On large solutions for the quasistatic problem in nonlinear viscoelasticity with the constitutive equations of Bodner-Partom

Krzysztof Chełmiński Technical University of Darmstadt and Instytut Matematyki Stosowanej i Mechaniki Uniwersytet Warszawski

Abstract

This work proves global in time existence of large solutions for a quasistatic problem in nonlinear viscoelasticity in the three-dimensional case. The basic idea is to apply the energy method for local in time solutions.

AMS classification : 35Q72, 73F99

1 Introduction

We shall consider the quasistatic problem for a system of equations modelling the nonelastic deformation of metals. The constitutive equations for this system have been proposed by R.S. Bodner and Y. Partom in [3]. We study the three-dimensional case with isotropic hardening. We show that the problem has global in time solutions for large initial data in Sobolev spaces. In the case with no hardening this problem has been studied by many authors, for example: G. Duvant and J.L. Lions in [6], P. Le Tallec in [8] and by myself in [5]. The dynamical case has been considered only in one space dimension. Global existence results are obtained by H.-D. Alber in [2] and by myself in [4].

Let Ω ⊂ ℝ^{3} be a bounded region with smooth boundary ∂Ω. The equations we consider are

\begin{aligned}div S(x, t) & = & F (x, t) \\ S(x, t) & = & D(E(x, t) − E^{n}(x, t)) \\ E(x, t) & = & \frac{1}{2}(∇u(x, t) + ∇^{T} u(x, t)) \\ s(x, t) & = & S(x, t) − \frac{1}{3}(tr S(x, t))I \\ ∂_{t}E^{n}(x, t) & = & G (\frac{|s(x, t)|}{z(x, t)}) \frac{s(x, t)}{|s(x, t)|} \\ ∂_{t}z(x, t) & = & m(z^{1} − z(x, t))|∂_{t}E^{n}(x, t)||s(x, t)| . \end{aligned}  (P)

We study problem (P) with Dirichlet boundary condition

u(x, t)_{|∂Ω} = g(x, t)  (D)

and with initial conditions

\begin{aligned}E^{n}(x, 0) & = & E^{n,0}(x) , \\ z(x, 0) & = & z^{0}(x) . \end{aligned}  (I)

Here u(x, t) is the displacement field; S(x, t) is the stress field; s(x, t) is the stress deviator; I is the identity matrix; E(x, t) is the strain field; E^{n}(x, t) is the inelastic part of the strain; the function F (x, t) describes body forces; z(x, t) is the hardening; m , z^{1} are positive material parameters and D is the elasticity tensor, which we assume to be constant, symmetric and positive definite.

In this paper we assume that the nonlinear constitutive function G satisfies the following

Assumptions:

1. G ∈ C^{1}([0, ∞); [0, ∞)) ,
2. \frac{G(·)}{(·)} ∈ C^{1}([0, ∞); [0, ∞)) and its derivative (\frac{G(·)}{(·)})′

is bounded ,

3. ∃_{G>0} ∀_{p≥0} G(p) ≤ G ,
4. ∃_{g>0} ∀_{p≥0} G′(p)p^{2} ≤ g .

(A)

R.S. Bodner and Y. Partom in [3] have proposed a function G of the following form

G(p) = \frac{2}{\sqrt{3}}D_{0} exp [− (\frac{n + 1}{n}) \frac{1}{p^{2n}} ]

where D_{0} > 0 and n ∈ ℕ are material parameters. It is easy to see that this function satisfies all assumptions (A). On the initial datum z^{0} we assume that

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

By assumption (A5) we have that the hardening is a positive L^{∞}-function. This follows directly from equation (P6). We integrate the equation and obtain

z(x, t) = z^{1} + (z^{0}(x) − z^{1}) exp {−m ∫_{0}^{t} |s(x, τ)||∂_{t}E^{n}(x, τ)| dτ } .  (1.1)

Equality (1.1) implies that

z^{∗} ≤ z(x, t) ≤ z^{1} for all (x, t) .

For tensors R = (r_{ij})_{i,j=1,2,3}, S = (s_{ij})_{i,j=1,2,3} we write

R · S = ∑_{i,j=1}^{3} r_{ij}s_{ij} and |R| = (R · R)^{\frac{1}{2}} .

By ℍ^{s}(Ω; ℝ^{k}) we denote the Sobolev space of functions defined on Ω with values in ℝ^{k}, with norm ∥ · ∥_{s} . We denote by ∥ · ∥ the norm in space ℍ^{0}(Ω; ℝ^{k}) = 𝕃^{2}(Ω; ℝ^{k}) and the scalar product by (·, ·).

Now we formulate the main result.

Theorem 1.1 Assume that initial data

\begin{aligned}E^{n,0} & ∈ & ℍ^{1}(Ω; ℝ_{sym}^{3×3}) , \\ z^{0} & ∈ & ℍ^{1}(Ω; ℝ_{+}) \end{aligned}  (1.2)

with tr E^{n,0}(x) = 0 for almost all x ∈ Ω . Moreover suppose that assumption (A5) holds and

\begin{aligned}g & ∈ & ℂ^{1}([0, ∞); ℍ^{\frac{3}{2}} (∂Ω; ℝ^{3})) , \\ F & ∈ & ℂ^{1}([0, ∞); 𝕃^{2}(Ω; ℝ^{3})) . \end{aligned}  (1.3)

Then there exists a global solution

(u, S, E^{n}, z) : Ω × ℝ_{+} ⟶ ℝ^{3} × (ℝ_{sym}^{3×3})^{2} × ℝ_{+}

of problem (P)+(D)+(I) with

∀_{T >0}  u ∈ ℍ^{1}([0, T]; ℍ^{2}(Ω; ℝ^{3})) ,
S, E^{n} ∈ ℍ^{1}([0, T] × Ω; ℝ_{sym}^{3×3}) ,
z ∈ ℍ^{1}([0, T] × Ω; ℝ_{+}) .

The proof of this theorem is based on energy estimates on local in time solutions of problem (P)+(D)+(I). The crucial point of the estimates is to show L^{2}-bounds for the spatial derivatives of the strain and the stress fields. We obtain this using that equation (P1) is elliptic.

Acknowledgement. This paper resulted from a cooperation with the Forschergruppe ”Inelastische Probleme und Probleme der Bruchmechanik” at the Technische Hochschule Darmstadt. The author thanks Hans-Dieter Alber for a long, helpful discussion and Piotr Gwiazda for remarks to the first version of this paper.

2 Local existence theorem

In this section we study the local in time existence of solutions of problem (P) in the Sobolev space ℍ^{1}([0, T] × Ω) . We work with a iteration process and apply the compact injection of ℍ^{1}([0, T] × Ω) in 𝕃^{2}([0, T] × Ω) .

Theorem 2.1 Let the initial data satisfy

\begin{aligned}E^{n,0} & ∈ & ℍ^{1}(Ω; ℝ_{sym}^{3×3}) , \\ z^{0} & ∈ & ℍ^{1}(Ω; ℝ_{+}) , \end{aligned}  (2.1)

and

\begin{aligned}g & ∈ & ℂ^{1}([0, ∞); ℍ^{\frac{3}{2}} (∂Ω; ℝ^{3})) , \\ F & ∈ & ℂ^{1}([0, ∞); 𝕃^{2}(Ω; ℝ^{3})) . \end{aligned}

Moreover suppose that tr E^{n,0} = 0 for almost all x ∈ Ω and assumption (A5) holds. Then there exists a local solution (u, S, E^{n}, z) : [0, T_{loc}] × Ω ⟶ ℝ^{3} × (ℝ_{sym}^{3×3})^{2} × ℝ_{+} of problem (P)+(D)+(I) in space

\begin{aligned}(u, S, E^{n}, z) & ∈ & ℍ^{1}([0, T_{loc}]; ℍ^{2}(Ω; ℝ^{3})) × (ℍ^{1}([0, T_{loc}] × Ω; ℝ_{sym}^{3×3})^{2} \\ & × & ℍ^{1}([0, T_{loc}] × Ω; ℝ_{+}) . \end{aligned}

Proof. Let K, α, β > 0 such that

∥E^{n,0}∥_{1} ≤ \frac{K}{2} , ∥z^{0}∥_{1} ≤ \frac{K}{2} , α < z^{∗} , β > z^{1} .

Let us define a nonlinear operator

P : A_{K} ⟶ A_{K}

where

A_{K} = {(W, z) ∈ 𝕃^{∞}([0, T]; ℍ^{1}(Ω; ℝ_{sym}^{3×3}) × 𝕃^{∞}([0, T]; ℍ^{1}(Ω; (α, β)) |
∂_{t}W ∈ 𝕃^{2}([0, T] × Ω; ℝ_{sym}^{3×3}) ,
∂_{t}z ∈ 𝕃^{2}([0, T] × Ω; ℝ_{+}) ,
sup_{t∈[0,T]} ∥W (t)∥_{1} ≤ K , sup_{t∈[0,T]} ∥z(t)∥_{1} ≤ K ,
∥∂_{t}W ∥_{𝕃^{2}([0,T]×Ω)} ≤ K , ∥∂_{t}z∥_{𝕃^{2}([0,T]×Ω)} ≤ K ,
∀_{(x,t)∈Ω×[0,T]} tr W (x, t) = 0 , W (0) = E^{n,0} , z(0) = z^{0}} .

First for (W, z) ∈ A_{K} we solve the following elliptic equation

div D (\frac{1}{2} (∇u_{W} (x, t) + ∇^{T} u_{W} (x, t))) = div DW (x, t) + F (x, t)  (2.2)

with Dirichlet boundary condition

u(x, t)_{|∂Ω} = g(x, t) .  (2.3)

We define now P(W, z) = (V, y) as a solution of the following system of ordinary differential equations

∂_{t}V = G (\frac{|s_{W} |}{z}) \frac{s_{W}}{|s_{W} |}  (2.4)

∂_{t}y = m(z^{1} − y)G (\frac{|s_{W} |}{z}) |s_{W} |  (2.5)

with initial conditions

V (0) = E^{n,0} , y(0) = z^{0}

where

\begin{aligned}E_{W} & = & \frac{1}{2} (∇u_{W} + ∇^{T} u_{W} ) , \\ S_{W} & = & D(E_{W} − W) , \\ s_{W} & = & S_{W} − \frac{1}{3}tr (S_{W}) · I . \end{aligned}

System (2.4)-(2.5) has solution (V, y) : Ω × [0, T] ⟶ ℝ^{9} × ℝ_{+} in the following form

\begin{aligned}V (t) & = & ∫_{0}^{t} G (\frac{|s_{W} |}{z}) \frac{s_{W}}{|s_{W} |} dτ + V (0) , \\ y(t) & = & z^{1} + (y(0) − z^{1})e^{−m ∫_{0}^{t} G(\frac{|s_{W} |}{z})|s_{W} |dτ} . \end{aligned}  (AS)

We will show that for suffitiently small T_{loc} ≤ T , P(W, z) ∈ A_{K} . From the symmetry of the matrix E^{n,0} we have that V (x, t) ∈ ℝ_{sym}^{3×3} . Moreover the assumption tr E^{n,0}(x) = 0 yields tr V (x, t) = 0 for almost all (x, t) ∈ Ω × [0, T] . From (AS) we obtain that the functions V, y are in 𝕃^{∞}(Ω × [0, T]) , and moreover z^{∗} ≤ y(x, t) ≤ z^{1} for a.a. (x, t) ∈ Ω × [0, T] . We estimate first the L^{2}-norms of the functions V and y . (AS1) yields

∥V (t)∥ ≤ ∥E^{n,0}∥ + ∥ ∫_{0}^{t} G (\frac{|s_{W} |}{z}) \frac{s_{W}}{|s_{W} |}dτ∥ ≤ ∥E^{n,0}∥ + tG|Ω|^{\frac{1}{2}}  (2.6)

Now from (2.5) we have

\frac{1}{2} \frac{d}{dt}∥y(t)∥^{2} = ∫_{Ω} m(z^{1} − y)yG (\frac{|s_{W} |}{z}) |s_{W} | dx ≤ mz^{1}G∥y(t)∥∥s_{W} (t)∥

∥y(t)∥ ≤ ∥z^{0}∥ + mz^{1}G ∫_{0}^{t} ∥s_{W} (τ)∥dτ  (2.7)

The function u_{W} is a solution of an elliptic system, thus we have for this function the strong estimates (see [7],[9])

∥u_{W} (t)∥_{2} ≤ C_{Ω}(∥div DW (t)∥ + ∥F (t)∥ + ∥g(t)∥_{\frac{3}{2} ;∂Ω})  (2.8)

where C_{Ω} > 0 is a constant, independent of W and t . Thus (2.8) implies

sup_{t∈[0,T]} ∥s_{W} (t)∥_{1} ≤ d( sup_{t∈[0,T]} ∥E_{W} (t)∥_{1} + sup_{t∈[0,T]} ∥W (t)∥_{1})
≤ dC_{Ω}(2K + sup_{t∈[0,T]} ∥F (t)∥ + sup_{t∈[0,T]} ∥g(t)∥_{\frac{3}{2} ;∂Ω}) (2.9)

where d = max_{i,j,k,l=1,2,3} d_{ijkl} and D = (d_{ijkl})_{i,j,k,l=1,2,3} . (2.7) with (2.9) imply

∥y(t)∥ ≤ ∥z^{0}∥ + tK^{∗}(T)  (2.10)

where with K^{∗}(T) we denote the right hand side of inequality (2.9). From (AS1) with assumptions (A) we can calculate the derivative ∂_{x_{i}}V for i = 1, 2, 3

\begin{aligned}∂_{x_{i}}V & = & ∫_{0}^{t} G′ (\frac{|s_{W} |}{z}) \frac{z∂_{x_{i}}|s_{W} | − |s_{W} |∂_{x_{i}}z}{z^{2}} \frac{s_{W}}{|s_{W} |}dτ \\ & + & ∫_{0}^{t} G (\frac{|s_{W} |}{z}) (\frac{∂_{x_{i}}s_{W}}{|s_{W} |} − \frac{s_{W}}{|s_{W} |} \frac{∂_{x_{i}}|s_{W} |}{|s_{W} |}) dτ + ∂_{x_{i}}E^{n,0} . \end{aligned}  (2.11)

Hence for the L^{2}-norm we obtain

∥∂_{x_{i}}V ∥ ≤ ∥∂_{x_{i}}E^{n,0}∥ + t(z^{∗})^{−1} (sup_{ξ≥0} G′(ξ) + 2 sup_{ξ≥0} \frac{G(ξ)}{ξ}) sup_{[0,t]} ∥∂_{x_{i}}s_{W} ∥

+t(z^{∗})^{−1} sup_{ξ≥0} G′(ξ)ξ sup_{[0,t]} ∥∂_{x_{i}}z∥ .

(2.12)

For the function ∂_{x_{i}}y we have from (2.5) the following inequality

\begin{aligned}|∂_{x_{i}}y| ∂_{t}|∂_{x_{i}}y| & ≤ & m(z^{1} − y)G′ (\frac{|s_{W} |}{z}) (\frac{|s_{W} |}{z}|∂_{x_{i}}|s_{W} || + (\frac{|s_{W} |}{z})^{2}|∂_{x_{i}}z|) |∂_{x_{i}}y| \\ & + & m(z^{1} − y)G (\frac{|s_{W} |}{z}) |∂_{x_{i}}|s_{W} |||∂_{x_{i}}y| . \end{aligned} (2.13)

We integrate (2.13) over (0, t) for t ≤ T

∥∂_{x_{i}}y(t)∥ ≤ m(z^{1} − z^{∗}) · t (sup_{ξ≥0} G′(ξ)ξ + sup_{ξ≥0} G(ξ)) sup_{[0,t]} ∥∂_{x_{i}}s_{W} ∥

+m(z^{1} − z^{∗}) · t sup_{ξ≥0} G′(ξ)ξ^{2} sup_{[0,t]} ∥∂_{x_{i}}z∥ + ∥∂_{x_{i}}z^{0}∥ .

(2.14)

Hence (2.6),(2.10),(2.12) and (2.14) imply that we can choose T_{loc}(K, g, F) ≤ T such that

sup_{t∈[0,T_{loc}]} ∥V (t)∥ ≤ K, sup_{t∈[0,T_{loc}]} ∥y(t)∥_{1} ≤ K .

We still need L^{2}-estimates for the t-derivatives of the functions V and y . From (2.4) we obtain immediately

∥∂_{t}V ∥_{𝕃^{2}([0,T]×Ω)} ≤ T |Ω|^{\frac{1}{2}} G .  (2.15)

Similarly (2.5) yields

∥∂_{t}y∥_{𝕃^{2}([0,T]×Ω)} ≤ T m(z^{1} − z^{∗})G sup_{[0,T]} ∥s_{W} ∥ .  (2.16)

Thus we can correct T_{loc}(K, g, F) so that ∥∂_{t}V ∥_{𝕃^{2}([0,T]×Ω)} ≤ K and ∥∂_{t}y∥_{𝕃^{2}([0,T]×Ω)} ≤ K . Also for T_{loc} small enough, P(W, z) ∈ A_{K} . Let us choose an element (V_{1}, y_{1}) ∈ A_{K} and define the sequence (V_{k}, y_{k}) = P(V_{k−1}, z_{k−1}) . This sequence is bounded in the space ℍ^{1}([0, T_{loc}]×Ω) . Moreover the sequence u_{V_{k}} is defined as the solution of equation (2.2) with the function V_{k} and is bounded in the space ℍ^{1}([0, T_{loc}]; ℍ^{2}(Ω)) , because for t-derivative of the function u_{V_{k}} we have an estimate similar to (2.8). Thus the sequence S_{V_{k}} is bounded in the space ℍ^{1}([0, T_{loc}] × Ω) . Therefore from the sequence (u_{V_{k}}, S_{V_{k}}, V_{k}, y_{k}) we can select a subsequence, which we denote by (u_{V_{k}}, S_{V_{k}}, V_{k}, y_{k}) again, such that

u_{V_{k}} ⟶ u weakly in ℍ^{1}([0, T_{loc}]; ℍ^{2}(Ω)) and strongly in 𝕃^{2}([0, T_{loc}]; ℍ^{1}(Ω)) ,
S_{V_{k}} ⟶ S weakly in ℍ^{1}([0, T_{loc}] × Ω) and strongly in 𝕃^{2}([0, T_{loc}] × Ω) ,
V_{k} ⟶ E^{n} weakly in ℍ^{1}([0, T_{loc}] × Ω) and strongly in 𝕃^{2}([0, T_{loc}] × Ω) ,
y_{k} ⟶ z weakly in ℍ^{1}([0, T_{loc}] × Ω) and strongly in 𝕃^{2}([0, T_{loc}] × Ω) .

(2.17)

The norm convergence in L^{2} implies that (u, S, E^{n}, z) is a solution of problem (P) in the weak sense. But (u, S, E^{n}, z) has the same regularity as the sequence (u_{V_{k}}, S_{V_{k}}, V_{k}, y_{k}) and therefore (u, S, E^{n}, z) is a solution of problem (P) in the strong sense. The proof of theorem 2.1 is complete.

3 Energy estimates and the global result

In this section we study the global in time existence of solutions of problem (P)+(D)+(I). We show that ℍ^{1}-Sobolev norms of the local solutions constructed in the last section are bounded in time. This result is obtained with the energy method. Let us start with the definition of the energy function. Let (u, S, E^{n}, z) be the local solution from theorem 2.1 and 0 < T ≤ T_{loc}. The energy function is a scalar function

E : [0, T] ⟶ ℝ
E(t) = \frac{1}{2}(S(t), E(t) − E^{n}(t)) .  (3.1)

In this paper we consider the quasistatic case and therefore the kinetic part of the energy is omitted. We will obtain that the energy function is bounded for all T > 0. This statement is equivalent to the boundness of the L^{2}-norm of the stress field, and of the elastic part of the strain field.

Lemma 3.1 Let the initial data (E^{n,0}, z^{0}) and the functions g, F satisfy assumptions of theorem 2.1. Then the solution (u, S, E^{n}, z) obtained in that theorem satisfies the following estimate

\begin{aligned}∃_{E^{∗}>0} ∀_{t∈[0,T]} E(t) & + & ∫_{0}^{t} ∫_{Ω} G (\frac{|s(x, t)|}{z(x, t)}) |s(x, t)| dxdt \\ & ≤ & E(0) + E^{∗} ∫_{0}^{t}\{∥F (t)∥ ∥∂_{t}u(t)∥ + ∥S(t)∥_{1} ∥∂_{t}g(t)∥_{0;∂Ω}\} dt \end{aligned} (3.2)

where

E(0) = \frac{1}{2}(S(0), E(0) − E^{n,0}) ,

the initial displacement field u(0) is the unique solution of boundary value problem

\begin{aligned}div D(\frac{1}{2} (∇u(0) + ∇^{T} u(0))) & = & div DE^{n,0} + F (0) \\ u(0)_{|∂Ω} & = & g(0) \end{aligned}

the initial strain E(0) field and the initial stress field S(0) are defined as follows

\begin{aligned}E(0) & = & \frac{1}{2} (∇u(0) + ∇^{T} u(0)) , \\ S(0) & = & E(0) − E^{n,0} . \end{aligned}

Proof. Let t ∈ [0, T] . Using the properties of the tensor D we find

\frac{d}{dt}E(t) = (S(t), ∂_{t}E(t)) − (S(t), ∂_{t}E^{n}(t)) = (S(t), ∇∂_{t}u(t)) − (s(t), ∂_{t}E^{n}(t))
= −(F (t), ∂_{t}u(t)) + ∫_{∂Ω} S(t) · n ∂_{t}g(t) dS(x)

− ∫_{Ω} G (\frac{|s(x, t)|}{z(x, t)}) |s(x, t)| dx

where n(x) is the exterior unit normal to ∂Ω .

The boundness of the function E follows now from L^{2}-estimates of the spatial derivatives of the stress field. This we prove in the next two lemmas. First we need L^{2}-bounds for the functions ∂_{x_{i}}z for i = 1, 2, 3 . These are given by :

Lemma 3.2 Let the initial data z^{0} satisfies assumption (A5). Under the assumptions of theorem 2.1 the local solution (u, S, E^{n}, z) satisfies the following inequality

∃_{Z_{T} >0} ∀_{t∈[0,T]} ∥∂_{x_{i}}z(t)∥^{2} ≤ Z_{T} exp(Z_{T} · t){∥∂_{x_{i}}z^{0}∥^{2} + ∫_{0}^{t} ∥∇S(τ)∥^{2} dτ} .  (3.3)

Proof. We can calculate the derivative ∂_{x_{i}}z from equation (1.1)

\begin{aligned}∂_{x_{i}}z & = & −m(z^{1} − z^{0}) exp\{−m ∫_{0}^{t} |s||∂_{t}E^{n}| dτ\}\{∫_{0}^{t}(∂_{x_{i}}|s||∂_{t}E^{n}| + |s|∂_{x_{i}}|∂_{t}E^{n}|) dτ\} \\ & & +∂_{x_{i}}z^{0} exp\{−m ∫_{0}^{t} |s||∂_{t}E^{n}| dτ\} . \end{aligned}

Thus for the L^{2}-norm we have

∥∂_{x_{i}}z∥^{2} ≤ 2∥∂_{x_{i}}z^{0}∥^{2} + 2(z^{1} − z^{∗})^{2}m^{2} ∫_{Ω} \{∫_{0}^{t} (|∂_{x_{i}}|s||G (\frac{|s|}{z}) + |s| |∂_{x_{i}}G (\frac{|s|}{z})|) dτ \}^{2} dx .

Using assumptions (A) we find

∥∂_{x_{i}}z∥^{2} ≤ 2∥∂_{x_{i}}z^{0}∥^{2} + 4(z^{1} − z^{∗})^{2}m^{2} · t(sup_{ξ≥0} G′(ξ)ξ^{2})^{2} ∫_{0}^{t} ∥∂_{x_{i}}z∥^{2} dτ
+ 4(z^{1} − z^{∗})^{2}m^{2} · t[(sup_{ξ≥0} G(ξ))^{2} + (sup_{ξ≥0} G′(ξ)ξ)^{2}] ∫_{0}^{t} ∥∇S∥^{2} dτ .

The statement of the lemma follows now by Gronwall’s lemma. We are now ready to estimate the L^{2}-norm of the spatial derivative of the stress field. Lemma 3.3 With the same assumptions as in the last lemma, the gradient of the stress field satisfies the following estimate

\begin{aligned}∃_{C_{T}^{1} ,C_{T}^{2} >0} ∀_{t∈[0,T]} ∥∇S(t)∥^{2} & ≤ & (∇S(0), ∇E(0) − ∇E^{n,0}) \\ & + & C_{T}^{1} ∫_{0}^{t} ∥∇S(τ)∥^{2} dτ + C_{T}^{2} ∫_{0}^{t} ∫_{0}^{τ} ∥∇S(η)∥^{2} dη dτ . \end{aligned}

(3.4)

The constants C_{T}^{1} and C_{T}^{2} depend on T, on the initial data and on the functions F, g . Proof. Velocity field ∂_{t}u(t) is the solution of the following strictly elliptic equation

div D\frac{1}{2} (∇∂_{t}u(t) + ∇^{T} ∂_{t}u(t)) = div D∂_{t}E^{n}(t) + ∂_{t}F (t)

with boundary condition

∂_{t}u(t)_{|∂Ω} = ∂_{t}g(t) .

Thus the velocity satisfies the following inequality ∥∂_{t}u(t)∥_{2}^{2} ≤ C_{Ω}(∥∂_{t}F (t)∥^{2} + ∥∂_{t}g(t)∥_{\frac{3}{2} ;∂Ω}^{2} + ∥div ∂_{t}E^{n}(t)∥^{2}) .

From equation (P5) we compute

∥div ∂_{t}E^{n}∥ ≤ ∥\frac{1}{z}∇ (\frac{G (\frac{|s|}{z})}{\frac{|s|}{z}} ) · s∥ + ∥−\frac{G (\frac{|s|}{z})}{\frac{|s|}{z}} \frac{∇z}{z^{2}} · s∥

+∥\frac{1}{z} \frac{G (\frac{|s|}{z})}{\frac{|s|}{z}}div s∥

≤ (z^{∗})^{−1} (sup_{ξ≥0} \frac{G(ξ)}{ξ}) ∥div s∥ + (z^{∗})^{−1} (sup_{ξ≥0} G′(ξ) + sup_{ξ≥0} \frac{G(ξ)}{ξ}) ∥∇|s|∥

+(z^{∗})^{−1} (2 sup_{ξ≥0} G(ξ) + sup_{ξ≥0} G′(ξ)ξ) ∥∇z∥ .

Hence there exist positive constants U^{1}, U^{2} such that

\begin{aligned}∥∂_{t}u(t)∥_{2}^{2} & ≤ & ∥∂_{t}F (t)∥^{2} + ∥∂_{t}g(t)∥_{\frac{3}{2} ;∂Ω}^{2} \\ & & +U^{1}∥∇S(t)∥^{2} + U^{2} ∫_{0}^{t} ∥∇S(τ)∥^{2} dτ . \end{aligned}  (3.5)

Now similarly as in lemma 3.1 we can calculate

\begin{aligned}\frac{d}{dt} \frac{1}{2}(∇S(t), ∇E(t) − ∇E^{n}(t)) & = & (∇S(t), ∇∂_{t}E(t)) − (∇S(t), ∇∂_{t}E^{n}(t)) \\ & = & (∇S(t), ∇∇∂_{t}u(t)) − (∇s(t), ∇∂_{t}E^{n}(t)) . \end{aligned} (3.6)

Again from equation (P5) we find for i = 1, 2, 3

−(∂_{x_{i}}s, ∂_{x_{i}}∂_{t}E^{n}) = −∫_{Ω} \frac{1}{z}G′ (\frac{|s|}{z}) (∂_{x_{i}}|s|)^{2} dx

−∫_{Ω} \frac{1}{|s|}G (\frac{|s|}{z}) (|∂_{x_{i}}s|^{2} − (∂_{x_{i}}|s|)^{2}) dx

+∫_{Ω} \frac{|s|}{z^{2}}G′ (\frac{|s|}{z}) ∂_{x_{i}}z∂_{x_{i}}|s| dx .

Thus assumptions (A) yield

− (∂_{x_{i}}s, ∂_{x_{i}}∂_{t}E^{n}) ≤ (z^{∗})^{−1} sup_{ξ≥0} G′(ξ)ξ ∥∂_{x_{i}}z∥ ∥∂_{x_{i}}S∥ .  (3.7)

We insert (3.7) into (3.6), integrate the inequality over (0, t) for t ≤ T and obtain the following estimate

\frac{1}{2}(∇S(t), ∇E(t) − ∇E^{n}(t)) ≤ \frac{1}{2}(∇S(0), ∇E(0) − ∇E^{n,0})

+∫_{0}^{t} ∥∇S(τ)∥ ∥∂_{t}u(τ)∥_{2} dτ

+(z^{∗})^{−1} sup_{ξ≥0} G′(ξ)ξ ∑_{i=1}^{3} ∫_{0}^{t} ∥∂_{x_{i}}z(τ)∥ ∥∂_{x_{i}}S(τ)∥ dτ .

The statement of the lemma follows then from lemma 3.2 and from estimate (3.5). References

[1] Alber, H.D. - ”On the system of equation from the theory of nonlinear viscoplasticity”. Preprint TH Darmstadt 1989

[2] Alber, H.D. - ”Global existence and boundedness of large solutions to nonlinear equations of viscoplasticity with hardening”. Preprint TH Darmstadt 1993

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

[4] Chełmiński, K. - ”Global in time existence of solutions to the constitutive model of Bodner-Partom with isotropic hardening”. Preprint TH Darmstadt 1993

[5] Chełmiński, K. - ”Existence of large solutions to the quasistatic problem for a threedimensional Maxwell material”. Preprint TH Darmstadt 1993

[6] Duvaut, G.,Lions, J.L. -”Inequalities in Mechanics and Physics”. Paris: Dunod 1972 [7] Egorow, Yu.V.,Shubin, M.A. - ”Partial differential equations I”. Encyclopaedia Mathematical Sciences Vol.30 Springer-Verlag Berlin 1992

[8] Le Tallec, P. - ”Numerical analysis of viscoelastic problems”. Masson Editeur Paris 1990

[9] Valent, T. - ”Boundary value problems of finite elasticity”. Springer Tracts in Natural Philosophy Vol.31 New York 1988 Krzysztof Chełmiński Institute of Applied Mathematics and Mechanics Banacha 2 PL–00–913 Warsaw

and

Fachbereich Mathematik Technische Hochschule Darmstadt Schloßgartenstr. 7 D–64289 Darmstadt