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On a rolling problem for porous materials
Pergamon
Mechanics Research Communications, Vol. 27, No. 6, pp. 637-642, 2000 Copynght© 2000 Elsevier Science Ltd Printed in the USA. All nghts reserved 0093-64131001$.-see front matter
PII: S0093-6413(00)00140.3
ON A ROLLING PROBLEM FOR POROUS MATERIALS
T A. Angelov Institute of Mechanics, Bulgarian Academy of Sciences, "Acad.G.Bonchev" str., bl.4, Sofia 1113, Bulgaria
(Received 22 March 2000, accepted for print 21 September 2000)
Introduction
There is a continuous interest in metal working industry to metal powder forming pro c~ sses. Metal powders are usually compacted in dies and sintered at elevated temperatures. Obtained metal powder preforms are further finished by various forming processes such a~s compaction, forging, extrusion, rolling. Powder preforms exhibit porosity and thus are more susceptible to fi-acture than solid metals. It is now well known that the flow theory of plasticity with rigid-plastic material model describes well the material behaviour at forming processes [1-3]. An extension of this theory to porous materials is given in [4-7]. Its applicability to different forming processes is further computationally examined by finite element method and Newton's type algorithms [8-10]. In the recent years convergent numerical methods were established for hot rolling problems, as the effects of strain rate sensitivity, nonlinear friction, temperature, damage and wear were taken into account [11-13]. In this work following [12, 13], a non-steady hot-strip rolling problem for powder based compressible materials, taking into account fi'iction, strain rate sensitivity, strain hardening and relative d, nsity effects, is stated. The material model, proposed by Mori and Osakada [:3, 8], is employed. A density evolution model, based on the mass conservation law, is introduced. A
638
T.A. ANGELOV
variational formulation is given and all existence and uniqueness theorem is stated. A convergent numerical method, analogous to the vaa'iable stiffness parameters method, proposed by l(orneev and Langer [14] for solving plastic flow theory problems, is presented.
Statement of the problem
We consider an isothermal, non-steady, hot-strip rolling process in a fixed domain until steady-state is reached at a time T E [0, oo). The workpiece is supposed isotropic, rigidp'astic, compressible metallic body occupying the domain ftC with
sufficiently regular
6 Fro. The b o u n d a r y F = O,n=l
boundary
Rk(k = 2, 3) (Fig.1 in [12]), ['1 O F2 U F4 U F5 is assumed
tractions flee, F:~ is the contact boundary and Ps is the boundary of symmetry. The points off~ = f~ U P are identified by their cartesian coordinates x = {x,} with respect to a fixed coordinate system, and the standard indicial notation and the summation convention over repeated indices is used. Let u(x, t) = {u,(x, t)}, ¢ ( x , t ) = {¢,j(x, t)}, k(x, t ) = {e,a(x, t)}, (1 _< i,j
< k), denote the velocity vector, stress tensor and strain rate tensor respectively. We denote by p(x, t) = pv(x, t)/p,(x, t) the relative density, where pv(x, t) and p,(x, t) are
the porous and solid material densities. Let , 1 .ff~ ~--- p
-('~- ) V 2
c~ 1 /2" Si35ia
+ da~
,
i- =
p(
-
.
1%
)V~gi3gi3+ -d- %, -
e(x,t) =~(x,0)+
1
2(X,T)~IT, (I)
be the equivalent stress, equivalent strain rate and equivalent strain, where a > 1 is a constant, d _> 0 is a function of the relative density, associated with the material compressibility, such that d -+ 0 when p -+ 1 and the classical yon Mises yield criterion and flow rule for
eij
~vS,j/3 are the components of the deviatoric stress and the strain rate tensors, a n = a,/3 and ~ = sii inconlpressible materials is obtained, sia = aij - an50
and
•
~i3 - -
are the hydrostatic pressure and the volume dilatation strain rate. Consider the following problem: - Find the velocity, u, stress ~r equivaleilt strain g and relative density p fields, satisfying the following equations and relations: - equations of motion cgui
a~a,j=pp,-~
in
f~x(0,7'),
(2)
- strain rate - velocity relations 1
(3)
ROLLING FOR POROUS MATERIALS
639
- yield criterion and flow rule
F(~,,~, p,:,~) -- a~ - ? ~ ( ~ ,
(4)
~) = 0,
- equivalent strain evolution equation
O~(x,t) _ h(x,t),
Ot
in
Q × (0, T),
(5)
- relative density evolution equation
Op(x,t) _
p(x t)i~,
in
Ot
f~ × (O,T),
(6)
- boundary conditions o'zff~a = 0
on
0%, = O,
El U F2 U F4 U F5 x (0, T ) , u~ = 0
'u~ = 0 if lar(U)l < r/(ph,eh, u), if
on
['6 x (0, T ) ,
(r) (s)
and then
u~ - UrR = 0,
I~rr(U)l=rf(ph,gh, u), then 3 const. A>_0,
such
that
Ur--Ur~=--~Crr(U) on
F a x ( 0 , T),
(9)
- initial conditions
u(x,O) = uo(×),
e(x,O) = Co(x),
p(x,O) = po(x).
(1o)
Here (~,j is Kronecker symbol; n = {ni} is the unit normal vector outward to F; u N, u r and a N, ~rr,, are the normal and tangential components of the velocity and the stress vector; Ur~~ is the rolling velocity; the shear strength limit for the material of Fa is given by the following Coulomb-Siebel friction law:
r/(ph, gh, u) = min t~jl~h(u)l, ~ip~,p(e~,~(u))}; ,~/(x) and 7n/(x) are the friction coefficient and factor; rv(~h, gh) = Cp(~h, g,,)/X/~
(11) is the
shear yield limit for the luaterial of F3. The convective terms are excluded from the corresponding equations, since their contribution to the material deformation is supposed insignificant. A natural assumption for the relative density is 0 < p0(x) _< p ( x , t ) < pl(x) < 1, where at p = pl the slightly compressible material behaviour is simulated [11]. Since the solid lnetal is incompressible, its density is constant in time during the process and assuming also that it is constant in spatial coordinates, without loss of generality we may take p, -- l [15]. The subindex h denotes that the corresponding quantities are mollified on F~. The strain
640
T.A. ANGELOV
and strain rate dependent uniaxial yield limit ap(e, f) is assumed monotonically increasing, almost everywhere differentiable function on both variables with bounded derivatives and such that: --
f
--
d(p) is also assumed differentiable with bounded derivative. Usually the following empirical expressions for d(p) where ~]a and 772 are positive constants. The relative density function and ap(e, g) are accepted
d(p) = O~o(1- p ) ~ ' for p C [po,p,j and d(p) = d(po) for p E [0,p0), d(p) = d(pl) for p E (px,o¢), ~(~,~) = c o ( l + c , W ~ ( h ~ for ~z [~,,&], ~ ~(~,~) = ~(~)2,
~(~) = ~(~'~q) 8~
(13)
[0,~],
for ~ [ o , ~ , ] u [ & , o o ) ,
q=l,~_,
(14)
where ao, al,c~,O <_ I <_ 3, are positive material constants, r/q(8) = r/q(g2) for g _2> ~2,
111 171(g2), 772 = -~-
772(0 ). Here g2, ~1, and g2 are the equivalent strain and strain rate model
bounds.
VariationM formulation
Let
v = {v:
v E (H'(a//k,
vN = 0 on r3 u F6},
H°(~/-
L~Ia),
be the spaces of admissible velocities, equivalent strains and densities. Let the following dense and continuous embeddings hold V C tt - H' C V', where H = (H°(~)) k, H' and V ~ are the topological dual spaces of H and V. Consider the following variational formulation associated with problem (2)-(10): - Find u(t) G V, ~(t), p(t) E H°(fl), satisfying for all t G [0, T], 0u
(p-~,v-u)+a(p,g,u;u,v-u)+j(p,g,u;vl-j(p,g,u;u)>O, Og
LOP -~(d:c = - fapfv~dz,
VveV
(15)
(16)
V( E H°(f~)
(17)
ROLLING FOR POROUS MATERIALS
641
~md the initial conditions u(0) = u0, ~(0) = c0, p(0) = P0. Here above the following notations are used: ,)
(18)
= t.
u.
ldr.
(19)
Jl
Let
u0(x) ~ v , %~(x) ~ (L.~. I':~))k, e0(x), p0(x), r,,(x) ~ L~(9.),
(')0)
aml let the friction coeificients # : ( x ) , m : ( x ) E L~.(F3) be sufficiently small. Then under the assumptions and propositions made, the following result holds. Theorem: The problem (15) - (17) has a unique solution, such that
Ou
uEL~(O,T;H) NL2(O,T;V), -:~EL2(O,T:V'),
e,/,EI,.~.(O,T;ff°(Ft)).
(21)
The proof of this theorem is based on the following au×ilia, ry problem. Let tbr almost all
[)n(t) E H°(f~), ,z = 0, J ..... be g,,+,(t), p,,+l(t) C lt°(ft), satisfying
I E [0, T], Urz(t) E V, Cn(/), Find u,,.+t(/) C V,
(p,~-,
v - un+l) +
a(p,,,~_,~,u,,;u,~+l,v
+j(p,,,g,,,u,,;v)-j(p~,e~,u~;u,~+~)>_O, -=~Tda"
=
e,~+~ d,~',
known. Then
--
u,~+l)+
VvEV
V~ ~ H°(f~)
L a,~..~ { dz = - J~ p,~+,i~,~+, { da',
V{ E H°(~l)
(22)
(23)
(24)
and the initial conditions u,~+l(0) = u0, g~+l(0) = #~0, P~+, (0) = p0. Following [14, 15], it can be shown that for any rz, this problem has a unique solution. It can be further shown that the obtained sequence {u,~(t),e,,(t),p,~(t)} converges strongly, at ,-, --+ ec., to a solution {u(t),g(t),p(t)} of (15) - (17) and this solution is unique. The above auxiliary problem describes a method of successive linearizations, called here variable stiffness parameters method, by an analogy with the method proposed in [141 for solving small strains plastic flow theory problems with hardening.
This method, combined with
an approl)riate time discretization scheme finally gives the time T, at which the process becomes steady-state, as well as the steady-state solution of (15) - (17). It is clear that the time retardation technique, proposed in [12,
13] is also
applicable. An extension of the above
considerations including temperature and damage effects is straightforward.
642
T.A. ANGELOV
References
[1] Zienkiewicz, O. (;., Flow formulation for numerical solution of forming processes, pp. 1 44, in PittmaIl, ,].F.T. et ah (eds.), N~zmerical analysis of jbrmmg processes, Wiley, (!hichester, U.K. (1984). I2] Iiobayashi, S., Oil, S.-I., Altan, T., Metal forming and the finite element method. Oxford University Press, Oxford (1989). [3] Mori, f,2., Rigid-plastic finite element solution of forming processes, pp. 73-79, in Pietrzyk, M. et al. (eds.), Huber's yield c,'iterio~z ir~ plasticity, AGH, Krakow (1994). [4] l{/~hn, H. A., Downey. C. L., Deformation characteristics and plasticity theory of sintered powder materials, Int. J, Powder Mater., 7, 15-25 (1971). [5] C.reen, R. 3., A plasticity theory for porous metals, h~t. d. Mech. ,5'ci., 14, 21.5-224 (1972). [6] Shima, S., Oyane, M., Plasticity theory for porous metals, h~t. d. Mech. 5'ci., 18, 285-291 (1976). [7] Do,'aivelu, S. M., Gegel, [t. L., Gunasekera, J. S., Malas, .l. (:., Morgan, J. T., A yield function for compressible P/M materials, h~t..]. ,~4ech..5'ci., 26, 527-535 (198:1). [8] Mori, K., Osakada, K., Analysis of the forming process of sintered powder metals by a rigid-plastic finite element method, Int. J. Mech, Sci., 29, No.4, 229-238 (1987). [9] Zienkiewicz, O. ('., [Iuang, (]. C., Lit,, Y. C., Adaptive FEM compntatiou of formiug processes application lo porous and nonq)orous materials, h~t. J. .\:~tm. Meth. f~gn9., 30, 1527-1553 (1990). [10] Chenot, J.-L., Bay, F., Fourment, L., Finite element simulation of metal powder forming, Int. d. Num. Meth. Engng., 30, 1649-1674 (1990). It 1] :\ngelov, T., A secant-modulus method for a rigid plastic rolling problem, 1~. J. NoT~-lmear Mech., 30, 169-178 (1995). [12] Angelov, T., On a rolling problem with damage and wear, Mech. Res. Comm., 26, No. 3, 281-286 (1999). [13] Angelov, T., A thermomechanically coupled rolling problem with damage, Mech. Re,,. Comm., 26, No. 3, 287-293 (1999). [14] Korneev. V. G., Langer, 17., Approximate solution of plastic flow theory problelns, Teubner-Te:rte zur Mathematik, Band 69, Teubner, Leipzig (1984). [15] Duvaut, G., Lions, J.-L., Les inequations en mecanique et en physique, Dunod, Paris (1972).
Mechanics Research Communications, Vol. 27, No. 6, pp. 637-642, 2000 Copynght© 2000 Elsevier Science Ltd Printed in the USA. All nghts reserved 0093-64131001$.-see front matter
PII: S0093-6413(00)00140.3
ON A ROLLING PROBLEM FOR POROUS MATERIALS
T A. Angelov Institute of Mechanics, Bulgarian Academy of Sciences, "Acad.G.Bonchev" str., bl.4, Sofia 1113, Bulgaria
(Received 22 March 2000, accepted for print 21 September 2000)
Introduction
There is a continuous interest in metal working industry to metal powder forming pro c~ sses. Metal powders are usually compacted in dies and sintered at elevated temperatures. Obtained metal powder preforms are further finished by various forming processes such a~s compaction, forging, extrusion, rolling. Powder preforms exhibit porosity and thus are more susceptible to fi-acture than solid metals. It is now well known that the flow theory of plasticity with rigid-plastic material model describes well the material behaviour at forming processes [1-3]. An extension of this theory to porous materials is given in [4-7]. Its applicability to different forming processes is further computationally examined by finite element method and Newton's type algorithms [8-10]. In the recent years convergent numerical methods were established for hot rolling problems, as the effects of strain rate sensitivity, nonlinear friction, temperature, damage and wear were taken into account [11-13]. In this work following [12, 13], a non-steady hot-strip rolling problem for powder based compressible materials, taking into account fi'iction, strain rate sensitivity, strain hardening and relative d, nsity effects, is stated. The material model, proposed by Mori and Osakada [:3, 8], is employed. A density evolution model, based on the mass conservation law, is introduced. A
638
T.A. ANGELOV
variational formulation is given and all existence and uniqueness theorem is stated. A convergent numerical method, analogous to the vaa'iable stiffness parameters method, proposed by l(orneev and Langer [14] for solving plastic flow theory problems, is presented.
Statement of the problem
We consider an isothermal, non-steady, hot-strip rolling process in a fixed domain until steady-state is reached at a time T E [0, oo). The workpiece is supposed isotropic, rigidp'astic, compressible metallic body occupying the domain ftC with
sufficiently regular
6 Fro. The b o u n d a r y F = O,n=l
boundary
Rk(k = 2, 3) (Fig.1 in [12]), ['1 O F2 U F4 U F5 is assumed
tractions flee, F:~ is the contact boundary and Ps is the boundary of symmetry. The points off~ = f~ U P are identified by their cartesian coordinates x = {x,} with respect to a fixed coordinate system, and the standard indicial notation and the summation convention over repeated indices is used. Let u(x, t) = {u,(x, t)}, ¢ ( x , t ) = {¢,j(x, t)}, k(x, t ) = {e,a(x, t)}, (1 _< i,j
< k), denote the velocity vector, stress tensor and strain rate tensor respectively. We denote by p(x, t) = pv(x, t)/p,(x, t) the relative density, where pv(x, t) and p,(x, t) are
the porous and solid material densities. Let , 1 .ff~ ~--- p
-('~- ) V 2
c~ 1 /2" Si35ia
+ da~
,
i- =
p(
-
.
1%
)V~gi3gi3+ -d- %, -
e(x,t) =~(x,0)+
1
2(X,T)~IT, (I)
be the equivalent stress, equivalent strain rate and equivalent strain, where a > 1 is a constant, d _> 0 is a function of the relative density, associated with the material compressibility, such that d -+ 0 when p -+ 1 and the classical yon Mises yield criterion and flow rule for
eij
~vS,j/3 are the components of the deviatoric stress and the strain rate tensors, a n = a,/3 and ~ = sii inconlpressible materials is obtained, sia = aij - an50
and
•
~i3 - -
are the hydrostatic pressure and the volume dilatation strain rate. Consider the following problem: - Find the velocity, u, stress ~r equivaleilt strain g and relative density p fields, satisfying the following equations and relations: - equations of motion cgui
a~a,j=pp,-~
in
f~x(0,7'),
(2)
- strain rate - velocity relations 1
(3)
ROLLING FOR POROUS MATERIALS
639
- yield criterion and flow rule
F(~,,~, p,:,~) -- a~ - ? ~ ( ~ ,
(4)
~) = 0,
- equivalent strain evolution equation
O~(x,t) _ h(x,t),
Ot
in
Q × (0, T),
(5)
- relative density evolution equation
Op(x,t) _
p(x t)i~,
in
Ot
f~ × (O,T),
(6)
- boundary conditions o'zff~a = 0
on
0%, = O,
El U F2 U F4 U F5 x (0, T ) , u~ = 0
'u~ = 0 if lar(U)l < r/(ph,eh, u), if
on
['6 x (0, T ) ,
(r) (s)
and then
u~ - UrR = 0,
I~rr(U)l=rf(ph,gh, u), then 3 const. A>_0,
such
that
Ur--Ur~=--~Crr(U) on
F a x ( 0 , T),
(9)
- initial conditions
u(x,O) = uo(×),
e(x,O) = Co(x),
p(x,O) = po(x).
(1o)
Here (~,j is Kronecker symbol; n = {ni} is the unit normal vector outward to F; u N, u r and a N, ~rr,, are the normal and tangential components of the velocity and the stress vector; Ur~~ is the rolling velocity; the shear strength limit for the material of Fa is given by the following Coulomb-Siebel friction law:
r/(ph, gh, u) = min t~jl~h(u)l, ~ip~,p(e~,~(u))}; ,~/(x) and 7n/(x) are the friction coefficient and factor; rv(~h, gh) = Cp(~h, g,,)/X/~
(11) is the
shear yield limit for the luaterial of F3. The convective terms are excluded from the corresponding equations, since their contribution to the material deformation is supposed insignificant. A natural assumption for the relative density is 0 < p0(x) _< p ( x , t ) < pl(x) < 1, where at p = pl the slightly compressible material behaviour is simulated [11]. Since the solid lnetal is incompressible, its density is constant in time during the process and assuming also that it is constant in spatial coordinates, without loss of generality we may take p, -- l [15]. The subindex h denotes that the corresponding quantities are mollified on F~. The strain
640
T.A. ANGELOV
and strain rate dependent uniaxial yield limit ap(e, f) is assumed monotonically increasing, almost everywhere differentiable function on both variables with bounded derivatives and such that: --
f
--
d(p) is also assumed differentiable with bounded derivative. Usually the following empirical expressions for d(p) where ~]a and 772 are positive constants. The relative density function and ap(e, g) are accepted
d(p) = O~o(1- p ) ~ ' for p C [po,p,j and d(p) = d(po) for p E [0,p0), d(p) = d(pl) for p E (px,o¢), ~(~,~) = c o ( l + c , W ~ ( h ~ for ~z [~,,&], ~ ~(~,~) = ~(~)2,
~(~) = ~(~'~q) 8~
(13)
[0,~],
for ~ [ o , ~ , ] u [ & , o o ) ,
q=l,~_,
(14)
where ao, al,c~,O <_ I <_ 3, are positive material constants, r/q(8) = r/q(g2) for g _2> ~2,
111 171(g2), 772 = -~-
772(0 ). Here g2, ~1, and g2 are the equivalent strain and strain rate model
bounds.
VariationM formulation
Let
v = {v:
v E (H'(a//k,
vN = 0 on r3 u F6},
H°(~/-
L~Ia),
be the spaces of admissible velocities, equivalent strains and densities. Let the following dense and continuous embeddings hold V C tt - H' C V', where H = (H°(~)) k, H' and V ~ are the topological dual spaces of H and V. Consider the following variational formulation associated with problem (2)-(10): - Find u(t) G V, ~(t), p(t) E H°(fl), satisfying for all t G [0, T], 0u
(p-~,v-u)+a(p,g,u;u,v-u)+j(p,g,u;vl-j(p,g,u;u)>O, Og
LOP -~(d:c = - fapfv~dz,
VveV
(15)
(16)
V( E H°(f~)
(17)
ROLLING FOR POROUS MATERIALS
641
~md the initial conditions u(0) = u0, ~(0) = c0, p(0) = P0. Here above the following notations are used: ,)
(18)
= t.
u.
ldr.
(19)
Jl
Let
u0(x) ~ v , %~(x) ~ (L.~. I':~))k, e0(x), p0(x), r,,(x) ~ L~(9.),
(')0)
aml let the friction coeificients # : ( x ) , m : ( x ) E L~.(F3) be sufficiently small. Then under the assumptions and propositions made, the following result holds. Theorem: The problem (15) - (17) has a unique solution, such that
Ou
uEL~(O,T;H) NL2(O,T;V), -:~EL2(O,T:V'),
e,/,EI,.~.(O,T;ff°(Ft)).
(21)
The proof of this theorem is based on the following au×ilia, ry problem. Let tbr almost all
[)n(t) E H°(f~), ,z = 0, J ..... be g,,+,(t), p,,+l(t) C lt°(ft), satisfying
I E [0, T], Urz(t) E V, Cn(/), Find u,,.+t(/) C V,
(p,~-,
v - un+l) +
a(p,,,~_,~,u,,;u,~+l,v
+j(p,,,g,,,u,,;v)-j(p~,e~,u~;u,~+~)>_O, -=~Tda"
=
e,~+~ d,~',
known. Then
--
u,~+l)+
VvEV
V~ ~ H°(f~)
L a,~..~ { dz = - J~ p,~+,i~,~+, { da',
V{ E H°(~l)
(22)
(23)
(24)
and the initial conditions u,~+l(0) = u0, g~+l(0) = #~0, P~+, (0) = p0. Following [14, 15], it can be shown that for any rz, this problem has a unique solution. It can be further shown that the obtained sequence {u,~(t),e,,(t),p,~(t)} converges strongly, at ,-, --+ ec., to a solution {u(t),g(t),p(t)} of (15) - (17) and this solution is unique. The above auxiliary problem describes a method of successive linearizations, called here variable stiffness parameters method, by an analogy with the method proposed in [141 for solving small strains plastic flow theory problems with hardening.
This method, combined with
an approl)riate time discretization scheme finally gives the time T, at which the process becomes steady-state, as well as the steady-state solution of (15) - (17). It is clear that the time retardation technique, proposed in [12,
13] is also
applicable. An extension of the above
considerations including temperature and damage effects is straightforward.
642
T.A. ANGELOV
References
[1] Zienkiewicz, O. (;., Flow formulation for numerical solution of forming processes, pp. 1 44, in PittmaIl, ,].F.T. et ah (eds.), N~zmerical analysis of jbrmmg processes, Wiley, (!hichester, U.K. (1984). I2] Iiobayashi, S., Oil, S.-I., Altan, T., Metal forming and the finite element method. Oxford University Press, Oxford (1989). [3] Mori, f,2., Rigid-plastic finite element solution of forming processes, pp. 73-79, in Pietrzyk, M. et al. (eds.), Huber's yield c,'iterio~z ir~ plasticity, AGH, Krakow (1994). [4] l{/~hn, H. A., Downey. C. L., Deformation characteristics and plasticity theory of sintered powder materials, Int. J, Powder Mater., 7, 15-25 (1971). [5] C.reen, R. 3., A plasticity theory for porous metals, h~t. d. Mech. ,5'ci., 14, 21.5-224 (1972). [6] Shima, S., Oyane, M., Plasticity theory for porous metals, h~t. d. Mech. 5'ci., 18, 285-291 (1976). [7] Do,'aivelu, S. M., Gegel, [t. L., Gunasekera, J. S., Malas, .l. (:., Morgan, J. T., A yield function for compressible P/M materials, h~t..]. ,~4ech..5'ci., 26, 527-535 (198:1). [8] Mori, K., Osakada, K., Analysis of the forming process of sintered powder metals by a rigid-plastic finite element method, Int. J. Mech, Sci., 29, No.4, 229-238 (1987). [9] Zienkiewicz, O. ('., [Iuang, (]. C., Lit,, Y. C., Adaptive FEM compntatiou of formiug processes application lo porous and nonq)orous materials, h~t. J. .\:~tm. Meth. f~gn9., 30, 1527-1553 (1990). [10] Chenot, J.-L., Bay, F., Fourment, L., Finite element simulation of metal powder forming, Int. d. Num. Meth. Engng., 30, 1649-1674 (1990). It 1] :\ngelov, T., A secant-modulus method for a rigid plastic rolling problem, 1~. J. NoT~-lmear Mech., 30, 169-178 (1995). [12] Angelov, T., On a rolling problem with damage and wear, Mech. Res. Comm., 26, No. 3, 281-286 (1999). [13] Angelov, T., A thermomechanically coupled rolling problem with damage, Mech. Re,,. Comm., 26, No. 3, 287-293 (1999). [14] Korneev. V. G., Langer, 17., Approximate solution of plastic flow theory problelns, Teubner-Te:rte zur Mathematik, Band 69, Teubner, Leipzig (1984). [15] Duvaut, G., Lions, J.-L., Les inequations en mecanique et en physique, Dunod, Paris (1972).