) are solutions to the noncoercive problem.

5. References

1 Alber, H.D.: Initial-Boundary Value Problems for the Inelastic Material Behaviour of Metals, Lecture Notes in Mathe-

matics, Springer (in preparation). 2 Brezis, H.: Operateurs Maximaux Monotones, Mathematics Studies No. 5, North-Holland, (1973). 3 Chełmiński, K. – Gwiazda, P.: On the Model of Bodner-Partom with Nonhomogeneous Boundary Data, preprint TH

Darmstadt (in preparation).

Addresses: Dr. Chełmiński Krzysztof Technische Hochschule Darmstadt, Schloßgartenstr. 7, 64289 Darmstadt, Deutschland, e-mail chelminski@mathematik.th-darmstadt.de