On Self-Controlling Models in the Theory of Nonelastic Material Behavior of Metals

Krzysztof Chełmiński ^{∗}

Technical University of Darmstadt

Abstract

We discuss here solvability of systems modelling the nonelastic material behavior of metals under the assumption of monotonicity of the constitutive function and of the self-controlling property of the model under consideration. The main idea is to use the conception of coercive limits ([9, 10, 12]) and to prove a suitable convergence result. Examples of self-controlling models are presented at the end of the article.

AMS classification: 73F15, 73E60, 73G25, 35Q72

1 Introduction

This article presents existence and uniqueness of large, global in time solutions for dynamic equations in the theory of inelastic material behavior of metals. On the basis of the book [1] this paper summarizes results obtained by the author in [9, 10] and with P. Gwiazda in [11, 12]. At the end of the article we give examples of plastic constitutive equations, for which the presented theory works.

In this work we study systems of differential equations modelling the nonelastic material behavior of metals. Such systems consist of partial differential equations resulting from mechanical balance laws and of other equations, so called constitutive equations, which describe special properties of the material under consideration. Let us assume that an inelastic material occupies the bounded region Ω ⊂ ℝ^{3} with the smooth boundary ∂Ω . The system of equations describing the deformation process of the material can be written in the form:

\begin{aligned}ρu_{tt}(x, t) & = & div_{x} T (x, t) , \\ T (x, t) & = & D(ε(x, t) − ε^{p}(x, t)) , \\ z_{t}(x, t) & = & f(ε(x, t), z(x, t)) . \end{aligned}  (GS)

where u : Ω × (0, T) → ℝ^{3} is the displacement vector, T : Ω × (0, T) → S^{3} = ℝ_{sym}^{3×3} is the stress tensor, ρ > 0 is the constant mass density, ε = \frac{1}{2}(∇u + ∇^{T} u) is the linearized strain tensor, D : S^{3} → S^{3} is the elasticity tensor, which we assume to be symmetric, positive definite and independent of x and t, ε^{p} : Ω × (0, T) → S^{3} is the plastic strain tensor, z = (ε^{p}, z̃) : Ω × (0, T) → ℝ^{N} is the vector of internal variables, and f : D(f) ⊂ S^{3} × ℝ^{N} → ℝ^{N} is a given constitutive function. Thus the general system (GS) consists of the balance of momentum (GS1) and of two constitutive equations. The first one (GS2) is an algebraic equation, which describe the linear stress - elastic strain relation and the second one (GS3) is called the plastic constitutive equation and is proposed in the form of an ordinary differential equation. Note that the system (GS) is nonlinear and all nonlinearities of the system are contained within the given constitutive function f. Therefore properties of f such continuity, differentiability, monotonicity, growth conditions influence existence, uniqueness and regularity of solutions for the system (GS). Thus we should study the class of admissible constitutive functions f. The first restriction for the class results from thermodynamical considerations. We require that there exists a function ψ : D(f) ⊂ S^{3} × ℝ^{N} → ℝ_{+} satisfying for all (ε, z) ∈ D(f) :

∗Supported by Sonderforschungsbereich 298.

ρ∇_{ε}ψ(ε, z) = T ,  (1.1)
ρ∇_{z}ψ(ε, z) · f(ε, z) ≤ 0 .  (1.2)

The function ψ is called free energy and the inequality (1.2) is called the dissipation inequality. Note that the inequality (1.2) restricts the class of constitutive functions f. Moreover the existence of the free energy satisfying (1.1) and (1.2) yields dissipation of the total energy for solutions of the system (GS), provided homogeneous boundary conditions of Dirichlet or Neumann type are prescribed. Let us assume that (u, z) is a smooth solution for the system (GS) with u(x, t) = 0 for (x, t) ∈ ∂Ω×[0, T) or D(ε(x, t)−ε^{p}(x, t))·n(x) = 0 (here n(x) denotes the exterior unit normal vector to ∂Ω at x ∈ ∂Ω ). Then we have

\frac{d}{dt} ∫_{Ω} ρ\{\frac{1}{2}|u_{t}(x, t)|^{2} + ψ(ε(x, t), z(x, t))\} dx

= ∫_{Ω} {ρu_{t}(x, t) · u_{tt}(x, t) + ρ∇_{ε}ψ(ε(x, t), z(x, t)) · ε_{t}(x, t)

+ ρ∇_{z}ψ(ε(x, t), z(x, t)) · z_{t}(x, t)} dx
= ∫_{Ω} {ρu_{t}(x, t) · u_{tt}(x, t) + T (x, t) · ∇u_{t}(x, t)  (1.3)
+ ρ∇_{z}ψ(ε(x, t), z(x, t)) · z_{t}(x, t)} dx

= ∫_{∂Ω} T (x, t) n(x) · u_{t}(x, t) dS(x)

+ ∫_{Ω} ρ∇_{z}ψ(ε(x, t), z(x, t)) · z_{t}(x, t) dx ≤ 0

which denotes that the sum of the kinetic energy and the free energy does not increase in time. The constitutive equation (GS2) and the requirement (1.1) imply that the free energy function ψ must be of the form

ρψ(ε, z) = \frac{1}{2}[D(ε − ε^{p})] · (ε − ε^{p}) + ψ_{1}(z) .  (1.4)

where the function ψ_{1} should be chosen so, that (1.2) holds. The next restriction for the class of admissible constitutive functions is to suppose that the function ψ_{1} is a positive semi-definite quadratic form and thus the free energy ψ is quadratic in all variables:

ρψ(ε, z) = \frac{1}{2}[D(ε − Bz)] · (ε − Bz) + \frac{1}{2}(Lz) · z ,  (1.5)

where B : ℝ^{N} → S^{3} is the linear operator defined by Bz = ε^{p} and L : ℝ^{N} → ℝ^{N} is a linear symmetric positive semi-definite operator. The assumption (1.5) implies that the gradient of ψ is linear and the gradient with respect to z has the form

−ρ∇_{z}ψ(ε, z) = B^{T} D(ε − Bz) − Lz =^{df} \overline{L}ε − Mz

with linear operators \overline{L} : S^{3} → ℝ^{N} , M : ℝ^{N} → ℝ^{N} defined by \overline{L}ε = B^{T} Dε and Mz = B^{T} DBz + Lz. Note that the operator M is always symmetric and positive semi-definite

Mz · z = B^{T} DBz · z + Lz · z = DBz · Bz + Lz · z ≥ 0 .  (1.6)

Now we will try to prove an estimate similar to (1.3) for differences of two smooth solutions for the system (GS). Let us assume that (u, z) and (\overline{u}, \overline{z}) are two smooth solutions of the system (GS) satisfying boundary condition of Dirichlet or Neumann type with the same boundary data. Then the linearity of the gradient of the free energy ψ yields

\frac{d}{dt} ∫_{Ω} ρ(\frac{1}{2}|u_{t} − \overline{u}_{t}|^{2} + ψ(ε − \overline{ε}, z − \overline{z})) dx

= − ∫_{Ω}[−ρ∇_{z}ψ(ε, z) + ρ∇_{z}ψ(\overline{ε}, \overline{z})] · (f(ε, z) − f(\overline{ε}, \overline{z})) dx .  (1.7)

Thus if we suppose that the constitutive function f has the form

f(ε, z) = g( − ρ∇_{z}ψ(ε, z))

with a monotone function g : ℝ^{N} → ℝ^{N} satisfying g(0) = 0 then first: the dissipation inequality holds

ρ∇_{z}ψ(ε, z) · g( − ρ∇_{z}ψ(ε, z)) = −( − ρ∇_{z}ψ(ε, z) − 0) · (g( − ρ∇_{z}ψ(ε, z)) − g(0)) ≤ 0

and second: the right hand side of the equality (1.7) is non-positive

\frac{d}{dt} ∫_{Ω} ρ(\frac{1}{2}|u_{t} − \overline{u}_{t}|^{2} + ψ(ε − \overline{ε}, z − \overline{z})) dx ≤ 0 .  (1.8)

If we additionally suppose that (u, z) and (\overline{u}, \overline{z}) are solutions to the same initial data then we obtain

∫_{Ω} |(u_{t} − \overline{u}_{t})|^{2} dx = 0 ,  ∫_{Ω} ψ(ε − \overline{ε}, z − \overline{z}) dx = 0 .  (1.9)

The first statement denotes the uniqueness of the velocity vector u_{t} = \overline{u}_{t} and consequently the uniqueness of the displacement u = \overline{u} and of the strain ε = \overline{ε}. We insert the last information into the second equality in (1.9) and obtain

0 = ∫_{Ω} ρψ(0, z − \overline{z}) dx = ∫_{Ω} \frac{1}{2}[(B^{T} DB + L)(z − \overline{z})] · (z − \overline{z}) dx
= ∫_{Ω} \frac{1}{2}[M(z − \overline{z})] · (z − \overline{z}) dx .

Thus we can prove the uniqueness of the vector z, provided that the operator M is positive definite. Note that M is defined as the sum of two symmetric positive semi-definite operators B^{T} DB and L. The assumption that M is positive definite implies that ker B ∩ ker L = {0}. From the symmetry of L we conclude additionally that range (L)⊥ ker L and consequently ker L ⊂ {(ε^{p}, 0) | ε^{p} ∈ S^{3}} = range (B) × {0}. Thus the operator L must be positive definite on the subspace {(ε^{p}, z̃) | ε^{p} = 0} and can only be degenerate in the “plastic direction”.

2 Monotone plastic constitutive equations

In this section we start with the definition of plastic constitutive equations of monotone type, which was introduced in [1]. We follow the notations from the book [1] and have:

Definition 2.1 a)We say that the plastic constitutive equation

z_{t} = f(ε, z)

where f : D(f) ⊂ S^{3} × ℝ^{N} → ℝ^{N} is of pre-monotone type, if and only if there exists a symmetric, positive definite N × N matrix M with the property that the symmetric N × N matrix L =^{df} M − B^{T} DB is positive semi-definite, and there exists a function g : D(g) ⊂ ℝ^{N} → ℝ^{N} such that

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 .

b) If additionally the function g is monotone and satisfies

∀_{z∈D(g)} g(z) · z ≥ 0 ,  (2.1)

then we say that the plastic constitutive equation is of monotone type or belongs to the class M.

Note that the condition (2.1) for continuous, monotone functions g with 0 ∈ D(g) is equivalent to 0 = g(0). Let us set v = u_{t}. Then the initial-boundary value problem with plastic constitutive equation of monotone type can be written in the form:

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

with the free energy

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

where L is a symmetric positive semi-definite operator L : ℝ^{N} → ℝ^{N} and the operator M = B^{T} DB +L is positive definite. Moreover we assume that the homogeneous Dirichlet boundary condition

v(x, t) = 0 for x ∈ ∂Ω and t ≥ 0 ,  (2.2)

or the homogeneous Neumann boundary condition

D(ε − Bz)(x, t) · n(x) = 0 for x ∈ ∂Ω and t ≥ 0  (2.3)

is satisfied, where n(x) denotes the exterior unit normal vector to ∂Ω at x ∈ ∂Ω . Finally the initial conditions are

\begin{aligned}v(x, 0) & = & v^{0}(x) , \\ ε(x, 0) & = & ε^{0}(x) , \\ z(x, 0) & = & z^{0}(x) \end{aligned}  (IC)

with given functions v^{0} : Ω → ℝ^{3} , ε^{0} : Ω → S^{3} , z^{0} : Ω → ℝ^{N}. Let us denote by C the operator generated by the right hand side of the system (MS)

C(v, ε, z) = (\frac{1}{ρ}divD(ε − Bz), \frac{1}{2}(∇v + ∇^{T} v), g(−ρ∇_{z}ψ(ε, z)))

with the domain

D_{D} = {(v, ε, z) ∈ 𝕃^{2}(Ω; ℝ^{3} × S^{3} × ℝ^{N}) | v ∈ ℍ_{0}^{1}(Ω; ℝ^{3})
and the right hand side of (MS) belongs to 𝕃^{2}(Ω; ℝ^{3} × S^{3} × ℝ^{N})}

for the homogeneous Dirichlet boundary condition (here ℍ_{0}^{1}(Ω; ℝ^{3}) denotes the closed subspace of the Sobolev space ℍ^{1}(Ω; ℝ^{3}) containing all functions with zero boundary value) or with

D_{N} = {(v, ε, z) ∈ 𝕃^{2}(Ω; ℝ^{3} × S^{3} × ℝ^{N}) | v ∈ ℍ^{1}(Ω; ℝ^{3}) ,

the right hand side of (MS) belongs to 𝕃^{2}(Ω; ℝ^{3} × S^{3} × ℝ^{N})

and the weak divergence divD(ε − Bz) satisfies

∀ u ∈ ℍ^{1}(Ω; ℝ^{3}) (divD(ε − Bz), u) = −(D(ε − Bz), ∇u)} .

for the homogeneous Neumann boundary data. Moreover by ⟨·, ·⟩_{ψ} we denote the following bilinear form generated by the free energy function ψ

⟨(v, ε, z), (v̄, ε̄, z̄)⟩_{ψ} = ∫_{Ω} (ρv · v̄ + [D(ε − Bz)] · (ε̄− Bz̄) + Lz · z̄) dx  (2.4)

and by |(v, ε, z)|_{ψ} = \sqrt{⟨(v, ε, z), (v, ε, z)⟩_{ψ}} the semi-norm defined by the bilinear form (2.4). The fundamental property of the system (MS) is monotonicity of the operator −C in the bilinear form ⟨·, ·⟩_{ψ}.

⟨C(v, ε, z) − C(\overline{v}, \overline{ε}, \overline{z}), (v, ε, z) − (\overline{v}, \overline{ε}, \overline{z})⟩_{ψ}

= ∫_{Ω} div(D(ε − Bz) − D(\overline{ε} − B\overline{z})) · (v − \overline{v}) dx

+ ∫_{Ω} (D(\frac{1}{2}(∇(v − \overline{v}) + ∇^{T} (v − \overline{v})) − Bg(−ρ∇_{z}ψ(ε, z)) + Bg(−ρ∇_{z}ψ(\overline{ε}, \overline{z}))))

·(ε − \overline{ε} − Bz + B\overline{z}) dx + ∫_{Ω} L(g(−ρ∇_{z}ψ(ε, z)) − g(−ρ∇_{z}ψ(\overline{ε}, \overline{z}))) · (z − \overline{z}) dx

(by the symmetry of D and by the definition of D_{D} or D_{N} )

= − ∫_{Ω} B^{T} D(ε − \overline{ε} − Bz + B\overline{z}) · (g(−ρ∇_{z}ψ(ε, z)) − g(−ρ∇_{z}ψ(\overline{ε}, \overline{z}))) dx + ∫_{Ω} L(g(−ρ∇_{z}ψ(ε, z)) − g(−ρ∇_{z}ψ(\overline{ε}, \overline{z}))) · (z − \overline{z}) dx = (by the symmetry of L)

= − ∫_{Ω} (B^{T} D(ε − \overline{ε}) − B^{T} DB(z − \overline{z}) − L(z − \overline{z}))
· (g(−ρ∇_{z}ψ(ε, z)) − g(−ρ∇_{z}ψ(\overline{ε}, \overline{z}))) dx
= − ∫_{Ω} ( − ρ∇_{z}ψ(ε, z) + ρ∇_{z}ψ(\overline{ε}, \overline{z})) · (g(−ρ∇_{z}ψ(ε, z)) − g(−ρ∇_{z}ψ(\overline{ε}, \overline{z}))) dx
(by the monotonicity of g) ≤ 0 .  (2.5)

Thus using the standard monotonicity method, (2.5) yields for two solutions (v, ε, z), (\overline{v}, \overline{ε}, \overline{z}) of the system (MS) which belongs to the domain of C:

\frac{d}{dt}|(v, ε, z)(t) − (\overline{v}, \overline{ε}, \overline{z})(t)|_{ψ}^{2}

= ⟨(v_{t}, ε_{t}, z_{t})(t) − (\overline{v}_{t}, \overline{ε}_{t}, \overline{z}_{t})(t), (v, ε, z)(t) − (\overline{v}, \overline{ε}, \overline{z})(t)⟩_{ψ} = ⟨C(v, ε, z)(t) − C(\overline{v}, \overline{ε}, \overline{z})(t), (v, ε, z)(t) − (\overline{v}, \overline{ε}, \overline{z})(t)⟩_{ψ} ≤ 0

and denotes that the semi-distance defined by the semi-norm | · |_{ψ} between trajectories does not increase in time

|(v, ε, z)(t) − (\overline{v}, \overline{ε}, \overline{z})(t)|_{ψ} ≤ |(v, ε, z)(0) − (\overline{v}, \overline{ε}, \overline{z})(0)|_{ψ} .  (2.6)

If we choose (\overline{v}, \overline{ε}, \overline{z})(t) = (v, ε, z)(t + h) with the initial data (v, ε, z)(h) for h > 0 then (2.6) implies

|(v, ε, z)(t + h) − (v, ε, z)(t)|_{ψ} ≤ |(v, ε, z)(h) − (v, ε, z)(0)|_{ψ} .  (2.7)

Dividing (2.7) by h and letting h tend to 0 we obtain

|(v_{t}, ε_{t}, z_{t})(t)|_{ψ} ≤ |(v_{t}, ε_{t}, z_{t})(0)|_{ψ}  (2.8)

and the left side of the system (MS) is controlled in the semi-norm | · |_{ψ}. If the quadratic form ψ is positive definite then the semi-norm | · |_{ψ} becomes a norm equivalent to the standard norm in the space 𝕃^{2}(Ω; ℝ^{3} × S^{3} × ℝ^{N}) and (2.8) implies boundedness of all components of the left hand side of the system (MS) in 𝕃^{2}(Ω; ℝ^{3} ×S^{3} ×ℝ^{N}). Such models for which the quadratic form ψ is positive definite we call coercive. Then for coercive models with the plastic constitutive equation from the class M we have the following existence theorem proved in [1].

Theorem 2.2 Let us assume that the quadratic form ψ is positive definite and the monotone function g : ℝ^{N} → ℝ^{N} is continuous with g(0) = 0. Then for all initial data (v^{0}, ε^{0}, z^{0}) from the domain of the operator C there exists a unique global in time solution (v, ε, z)(t) satisfying

|(v_{t}, ε_{t}, z_{t})(t)|_{ψ} ≤ |C(v^{0}, ε^{0}, z^{0})|_{ψ} .  (2.9)

Let us now study the noncoercive case. The main idea is to approximate noncoercive models by a sequence of coercive models. The fundamental question is: how should we choose a “good” approximation process? Here the mechanical considerations help us to answer the question. We propose two types of coercive approximations.

A. Internal coercive approximation. Noncoercive models with the rheological diagram presented by the figure below (ker L = range (B) × {0})

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ψ-positive semi-definite

(here σ = P T = T − \frac{1}{3}(trT) · I) we approximate with coercive models with the rheological diagram

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ψ-positive definite

(here σ^{p,k} = P Dε^{p,k}). Thus the system of equations for the coercive approximation is in the form

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

and the quadratic form ψ^{k} is defined by

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} DBz^{k} · Bz^{k} . (2.10)

Moreover we assume that the homogeneous Dirichlet or Neumann boundary condition holds and the initial conditions are

v^{k}(0) = v^{0} , ε^{k}(0) = ε^{0} , z^{k}(0) = (\frac{k}{k + 1}ε^{p,0}, z̃^{0}) .  (2.11)

Note that we have changed the initial data only for the plastic strain. The quadratic form ψ^{k} is positive definite and Theorem 2.2 yields the existence of global in time solutions for all k ∈ ℕ \ {0} .

B. External coercive approximation. The idea of the external coercive approximation is motivated by the following rheological diagram:

...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................  ............................................................................................................................................................................................................................................................................................................................................................................................................................... ........................................................................................................................................................................................................ general noncoercive model external coercive approximation \frac{1}{k} Dε^{k} ..........................................................................................................................................................................................................................................................................................................................................................................................................

Therefore we require that the sequence (v^{k}, ε^{k}, z^{k}) satisfies the system

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

with the homogeneous Dirichlet boundary condition

v^{k}(x, t) = 0 for x ∈ ∂Ω and t ≥ 0 ,

or with the homogeneous Neumann boundary condition

D(ε^{k} − Bz^{k} + \frac{1}{k}ε^{k})(x, t) · n(x) = 0 for x ∈ ∂Ω and t ≥ 0

and with the initial conditions

v(0) = v^{0} , ε(0) = ε^{0} , z(0) = z^{0} .

The quadratic form ψ^{k} is defined now by

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} (D(ε^{k})) · ε^{k} . (2.12)

Note that for the external approximation process the gradient ∇_{z}ψ^{k} is “constant” for all k > 0 (∇_{z}ψ^{k}(ε^{k}, z^{k}) = ∇_{z}ψ(ε^{k}, z^{k})) and therefore the argument of the nonlinear function g is the same as in the noncoercive problem. The quadratic form ψ^{k} is positive definite on S^{3} × ℝ^{N} but we cannot use Theorem 2.2. The first equation (ECA1) contains the perturbation term \frac{1}{k}divDε^{k} and is not in the form of equation (MS1). However we can try to prove an analogous theorem to Theorem 2.2 for the system (ECA). This is done in the article [10] and thus for all k > 0 the problem (ECA) possesses global in time unique solutions.

Next we want to study the limit case k → ∞. In general the convergence of the process (ICA) or (ECA) to a solution of the system (MS) is an open problem. We can prove that this holds for a special subclass of monotone problems. Therefore we define the class of self-controlling models.

Definition 2.3 We say that the model (MS) is self-controlling if and only if there exists a function F : ℝ_{+}^{2} → ℝ_{+} mapping sets, bounded in at least one variable, into bounded sets such that for all functions y ∈ 𝕃^{2}(Ω; ℝ^{N}) with |y(x)| ≥ 1 for a.e. x ∈ Ω it holds that

∥Bg(y)∥ ≤ F(∥Lg(y)∥, ∥y∥) .  (2.13)

Inequality (2.13) says that we can control the right hand side of the ordinary differential equation for the plastic strain by the right hand side of the equations for the variables Lz or by the argument of the nonlinear function g. Note that all bounded functions Bg satisfy (2.13) automatically with constant F and lipschitz functions Bg satisfy (2.13) with F in the form F(a, b) = c · b + d with positive constants c, d > 0. The main result for the self-controlling models is the following:

Theorem 2.3 For self-controlling models the approximate sequence (v^{k}, ε^{k}, z^{k}) defined by the process (ICA) or (ECA) contains a subsequence which converges to a solution of the system (MS), provided the initial data satisfies: div Dε^{0} ∈ 𝕃^{2}(Ω; ℝ^{3}) and for the Neumann boundary condition Dε^{0}(x) · n(x) = 0 for x ∈ ∂Ω.

Theorem 2.3 is proved in paragraphs 5 and 6 of the article [10]. Therefore we sketch only the proof for the process (ICA). For the process (ECA) the main idea of the proof is the same.

Sketch of the proof for the process (ICA). From Theorem 2.2 we obtain

|(v_{t}^{k}(t), ε_{t}^{k}(t), z_{t}^{k}(t)|_{ψ^{k}} ≤ |C^{k}(v^{0}, ε^{0}, z^{0,k})|_{ψ^{k}}  (2.14)

where the operator C^{k} is defined similar to the operator C. Using the assumption divDε^{0} ∈ 𝕃^{2}(Ω; ℝ^{3}) we prove that the right hand side of (2.14) is bounded. Therefore there exists a positive constant C independent of k such that

\{∥v_{t}^{k}(t)∥, ∥ε_{t}^{k}(t) − Bz_{t}^{k}(t)∥, ∥Lz_{t}^{k}(t)∥, \frac{1}{\sqrt{k}} ∥Bz_{t}^{k}(t)∥\} ≤ C .  (2.15)

Now by the self-controlling property of the model (MS) we conclude that

∥Bz_{t}^{k}(t)∥^{2} = ∥Bg(y^{k}(t))∥^{2} ≤ C_{∗}^{2}|Ω| + F^{2}(∥Lg(y^{k}(t))∥, ∥y^{k}(t)∥)

where y^{k}(t) = −ρ∇_{z}ψ^{k}(ε^{k}(t), z^{k}(t)) , C_{∗} = sup_{y∈B_{N} (1)} |Bg(y)| and B_{N}(1) denotes the unit ball in ℝ^{N} . Inequality (2.15) yields that the arguments ∥Lg(y^{k}(t))∥ and ∥y^{k}(t)∥ of the function F are bounded and we can remove the factor \frac{1}{\sqrt{k}} in inequality (2.15). Next using the boundedness of the sequence Bz_{t}^{k} we prove with the energy method that the sequence of arguments −ρ∇_{z}ψ^{k}(ε^{k}(t), z^{k}(t)) of the nonlinear function g converges strongly. The last statement with the weak convergence of the left hand sides of the system (ICA) ends the proof.

3 Lipschitz perturbations of the class M

Mechanical models, which we find in the literature are often of pre-monotone type (see examples in the next section). Thus the plastic constitutive equation is of the form

z_{t} = g( − ρ∇_{z}ψ(ε, z))  (3.1)

with continuous g : ℝ^{N} → ℝ^{N} satisfying g(0) = 0 and with the free energy ψ given by the quadratic form

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

with a symmetric positive semi-definite operator L : ℝ^{N} → ℝ^{N} and with the positive definite operator M = B^{T} DB + L. Unfortunately the very strong requirement that g is monotone so that the model belongs to the class M, is seldom satisfied in practice. Therefore we want to enlarge the class M using the simple idea of lipschitz perturbations. We want to replace the strong monotonicity assumption of the function g by a weaker requirement.

Definition 3.1 We say that the plastic constitutive equation (3.1) belongs to the class LM if and only if it is of pre-monotone type and there exists a lipschitz operator L : ℝ^{N} → ℝ^{N} with L(0) = 0 such that the function g + L is monotone.

Definition 3.1 implies immediately that M ⊂ LM. Moreover if the model under consideration belongs to the class LM then we can study the lipschitz perturbed plastic constitutive equation

z_{t} = g( − ρ∇_{z}ψ(ε, z)) + L( − ρ∇_{z}ψ(ε, z))  (3.2)

which is of monotone type.

Corollary 3.2 The plastic constitutive equation (3.1) belongs to the class LM if and only if

∃_{l>0} ∀_{z,z′∈ℝ^{N}}  (g(z) − g(z′)) · (z − z′) ≥ −l|z − z′|^{2} .  (3.3)

Proof. ⇒ Let L : ℝ^{N} → ℝ^{N} be a lipschitz operator such that the function g + L is monotone. Then

∀_{z,z′∈ℝ^{N}} (g(z) − g(z′)) · (z − z′) ≥ −(L(z) − L(z′)) · (z − z′) ≥ −l|z − z′|^{2}

where l > 0 is the lipschitz constant of the operator L . ⇐ Let us put L = l · I . Then L(0) = 0 and (3.3) yields the monotonicity of the function g + L.

Corollary 3.2 implies that the general lipschitz perturbation L can be always replaced by a homothety l · I with a positive constant l. Let us suppose that (v, ε, z)(t) is a smooth solution for the system (MS) with the plastic constitutive equation from the class LM satisfying the homogeneous Dirichlet or Neumann boundary condition. Then we have

\frac{d}{dt} ∫_{Ω} ρ\{\frac{1}{2}|v(x, t)|^{2} + ψ(ε(x, t), z(x, t))\} dx

= − ∫_{Ω} ( − ρ∇_{z}ψ(ε(x, t), z(x, t))) · g( − ρ∇_{z}ψ(ε(x, t), z(x, t))) dx (3.4)
≤ (by the inequality (3.3)) ≤ l ∫_{Ω} | − ρ∇_{z}ψ(ε(x, t), z(x, t))|^{2}dx .

Using that the quadratic form ψ is positive semi-definite we obtain that there exists a positive constant C such that

| − ρ∇_{z}ψ(ε, z)|^{2} = |B^{T} D(ε − Bz) − Lz|^{2} ≤ Cρψ(ε, z) .  (3.5)

Inserting estimate (3.5) into (3.4) we get

\frac{d}{dt} ∫_{Ω} ρ(|v|^{2} + ψ(ε, z)) dx ≤ C ∫_{Ω} ρψ(ε, z) dx ≤ C ∫_{Ω} ρ(|v|^{2} + ψ(ε, z)) dx .  (3.6)

The inequality (3.6) yields that the total energy for models from the class LM can increase exponentially, but does not blow up in finite time. In the same manner as in the previous section we can verify uniqueness of smooth solutions for the system (MS) with constitutive plastic equations from the class LM. Next we study the existence theory for such systems.

Theorem 3.3 (existence theorem for coercive models) Let the plastic constitutive equation (3.1) belong to the class LM with continuous function g : ℝ^{N} → ℝ^{N} satisfying g(0) = 0 and the quadratic form ψ be positive definite. Then for all initial data from the domain of the operator C the system (MS) possesses unique global solutions.

Proof. From Corollary 3.2 there exists l > 0 such that the problem

v_{t}(x, t) = \frac{1}{ρ}divD(ε(x, t) − Bz(x, t))
ε_{t}(x, t) = \frac{1}{2}(∇v(x, t) + ∇^{T} v(x, t))  (3.7)
z_{t}(x, t) = g( − ρ∇_{z}ψ(ε(x, t), z(x, t))) − lρ∇_{z}ψ(ε(x, t), z(x, t))

with the boundary conditions (2.2) or (2.3) and with the initial conditions (IC) is of monotone type. Therefore we can use Theorem 2.2 and obtain that the system (3.7) has unique global solutions to all initial data from the domain of the operator C_{l} generated by the right hand side of (3.7). It is easy to see that that the operator C from the problem (MS) can be written as the sum

C(v, ε, z) = C_{l}(v, ε, z) + T_{l}(v, ε, z)

with the lipschitz operator

T_{l}(v, ε, z) = (0, 0, −l(\overline{L}ε − Mz)) .  (3.8)

Thus the domain of the operator C is equal to the domain of the operator C_{l}. Then the general theory of lipschitz perturbations of the monotone operators presented in [4] (theorem 3.17 and remark 3.14) completes the proof.

For noncoercive models with the plastic constitutive equation from the class LM we use again the idea of coercive limits, provided that the system (MS) is self-controlling.

Theorem 3.4 (existence theorem for self-controlling models) Let the plastic constitutive equation (3.1) belong to the class LM with continuous function g : ℝ^{N} → ℝ^{N} satisfying g(0) = 0. Moreover suppose that the model (MS) has the selfcontrolling property (2.13). Then for all initial data from the domain of the operator C which satisfy

i) div Dε^{0} ∈ 𝕃^{2}(Ω; ℝ^{3})
ii) for the Neumann boundary data Dε^{0}(x) · n(x) = 0 for a.e. x ∈ ∂Ω

the system (MS) possesses unique global solutions.

Proof. For the system (3.7), which is monotone but noncoercive, we use the external coercive approximation process. Thus the system of equations is now of the form

v_{t}(x, t) = \frac{1}{ρ}divD(ε(x, t) − Bz(x, t) + \frac{1}{k}ε(x, t))
ε_{t}(x, t) = \frac{1}{2}(∇v(x, t) + ∇^{T} v(x, t))  (3.9)
z_{t}(x, t) = g( − ρ∇_{z}ψ^{k}(ε(x, t), z(x, t))) − lρ∇_{z}ψ^{k}(ε(x, t), z(x, t))

for natural numbers k > 0. As usual we assume that the boundary condition (2.2) or (2.3) holds and the initial conditions are given by (IC). Following the idea of the external coercive approximation the free energy function ψ^{k} is defined by

ρψ^{k}(ε, z) = \frac{1}{2}[D(ε − Bz)] · (ε − Bz) + \frac{1}{2}(Lz) · z + \frac{1}{2k} (Dε) · ε .  (3.10)

The system (3.9) is written in the same form as the problem (ECA) (the function g + l · I is monotone) and thus for initial data from the domain of the operator C_{l,k} generated by the right hand side of (3.9) we obtain the existence of unique global solutions. Next we perturb the system (3.9) by the the lipschitz operator T_{l} defined by (3.8) and obtain the same result for the system

v_{t}(x, t) = \frac{1}{ρ}divD(ε(x, t) − Bz(x, t) + \frac{1}{k}ε(x, t))
ε_{t}(x, t) = \frac{1}{2}(∇v(x, t) + ∇^{T} v(x, t))  (3.11)
z_{t}(x, t) = g( − ρ∇_{z}ψ^{k}(ε(x, t), z(x, t))) .

The last step in the proof is to use for the system (3.11) the idea of coercive limits for the self-controlling models, presented in Theorem 2.3.

4 Examples

In the last section we discuss the applicability of the theory presented in this work. We show that several plastic constitutive equations which are used in engineering practice belong to the classes defined above.

1. The monotone model of Bodner-Partom In this model the vector z of internal variables consists of the six components of the inelastic strain tensor ε^{p}, and of one positive variable y, which describes isotropic hardening. The plastic constitutive equation is in the form

\begin{aligned}ε_{t}^{p} & = & F (\frac{|σ|}{y}) \frac{σ}{|σ|} , \\ y_{t} & = & \frac{|σ|}{y} F (\frac{|σ|}{y}) , \end{aligned}  (4.1)

with a given function F : ℝ_{+} → ℝ_{+} . Here σ = T − \frac{1}{3} (tr T) · I is the stress deviator. It is proved in [1], that this model is of monotone type only for F (s) = ds^{β}, d > 0, β > 1 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} .

Note that the last coefficient in the quadratic form ψ depends on the power of the function F. The operator L(ε^{p}, y) = (0, \frac{β}{β+1} y) is only positive semi-definite and the operator M(ε^{p}, y) = (B^{T} DB + L)(ε^{p}, y) = (Dε^{p}, \frac{β}{β+1} y) is positive definite and the model is noncoercive. The simple calculation

∥ε_{t}^{p}(t)∥^{2} = d^{2} ∫_{Ω} (\frac{|σ(t)|}{y(t)})^{2β}dx
≤ d^{2}\{ ∫_{Ω} (\frac{|σ(t)|}{y(t)})^{2β+2}dx\}^{\frac{β}{β+1}} · |Ω|^{\frac{1}{β+1}}
= d^{\frac{2}{β+1}} ∥y_{t}(t)∥^{\frac{2β}{β+1}} · |Ω|^{\frac{1}{β+1}} .

verifies that the plastic constitutive equation is self-controlling (we can choose the function F in the form F(a, b) = c · a^{\frac{β}{β+1}} with a suitable positive constant c).

2. The true model of Bodner-Partom This model proposed by S.R. Bodner and Y. Partom in [3] has similar structure to (4.1) and thus the vector of internal variables z contains again the same components (ε^{p}, y). The plastic constitutive equation are given by the system

\begin{aligned}ε_{t}^{p} & = & d exp ( − α(\frac{y}{|σ|})^{n})\frac{σ}{|σ|} \\ y_{t} & = & m(y_{1} − y)|ε_{t}^{p}| |σ| − Ay_{1}(\frac{y − y_{2}}{y_{1}} )^{r} \end{aligned}  (4.2)

where σ = D(ε − ε^{p}) − \frac{1}{3}tr(D(ε − ε^{p})) · I and n, r > 1, α, d, m > 0, A ≥ 0, y_{1} > y_{2} > 0 are constants. We see that the right hand side depends only on the stress deviator σ and on the isotropic hardening function y. Therefore we define the free energy associated with the model as follows:

ρψ(ε, z) = ρψ(ε, ε^{p}, y) = \frac{1}{2}[D(ε − ε^{p})] · (ε − ε^{p}) + \frac{1}{2}y^{2} .  (4.3)

Note that now the coefficient for y^{2} does not depend of the special form of nonlinear functions used in (4.2). The operator L(ε^{p}, y) = (0, y) is positive semi-definite and the operator M(ε^{p}, y) = (B^{T} DB + L)(ε^{p}, y) = (Dε^{p}, y) is positive definite on S^{3} × ℝ. Then the model is of pre-monotone type and in the article [11] it is proved that the system (4.2) belongs to the class LM. The main idea of the proof is to use the a priori estimates for the system (4.2) and the special properties of the exponential function from the equation for the plastic strain. The coercivity of the model fails, but the self-controlling inequality is elementary satisfied (the right hand side of the equation for Bz = ε^{p} is bounded) and we can use Theorem 3.4.

3. The model of Chan-Bodner-Lindholm In [7] K.S. Chan, S.R. Bodner and U.S. Lindholm generalize the model from example 2 introducing the new internal variable β ∈ S^{3} which describes the directional hardening. The plastic constitutive equation is of the form

\begin{aligned}ε_{t}^{p} & = & d exp ( − α(\frac{y_{c}}{|σ|})^{n})\frac{σ}{|σ|} \\ y_{t} & = & m(y_{1} − y)|ε_{t}^{p}| |σ| − Ay_{1}(\frac{y − y_{2}}{y_{1}} )^{r_{1}} \\ β_{t} & = & md(r\frac{σ}{|σ|} − β) exp ( − α(\frac{y_{c}}{|σ|})^{n}) − C|β|^{r_{2}−1}β \end{aligned}  (4.4)

where y_{c} = y + \frac{σ}{|σ|}β and A , C ≥ 0, n , r_{1} , r_{2} > 1, α , r , d , m > 0 and y_{1} > y_{2} > 0 are material constants. To the system (4.4) we define the quadratic form

ρψ(ε, z) = ρψ(ε, ε^{p}, y) = \frac{1}{2}[D(ε − ε^{p})] · (ε − ε^{p}) + \frac{1}{2}y^{2} + \frac{1}{2}|β|^{2} .  (4.5)

As in example 3 the operator L(ε^{p}, y, β) = (0, y, β) is only positive semi-definite and M(ε^{p}, y, β) = (Dε^{p}, y, β) is positive definite. Thus the model is pre-monotone but noncoercive. In the work [12] the author verifies that the plastic constitutive equation (4.4) belongs to the the class LM. It is clear again that the system (4.4) is self-controlling and the model of Chan-Bodner-Lindholm satisfies all assumptions of Theorem 3.4.

Remark All models with the constitutive equation for the isotropic hardening in the form

y_{t} = f(y)|ε_{t}^{p}|  (4.6)

where the function f is positive and: bounded (for example the model of Bruhns [5]), or monotone decreasing with f(y_{1}) = 0 for some y_{1} (for example the model of Chaboche [6]) are self-controlling.

Unfortunately the property of belonging to the class LM depends very strongly on the whole form of the plastic constitutive equation and is not simple to verify.

At the end of the article we present a model which does not satisfy the self-controlling requirement.

4. The plastic power law We assume that the vector z of internal variables contains only the plastic strain ε^{p} and the plastic constitutive equation is given by the following power law

ε_{t}^{p} = d|σ|^{β}σ  (4.7)

with positive constants d, β > 0. The free energy function associated with (4.7) is defined by

ρψ(ε, ε^{p}) = \frac{1}{2}[D(ε − ε^{p})] · (ε − ε^{p})

and thus the operator Mε^{p} = Dε^{p} , is positive definite and the operator Lε^{p} = 0 . Then we conclude again that the model is noncoercive. The monotonicity of the power function implies that equation (4.7) belongs to the class M. The right hand side of the equation is not controlled by the argument σ in the 𝕃^{2}–topology (note β > 0) , and the model does not belong to the class of self-controlling models. However equation (4.7) satisfies

∥ε_{t}^{p}∥_{𝕃^{(β+2)/(β+1)}}^{(β+2)/(β+1)} ≤ C∥σ∥_{𝕃^{β+2}}^{β+2}  (4.8)

which is a similar inequality to the self-controlling requirement. Estimate (4.8) is the fundamental tool in the proof of existence of global solutions for the system (MS) with the plastic constitutive equation (4.7) (see for details paragraph 7 in [10]).

References

[1] H.D. Alber : “Initial-Boundary Value Problems for the Inelastic Material Behaviour of Metals” – Lecture Notes in Mathematics, Springer (in preparation)

[2] H.D. Alber : “Mathematische Theorie des inelastischen Materialverhaltens von Metallen” – Preprint Nr. 1682 FB Mathematik TH Darmstadt 1994

[3] S.R. Bodner, Y. Partom : “Constitutive Equations for Elastic–Viscoplastic Strain– Hardening Materials” – J. of Appl Mech. p. 385-389, 1975

[4] H. Brézis : “Operateurs maximaux monotones”– North Holland, Amsterdam, 1973

[5] O.T. Bruhns, B. Boecke, F. Link: “ The constitutive relations of elastic-plastic materials at small strains”– Nucl. Eng. and Design, 85 (1984), 325–331

[6] J.L. Chaboche: “ Viscoplastic constitutive equations for the description of cyclic and anisotropic behavior of metals” – Bull. Acad. Polonaise Sci., Série Sci. Tech. 25 (1977), 33–39, 42–48

[7] Chan, K.S. Bodner, S.R. Lindholm, U.S. : “Phenomenological Modeling of Hardening, and Thermal Recovery in Metals” – J. of Engineering Materials and Technology 1988, Vol. 110/1

[8] K. Chełmiński, H.D. Alber : “Existence theory for the equations of inelastic material behavior of metals – Transformation of interior variables and energy estimates” – Roczniki PTM: Appl. Math. 39 (1996), 1–15 [9] K. Chełmiński : “On initial–boundary value problems for the inelastic material be-

haviour of metals”– to appear in Z. Angew. Math. Mech.

[10] K. Chełmiński : “Coercive limits for a subclass of monotone constitutive equations in the theory of inelastic material behavior of metals” – accepted for publication in Roczniki PTM: Appl. Math. 40

[11] K. Chełmiński, P. Gwiazda : “On the model of Bodner-Partom with nonhomogeneous boundary data”– submitted to Math. Nach.

[12] K. Chełmiński, P. Gwiazda : “The Model of Bodner-Partom and the Method of Monotone Operators”– submitted to Bull. of Pol. Acad. Sci.: Tech. Sci.

[13] P. Gwiazda : “Existence and Uniqueness Theorem for the Chan-Bodner-Linholm Model”– Preprint TU Darmstadt in preparation

Addresses: Krzysztof Chełmiński Fachbereich Mathematik Technische Hochschule Darmstadt Schloßgartenstr. 7 64289 Darmstadt, Germany e-mail: chel@mathematik.th-darmstadt.de