Stress L^{∞}– estimates and the uniqueness problem for the quasistatic equations to the model of Bodner–Partom

Krzysztof Chełmiński

Technical University of Darmstadt

Dedicated to Rolf Leis

Abstract

This paper proves the uniqueness result for global in time large solutions of quasistatic equations to an inelastic model of material behavior of metals, provided that an apriori L^{∞}–estimation for the Cauchy stress tensor holds.

AMS classifications: 35Q72, 73F99

1 Introduction

In this work we consider a system of equations, which describes an inelastic material behavior of metals. The system consists of linear partial differential equations and of nonlinear ordinary differential equations for internal variables. Here we are interested in studying the uniqueness problem for the quasistatic case. Thus the system of equations can be written in the following form

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

for (x, t) ∈ Ω × ℝ_{+} , where Ω ⊆ ℝ^{3} is an open, connected set with smooth boundary ∂Ω (this set represents the masspoints of the body). Here u : Ω × ℝ_{+} ⟶ ℝ^{3} is the displacement vector, T : Ω×ℝ_{+} ⟶ ℝ_{sym}^{3×3} is the Cauchy stress tensor, ε(∇_{x}u(x, t)) = \frac{1}{2}(∇_{x}u(x, t) + (∇_{x}u(x, t))^{T} ) is the linear strain tensor, ε_{p} : Ω × ℝ_{+} ⟶ ℝ_{sym}^{3×3} is the

plastic strain tensor, z(x, t) = (ε_{p}(x, t), z̃(x, t)) ∈ ℝ_{sym}^{3×3} × ℝ^{N−9} ⊆ ℝ^{N} (where N > 9 denote the quantity of ordinary differential equations) is the vector of internal variables and F : Ω × ℝ_{+} ⟶ ℝ^{3} is a given external force. The matrix D is the elasticity tensor, which we assume to be positive definite and independent of x and t. Finally the given function f describes the evolution of the internal variables. The equation (P1) is the balance of momentum and the equations (P2), (P3) are called constitutive equations. (P2) describes the linear stress-elastic strain relation and is independent of the considered material. (P3) is the plastic constitutive relation. This equation depends on the material and yields nonlinearities to the system. Thus we can say, that the function f defines a model for the metal under consideration. For models of monotone type (see definition in [4]) similarly as in [4] we can prove the uniqueness of strong solutions to the problem (P). In the book [10] the uniqueness theorem has been proved under the strong assumption, that the function f is a global Lipschitz function. On the other hand in the engineering literature there can be found nonlipschitz models, which are not of monotone type and which cannot be transformed to monotone type ([4]). Therefore we must study such models separatly. This work considers the model of Bodner-Partom. In the model the vector z of internal variables consists of the nine components of the plastic strain tensor ε_{p}, and of one positive variable y describing isotropic hardening of the metal. Thus N = 10 and the plastic constitutive equations are in the form

\frac{∂}{∂t} ε_{p} = G(\frac{|σ|}{y}) \frac{σ}{|σ|}
\frac{∂}{∂t} y = m(y_{1} − y)G(\frac{|σ|}{y}) |σ|  (C)

where σ = T − \frac{1}{3}(tr T) · I is the stress deviator, G : ℝ_{+} → ℝ_{+} is a given function and m > 0, y_{1} > 0 are given constants. In [5] S.R. Bodner and Y. Partom have proposed the function G to be of the form

G_{BP} (p) = A exp (−\frac{B}{p^{2k}} ) , A, B, ∈ ℝ_{+} k > \frac{1}{2}.

In this paper we assume that the function G satisfies:

Assumptions

1. G(p), \frac{G(p)}{p} are ℂ^{∞}(ℝ_{+}, ℝ_{+}) functions, (A1)

2. ∃_{g>0} ∀_{p∈ℝ_{+}} G(p) < g , (G is bounded) (A2)

3. ∀_{p∈ℝ_{+}} G′(p) ≥ 0 (G is nondecreasing) (A3)

4. ∃_{g′>0} ∀_{p∈ℝ_{+}} G′(p)p^{2} < g′ . (A4)

An easy computation shows that the function G_{BP} satisfies all assumptions (A1) − (A4) . It is easy to see that the function G is a global Lipschitz function, but the right hand side of the equation (C2) is not a global Lipschitz function. Thus we study the following system of equations

\begin{aligned}div_{x} T (x, t) & = & F (x, t) \\ T (x, t) & = & D(ε(∇_{x}u(x, t)) − ε_{p}(x, t)) \\ ε(∇_{x}u(x, t)) & = & \frac{1}{2}(∇_{x}u(x, t) + (∇_{x}u(x, t))^{T} ) \\ σ(x, t) & = & T (x, t) − \frac{1}{3}(tr T (x, t) · I \\ ∂_{t}ε_{p}(x, t) & = & G(\frac{|σ(x, t)|}{y(x, t)}) \frac{σ(x, t)}{|σ(x, t)|} \\ ∂_{t}y(x, t) & = & m(y_{1} − y(x, t))G(\frac{|σ(x, t)|}{y(x, t)})|σ(x, t)| , \end{aligned}  (BP)

with the Dirichlet boundary condition

u(x, t)_{|∂Ω} = g(x, t)

and with the initial conditions

\begin{aligned}ε_{p}(x, 0) & = & ε_{p}^{(0)}(x) , \\ y(x, 0) & = & y^{(0)}(x) . \end{aligned}

where g : ∂Ω × ℝ_{+} ⟶ ℝ^{3}, ε_{p}^{(0)} : Ω ⟶ ℝ_{sym}^{3×3}, y^{(0)} : Ω ⟶ ℝ_{+} are given functions. Moreover for the initial data y^{(0)} we assume that

∃_{y^{∗}>0} ∀_{x∈Ω} y^{∗} ≤ y^{(0)}(x) ≤ y_{1} .  (A5)

By (A5) the isotropic hardening function y(x, t) is a positive 𝕃^{∞}–function satisfying

y^{∗} ≤ y(x, t) ≤ y_{1}

for all (x, t) ∈ Ω × ℝ_{+}. In [8] existence of strong global in time solutions (u(t), T (t), ε_{p}(t), y(t)) ∈ ℍ^{2}(Ω; ℝ^{3}) × ℍ^{1}(Ω; ℝ_{sym}^{3×3})^{2} × ℍ^{1}(Ω; ℝ_{+}) to the problem (BP) for all initial data (ε_{p}^{(0)}, y^{(0)}) ∈ ℍ^{1}(Ω; ℝ_{sym}^{3×3}) × ℍ^{1}(Ω; ℝ_{+}) is proved (ℍ^{s} denotes the usual Sobolev space). Moreover existence results for another models can be found in [1,2,3,6,7,11,12]. In this paper we will study the uniqueness of solutions to for the problem (BP).

2 Uniqueness theorem

In this section we prove the uniqueness result for the problem (BP), provided an apriori 𝕃^{∞}–estimation for the stress deviator σ holds. Thus we replace the uniqueness problem for the system (BP) with a regularity question.

Theorem 1 Assume that the initial data (ε_{p}^{(0)}, y^{(0)}) satisfy

ε_{p}^{(0)} ∈ ℍ^{1}(Ω; ℝ_{sym}^{3×3}) , y^{(0)} ∈ ℍ^{1}(Ω; ℝ_{+})

and tr ε_{p}^{(0)} = 0 for almost all x ∈ Ω . Moreover, suppose that the assumptions (A1) – (A5) hold and

\begin{matrix}g ∈ 𝕃^{∞}(ℝ_{+}; ℍ^{\frac{3}{2}} (∂Ω; ℝ^{3}) , \\ F ∈ 𝕃^{∞}(ℝ_{+}; 𝕃^{2}(Ω; ℝ^{3}) . \end{matrix}

Let (u^{i}, T^{i}, ε_{p}^{i}, y^{i}) i = 1, 2 be two solutions of the problem (BP) to the same data g , F , ε_{p}^{(0)} , y^{(0)}. If for i = 1, 2 the following inequality

∀_{t^{∗}≥0} ∃_{C_{i}(t^{∗})} sup_{t≤t^{∗}} ∥σ(t)∥_{𝕃^{∞}} ≤ C_{i}(t^{∗})

holds, then (u^{1}, T^{1}, ε_{p}^{1}, y^{1}) = (u^{2}, T^{2}, ε_{p}^{2}, y^{2}) .

Proof. Let us define (u, T, ε_{p}, y) = (u^{1}, T^{1}, ε_{p}^{1}, y^{1}) − (u^{2}, T^{2}, ε_{p}^{2}, y^{2}). The functions (u, T, ε_{p}, y) satisfy the following system of equations

div_{x} T (x, t) = 0
T (x, t) = D(ε(∇_{x}u(x, t)) − ε_{p}(x, t))
ε(∇_{x}u(x, t)) = \frac{1}{2}(∇_{x}u(x, t) + (∇_{x}u(x, t))^{T} )
σ(x, t) = T (x, t) − \frac{1}{3}(tr T (x, t) · I
∂_{t}ε_{p}(x, t) = G(\frac{|σ^{1}(x, t)|}{y^{1}(x, t)}) \frac{σ^{1}(x, t)}{|σ^{1}(x, t)|} +

−G(\frac{|σ^{2}(x, t)|}{y^{2}(x, t)}) \frac{σ^{2}(x, t)}{|σ^{2}(x, t)|}

∂_{t}y(x, t) = m(y_{1} − y^{1}(x, t))G(\frac{|σ^{1}(x, t)|}{y^{1}(x, t)})|σ^{1}(x, t)| +

−m(y_{1} − y^{2}(x, t))G(\frac{|σ^{2}(x, t)|}{y^{2}(x, t)})|σ^{2}(x, t)|

where σ^{i}(x, t) = T^{i}(x, t) − \frac{1}{3}(tr T^{i}(x, t) · I , i = 1, 2 , the homogeneous Dirichlet boundary condition

u(x, t)_{|∂Ω} = 0

and the homogeneous initial conditions

\begin{aligned}ε_{p}(x, 0) & = & 0 , \\ y(x, 0) & = & 0 . \end{aligned}

Let us define an energy function

E(t) = \frac{1}{2}(T (t), ε(∇_{x}u(t)) − ε_{p}(t))

where (·, ·) denotes the 𝕃^{2}–scalar product. The assumptions for the elastic tensor D imply that the function E is not negative. For the time derivative of the energy function by the symmetry of the stress tensor T and by tr ε_{p}(t) = 0 we have

\frac{d}{dt} E(t) = (T (t), \frac{∂}{∂t}ε(∇_{x}u(t))) − (T (t), \frac{∂}{∂t}ε_{p}(t))
= (T (t), ∇_{x} \frac{∂}{∂t}u(t)) − (σ(t), \frac{∂}{∂t}ε_{p}(t))
= −(div T (t), \frac{∂}{∂t}u(t)) − (σ(t), \frac{∂}{∂t}ε_{p}(t))
= −(σ(t), \frac{∂}{∂t}ε_{p}(t)) .

We substitute the right hand side of the ordinary differential equation for ε_{p} in the last equality and we obtain

\frac{d}{dt} E(t) = − (σ^{1}(t) − σ^{2}(t), G(\frac{|σ^{1}(t)|}{y^{1}(t)}) \frac{σ^{1}(t)}{|σ^{1}(t)|} − G(\frac{|σ^{2}(t)|}{y^{2}(t)}) \frac{σ^{2}(t)}{|σ^{2}(t)|})
= −∫_{Ω} G (\frac{|σ^{1}(t)|}{y^{1}(t)}) |σ^{1}(t)| dx − ∫_{Ω} G (\frac{|σ^{2}(t)|}{y^{2}(t)}) |σ^{2}(t)| dx +

+∫_{Ω} G (\frac{|σ^{1}(t)|}{y^{1}(t)}) \frac{σ^{1}(t) · σ^{2}(t)}{|σ^{1}(t)|} dx + ∫_{Ω} G(\frac{|σ^{2}(t)|}{y^{2}(t)}) \frac{σ^{1}(t) · σ^{2}(t)}{|σ^{2}(t)|} dx

≤ ∫_{Ω} \{G (\frac{|σ^{1}(t)|}{y^{1}(t)}) − G (\frac{|σ^{2}(t)|}{y^{2}(t)})\} (|σ^{1}(t)| − |σ^{2}(t)|) dx

The last inequality shows that the isotropic hardening function violates the monotonicity of the problem (BP). Let us fix t^{∗} > 0. Thus for all t ≤ t^{∗} we can write

\begin{aligned}\frac{d}{dt} E(t) & ≤ & C∫_{Ω} |\frac{|σ^{1}(t)|}{y^{1}(t)} − \frac{|σ^{2}(t)|}{y^{2}(t)}|||σ^{1}(t)| − |σ^{2}(t)|| dx \\ & ≤ & C (∫_{Ω} |σ(t)|^{2} \frac{1}{y^{1}(t)} dx + ∫_{Ω} \frac{1}{y^{1}(t)y^{2}(t)}|y(t)||σ^{2}(t)||σ(t)| dx) \\ & ≤ & C (\frac{1}{y^{∗}} ∫_{Ω} |σ(t)|^{2} dx + \frac{C_{2}(t^{∗})}{(y^{∗})^{2}} ∫_{Ω}|y(t)||σ(t)| dx) \\ & ≤ & C̃(E(t) + ∥y(t)∥_{𝕃^{2}}^{2}) \end{aligned} (2.1)

where C is the global Lipschitz constant of the function G. Now we try to prove a similar inequality as above for the function ∥y(t)∥_{𝕃^{2}}^{2}.

\frac{d}{dt} \frac{1}{2}∥y(t)∥_{𝕃^{2}}^{2} = (y(t), \frac{∂}{∂t}y(t))
= my_{1}∫_{Ω} \{G (\frac{|σ^{1}(t)|}{y^{1}(t)}) |σ^{1}(t)| − G (\frac{|σ^{2}(t)|}{y^{2}(t)}) |σ^{2}(t)|\} y(t) dx

−m∫_{Ω} y(t)(y^{1}(t) − y^{2}(t))G (\frac{|σ^{1}(t)|}{y^{1}(t)}) |σ^{1}(t)| dx

−m ∫_{Ω} y(t)y^{2}(t) \{G (\frac{|σ^{1}(t)|}{y^{1}(t)}) |σ^{1}(t)| − G (\frac{|σ^{2}(t)|}{y^{2}(t)}) |σ^{2}(t)|\} dx

≤ 2my_{1}C∫_{Ω} |\frac{|σ^{1}(t)|}{y^{1}(t)} − \frac{|σ^{2}(t)|}{y^{2}(t)}| |σ^{1}(t)||y(t)| dx + my_{1}g ∫_{Ω} |σ(t)||y(t)| dx

+ mgC_{1}(t^{∗})∫_{Ω} |y(t)|^{2} dx + my_{1}g ∫_{Ω} |σ(t)||y(t)| dx .

Thus similarly as for the energy function we obtain

\frac{d}{dt} \frac{1}{2}∥y(t)∥_{𝕃^{2}}^{2} ≤ C̃(E(t) + ∥y(t)∥_{𝕃^{2}}^{2})  (2.2)

We sum (2.1) and (2.2). By E(0) = 0 and ∥y(0)∥_{𝕃^{2}} = 0 we have for all t ≤ t^{∗} E(t) = ∥y(t)∥_{𝕃^{2}} = 0. Also σ(t) = 0. Thus the ordinary differential equation for ε_{p} yields ε_{p}(t) = 0. Moreover T (t) = 0 implies ε(t) − ε_{p}(t) = 0 and by the Korn inequality we finally have u(t) = 0. The proof of the theorem is complete.

3 W^{1,q}– estimates

By theorem 1 the problem (BP) has a unique solution if we prove an apriori 𝕃^{∞}– estimation for the stress deviator σ. Thus we can try and prove the following regularity result: if the given data are more regular then all solutions to these data are more regular too. Let us denote by W^{1,q} the usual Sobolev space over 𝕃^{q}.

Theorem 2 Assume that the initial data (ε_{p}^{(0)}, y^{(0)}) satisfy

∃_{q>3} ε_{p}^{(0)} ∈ W^{1,q}(Ω; ℝ_{sym}^{3×3}) , y^{(0)} ∈ W^{1,q}(Ω; ℝ_{+})

with tr ε_{p}^{(0)} = 0 for all x ∈ Ω . Moreover, suppose that the assumptions (A1) – (A5) hold and

\begin{matrix}g ∈ 𝕃^{∞}(ℝ_{+}; W^{2−1/q,q}(∂Ω; ℝ^{3}) , \\ F ∈ 𝕃^{∞}(ℝ_{+}; 𝕃^{q}(Ω; ℝ^{3}) . \end{matrix}

Then a solution of the problem (BP) satisfies the following estimation

∀_{t^{∗}≥0} ∃_{C^{∗}(t^{∗})} sup_{t≤t^{∗}} ∥T (t)∥_{W^{1,q}} ≤ C^{∗}(t^{∗}) .

Proof. We start with the 𝕃^{q}– estimation for the gradient of the stress tensor T. From the constitutive equation for T we obtain

∥∇T (t)∥_{𝕃^{q}}^{q} ≤ C_{q} (∥u(t)∥_{W^{2,q}}^{q} + ∥∇ε_{p}(t)∥_{𝕃^{q}}^{q}) .

The displacement vector u satisfies the following linear elliptic boundary value problem

\begin{aligned}div Dε(∇_{x}u(x, t)) & = & div Dε_{p}(x, t) + F (x, t) \\ u(x, t)_{|∂Ω} & = & g(x, t) . \end{aligned}

Thus by the Sobolev–regularity theorem for solutions of linear elliptic equations we have

∃_{C>0} ∥u(t)∥_{W^{2,q}} ≤ C (∥div Dε_{p}(t)∥_{𝕃^{q}} + ∥F (t)∥_{𝕃^{q}} + ∥g(t)∥_{W^{2−1/q,q}})  (3.1)

To prove the boundedness of the 𝕃^{q}– norm of the stress tensor T we must estimate the 𝕃^{q}– norm of the gradient of the plastic strain tensor ε_{p}. Thus we compute \frac{∂}{∂x_{i}} ε_{p}(x, t) i = 1, 2, 3 from the ordinary differential equation for ε_{p}.

\begin{aligned}\frac{∂}{∂x_{i}} ε_{p}(t) & = & \frac{∂}{∂x_{i}} ε_{p}^{(0)} + ∫_{0}^{t} \{G′ (\frac{|σ(τ)|}{y(τ)}) \frac{σ(τ)}{y(τ)|σ(τ)|} \frac{∂}{∂x_{i}} |σ(τ)| + \\ & & −G′ (\frac{|σ(τ)|}{y(τ)}) \frac{σ(τ)}{y^{2}(τ)} \frac{∂}{∂x_{i}} y(τ) + G (\frac{|σ(τ)|}{y(τ)}) ( \frac{\frac{∂}{∂x_{i}} σ(τ)}{|σ(τ)|} − \frac{σ(τ)\frac{∂}{∂x_{i}} |σ(τ)|}{|σ(τ)|^{2}} ) \} dτ . \end{aligned}

Let us fix t^{∗} > 0. Also for all t ≤ t^{∗} the last equality yields: there exists a positive constant C_{q}(t^{∗}) such that the following inequality

∥\frac{∂}{∂x_{i}} ε_{p}(t)∥_{𝕃^{q}}^{q} = C_{q}(t^{∗}) \{∥\frac{∂}{∂x_{i}} ε_{p}^{(0)}∥_{𝕃^{q}}^{q}+

+ ((y^{∗})^{−1} sup_{p∈ℝ}G′(p))^{q} ∫_{0}^{t} ∥\frac{∂}{∂x_{i}} σ(τ)∥_{𝕃^{q}}^{q} dτ+

+ ((y^{∗})^{−1} sup_{p∈ℝ}G(p)/p)^{q} ∫_{0}^{t} ∥\frac{∂}{∂x_{i}} σ(τ)∥_{𝕃^{q}}^{q} dτ+

+ ((y^{∗})^{−1} sup_{p∈ℝ}G′(p)p)^{q} ∫_{0}^{t} ∥\frac{∂}{∂x_{i}} y(τ)∥_{𝕃^{q}}^{q} dτ \}

(3.2)

holds for i = 1, 2, 3. Now we will try to estimate the 𝕃^{q}– norm of the gradient of the isotropic hardening function y. We integrate the ordinary differential equation for y and we obtain

y(t) = y_{1} + (y^{(0)} − y_{1}) exp \{−m ∫_{0}^{t} G (\frac{|σ(τ)|}{y(τ)}) |σ(τ)| dτ \} .  (3.3)

From the last formula we can compute the spatial derivatives of the function y

\frac{∂}{∂x_{i}} y(t) = \frac{∂}{∂x_{i}} y^{(0)} exp \{−m ∫_{0}^{t} G (\frac{|σ(τ)|}{y(τ)}) |σ(τ)| dτ \}+

(y^{(0)} − y_{1})exp \{−m ∫_{0}^{t} G (\frac{|σ(τ)|}{y(τ)}) |σ(τ)| dτ \}×

\{−m ∫_{0}^{t} (\frac{∂}{∂x_{i}} G (\frac{|σ(τ)|}{y(τ)}) |σ(τ)| + G (\frac{|σ(τ)|}{y(τ)}) \frac{∂}{∂x_{i}} |σ(τ)|) dτ \}

for i = 1, 2, 3. Again by assumptions (A1)-(A5) we conclude that for all t ≤ t^{∗} there exists a positive constant C_{q}^{∗}(t^{∗}) such that for i = 1, 2, 3

∥\frac{∂}{∂x_{i}} y(t)∥_{𝕃^{q}}^{q} = C_{q}^{∗}(t^{∗}) \{∥\frac{∂}{∂x_{i}} y^{(0)}∥_{𝕃^{q}}^{q}+

+ (sup_{p∈ℝ}G(p))^{q} ∫_{0}^{t} ∥\frac{∂}{∂x_{i}} σ(τ)∥_{𝕃^{q}}^{q} dτ+

+ (sup_{p∈ℝ}G′(p)p)^{q} ∫_{0}^{t} ∥\frac{∂}{∂x_{i}} σ(τ)∥_{𝕃^{q}}^{q} dτ+

+ (sup_{p∈ℝ}G′(p)p^{2})^{q} ∫_{0}^{t} ∥\frac{∂}{∂x_{i}} y(τ)∥_{𝕃^{q}}^{q} dτ \}

(3.4)

We substitute (3.2) in (3.1) and add (3.4). Thus by the Gronwall lemma we obtain that the 𝕃^{q}– norm of the gradients of the stress tensor T and of the isotropic hardening function y do not blow up in time. To end this proof we need 𝕃^{q}–estimation for the function T. Again the constitutive equation for the stress tensor T yields

∥T (t)∥_{𝕃^{q}} ≤ C_{q} (∥∇u(t)∥_{𝕃^{q}} + ∥ε_{p}(t)∥_{𝕃^{q}}) .

The plastic strain tensor ε_{p} is a 𝕃^{∞}– function and we can estimate the 𝕃^{q}– norm of the gradient of u by the right hand side of the inequality (3.1). Thus the W^{1,q}– norm of the stress tensor T does not blow up in time.

By theorem 2 and by the Sobolev embending theorem we can prove now an apriori 𝕃^{∞}– estimation for the stress tensor T for data, which satisfy the assumptions of theorem 2. Thus for such data by theorem 1 we have, that the problem (BP) has a unique solution.

Remark The method presented in this paper does not work for the dynamical case. In this case a regularity result such as the inequality (3.1) is false. The uniqueness problem for the dynamic Bodner-Partom equations in the two dimensional case is studied in [7].

References

[1] Alber, H.D. : “On a system of equations from the theory of nonlinear viscoplasticity” – Preprint Nr.1265 FB Mathematik TH Darmstadt 1989

[2] Alber, H.D. : “Global existence and boundedness of large solutions to nonlinear equations of viscoelasticity with hardening” – Commun. Math. Phys. 166, p. 565–601, 1995

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

[4] Alber, H.D. :“Initial-boundary value problem of inelastic material behavior of metals” – manuscript in preparation

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

[6] Chelminski, K. : “Global in time existence of solutions to the constitutive model of Bodner–Partom with isotropic hardening” – Demonstratio Mathematica v.28 p. 667–688 1995

[7] Chelminski, K. : “Stress 𝕃^{∞}– estimates and the uniqueness problem to the model of Bodner-Partom” – Preprint TH Darmstadt in preparation

[8] Chelminski, K. : “On large solutions for the quasistatic problem in nonlinear viscoelasticity with the constitutive equations of Bodner–Partom” – shall appear in Math. Meth. in the App. S. 1996

[9] Chelminski, K. : “Energy estimates and global in time results for a problem from nonlinear viscoelasticity ” – shall appear in Bull. of the Pol. Acad. of Sci.: Math. 1996

[10] Ionescu, I.R., Sofonea, M. : “Functional and numerical methods in viscoplasticity” – Oxford, New York Tokio: Oxford University Press, 1993

[11] Klaus, F. : “Existenz– und Eindeutigkeitssatz für die Millerschen Gleichungen zur nichtlinearen Viskoelastizität mit Härtung” – Dissertation, FB Mathematik, TH Darmstadt 1994

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