Monotonicity of Operators of Viscoplastic Response: Application to the Model of Bodner-Partom

Krzysztof Chełmiński ^{∗}

Technical University of Darmstadt

and

Piotr Gwiazda^{†}

Warsaw University

Abstract

A class of viscoplastic constitutive models relating strain rate to stress and state variables (of the type proposed by S.R. Bodner and Y. Partom) is considered and the monotonicity of the related operator is studied. It was shown earlier ([1]) that the operator considered is not of monotone type but the extended class of the monotone constitutive equations is introduced (class LM) and evolution of total energy of the viscoplastic body from this class is studied. It is shown that assuming vanishing external forces, homogeneous boundary conditions and non vanishing initial state in the dissipative system the total energy cannot blow up in finite time. The noncoercive model of Bodner-Partom is approximated (using the idea from [7]) by a sequence of coercive plastic constitutive equations.

AMS classification: 35Q72, 73E60, 73F99

1 Introduction

The evolution of inelastic deformation processes is of primary importance for analysis of undergoing viscous or viscoplastic deformation with associated flow rule of hardening and damage. The fundamental question is associated with convergence process starting from different initial states but subjected to the same boundary loading or displacement. The selected problem can be passed for body with homogeneous boundary condition (vanishing displacement or traction on the boundary) but possessing non vanishing initial stress and state variables. The evolution of such body which be discussed in this paper for a specific constitutive model proposed by S.R. Bodner and Y. Patrom [3]. The monotonicity property is studied and the modified definition (the class LM) is introduced. An extensive review of inelastic constitutive models was provided in [1], but we consider only specific model.

∗Supported by Sonderforschungsbereich 298.

†Supported by DAAD.

The nonelastic deformation of a metal can be study using the notion of the vector of internal variables. We suppose that this vector consists of plastic strain ε^{p} ∈ S^{3} (the set of 3 × 3 symmetric real matrixes) and of other components z̃, which describe various properties of material under consideration. Thus we assume that z = (ε^{p}, z̃) ∈ ℝ^{N} . Let Ω ⊂ ℝ^{3} be an open, connected set with smooth boundary ∂Ω . Using the fundamental assumption of small deformations, the system of equations in the theory of nonelastic material behavior of metals can be written in the form

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

for unknown velocity field v(x, t), strain tensor ε(x, t) and vector of internal variables z(x, t) for x ∈ Ω and t ≥ 0 . Here D : S^{3} → S^{3} is the elasticity tensor, which we assume to be constant, symmetric and positive definite, ρ is the constant density, B is a given linear operator from ℝ^{N} into S^{3}, g : ℝ^{N} → ℝ^{N} is a given (usually nonlinear) constitutive function and ψ is the free energy function. In this paper we suppose that the free energy function is quadratic and positive semi-definite. The properties of the tensor D result that ψ is necessarily of the form

ρψ(ε, z) = \frac{1}{2}[D(ε − Bz)] · (ε − Bz) + \frac{1}{2}Lz · z  (1.2)

with a symmetric positive semi-definite operator L : ℝ^{N} → ℝ^{N} . Moreover we assume that the homogeneous Dirichlet boundary condition

v(x, t) = 0 for x ∈ ∂Ω and t ≥ 0 ,  (1.3)

or the homogeneous Neumann boundary condition

D(ε − Bz)(x, t) · n(x) = 0 for x ∈ ∂Ω and t ≥ 0  (1.4)

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}. The equation (MS1) is the balance of momentum with external forces equal to zero. (MS2) is a consequence of the assumption of small deformations, and the system of ordinary differential equations (MS3) is called the plastic constitutive equation. The assumption that the field of external forces vanish is not important from mathematical point of view. The fundamental mathematical tool, which we use here is the theory of evolution equations for monotone operators, and existence of external forces changes only the character of the equation from homogeneous to nonhomogeneous.

The linear operator B is in all mechanical models defined as the projector onto the plastic direction Bz = ε^{p} , but from the mathematical point of view it is convenient to study the general case B : ℝ^{N} → S^{3} . In rate independent models (for example in the perfect plasticity) the constitutive function g becomes a set-value function g : ℝ^{N} → P(ℝ^{N}) and the ordinary differential equation (MS3) becomes a differential inclusion. In this paper we study only the class of rate dependent models, which contains the model of Bodner-Partom defined below. Using (1.2) we see that the gradient of the energy function with respect to z is a linear operator

− ρ∇_{z}ψ(ε, z) = B^{T} D(ε − Bz) − Lz = L̃ε − Mz  (1.5)

where L̃ = B^{T} D : S^{3} → ℝ^{N} and M = B^{T} DB + L : ℝ^{N} → ℝ^{N} are linear operators. Thus we conclude that the evolution of the vector z depends only on the stress tensor T = D(ε − Bz) and on those components of the vector z which belong to the range of the operator L .

Following the author’s notations from the book [1] we say that the plastic constitutive equation (MS3) is of

1. pre-monotone type if and only if the symmetric operator M = B^{T} DB +L is positive definite,

2. monotone type if and only if it is of pre-monotone type and the constitutive function g : ℝ^{N} → ℝ^{N} is monotone with g(0) = 0 .

The whole class of plastic constitutive equations of monotone type we denote by M . If we study a several mechanical models in the theory of the nonelastic material behavior of metals then we can observe that usually they are of pre-monotone type, but seldom of monotone type. The requirement that the function g must be monotone is very strong. Therefore in [1] the author enlarges the class M using the idea of the transformation of internal variables, and defines the class T M . We say that the plastic constitutive equation belongs to this class if there exists a smooth transformation field H : ℝ^{N} → ℝ^{N} such that in the transformed system

\frac{∂}{∂t}v(x, t) = \frac{1}{ρ}divD(ε(x, t) − BH^{−1}h(x, t))
\frac{∂}{∂t}ε(x, t) = \frac{1}{2}(∇v(x, t) + ∇^{T} v(x, t))
\frac{∂}{∂t}h(x, t) = H′(H^{−1}h(x, t)) g(L̃ε(x, t) − MH^{−1}h(x, t))

(here h = H(z)) the last equation belongs to the class M . Moreover the operator BH^{−1} should be linear from ℝ^{N} into S^{3}. This requirement restricts the class of admissible transformation fields.

Let us now present the constitutive equations proposed in [3] by R.S. Bodner and Y. Partom

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

where σ = T − \frac{1}{3}trT · I and n, r > 1, α, d, m > 0, A ≥ 0, y_{1} > y_{2} > 0 are material constants. Thus the vector of internal variables z = (ε^{p}, y) except ε^{p} contains one scalar variable y which describe the isotropic hardening of the metal.

H.-D. Alber in the book [1] proves that the constitutive equations (BP) are not of monotone type and can be not transformed into the class M. Therefore the goal of the paper is to enlarge the class of monotone constitutive equations M using lipschitz perturbations method and to define a new class of plastic constitutive equations for which the theory of monotone operators can be used and which contains the constitutive equations (BP).

2 The class LM

Our idea is very simple. We want to replace the strong monotonicity assumption of the constitutive function g by the following weaker requirement:

Definition 2.1 We say that the plastic constitutive equation (MS3) 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} such that this constitutive equation with the constitutive function g + L is of monotone type.

Note that the assumption g(0) = 0 implies that the operator L in Definition 2.1 satisfies the same condition.

Corollary 2.2 The plastic constitutive equation (MS3) 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} .  (2.1)

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 by inequality (2.1) the function g + L : ℝ^{N} → ℝ^{N} is monotone and (g + L)(0) = 0 .

□

Thus Corollary 2.2 demonstrates that the lipschitz operator L in Definition 2.1 can be always chosen in the form l · I with a non-negative constant l. Moreover it is easy to see that M ⊂ LM (put L = 0) and the new class of plastic constitutive equations LM enlarges the class M.

Let us now define the total energy function (the sum of the kinetic energy and the free energy) connected with the system (MS):

E(v, ε, z)(t) = \frac{1}{2} ∫_{Ω} ρv^{2}(x, t) dx + ∫_{Ω} ρψ(ε(x, t), z(x, t)) dx

= \frac{1}{2} ∫_{Ω} ρv^{2}(x, t) dx + \frac{1}{2} ∫_{Ω} [D(ε(x, t) − Bz(x, t))] · (ε(x, t) − Bz(x, t)) dx (2.2)

+ \frac{1}{2} ∫_{Ω} Lz(x, t) · z(x, t) dx .

For models of monotone type, for which the dissipation inequality ([1]) is always satisfied, it is easy to verify that the total energy E(v, ε, z)(t) is a nonincreasing function and thus

E(v, ε, z)(t) ≤ E(v, ε, z)(0) .

(for details see for example [1, 5, 6]). The boundedness of the energy function implies 𝕃^{2}-estimates for the velocity field v, for the stress tensor D(ε − ε^{p}) and for these internal variables which belong to the range of the operator L. This is the basic information for several methods used in existence theorems. We will now demonstrate that for models from the class LM the energy increase can be controlled. For existence of global in time solutions such kind of statement is necessary. Let us assume that the plastic constitutive equation (MS3) belongs to the class LM and (v, ε, z) be a smooth (so that the integrals and derivatives below are well defined) solution for the system (MS), satisfying the boundary condition (1.3) or (1.4) and the initial condition (IC). Then using the energy method we obtain

\frac{d}{dt}E(v, ε, z)(t) = ∫_{Ω} [D(ε(x, t) − Bz(x, t))] · (ε_{t}(x, t) − Bz_{t}(x, t)) dx

+∫_{Ω} ρv(x, t) · v_{t}(x, t) dx + ∫_{Ω} Lz(x, t) · z_{t}(x, t) dx

(by the homogeneous boundary conditions (1.3) or (1.4))

= ∫_{Ω} v(x, t) · [ρv_{t}(x, t) − D(ε(x, t) − Bz(x, t))] dx

+ ∫_{Ω} [ − B^{T} Dε(x, t) + (B^{T} DB + L)z(x, t)] · z_{t}(x, t) dx

= − ∫_{Ω} (L̃ε(x, t) − Mz(x, t)) · g(L̃ε(x, t) − Mz(x, t)) dx
≤ ((using (2.1) and g(0) = 0) ≤ l ∫_{Ω} |L̃ε(x, t) − Mz(x, t)|^{2} dx

(2.3)

where the positive constant l > 0 exists from Corollary 2.2. (If l = 0 then the plastic constitutive equation is of monotone type and the total energy does not increase.) Next we will estimate the last integral in (2.3) by the free energy.

\begin{aligned}|L̃ε − Mz|^{2} & = & |B^{T} D(ε − Bz) − Lz|^{2} ≤ 2|B^{T} D(ε − Bz)|^{2} + 2|Lz|^{2} \\ & ≤ & C([D(ε − Bz)] · (ε − Bz) + Lz · z) = 2Cρψ(ε, z) \end{aligned}  (2.4)

where C > 0 is a positive constant. We have used here that D is symmetric and positive definite and that L is symmetric and positive semi-definite. Inserting (2.4) into (2.3) we obtain

\frac{d}{dt}E(v, ε, z)(t) ≤ 2Cρ ∫_{Ω} ψ(ε(x, t), z(x, t)) dx ≤ 2CρE(v, ε, z)(t) .  (2.5)

Then integrating (2.5) we have

E(v, ε, z)(t) ≤ e^{2Cρt}E(v, ε, z)(0)  (2.6)

and the total energy can increase exponential, but does not blow up in finite time. With similar estimates to (2.6) we can prove for differences of two smooth solutions for the system (MS) with the plastic constitutive equations from the class LM. Let us assume that (v^{1}, ε^{1}, z^{1}) and (v^{2}, ε^{2}, z^{2}) are two smooth solutions for the system (MS) with the same boundary conditions (1.3) or (1.4) and the same initial conditions (IC). Let us denote by (v, ε, z) the difference (v^{1} − v^{2}, ε^{1} − ε^{2}, z^{1} − z^{2}) . Then in the same manner as used in (2.3) we get

\frac{d}{dt}E(v, ε, z)(t) = − ∫_{Ω} (L̃ε(x, t) − Mz(x, t)) · z_{t}(x, t) dx

= − ∫_{Ω} (L̃ε(x, t) − Mz(x, t)) ·

· [g(L̃ε^{1}(x, t) − Mz^{1}(x, t)) − g(L̃ε^{2}(x, t) − Mz^{2}(x, t))] dx  (2.7)

≤ l ∫_{Ω} |L̃ε(x, t) − Mz(x, t)|^{2} dx .

Next using (2.4) and (2.5) we conclude that

E(v, ε, z)(t) = 0 for all t ≥ 0 .  (2.8)

This yields that v^{1} = v^{2} and ε^{1} = ε^{2} . Inserting the last result into (2.8) we have

\begin{aligned}0 & = & ∫_{Ω} \{[DBz(x, t)] · Bz(x, t) + Lz(x, t) · z(x, t)\} dx \\ & = & ∫_{Ω} (B^{T} DB + L)z(x, t) · z(x, t) dx = ∫_{Ω} Mz(x, t) · z(x, t) dx \end{aligned}  (2.9)

and the pre-monotonicity of the model (M > 0) implies z^{1} = z^{2} . Thus we have proved the uniqueness of smooth solutions for the system (MS) with plastic constitutive equations from the class LM .

Next we want to study the existence theory for the system (MS) with plastic constitutive equations from the class LM . In this theory the fundamental property of a model is the so-called coercivity assumption.

Definition 2.3 We say that the plastic constitutive equation (MS3) is coercive if and only if the free energy ψ associated with this equation is positive definite.

Note that the coercivity of (MS3) is equivalent to the requirement that the operator L is positive definite. The main idea in the existence theory is to use the method of monotone operators ([4]). We use here the standard notations: ℍ^{m}(Ω; ℝ^{d}) denotes the Sobolev space of functions from 𝕃^{2}(Ω; ℝ^{d}) which all weak derivatives up to order m belong to 𝕃^{2}(Ω; ℝ^{d}) . The scalar product and the norm in the space ℍ^{m}(Ω; ℝ^{d}) we denote by (·, ·)_{m} and by ∥·∥_{m} respectively. The scalar product and the norm in the space 𝕃^{2}(Ω; ℝ^{d}) we denote by (·, ·) and by ∥ · ∥ . Now we write the system (MS) as an evolution equation

\frac{d}{dt}w(t) = Cw(t) w(0) = w^{0}  (EE)

(here w = (v, ε, z) and w^{0} = (v^{0}, ε^{0}, z^{0})) with the operator C : 𝕃^{2}(Ω; ℝ^{3} × S^{3} × ℝ^{N}) → 𝕃^{2}(Ω; ℝ^{3} × S^{3} × ℝ^{N}) defined by

C(v, ε, z) = (\frac{1}{ρ}div(ε − Bz), \frac{1}{2}(∇v + ∇^{T} v), g(L̃ε − Mz)) .  (2.10)

The domain of the operator C depends on the boundary condition under consideration and is defined by

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 (2.10) belongs to 𝕃^{2}(Ω; ℝ^{3} × S^{3} × ℝ^{N})}

(2.11)

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

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 (2.10) belongs to 𝕃^{2}(Ω; ℝ^{3} × S^{3} × ℝ^{N})

and the weak divergence div(ε − Bz) satisfies

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

(2.12)

for the homogeneous Neumann boundary condition.

Theorem 2.4 (existence theorem for coercive models from the class LM) Let the plastic constitutive equation (MS3) be coercive and belong to the class LM. If the constitutive function g : ℝ^{N} → ℝ^{N} is continuous then the operator −C can be represented as a sum of a maximal monotone operator and a lipschitz operator in the space H = 𝕃^{2}(Ω; ℝ^{3} × S^{3} × ℝ^{N}) equipped with the scalar product

⟨(v, ε, z), (v̄, ε̄, z̄)⟩_{H} = ∫_{Ω} (ρv · v̄ + [D(ε − Bz)] · (ε̄− Bz̄) + Lz · z̄) dx .  (2.13)

Moreover for all initial data w^{0} = (v^{0}, ε^{0}, z^{0}) from the domain of C there exists a unique global in time solution (v, ε, z) of the equation (EE) satisfying

∥(v_{t}(t), ε_{t}(t), z_{t}(t))∥_{H} ≤ Le^{Lt}∥C(v^{0}, ε^{0}, z^{0})∥_{H}

with a positive constant L which does not depend of the initial data.

Proof. The main idea of the proof is to use the existence theorem (4.3.1) from the book [1] for coercive and monotone models and next the lipschitz perturbation method of maximal monotone operators (theorem (3.17) in [4]). Therefore we start by studying the following system

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

with the boundary condition (1.3) or (1.4) and with the initial conditions (IC). The constitutive function g + L satisfies all assumptions of theorem (4.2.1) from [1]. Thus the operator −C_{l} generated by the right side of the system (PMS) with the domain D_{D} or D_{N} is a maximal monotone operator in the space H. Next we write the system (MS) in the form

\frac{d}{dt}w(t) = C_{l}w(t) + T_{l}w(t) w(0) = w^{0}  (2.14)

where the operator T_{l} is defined by

T_{l}(v, ε, z) = (0, 0, −l(L̃ε − Mz))  (2.15)

with the domain D(T_{l}) = H. It is easy to see that T_{l} is a lipschitz operator and therefore we can use theorem (3.17) from [4]. This implies the global existence result for the system (MS) with plastic constitutive equations from the class LM, provided the coercivity of these equations and the continuity of the function g .

□

For noncoercive models the existence theory is more complicated. A general existence result for noncoercive, pre-monotone models is still an open problem. We want to use here the concept of coercive limits (see for details for example [6, 7]) to prove an existence theorem for a special subclass of noncoercive models. Therefore we cite definition 5.4 from [7]

Definition 2.1 We say that the plastic constitutive equation (MS3) is self-controlling or belongs to the class SC if and only if for all measurable functions y : Ω → ℝ^{N} with |y(x)| ≥ 1 for almost every x ∈ Ω, the following estimate holds:

∥Bg(y)∥ ≤ F(∥Lg(y)∥, ∥y∥)  (2.16)

where F : ℝ_{+}^{2} → ℝ_{+} is a function mapping sets, bounded in at least one variable, into bounded sets.

The estimate (2.16) says that for plastic constitutive equations from the class SC we can control the right hand side of the ordinary differential equations for the variables Bz by the right hand side of the ordinary differential equations for the variables Lz or by the argument of the constitutive function g.

Theorem 2.5 (existence theorem for models from the class LM ∩ SC) Let the plastic constitutive equation (MS3) belong to the class LM ∩ SC and the constitutive function g : ℝ^{N} → ℝ^{N} be continuous. Then for all initial data w^{0} = (v^{0}, ε^{0}, z^{0}) 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 equation (EE) possesses unique, global in time solutions in 𝕃^{2}(Ω; ℝ^{3} × S^{3} × ℝ^{N}) .

Proof. First note that now the formula (2.13) does not define a scalar product in the space 𝕃^{2}(Ω; ℝ^{3} × S^{3} × ℝ^{N}) and thus the global monotonicity of the operator −C_{l} fails. We can only conclude that −C_{l} is monotone in the bilinear form given by (2.13). Nevertheless the idea of the proof is again easy. We use the conception of the coercive approximation ([7]) and for all natural numbers k > 0 we study the system

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

where the free energy function 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 .  (2.17)

Moreover we suppose that the boundary condition (1.3) or (1.4) holds and the initial conditions are given by (IC). If we define by H_{k} the space 𝕃^{2}(Ω; ℝ^{3} × S^{3} × ℝ^{N}) with the scalar product

⟨(v, ε, z), (v̄, ε̄, z̄)⟩_{H_{k}} = ∫_{Ω} [ρv · v̄ + [D(ε − Bz)] · (ε̄− Bz̄) + Lz · z̄ + \frac{1}{k}(Dε) · ε̄] dx (2.18)

then theorem (6.1) from [7] yields that the right hand side of the system (CMS) generates a maximal monotone operator in the space H_{k} . Next in the same manner as in Theorem 2.4, we perturb the system (CMS) by the lipschitz operator T_{l} defined by (2.15). The operator T_{l} is linear and clearly lipschitz in H_{k} for all k, but the geometry of the space 𝕃^{2}(Ω; ℝ^{3}×S^{3}×ℝ^{N}) depends on k, which implies that the lipschitz constant of the operator T_{l} can depend on k. By the linearity of T_{l} it is enough to estimate only the differences T_{l}(v, ε, z) − T_{l}(0, 0, 0). Then by the definition of the norm in the space H_{k} we have

∥T_{l}(v, ε, z)∥_{H_{k}} = l^{2} ∫_{Ω} (DB(L̃ε − Mz) · B(L̃ε − Mz) + L(L̃ε − Mz) · (L̃ε − Mz)) dx

≤ C_{1}l^{2} ∫_{Ω} |L̃ε − Mz|^{2}dx = C_{1}l^{2} ∫_{Ω} |B^{T} D(ε − Bz) − Lz|^{2}dx

≤ C_{2}l^{2} ∫_{Ω} (|D(ε − Bz)|^{2} + |Lz|^{2}) dx ≤ (in the same manner as in(2.4))

≤ C_{3}l^{2} ∫_{Ω} (D(ε − Bz) · (ε − Bz) + Lz · z) dx

≤ C_{3}l^{2} ∫_{Ω} (ρv^{2} + D(ε − Bz) · (ε − Bz) + Lz · z + \frac{1}{k}(Dε) · ε) dx

= C_{3}l^{2}∥(v, ε, z)∥_{H_{k}}  (2.19)

where C_{1}, C_{2}, C_{3} are positive constants independent of k. (2.19) implies that the sequence of lipschitz constants of the operator T_{l} in the spaces H_{k} can be estimate by a constant, which does not depend of k. Thus we obtain that for initial data w^{0} = (v^{0}, ε^{0}, z^{0}) from the domain of the operator C the system

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

has a unique, global in time solution (v^{k}, ε^{k}, z^{k}) satisfying

∥(v_{t}^{k}, ε_{t}^{k}, z_{t}^{k})(t)∥_{H_{k}} ≤ Ce^{Ct}∥(v^{0}, ε^{0}, z^{0})(t)∥_{H_{k}}  (2.20)

where C is a positive constant which does not depend of k. (2.20) and the assumptions i) + ii) for the initial data imply the following 𝕃^{2}-bounds for the left hand side of the system (CPMS)

ess sup_{t∈[0,T]}(∥(v_{t}^{k}(t)∥ , ∥(ε_{t}^{k}(t) − Bz_{t}^{k}(t)∥ , ∥Lz_{t}^{k}(t)∥ , \frac{1}{\sqrt{k}} ∥(ε_{t}^{k}(t)∥) ≤ C(T) . (2.21)

Using the idea of self-controlling we can remove the factor \frac{1}{\sqrt{k}} in the inequality (2.21), and in the same manner as in corollary (6.2) from [7], we prove that the sequence (v^{k}, ε^{k}, z^{k}) converges to the unique, global solution of the system (MS).

□

3 The model of Bodner-Partom as an element in the class LM ∩ SC

In this section we prove that the non-monotone and non-transformable to monotone type model of Bodner-Partom (see section 8 in [1]) belongs to LM. Using the self-controlling effect of the model we obtain the existence theorem for the system (MS) with the plastic constitutive equations (BP). Let us rewrite the equations (BP)

\begin{aligned}\frac{∂}{∂t}ε^{p} & = & d exp ( − α(\frac{ŷ}{|σ|})^{n})\frac{σ}{|σ|} \\ \frac{∂}{∂t}y & = & m(y_{1} − y)|\frac{∂}{∂t}ε^{p}| |σ| − Ay_{1}(\frac{(y − y_{2})^{+}}{y_{1}} )^{r} \end{aligned}  (BP’)

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, ŷ = max(y_{2}, y) and the function (y − y_{2})^{+} is the positive part of the difference y − y_{2} . Note that the right hand side of the equations (BP’) is well defined for all z = (ε^{p}, y) ∈ S^{3} × ℝ and Lemma 3.1 shows that our changes are admissible. First we observe that the equations (BP’) can be written in the form (MS3). Therefore we define the free energy associated with the model:

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

Thus the operator B is defined by B(ε^{p}, y) = ε^{p} and the operator L(ε^{p}, y) = (0, y) is positive semi-definite. The gradient −ρ∇_{z}ψ has the form

− ρ∇_{(ε^{p},y)}ψ(ε, ε^{p}, y) = B^{T} D(ε − ε^{p}) − L(ε^{p}, y) = (D(ε − ε^{p}), −y) .  (3.2)

If we denote by P : S^{3} → S^{3} the projector P ε = ε − \frac{1}{3}tr ε · I onto the linear subspace {ε ∈ S^{3} | tr ε = 0} ⊂ S^{3} then the equations (BP’) can be written in the form

\frac{∂}{∂t}z = ( \begin{matrix}\frac{∂}{∂t}ε^{p} \\ \frac{∂}{∂t}y \end{matrix} ) = g( − ρ∇_{(ε^{p},y)}ψ(ε, ε^{p}, y)) = ( \begin{matrix}g_{1}((D(ε − ε^{p}), −y)) \\ g_{2}((D(ε − ε^{p}), −y)) \end{matrix} )  (3.3)

where

g_{1}((D(ε − ε^{p}), −y)) = d exp ( − α(\frac{ŷ}{|P (D(ε − ε^{p}))|})^{n})\frac{P (D(ε − ε^{p}))}{|P (D(ε − ε^{p}))|} ,
g_{2}((D(ε − ε^{p}), −y)) = m(y_{1} − y) |g_{1}((D(ε − ε^{p}), −y))| |P (D(ε − ε^{p}))|

− Ay_{1}(\frac{(y − y_{2})^{+}}{y_{1}} )^{r} .

Note that if the argument −ρ∇_{z}ψ(ε, z) of the function g is equal to zero than D(ε−ε^{p}) = 0 and y = 0 and consequently g((0, 0)) = (0, 0). Moreover the operator M(ε^{p}, y) = (B^{T} DB + L)(ε^{p}, y) = (Dε^{p}, y) is positive definite in S^{3} × ℝ and therefore the model of Bodner-Partom is of pre-monotone type. The free energy function ψ defined by (3.1) is only positive semi-definite on S^{3} × S^{3} × ℝ and the coercivity of the model fails. But the right hand side of the evolution equation for ε^{p} is bounded and thus the model (BP’) belongs to the class SC . Next we want to show that the equation (BP’) is from the class LM. To do this we need the following lemma about 𝕃^{∞}-estimates for the isotropic hardening function y.

Lemma 3.1 Assume that y(t) ∈ W^{1,∞}([0, T); 𝕃^{2}(Ω; ℝ^{1})) is a solution of the differential equation:

\frac{d}{dt} y(x, t) = g_{2}(D(ε(x, t) − ε^{p}(x, t)), −y(x, t)))  (3.4)

for the given function D(ε − ε^{p})− ∈ 𝕃^{2}([0, T) × Ω; S^{3}) . Moreover suppose that the initial data y^{0} satisfies

y_{2} ≤ y^{0}(x) ≤ y_{1} for almost all x ∈ Ω .

Then for each t ∈ [0, T), and for almost x ∈ Ω the isotropic hardening function y satisfies the same condition

y_{2} ≤ y(x, t) ≤ y_{1} .

Proof. First we prove that y(x, t) ≤ y_{1} . We multiply equation (3.4) by (y(x, t) − y_{1})^{+} and integrate over Ω. Then we obtain

∫_{Ω} \frac{d}{dt} y(x, t) · (y(x, t) − y_{1})^{+} dx = ∫_{Ω} g_{2}(D(ε(x, t) − ε^{p}(x, t)), −y(x, t))) · (y(x, t) − y_{1})^{+}dx,

and by definition of (y(x, t) − y_{1})^{+}:

\frac{1}{2} \frac{d}{dt} ∥(y(x, t) − y_{1})^{+}∥_{𝕃^{2}(Ω)}^{2} = ∫_{Ω} g_{2}(D(ε(x, t) − ε^{p}(x, t)), −y(x, t))) · (y(x, t) − y_{1})^{+}dx.

The next step is the integration with respect to t. Using that the initial data y^{0} belongs to the interval [y_{2}, y_{1}] we conclude that

∥(y(x, t) − y_{1})^{+}∥_{𝕃^{2}(Ω)}^{2} = 2 ∫_{0}^{t} ∫_{Ω} g_{2}(D(ε(x, t) − ε^{p}(x, t)), −y(x, t))) · (y(x, t) − y_{1})^{+}dx.

Next we observe that for y ≥ y_{1}

g_{2}((D(ε − ε^{p}), −y)) ≤ 0

and consequently the right hand side of the last integral is not positive. This implies that

∥(y(x, t) − y_{1})^{+}∥_{𝕃^{2}(Ω)} = 0 .

The proof for the inequality y_{2} ≤ y(x, t) runs analogously. We use that

g_{2}((D(ε − ε^{p}), −y)) ≥ 0 for y ≤ y_{2} .

□

By Lemma 3.1 we can solve our problem with a slightly different constitutive equation:

\begin{aligned}\frac{∂}{∂t}ε^{p} & = & g̃_{1}(D(ε − ε^{p}), −y)) & = & g_{1}(D(ε − ε^{p}), −η(y))) , \\ \frac{∂}{∂t}y & = & g̃_{2}(D(ε − ε^{p}), −y)) & = & g_{2}(D(ε − ε^{p}), −η(y))) , \end{aligned}  (BP”)

where by η we denote the projector on the interval [y_{2}, y_{1}]:

η(y) = y − (y − y_{1})^{+} + (y_{2} − y)^{+} .  (3.5)

The equation (BP”) is equivalent to equation (BP’) from our point of view: only initial data y^{0}(x) ∈ [y_{2}, y_{1}] for a.e. x ∈ Ω are physical admissible. Moreover for the function g̃ we have that g̃((0, 0)) = g((0, y_{2})) = 0. Now we prove that the constitutive equation (BP”) belongs to the class LM. To simplify the notation we introduce:

G(p) = d exp ( − α(\frac{1}{p})^{n}),

for each p ≥ 0. In the next lemmas we show that the ordinary differential equation (BP”) satisfies inequality (2.1) from Corollary 2.2. The first lemma shows that the right hand side of the equation (BP”1) is a global lipschitz function.

Lemma 3.2 Let g̃_{1} be as defined in (BP”) then for all ε − ε^{p}, \overline{ε} − \overline{ε^{p}} ∈ S^{3} and for all y, \overline{y} ∈ ℝ the following inequality holds:

\begin{aligned}|g̃_{1}(D(ε − ε^{p}), −y)) − g̃_{1}(D(\overline{ε} − \overline{ε}^{p}), −\overline{y}))| & = & |G(\frac{|σ|}{η(y)})\frac{σ}{|σ|} − G(\frac{|\overline{σ}|}{η(\overline{y})})\frac{\overline{σ}}{|\overline{σ}|}| \\ & ≤ & L_{1} (|σ − \overline{σ}| + |y − \overline{y}|) \end{aligned} (3.6)

where σ = P D(ε − ε^{p}), \overline{σ} = P D(\overline{ε} − \overline{ε^{p}}) and the positive constant L_{1} does not depend of the arguments of the function g̃_{1}.

Proof. We estimate the difference g̃_{1}(·) − g̃_{1}(\overline{·}) by:

|G(\frac{|σ|}{η(y)})\frac{σ}{|σ|} − G(\frac{|\overline{σ}|}{η(\overline{y})})\frac{\overline{σ}}{|\overline{σ}|}| ≤ |G(\frac{|σ|}{η(y)}) − G(\frac{|\overline{σ}|}{η(\overline{y})})| + G(\frac{|\overline{σ}|}{η(\overline{y})}) |\frac{σ}{|σ|} − \frac{\overline{σ}}{|\overline{σ}|}|
≤ |G(\frac{|σ|}{η(y)}) − G(\frac{|\overline{σ}|}{η(\overline{y})})|

+G(\frac{|\overline{σ}|}{η(\overline{y})})(\frac{1}{|\overline{σ}|} |σ − \overline{σ}| + |σ|\frac{||σ| − |\overline{σ}||}{|σ||\overline{σ}|})

≤ |G(\frac{|σ|}{η(y)}) − G(\frac{|\overline{σ}|}{η(\overline{y})})| + G(\frac{|\overline{σ}|}{η(\overline{y})})\frac{η(\overline{y})}{|\overline{σ}|} · \frac{2}{η(\overline{y})}|σ − \overline{σ}|
≤ |G(\frac{|σ|}{η(y)}) − G(\frac{|\overline{σ}|}{η(\overline{y})})| + sup_{p≥0} \frac{G(p)}{p} \frac{2}{y_{2}} |σ − \overline{σ}| .

It is not hard to notice that there exists a positive constant C_{1}(d, α, n) such that:

|G(p) − G(\overline{p})| = d| exp(−α(p)^{−n}) − exp(−α(\overline{p})^{−n})|
≤ \frac{C_{1}(d, α, n)}{1 + min\{p, \overline{p}\}}|p − \overline{p}| .

(3.7)

Inequality (3.7) is a stronger statement than the lipschitz estimate. We obtain by (3.7) that the difference |G(p) − G(\overline{p})| is small provided that the arguments are sufficiently large. (3.7) can be easily verified from the boundedness of the function pG′(p). Now using inequality (3.7) we estimate the difference G(·) − G(\overline{·}):

|G(\frac{|σ|}{η(y)}) − G(\frac{|\overline{σ}|}{η(\overline{y})})| ≤ \frac{C_{1}(d, α, n)}{1 + min \{\frac{|σ|}{η(y)}, \frac{|\overline{σ}|}{η(\overline{y})}\}} |\frac{|σ|}{η(y)} − \frac{|\overline{σ}|}{η(\overline{y})}|
≤ \frac{C_{1}(d, α, n)}{1 + min \{\frac{|σ|}{η(y)}, \frac{|\overline{σ}|}{η(\overline{y})}\}} |max \{\frac{|σ|}{η(y)}, \frac{|\overline{σ}|}{η(\overline{y})}\} − min \{\frac{|σ|}{η(y)}, \frac{|\overline{σ}|}{η(\overline{y})}\}|
≤ \frac{C_{1}(d, α, n)}{1 + min \{\frac{|σ|}{η(y)}, \frac{|\overline{σ}|}{η(\overline{y})}\}} max \{\frac{1}{η(y)}, \frac{1}{η(\overline{y})}\} ||σ| − |\overline{σ}||

+\frac{C_{1}(d, α, n)}{1 + min \{\frac{|σ|}{η(y)}, \frac{|\overline{σ}|}{η(\overline{y})}\}} min \{|σ|, |\overline{σ}|\} |\frac{1}{η(y)} − \frac{1}{η(\overline{y})}|

≤ \frac{C_{1}(d, α, n)}{y_{2}}(|σ − \overline{σ}| + \frac{y_{1}}{y_{2}} |η(y) − η(\overline{y})|) .

The contraction property of the projector η: |η(y) − η(\overline{y})| ≤ |y − \overline{y}| completes the proof.

□

The right hand side of the equation (BP”2) is not global lipschitz, but we demonstrate that the equation satisfies a similar inequality to (2.1).

Lemma 3.3 Let g̃_{2} be as defined in (BP”2) then for all ε − ε^{p}, \overline{ε} − \overline{ε^{p}} ∈ S^{3} and for all y, \overline{y} ∈ ℝ the following inequality holds:

\begin{matrix}[g̃_{2}(D(ε − ε^{p}), −η(y)) − g̃_{2}(D(\overline{ε} − \overline{ε}^{p}), −η(\overline{y}))] · (y − \overline{y}) \\ ≤ L_{2}(|P (D[(ε − ε^{p}) − (\overline{ε} − \overline{ε}^{p})])|^{2} + |y − \overline{y}|^{2}) \end{matrix}  (3.8)

with a positive constant L_{2} which does not depend of the arguments of the function g̃_{2}. Proof. By the definition of the function g̃_{2} we obtain

\begin{matrix}[g̃_{2}(ε − ε^{p}, −η(y)) − g̃_{2}(\overline{ε} − \overline{ε}^{p}, −η(\overline{y}))] · (y − \overline{y}) \\ = m[ (y_{1} − η(y)) G(\frac{|σ|}{η(y)})|σ| − (y_{1} − η(\overline{y})) G(\frac{|\overline{σ}|}{η(\overline{y})})|\overline{σ}|] · (y − \overline{y}) \\ −Ay_{1}((\frac{(η(y) − y_{2})^{+}}{y_{1}} )^{r} − (\frac{(η(\overline{y}) − y_{2})^{+}}{y_{1}} )^{r}) · (y − \overline{y}) \\ ≤ m [(y_{1} − η(y)) − (y_{1} − η(\overline{y}))] (y − \overline{y}) G(\frac{|σ|}{η(y)})|σ| \\ +m (y_{1} − η(y)) (G(\frac{|σ|}{η(y)})|σ| − G(\frac{|\overline{σ}|}{η(\overline{y})})|\overline{σ}|) · (y − \overline{y}) . \end{matrix}  (3.9)

By the monotonicity of the projector η

(η(y) − η(\overline{y})) · (y − \overline{y}) ≥ 0 for all y, \overline{y} ∈ ℝ

the first term in the right hand side of inequality (3.9) is non-positive. We have to estimate the second one. To this end the following inequality is necessary:

|G(p) − G(\overline{p})| = d| exp(−α(p)^{−n}) − exp(−α(\overline{p})^{−n})|
≤ \frac{C_{2}(d, α, n)}{1 + min\{(p)^{2}, (\overline{p})^{2}\}}|p − \overline{p}| .

(3.10)

Inequality (3.10) is similar to inequality (3.7) and results from the boundedness of the function p^{2}G′(p). Using (3.10) and following the min-max idea from the proof of Lemma 3.2, we obtain:

|G(\frac{|σ|}{η(y)})|σ| − G(\frac{|\overline{σ}|}{η(\overline{y})})|\overline{σ}|| ≤ max \{G(\frac{|σ|}{η(y)}), G(\frac{|\overline{σ}|}{η(\overline{y})})\} ||σ| − |\overline{σ}||

+min \{|σ|, |\overline{σ}|\} |G(\frac{|σ|}{η(y)}) − G(\frac{|\overline{σ}|}{η(\overline{y})})|

≤ (\frac{C_{2}(d, α, n)y_{1})}{(y_{2})^{2}} + sup_{p≥0} G(p))|σ − \overline{σ}|

+\frac{C_{2}(d, α, n)(y_{1})^{2}}{(y_{2})^{2}} |y − \overline{y}| .

Inserting the last inequality into (3.9), we complete the proof.

□

Finally we verify that Lemma 3.2 and Lemma 3.3 imply that the plastic constitutive equation (BP”) belongs to the class LM.

[g̃((D(ε − ε^{p}), −y)) − g̃((D(\overline{ε} − \overline{ε^{p}}), −\overline{y}))] · (D(ε − ε^{p}) − D(\overline{ε} − \overline{ε^{p}}), −y + \overline{y})

= [g_{1}((D(ε − ε^{p}), −η(y))) − g_{1}((D(\overline{ε} − \overline{ε^{p}}), −η(\overline{y})))] · (P D(ε − ε^{p}) − P D(\overline{ε} − \overline{ε^{p}}))

+ [g_{2}((D(ε − ε^{p}), −η(y))) − g_{2}((D(\overline{ε} − \overline{ε^{p}}), −η(\overline{y})))] · (\overline{y} − y)
≥ −L_{1} (|σ − \overline{σ}| + |y − \overline{y}|) · |σ − \overline{σ}| − L_{2} (|σ − \overline{σ}|^{2} + |y − \overline{y}|^{2})
≥ −l (|D(ε − ε^{p}) − D(\overline{ε} − \overline{ε^{p}})|^{2} + |y − \overline{y}|^{2})

with a suitable positive constant l (we have used here that the projector P is a contraction). Thus the inequality (2.1) from Corollary 2.2 is satisfied.

Remark The domain D_{D} of the operator C, generated by the right hand side of the system (MS) with the plastic constitutive equation (BP”), is equal to ℍ_{0}^{1}(Ω; ℝ^{3}) × 𝕃^{2}(Ω; S^{3} × ℝ^{N}). This is because the right hand side of equation (BP”1) is bounded, and in the right hand side of equation (BP”2), only the function |σ| does not belong to 𝕃^{∞}. In this situation we can drop the assumption div Dε^{0} ∈ 𝕃^{2}(Ω; ℝ^{3}) from Theorem 2.5. We approximate the initial data ε^{0} by a sequence of functions ε_{n}^{0} from the domain of the operator C for which div Dε_{n}^{0} ∈ 𝕃^{2}(Ω; ℝ^{3}). Such a sequence exists since the domain D_{D} has very simple structure. In the same manner we can omit the second assumption for the initial data from this theorem.

Conclusion: This work shows that the idea of lipschitz perturbations of monotone operators can be successful implemented in the theory of viscoplastic response with coercive plastic constitutive equations (Theorem 2.4) or with self-controlling plastic constitutive equations (Theorem 2.5). From this paper can be easy conclude the uniqueness of strong solutions for the Bodner-Partom model (or for a general model from the class LM ∩ SC) with nonhomogeneous boundary conditions.

References

[1] H.D. Alber : “Initial-Boundary Value Problems for the Inelastic Material Behaviour of Metals” – Lecture Notes in Mathematics No. 1682, Springer 1998

[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] 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

[6] K. Chełmiński : “On initial–boundary value problems for the inelastic material behaviour of metals”– to appear in Z. Angew. Math. Mech. 1998

[7] K. Chełmiński : “Coercive limits for a subclass of monotone constitutive equations in the theory of inelastic material behavior of metals” – to appear in Rocz.PTM: Mat. Stos. 1997

[8] K. Chełmiński : “On self-controlling models in the theory of inelastic material behavior of metals” – to appear in Continuum Mechanics and Thermodynamics 1998

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

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

[11] P. Le Tallec : “Numerical analysis of viscoelastic problems” – Paris: Masson. Berlin, Heidelberg, New York: Springer 1990

Addresses: Krzysztof Chełmiński Fachbereich Mathematik Technische Hochschule Darmstadt Schloßgartenstr. 7 64289 Darmstadt, Germany e-mail: chel@mathematik.th-darmstadt.de Piotr Gwiazda Institute of Applied Math. and Mech. Warsaw University Banacha 2 00–913 Warsaw, Poland e-mail: pgwiazda@appli.mimuw.edu.pl