On Monotone Plastic Constitutive Equations with Polynomial Growth Condition

Krzysztof Chełmiński ^{∗}

Technical University of Darmstadt

Abstract In the theory of inelastic behaviour of metals we study an example of a monotone plastic constitutive equation, which does not belong to the class of the selfcontrolling models ([Ch1]). Existence and uniqueness of solutions for this model is obtained in Orlicz spaces using the external coercive approximation ([Ch2]) and the Minty-Browder method.

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

1 Introduction

Under the theory of inelastic behaviour of metals we understand here the theory with internal variables. This theory assumes existence of the vector of internal variables, whose components uniquely determine the state of the material. Using the fundamental assumption of small deformations, the system of equations which describes an inelastic deformation of a metal, consists of the equations of motion (linear partial differential equations) and of plastic constitutive equations written as a system of ordinary differential equations for the vector of internal variables z. Thus if the material occupies a bounded domain Ω ⊂ ℝ^{3} with the smooth boundary ∂Ω then to find are not only the displacement vector u : Ω × [0, ∞) → ℝ^{3} and the stress tensor T : Ω × [0, ∞) → ℝ_{sym}^{3×3} = S^{3}, but additionally we have to find the vector of internal variables z = (ε^{p}, z̃) ∈ S^{3} × ℝ^{d}. This vector consists of the plastic strain tensor ε^{p} and of other internal variables which describe various properties of the metal (for example isotropic or kinematic hardening). In this work we will only study an example from the class M of monotone plastic constitutive equations. This class was introduced by H.-D. Alber in [Al]. The system of equations with plastic constitutive equation from the class M can be written in the form

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

∗Supported by Sonderforschungsbereich 298.

(the subscripts t and x denote the partial derivatives with respect to t and x respectively). Here v = u_{t} is the velocity vector, ε = \frac{1}{2}(∇_{x}u + ∇_{x}^{T} u) denotes the strain tensor, the constant ρ is the mass density, D : S^{3} → S^{3} is the elasticity tensor, which we assume to be constant, symmetric and positive definite, B : S^{3} × ℝ^{d} → S^{3} is a projector defined by Bz = B(ε^{p}, z̃) = ε^{p}, g : S^{3} × ℝ^{d} → S^{3} × ℝ^{d} is a given monotone vector field satisfying g(0) = 0 and ψ(ε, z) is the free energy function given by the formula

ρψ(ε, z) = \frac{1}{2}[D(ε − Bz)] · (ε − Bz) + \frac{1}{2}Lz · z  (1.1)

with a symmetric positive semi-definite matrix L. The matrix L must be so chosen that the matrix M = B^{T} DB + L is positive definite. The system (MS) will be considered either with the homogeneous Dirichlet boundary condition

v(x, t) = 0 for x ∈ ∂Ω and t ≥ 0 ,  (1.2)

or with the homogeneous Neumann boundary condition

D(ε − Bz)(x, t) · n(x) = 0 for x ∈ ∂Ω and t ≥ 0  (1.3)

(here n(x) denotes the exterior unit normal vector to ∂Ω at x ∈ ∂Ω ). Moreover the initial conditions are

v(x, 0) = v^{0}(x) ε(x, 0) = ε^{0}(x) z(x, 0) = z^{0}(x)  (IC)

with given functions v^{0} : Ω → ℝ^{3} , ε^{0} : Ω → S^{3} , z^{0} : Ω → S^{3} × ℝ^{d}. The properties of the matrix L and (1.1) yield that the free energy function is a semi-definite quadratic form in S^{3} × (S^{3} × ℝ^{d}). If this form is positive definite then we say that the model under consideration is coercive. In [Al] (Theorem 4.2.1 p.48) it was proved that for a continuous, monotone vector field g, coercive problem (MS) with boundary condition (1.2) or (1.3) can be written as an evolution problem to the maximal monotone operator A defined by the right hand side of (MS) with the natural 𝕃^{2} domain. Thus such systems possess unique global solutions to all initial data from the domain of the operator A. With this result the subclass of coercive and monotone models was studied completely and only the noncoercive case is left to investigation. The work [Ch1] was the first which gave a partial answer to the question: have noncoercive problems global, large solutions? The main idea of [Ch1] is contained in the definition of a coercive approximation process

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

with the free energy function

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} (Dε^{k}) · ε^{k}  (1.4)

and with the same boundary and initial conditions as for the system (MS). In [Ch1] is introduced the subclass SC of self-controlling models for which the coercive approximation (CA) converges in 𝕃^{2} to a solution of the system (MS). We say that the plastic constitutive equation (MS3) belongs to the class SC if the 𝕃^{2} norm of the right hand side Bg of the system (MS) can be controlled by the 𝕃^{2} norm of the right hand side Lg of the system (MS) or by the 𝕃^{2} norm of the arguments of Bg (Definition 5.4 in [Ch1] p.62). Thus another subclass of M was found, namely SC ∩ M, for which the problem of existence and uniqueness of global large solutions was solved.

In this article we study the following noncoercive problem

ρv_{t}(x, t) = div_{x} D(ε(x, t) − ε^{p}(x, t))
ε_{t}(x, t) = \frac{1}{2}(∇_{x}v(x, t) + ∇_{x}^{T} v(x, t))  (MP)
ε_{t}^{p}(x, t) = G(σ(x, t))

where σ = P T = P D(ε−ε^{p}) = T − \frac{1}{3}(trT)I denotes the stress deviator and G : S^{3} → P S^{3} is a continuous monotone vector field satisfying G(0) = 0. If |G(σ)| → ∞ faster then linear for |σ| → ∞ then the problem (MP) does not belong to the class SC. Note that (MP) is a generalization of the example from section 7 of [Ch1]. Using the theory of Orlicz spaces ([CS, KR]) and the Minty-Browder method ([Ev, Li]) and assuming that the vector field G(σ) satisfies a polynomial growth condition for |σ| → ∞, we prove existence and uniqueness of global large solutions for the system (MP). Thus this paper enlarges the class of monotone plastic constitutive equations for which the coercive approximation process (CA) converges and the limit solves the origin noncoercive problem.

Section 2 of this paper presents basic definitions and theorems from the theory of Orlicz spaces to serve as a reference for the following sections. Readers who are familiar with this theory can omit this section. In section 3 we define the main problem and prove that the coercive approximation (CA) converges in 𝕃^{1}. In section 4 we use the Minty-Browder method to conclude that the coercive limit solves the problem (MP). Section 5 contains remarks which compare presented result with other articles, and an open question.

2 Basic definitions and properties of Orlicz spaces

This section is devoted to listing briefly some basic definitions and results from the theory of Orlicz spaces. The reader desirous of more informations should consult, for example, the book [KR] or the work [CS].

Let M : ℝ → ℝ be an even, continuous and convex function. If M has the following properties

(1) M(u) > 0 for u ≠ 0 and M(0) = 0 ,

(2) lim_{u→0} \frac{M(u)}{u} = lim_{u→∞} \frac{u}{M(u)} = 0

then we call it an Orlicz function (in the book [KR] it is called a N-function).

Proposition 2.1 A real function M is an Orlicz function if and only if M admits of the representation

M(u) = ∫_{0}^{|u|} p(t) dt  (2.1)

where the function p is right-continuous, nondecreasing, p(t) > 0 for t > 0, p(0) = 0 and lim_{u→∞} p(u) = ∞.

The representation (2.1) and properties of the function p yield that for u > 0 the following useful inequalities hold

M(u) ≤ up(u) and M(2u) ≥ up(u) .  (2.2)

Let p has the properties from Proposition 2.1 then we call the function q defined by q(s) = sup{t | p(t) ≤ s} the right-inverse function of p. Let M be an Orlicz function, p be the right derivative of M and q be the right-inverse of p. Then we define by

M^{∗}(u) = ∫_{0}^{|u|} q(s) ds  (2.3)

the complementary Orlicz function of M.

Proposition 2.2 For an Orlicz function M hold

a) (M^{∗})^{∗} = M (M and M^{∗} are mutually complementary Orlicz functions),

b) (Young Inequality) ∀_{u,v ∈ℝ} uv ≤ M(u) + M^{∗}(v) .

If for any two Orlicz functions M_{1}, M_{2} there exist positive numbers u_{0} and k such that M_{1}(u) ≤ M_{2}(ku) for all u ≥ u_{0} then we write M_{1} ≺ M_{2}. If M_{1} ≺ M_{2} and M_{2} ≺ M_{1} then we write M_{1} ∼ M_{2} and say that the Orlicz functions M_{1} and M_{2} are equivalent. From the definition of the comparison relation we have that if M_{1} ≺ M_{2} then M_{2}^{∗} ≺ M_{1}^{∗} and consequently if M_{1} ∼ M_{2} then M_{1}^{∗} ∼ M_{2}^{∗}. A very important subclass of Orlicz functions is the class satisfying the Δ_{2} condition. We say that an Orlicz function M satisfies the Δ_{2} condition if

∃_{u_{0}>0} ∃_{k>0} ∀_{u≥u_{0}} M(2u) ≤ kM(u) ,  (2.4)

and we write shortly M ∈ Δ_{2}. If both M and M^{∗} belong to this class we write M ∈ Δ_{2} ∩ ∇_{2}.

Proposition 2.3 If an Orlicz function satisfies the Δ_{2} condition then

∃_{α>1} ∃_{u_{0}>0} ∃_{k>0} ∀_{u≥u_{0}}  M(u) ≤ ku^{α} .  (2.5)

Thus the Orlicz functions from the class Δ_{2} do not increase faster as any polynomial. Let A ⊂ ℝ^{n} be a bounded Lebesgue measurable set. If M is an Orlicz function then the Orlicz class L_{M}(A) consists of all (equivalence classes of) Lebesgue integrable real-valued functions for which

∫_{A} M(f(x)) dx < ∞ .  (2.6)

The Orlicz space L_{M}(A) is the linear hull of the Orlicz class L_{M}(A). It can be shown that

L_{M}(A) = {f ∈ 𝕃^{1}(A) | ∃_{λ>0} ∫_{A} M(λf(x)) dx < ∞ } .  (2.7)

For any two comparable Orlicz functions M_{1} ≺ M_{2} we have L_{M_{1}(A)} ⊂ L_{M_{2}(A)}. In the Orlicz space L_{M}(A) we define the Orlicz norm by

∥f∥_{M} = inf_{λ>0} \frac{1}{λ}[1 + ∫_{A} M(λf(x)) dx] .  (2.8)

The Orlicz space L_{M}(A) equipped with the Orlicz norm is a Banach space. A very important closed linear subspace of the space L_{M}(A) is defined by

E_{M}(A) = {f ∈ 𝕃^{1}(A) | ∀_{λ>0} ∫_{A} M(λf(x)) dx < ∞ } .  (2.9)

E_{M}(A) is the closure of the set of bounded functions in the Orlicz norm. In the space L_{M}(A) can be defined a second norm, called the Luxemburg norm

∥f∥_{M,L} = inf \{λ > 0 | ∫_{A} M(\frac{1}{λ}f(x)) dx ≤ 1\} .  (2.10)

The norms ∥ · ∥_{M} and ∥ · ∥_{M,L} are equivalent and the following Hölder type inequality holds

∀_{f∈L_{M} (A)} ∀_{g∈L_{M^{∗}}(A)} ∫_{A} f(x)g(x) dx ≤ ∥f∥_{M} ∥g∥_{M^{∗},L} .  (2.11)

Proposition 2.3

a) (L_{M}(A), ∥ · ∥_{M}) = (E_{M^{∗}}(A), ∥ · ∥_{M^{∗},L})^{∗} ,

b) L_{M}(A) = E_{M}(A) if and only if M ∈ Δ_{2} ,

c) L_{M}(A) is reflexive if and only if M ∈ Δ_{2} ∩ ∇_{2} .

Remark Orlicz spaces can be defined for vector valued functions. We say that f : A → ℝ^{k} belongs to L_{M}(A; R^{k}) if all components f_{i} i = 1, …, k belong to L_{M}(A). The norm in the space L_{M}(A; R^{k}) is defined by ∥f∥_{M} = ∑_{i=1}^{k} ∥f_{i}∥_{M} .

3 Coercive approximation and existence of the limit in 𝕃^{1}

This section starts with the investigation of the main problem

ρv_{t}(x, t) = div_{x} D(ε(x, t) − ε^{p}(x, t))
ε_{t}(x, t) = \frac{1}{2}(∇_{x}v(x, t) + ∇_{x}^{T} v(x, t))  (MP)
ε_{t}^{p}(x, t) = G(σ(x, t))

where σ = P T = P D(ε − ε^{p}) is the stress deviator and the given vector field G maps the space S^{3} into the subspace P S^{3} = {σ ∈ S^{3} | tr σ = 0}. P denotes the orthogonal projector on this subspace. Moreover we assume that the homogeneous Dirichlet boundary condition (1.2) or the homogeneous Neumann boundary condition (1.3) holds and the initial conditions are

v(x, 0) = v^{0}(x) ε(x, 0) = ε^{0}(x) ε^{p}(x, 0) = ε^{p,0}(x)  (IC)

with given functions v^{0} : Ω → ℝ^{3} , ε^{0}, ε^{p,0} : Ω → S^{3}. Thus in the model (MP) the vector of internal variables z contains only the plastic strain ε^{p} and the free energy function ψ(ε, z) = ψ(ε, ε^{p}) describes the elastic work

ρψ(ε, ε^{p}) = \frac{1}{2}[D(ε − ε^{p})] · (ε − ε^{p}) .  (3.1)

The quadratic form (3.1) is only positive semi-definite on S^{3} × S^{3} and therefore the problem (MP) is not coercive. Moreover if |G(σ)| → ∞ faster then linear for |σ| → ∞ we cannot control the 𝕃^{2} norm of G(σ) by the 𝕃^{2} norm of σ a priori, and the model (MP) does not belong to the class SC of self-controlling problems. To prove that (MP) is of monotone type we need that G satisfies:

Monotonicity Assumption

∀_{σ_{1},σ_{2}∈S^{3}} (G(σ_{1}) − G(σ_{2})) · (σ_{1} − σ_{2}) ≥ 0 , G(0) = 0 and G is continuous. (A1)

The right hand side G(σ) of the equation (MP3) can be written in the form

G(σ) = G(P T) =^{df} g(T) = g( − ρ∇_{ε^{p}}ψ(ε, ε^{p})) .

The assumption (A1) implies the monotonicity of g: namely for T_{1}, T_{2} ∈ S^{3}

\begin{aligned}(g(T_{1}) − g(T_{2})) · (T_{1} − T_{2}) & = & (G(P T_{1}) − G(P T_{2})) · (T_{1} − T_{2}) \\ & = & (G(P T_{1}) − G(P T_{2})) · (P T_{1} − P T_{2}) ≥ 0 , \end{aligned}

moreover it holds that g(0) = G(P 0) = G(0) = 0. Thus the model (MP) belongs to the class M of monotone problems. (Note that the operator B^{T} DB + L = D is positive definite on S^{3}.) In the same manner as in [Ch1] we approximate the problem (MP) by a sequence of monotone and coercive problems. Let k > 0 be a natural number, then the approximation is defined by

ρv_{t}^{k}(x, t) = div_{x} D(ε^{k}(x, t) − ε^{p,k}(x, t) + \frac{1}{k}ε^{k}(x, t))
ε_{t}^{k}(x, t) = \frac{1}{2}(∇_{x}v^{k}(x, t) + ∇_{x}^{T} v^{k}(x, t))  (CA)
ε_{t}^{p,k}(x, t) = G(σ^{k}(x, t)) = G(P D(ε^{k}(x, t) − ε^{p,k}(x, t)))

with the free energy function

ρψ^{k}(ε^{k}, z^{k}) = \frac{1}{2}[D(ε^{k} − ε^{p,k})] · (ε^{k} − ε^{p,k}) + \frac{1}{2k} (Dε^{k}) · ε^{k} .  (3.2)

Moreover we assume that the homogeneous Dirichlet boundary condition

v^{k}(x, t) = 0 for x ∈ ∂Ω and t ≥ 0 ,  (3.3)

or the homogeneous Neumann boundary condition holds

D(ε^{k} − ε^{p,k} + \frac{1}{k}ε^{k})(x, t) · n(x) = 0 for x ∈ ∂Ω and t ≥ 0  (3.4)

(here n(x) denotes the exterior unit normal vector to ∂Ω at x ∈ ∂Ω ). Finally the initial conditions are given by

v^{k}(x, 0) = v^{0}(x) ε^{k}(x, 0) = ε^{0}(x) ε^{p,k}(x, 0) = ε^{p,0}(x)  (IC)

with the same given initial data (v^{0}, ε^{0}, ε^{p,0}) as for the problem (MP).

For the initial data (v^{0}, ε^{0}, ε^{p,0}) ∈ 𝕃^{2}(Ω; ℝ^{3} × S^{3} × S^{3}) we assume that ID1. (div D(ε^{0} − ε^{p,0} + \frac{1}{k}ε^{0}), \frac{1}{2}(∇v^{0} + ∇^{T} v^{0}), G(P D(ε^{0} − ε^{p,0}))) ∈ 𝕃^{2}(Ω; ℝ^{3} × S^{3} × S^{3})

and tr ε^{p,0}(x) = 0 for a.e. x ∈ Ω ,

ID2. for the Dirichlet boundary condition v^{0} ∈ ℍ_{1}^{∘}(Ω; ℝ^{3}) and for the Neumann boundary value problem it holds that

∀_{u∈ℍ_{1}(Ω;ℝ^{3})} (div D(ε^{0} − ε^{p,0}), u) = −(D(ε^{0} − ε^{p,0}), ∇u) ,

ID3. div Dε^{0} ∈ 𝕃^{2}(Ω; ℝ^{3}) and for the Neumann boundary condition Dε^{0}(x) · n(x) = 0 for x ∈ ∂Ω .

(By ℍ_{1}(Ω; ℝ^{k}) we denote the usual Sobolev space of all functions in 𝕃^{2}(Ω; ℝ^{k}) , which have weak derivatives in 𝕃^{2}(Ω; ℝ^{k}). Moreover ℍ_{1}^{∘}(Ω; ℝ^{k}) denotes the closed subspace of the space ℍ_{1}(Ω; ℝ^{k}) which consists of all functions f satisfying f(x) = 0 for x ∈ ∂Ω in the weak sense. The norm and the scalar product in the space 𝕃^{2}(Ω; ℝ^{k}) will be denoted by ∥·∥ and (·, ·) respectively.) Observe that two first assumptions yield that the initial data (v^{0}, ε^{0}, ε^{p,0}) belongs to the domain of the operator generated by the right hand side of the system (MP), and the third assumption gives that for all k the data belongs to the domain of the operator generated by the right hand side of the system (CA).

Lemma 3.1 For all initial data (v^{0}, ε^{0}, ε^{p,0}) which satisfy the assumptions ID1-ID3, the problem (CA) for all natural numbers k > 0 possesses global in time solutions with the property

∀_{T >0} (v^{k}, ε^{k}, ε^{p,k}) ∈ 𝕎^{1,∞}((0, T); 𝕃^{2}(Ω; ℝ^{3} × S^{3} × S^{3})) .  (3.5)

(the space 𝕎^{1,∞}(0, T) consists of all bounded functions which weak derivatives are also bounded). Proof: This lemma is a special case of the problem from section 6 of [Ch1] and therefore the proof will be omitted.

Before we start to study the limit case k → ∞, we require additionally that the vector field G satisfies the

Nondegeneration Assumption

∃_{R>0} ∃_{α∈(0,1]} ∀_{σ∈S^{3}} G(σ) · σ ≥ α |G(σ)| |σ| . (A2)

This assumption says that the direction of evolution of the plastic strain should be “near” to the direction of the stress deviator. In the practice this assumption is always satisfied. The standard model which can be found in the literature, has the form

G(σ) = G(|σ|)\frac{σ}{|σ|} ,  (3.6)

and the assumption (A2) holds for it with α = 1.

Theorem 3.2 Let the initial data (v^{0}, ε^{0}, ε^{p,0}) satisfy the assumptions ID1-ID3. Then the approximation sequence (v^{k}, ε^{k}, ε^{p,k}) contains a subsequence (v^{k_{l}}, ε^{k_{l}}, ε^{p,k_{l}}) for which holds: for all T > 0

v^{k_{l}} ⇀^{∗} v in 𝕎^{1,∞}((0, T); 𝕃^{2}(Ω; ℝ^{3})) ,
ε^{k_{l}}⇀ε in 𝕃^{1}(Ω × (0, T); S^{3}) ,
ε^{p,k_{l}}⇀ε^{p} in 𝕃^{1}(Ω × (0, T); S^{3}) ,
σ^{k_{l}} ⇀^{∗} σ in 𝕎^{1,∞}((0, T); 𝕃^{2}(Ω; S^{3}))

where σ = P D(ε − ε^{p}).

Proof: Let us fix T > 0. Using the general theory of evolution equations to monotone operators ([Br]) we obtain that exists a positive constant C such that

esssup_{t ∈ [0, T]} \{∥v_{t}^{k}(t)∥ , ∥ε_{t}^{k}(t) − ε_{t}^{p,k}(t)∥ , \frac{1}{\sqrt{k}} ∥ε_{t}^{k}(t)∥\} ≤ C ,  (3.7)

and C does not depend on k and T. Thus there exists an independent on k constant C_{T} > 0 for which

esssup_{t ∈ [0, T]} \{∥v^{k}(t)∥ , ∥ε^{k}(t) − ε^{p,k}(t)∥ , \frac{1}{\sqrt{k}} ∥ε^{k}(t)∥\} ≤ C_{T} .  (3.8)

Thus (3.7) and (3.8) yield that the sequences of velocities {v^{k}} and stresses {T^{k}} and those time derivatives are bounded in 𝕃^{∞}((0, T); 𝕃^{2}). To end the proof we must only show that the sequence {ε^{k}} or the sequence {ε^{p,k}} possesses a convergent subsequence in 𝕃^{1}(Ω × (0, T); S^{3}). Let us define the total energy for the system (CA) by

\begin{aligned}E^{k}(t) & = & \frac{ρ}{2} ∥v^{k}(t)∥^{2} + \frac{1}{2} ∫_{Ω} [D(ε^{k}(t) − ε^{p,k}(t))] · (ε^{k}(t) − ε^{p,k}(t)) dx \\ & & +\frac{1}{2k} ∫_{Ω}(Dε^{k}(t)) · ε^{k}(t) dx \end{aligned}  (3.9)

then using the standard energy method we obtain

\frac{d}{dt} E^{k}(t) = ∫_{Ω} ρv^{k}(t)v_{t}^{k}(t) dx + ∫_{Ω} [D(ε^{k}(t) − ε^{p,k}(t))] (ε_{t}^{k}(t) − ε_{t}^{p,k}(t)) dx
+\frac{1}{k} ∫_{Ω}(Dε^{k}(t)) · ε_{t}^{k}(t) dx  (3.10)
= − ∫_{Ω} [D(ε^{k}(t) − ε^{p,k}(t))] · ε_{t}^{p,k}(t) dx = − ∫_{Ω} G(σ^{k}(t)) · σ^{k}(t) dx .

Integration with respect to t on the interval (0, t) yields

\begin{aligned}E^{k}(t) + ∫_{0}^{t} ∫_{Ω} G(σ^{k}(τ)) · σ^{k}(τ) dx dτ & = & E^{k}(0) ⟶^{k→∞}  \frac{ρ}{2} ∥v^{0}∥^{2} \\ & & +\frac{1}{2} ∫_{Ω} [D(ε^{0} − ε^{p,0})] · (ε^{0} − ε^{p,0}) dx . \end{aligned} (3.11)

Hence the energy estimation (3.11) yields not only the boundedness of the energy function but moreover we know that the sequence {G(σ^{k})·σ^{k}} belongs to the space 𝕃^{1}(Ω×(0, t); ℝ)) for all t > 0. The calculation (3.10) gives additionally

esssup_{t ∈ [0, T]} ∫_{Ω} G(σ^{k}) · σ^{k} dx = esssup_{t ∈ [0, T]} { − ∫_{Ω} ρv^{k}v_{t}^{k} dx

− ∫_{Ω} [D(ε^{k} − ε^{p,k})] (ε_{t}^{k} − ε_{t}^{p,k}) dx − \frac{1}{k} ∫_{Ω}(Dε^{k}) · ε_{t}^{k} dx\}

≤ c · esssup_{t ∈ [0, T]} {∥v^{k}∥ · ∥v_{t}^{k}∥  (3.12)

+ ∥ε^{k} − ε^{p,k}∥ · ∥ε_{t}^{k} − ε_{t}^{p,k}∥ + \frac{1}{\sqrt{k}} ∥ε^{k}∥ · \frac{1}{\sqrt{k}} ∥ε_{t}^{k}∥\}

≤ (by (3.6)) ≤ C(T) .

Using the assumption (A2) we conclude

esssup_{t ∈ [0, T]} ∥ε_{t}^{p,k}∥_{𝕃^{1}} = esssup_{t ∈ [0, T]} ∫_{Ω} |G(σ^{k})| dx
≤ esssup_{t ∈ [0, T]} { ∫_{Ω∩A_{t}^{k}} |G(σ^{k})| dx + ∫_{Ω\A_{t}^{k}} |G(σ^{k})| |σ^{k}| dx}
(where A_{t}^{k} = {x ∈ Ω | |σ^{k}(x, t)| ≤ 1})  (3.13)
≤ |Ω| sup_{|σ|≤1} |G(σ)| + R|Ω| sup_{|σ|≤R} |G(σ)| + esssup_{t ∈ [0, T]} ∫_{Ω} G(σ^{k}) · σ^{k} dx .

This implies the boundedness of the sequence {ε_{t}^{p,k}} in the space 𝕃^{∞}((0, T); 𝕃^{1}(Ω; S^{3})). Unfortunately the space 𝕃^{1} is not a dual space to a Banach space and boundedness in general does not imply weak precompactness. Let us define by

Ĝ(|σ|) = sup_{|η|≤|σ|} |G(η)|  (3.14)

the upper bound of the vector field G. Note that for all σ it holds Ĝ(|σ|) ≥ |G(σ)|. The function Ĝ is continuous, nondecreasing and satisfies Ĝ(0) = 0. Moreover we can assume that Ĝ(p) → ∞ for p → ∞. (If it is not the case then the function |G(σ)| is bounded and the model (MP) belongs to the class SC.) Thus Ĝ possesses the left inverse function Ĝ^{−1}(q) = inf{p | Ĝ(p) ≥ q} for which it holds: Ĝ^{−1} is nondecreasing, Ĝ^{−1}(q) → ∞ for q → ∞ and Ĝ^{−1}(Ĝ(p)) ≤ p. To obtain that the set {ε_{t}^{p,k}} is weakly precompact in 𝕃^{1}(Ω × (0, T); S^{3})) we should prove that this set is uniformly integrable (Theorem of Dunford-Pettis p.274 in [Ed]). To do this we use that the property “uniformly integrable” on bounded sets is equivalent to the property “higher integrable” (Theorem 22 p.24 in [DM]).

∫_{0}^{T} ∫_{Ω} |ε_{t}^{p,k}|Ĝ^{−1}(|ε_{t}^{p,k}|) dx dt = ∫_{0}^{T} ∫_{Ω} |G(σ^{k})|Ĝ^{−1}(|G(σ^{k})|) dx dt
≤ (by properties of the function Ĝ^{−1}and by (A2))  (3.15)

≤ ∫_{0}^{T} ∫_{Ω} |G(σ^{k})| |σ^{k}| dx dt ≤ \frac{1}{α} ∫_{0}^{T} ∫_{Ω} G(σ^{k}) · σ^{k} dx dt + C_{T,R}

where the positive constant C_{T,R} does not depend on k. Thus the sequences {ε_{t}^{p,k}} and {ε^{p,k}(t) = ∫_{0}^{t} ε_{t}^{p,k}(τ) dτ + ε^{p,0}} are weakly precompact in 𝕃^{1}(Ω × (0, T); S^{3})). Using (3.7) and (3.8) the same result holds for the sequences {ε_{t}^{k}} and {ε^{k}}.

The last theorem implies immediately that the limit functions (v, ε, ε^{p}) satisfy the following system of equations

ρv_{t}(x, t) = div_{x} D(ε(x, t) − ε^{p}(x, t))
ε_{t}(x, t) = \frac{1}{2}(∇_{x}v(x, t) + ∇_{x}^{T} v(x, t))  (3.16)
ε_{t}^{p}(x, t) = w − lim_{k→∞} G(σ^{k}(x, t)) =^{df} χ(x, t)

with the same boundary and initial conditions as for the problem (MP). What is left to be shown is that χ(x, t) = G(σ(x, t)) for a.e. (x, t) ∈ Ω × (0, T) where σ is the weak 𝕃^{2} limit of the sequence (or of a subsequence of the sequence) {σ^{k}}. If we denote by μ_{(x,t)} the Young measure generated by the sequence {σ^{k}} , then using the Theorem from [Ba] the limit function χ can be represented by

χ(x, t) = ∫_{ℝ^{9}} G(λ)dμ_{(x,t)}(λ) for a.e. (x, t) ∈ Ω × (0, T) .

Therefore our problem can be reduced to the question μ_{(x,t)} =^{?} δ_{σ(x,t)} where δ_{σ(x,t)} is the Dirac delta concentrated at the point σ(x, t).

4 The Minty-Browder method

This section is devoted to answer the question χ(x, t) =^{?} G(σ(x, t)) for a.e. (x, t) ∈ Ω × (0, T). We use here the famous procedure presented by J.L. Lions in [Li]. L.C. Evans called this method in [Ev] the Minty-Browder method. In the previous section we have not used that the vector field G satisfies any growth condition. In this section we require additionally that G has the following property

Polynomial Growth Condition

\begin{matrix}There exist continuous, nondecreasing functions G̃, G : ℝ_{+} → ℝ_{+} \\ with the properties: G̃(0) = G(0) = 0 lim_{p→∞} G̃(p) = lim_{p→∞} G(p) = ∞ \\ and & lim_{p→∞} \frac{G̃(p)}{G(p)} = g^{∗} > 0 & such that \\ ∃_{R>0} ∀_{|σ|≥R} G̃(|σ|) ≤ |G(σ)| ≤ G(|σ|) \\ and the Orlicz function M(t) =^{df} ∫_{0}^{|t|} G(s) ds belongs to the class Δ_{2} ∩ ∇_{2}. \end{matrix} \} (A3)

The assumption that lim_{p→∞} \frac{G̃(p)}{G(p)} exists and is positive yields that the Orlicz functions defined by G̃ and by G are equivalent (Lemma 3.2 p.18 in [KR]). We can additionally assume that the function G is strictly increasing, because if it is not the case then there exists an Orlicz function which is equivalent to M, and whose derivative is continuous and strictly increasing (Theorem 1.18 p.12 in [CS]). Thus the function G possesses the inverse function G^{−1}. The Polynomial Growth Condition says that for large arguments |σ| the function |G(σ)| grows “uniformly” as a derivative of an Orlicz function and does not growth faster then any polynomial (Proposition 2.3). For the example presented in (3.6) we have G̃(p) = G(p) and the assumption (A3) is satisfied if G has a polynomial growth. Without loss of generality we can assume that the result from the previous section is true for the whole approximation sequence (v^{k}, ε^{k}, ε^{p,k}). Hence it holds that v^{k} ⇀^{∗} v in 𝕃^{∞}((0, T); 𝕃^{2}(Ω; ℝ^{3})), ε^{k} ⇀ ε , ε^{p,k} ⇀ ε^{p} in 𝕃^{1}(Ω × (0, T); S^{3}) and moreover σ^{k} = D(ε^{k} − ε^{p,k}) ⇀^{∗} σ = D(ε − ε^{p}) in 𝕃^{∞}((0, T); 𝕃^{2}(Ω; S^{3})). Let us denote by χ the weak limit w − lim_{k→∞} ε^{p,k} = w − lim_{k→∞} G(σ^{k}). The first step of the Minty-Browder method is the proof of the integrability of the function χ · σ.

Theorem 4.1 For the approximation sequence (v^{k}, ε^{k}, ε^{p,k}) defined as solutions of the problem (CA) it holds

χ · σ ∈ 𝕃^{∞}((0, T); 𝕃^{1}(Ω; ℝ)) .  (4.1)

Proof: Let us denote by L_{M}(Ω; S^{3}) the Orlicz space generated by the Orlicz func-

tion M(t) = ∫_{0}^{|t|} G(s) ds and by L_{M^{∗}}(Ω; S^{3}) the Orlicz space generated by the function

M^{∗}(t) = ∫_{0}^{|t|} G^{−1}(s) ds. It is enough to show that σ ∈ 𝕃^{∞}((0, T); L_{M}(Ω; S^{3})) and

χ ∈ 𝕃^{∞}((0, T); L_{M^{∗}}(Ω; S^{3})), because the Hölder inequality (2.11) will then complete the proof. Using assumption (A3) we conclude that

∫_{Ω} G̃(|σ^{k}|)|σ^{k}| dx ≤ ∫_{Ω} |G(σ^{k})| |σ^{k}| dx + C_{R} ,  (4.2)
∫_{Ω} G(|σ^{k}|)|σ^{k}| dx ≤ C_{g^{∗},R}( ∫_{Ω} G̃(|σ^{k}|)|σ^{k}| dx + 1)  (4.3)

where the positive constants C_{R} and C_{g^{∗},R} do not depend on k. Consequently we obtain that the sequence {G(|σ^{k}|)|σ^{k}|} is bounded in 𝕃^{∞}((0, T); 𝕃^{1}(Ω; ℝ)). Hence using the definition of the Orlicz norm and (2.2) we have

\begin{aligned}∥σ^{k}∥_{M} & = & ∑_{i,j=1}^{3} ∥σ_{i j}^{k}∥_{M} ≤ ∑_{i,j=1}^{3} (1 + ∫_{Ω} M(|σ_{i j}^{k}|) dx) \\ & ≤ & 9(1 + ∫_{Ω} M(|σ^{k}|) dx) ≤ 9(1 + ∫_{Ω} G(|σ^{k}|)|σ^{k}| dx) . \end{aligned}  (4.4)

In the same manner, for the sequence {ε_{t}^{p,k}} we derive

\begin{aligned}∥ε_{t}^{p,k}∥_{M^{∗}} & ≤ & 9(1 + ∫_{Ω} M^{∗}(|ε_{t}^{p,k}|) dx) ≤ 9(1 + ∫_{Ω} G^{−1}(|ε_{t}^{p,k}|)|ε_{t}^{p,k}| dx) \\ & ≤ & 9(1 + |Ω| sup_{|σ|≤R} G^{−1}(|G(σ)|)|G(σ)| + ∫_{Ω} G(|σ^{k}|)|σ^{k}| dx) . \end{aligned}  (4.5)

Thus the sequence {σ^{k}} is bounded in the space 𝕃^{∞}((0, T); L_{M}(Ω; S^{3})) and the sequence {ε_{t}^{p,k}} is bounded in the space 𝕃^{∞}((0, T); L_{M^{∗}}(Ω; S^{3})). These observations and the assumption (A3) complete the proof.

The second step of the Minty-Browder method is the comparison of the energy estimates for the approximate problem (CA) with the energy estimates for the system (3.16) which is satisfied by the limit functions. Theorem 4.2 For the limit functions χ and σ it holds

∀_{t∈[0,T]} lim inf_{k→∞} ( ∫_{0}^{t} ∫_{Ω} G(σ^{k}) · σ^{k} dx dτ ) ≤ ∫_{0}^{t} ∫_{Ω} χ · σ dx dτ .  (4.6)

Proof: Let us denote by

E(t) = \frac{ρ}{2}∥v(t)∥^{2} + \frac{1}{2} ∫_{Ω} (D(ε(t) − ε^{p}(t))) · (ε(t) − ε^{p}(t))dx  (4.7)

the total energy function for the system (3.16). Then using Theorem 4.1 we conclude that

\begin{aligned}E(t) & + & ∫_{0}^{t} ∫_{Ω} χ(τ) · σ(τ) dx dτ = E(0) \\ & = & \frac{ρ}{2}∥v^{0}∥^{2} + \frac{1}{2} ∫_{Ω} [D(ε^{0} − ε^{p,0})](ε^{0} − ε^{p,0})dx . \end{aligned}  (4.8)

From the comparison of the initial energies

E^{k}(0) = E(0) + \frac{1}{2k} ∫_{Ω}(Dε^{0}) · ε^{0} dx

and from the energy estimate (3.11) for the problem (CA) we obtain

\frac{ρ}{2}∥v^{k}(t)∥^{2} + \frac{1}{2} ∫_{Ω} (D(ε^{k}(t) − ε^{p,k}(t))) · (ε^{k}(t) − ε^{p,k}(t))dx

+ ∫_{0}^{t} ∫_{Ω} G(σ^{k}(τ)) · σ^{k}(τ) dx dτ − \frac{1}{2k} ∫_{Ω}(Dε^{0}) · ε^{0} dx  (4.9)

≤ E(t) + ∫_{0}^{t} ∫_{Ω} χ(τ) · σ(τ) dx dτ .

Because the convexity of the energy function E(t) yields

lim inf_{k→∞} (\frac{ρ}{2}∥v^{k}(t)∥^{2} + \frac{1}{2} ∫_{Ω} [D(ε^{k}(t) − ε^{p,k}(t))] · (ε^{k}(t) − ε^{p,k}(t))dx) ≥ E(t) .

we deduce that

lim inf_{k→∞} ( ∫_{0}^{t} ∫_{Ω} G(σ^{k}) · σ^{k} dx dτ − \frac{1}{2k} ∫_{Ω}(Dε^{0}) · ε^{0}dx)

= lim inf_{k→∞} ( ∫_{0}^{t} ∫_{Ω} G(σ^{k}) · σ^{k} dx dτ ) ≤ ∫_{0}^{t} ∫_{Ω} χ · σ dx dτ .  (4.10)

The third step of the method is the main step which bases on the monotonicity of the vector field G and on the reflexivity of the Orlicz spaces L_{M} and L_{M^{∗}}. Theorem 4.3

χ(x, t) = G(σ(x, t)) for a.e. (x, t) ∈ Ω × (0, T)

Proof: Let us choose w ∈ 𝕃^{∞}(Ω × (0, T); S^{3}) and denote by α = σ − λw with a positive number λ. The function λw as a bounded function belongs to the space 𝕃^{∞}((0, T); L_{M}(Ω; S^{3})), hence for the function α we obtain the same result. Next we want to show that G(α) is an element from 𝕃^{∞}((0, T); L_{M^{∗}}(Ω; S^{3})).

G^{−1}(|G(α)|)|G(α)| ≤ |α|G(|α|) + C_{R} ≤ |σ|G(|σ|) + C_{λ,w,R}(|σ| + G(|σ|) + 1) (4.11)

where the positive constant C_{λ,w,R} depends on λ , R and on the 𝕃^{∞} norm of the function w. Then (4.11) implies

∫_{Ω} M^{∗}(|G(α)|) dx ≤ ∫_{Ω} G^{−1}(|G(α)|)|G(α)| dx ≤ ∫_{Ω} M(2|σ|) dx + C_{t}

and the Δ_{2} property of the function M proves the last statement. Now we are ready to use the main idea of the Minty-Browder method. The monotonicity of the vector field G gives

∫_{0}^{T} ∫_{Ω} (G(σ^{k}) − G(α)) · (σ^{k} − α) dx dt ≥ 0 .  (4.12)

Then using the Theorem 4.2 we conclude that

∫_{0}^{T} ∫_{Ω} (χ − G(α)) · (σ − α) dx dt ≥ lim inf_{k→∞} ∫_{0}^{T} ∫_{Ω} (G(σ^{k}) − G(α)) · (σ^{k} − α) dx dt . (4.13)

Inserting into (4.13) the definition of α and dividing by λ we obtain

∫_{0}^{T} ∫_{Ω} (χ − G(σ − λw)) · w dx dt ≥ 0 .  (4.14)

Let us observe that |G(σ − λw) · w| ≤ C_{w}G(|σ| + C_{λ ,w,R}) and the sequence of constants C_{λ ,w,R} is bounded for λ → 0^{+}. Then by the Dominated Convergence Theorem in the limit λ → 0^{+}, (4.14) is transformed to

∫_{0}^{T} ∫_{Ω} (χ(x, t) − G(σ(x, t))) · w(x, t) dx dt ≥ 0 .  (4.15)

Replacing w by −w completes this proof.

5 Conclusions

Remark 5.1 (Uniqueness of the solutions)

Let us denote by u^{0}(x) the displacement vector u(x, t) at time t = 0, and assume that (v, ε, ε^{p}), (v̄, ε̄, ε̄^{p}) from the space 𝕎^{1,∞}((0, T); 𝕃^{2}(Ω; ℝ^{3}) × L_{M^{∗}}(Ω; S^{3}) × L_{M^{∗}}(Ω; S^{3})) are two solutions for the problem (MP) satisfying the boundary condition (1.2) or (1.3) and initial conditions to the same initial data. Moreover, assume that the differences (ε − ε^{p}), (ε̄− ε̄^{p}) belong to the space 𝕎^{1,∞}((0, T); L_{M}((Ω; S^{3})). Using the energy method we have

\begin{aligned}\frac{d}{dt}(\frac{ρ}{2}∥v − v̄∥^{2} & + & \frac{1}{2} ∫_{Ω} (D(ε − ε^{p} − ε̄+ ε̄^{p})) · (ε − ε^{p} − ε̄+ ε̄^{p}) dx) \\ & = & − ∫_{Ω} (G(σ) − G(σ̄)) · (σ − σ̄) dx ≤ 0 \end{aligned}

where σ = P D(ε − ε^{p}) and σ̄ = P D(ε̄ − ε̄^{p}). Then v = v̄ and u = ∫_{0}^{t} v(τ) dτ + u^{0} = ∫_{0}^{t} v̄(τ) dτ + u^{0} = ū. Consequently ε = ε̄ and ε^{p} = ε̄^{p}.

Remark 5.2 In the article [ChN] it was proved that for the models with monotone plastic constitutive equation of the gradient type (the vector field g can be represented as the gradient of a convex function) the problem (MS) possesses a global large solution. Note that the model

G(σ) = G(|σ|)\frac{σ}{|σ|}

with a smooth, nondecreasing function G which satisfies G(0) = 0 is of the monotonegradient type (without any (!) growth assumptions for G). In [ChN] has be shown that plastic strain ε^{p} and the strain ε belong to the space 𝕃^{1}(Ω × (0, T); S^{3}). The method used in [ChN] works only for models of the monotone-gradient type.

At the end we want to summarize the results for models with plastic constitutive equations of monotone type:

monotone model  ε ε^{p}  reference
coercive  𝕃^{2}(Ω)  [Al, BLeTR, LeT, Zei]
self-controlling  𝕃^{2}(Ω)  [Ch1, Ch2, ChG]
c|σ|^{α} ≤ |G(σ)| ≤ C|σ|^{α}  𝕃^{\frac{α+1}{α}} (Ω)  [Ch1, LeT]
G̃(|σ|) ≤ |G(σ)| ≤ G(|σ|)
and MG̃ ∼ M_{G} ∈ Δ_{2} ∩ ∇_{2}  L_{M^{∗}}(Ω)  this work
G(σ) = G(|σ|)\frac{σ}{|σ|}  𝕃^{1}(Ω)  [ChN]
G(σ) = ∂I_{K}(σ)  M(Ω)
K = {|σ| ≤ d} (perfect elastoplasticity) (bounded measures)  [AL]
general monotone model  M(Ω) (?)

References

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[AL] G. Anzellotti, S. Luckhaus: Dynamical Evolution of Elasto-Perfectly Plastic Bodies App. Math. Optim. 15 (1987) 121-140

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[Ch1] K. Chełmiński: Coercive Limits for a Subclass of Monotone Constitutive Equations in the Theory of Inelastic Material Behaviour of Metals Roczniki PTM: App. Math. 40 (1997) 41-81

[Ch2] K. Chełmiński: On Self-Controlling Models in the Theory of Inelastic Material Behaviour of Metals Cont. Mech. Therm. 10 (1998) 121-133

[ChG] K. Chełmiński, P. Gwiazda: Monotonicity of Operators of Viscoplastic Response: Application to the Model of Bodner-Partom to appear in Bull. of Pol. Acad. of Sci.: Tech. Sci. (1998)

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[DM] C. Dellacherie, P.-A. Meyer: Probabilities and Potential Mathematics Studies 29, North Holland (1978)

[Ed] R.E. Edwards: Functional Analysis Holt, Rinehart and Winston, New York, (1965)

[Ev] L.C. Evans: Weak Convergence Methods for Partial Differential Equations Conference Board of the Math Sci. 74 AMS, Providence, (1990)

[KR] M.A. Krasnosielskii, Y.B. Rutickii: Convex Functions and Orlicz Spaces P.Noordhoff Ltd.-Groningen (1961)

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[Zei] E. Zeidler: Nonlinear functional analysis and its applications, IV. (Applications to mathematical physics.) New York: Springer (1988)

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