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On a rolling problem with damage and wear
Mechanics Research Communications, Vol. 26, No. 3, pp. 281-286, 1999
Copyright © 1999 Elsevier Science Ltd Printed in the USA. All rights reserved 0093-6413/99IS-see front matter
Pergamon PII S0093-6413(99)00025-7
ON A ROLLING PROBLEM WITH DAMAGE AND WEAR
T. A. Angelov Institute of Mechanics, Bulgarian Academy of Sciences, "Acad.G.Bonchev" str., bl.4, Sofia 1113, Bulgaria
(Received 17 March 1998; accepted for print 9 March 1999)
Introduction
The late 1970s and early 1980s were noted with significant achievements in the field of mechanical modelling and computer simulation of rolling processes. The flow theory of plasticity with rigid-plastic(viscoplastic), incompressible or slightly compressible material model was found to describe quite well the mechanical behaviour of the workpiece at a hot rolling process. Steady-state, transient and thermomechanically coupled problems were stated, taking also into account different material effects as rate sensitivity and hardening. A constant Coulomb or Siebel friction law for the roll-workpiece interface was usually accepted. The computational experiments are usually based on the finite element method applied to the virtual power principle, combined with either a penalty function or a Lagrange multiplier method. Different iterative computational procedures are proposed and used for solving the obtained systems of strongly nonlinear equations. The increasing requirements for quality of the rolling products states the future research trends namely in developing adequate material damage, wear and friction models. Corresponding problems have to be stated and studied, for which efficient numerical methods have to be developed.
281
282
T. ANGELOV
Recently, an isothermal steady-state rolling problem with a rigid-plastic, rate-sensitive at a constant strain, slightly compressible material model and a velocity dependent Coulomb or Siebel friction law, was stated and studied in [1, 2]. Existence and uniqueness results were obtained and the convergence of an iterative numerical method was proved. A basically new approach for treating transient problems, taking into account strain hardening, was presented in [3]. in this work following [1 - 3], a non-steady hot-strip rolling problem for incompressible materials, taking into account strata hardening, damage and wear effects, is stated. The material damage is introduced by a Lemaitre type ductile damage model [4]. The roll wear, interpreted as a change of the radius, due to a loss of roll material, or a gain of workpiece material, is introduced by Archard wear law [5, 6]. A variational formulation is given and an existence and uniqueness theorem is stated. The main steps for a computational algorithm are outlined.
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, oc) [:3]. The workpiece is supposed isotropM rigidplastic, incompressible metallic body occupying the domain ~ C Rk(k = 2, 3) (Fig.l), with 6 sufficiently regular boundary P = Urn= l[ • .,.
F1 U P2 U F4 U F5
is the contact boundary and F6 is the boundary of symmetry.
q v,
gl
FIG. 1 Workpiece geometry at rolling
is assumed tractions free, r3
ROLLING WITH DAMAGE AND WEAR
283
The points of (~ = Q U P are identified by their cartesian coordinates x = { x i } , and the standard indicial notation is used. T h e superposed dot denotes partial time derivative. Let u ( x , t ) = {ui(x,t)}, ~r(x,t) = { a i j ( x , t ) } ,
k ( x , t ) = {&j(x,t)}, (1 < i , j <_ k), D ( x , t ) and
w(x, t) denote the velocity vector, stress and strain rate tensors, damage and wear variables respectively. A notation as e.g. u(t) = u ( . , t ) will be also used. Let
2. o
~ -~- ~Eij6ij ,
@ .--__~ S i j S i j '
be the equivalent stress and strain rate, where sij = oij
(l) eij = eij -- ~ i j / 3 ,
-- O'H~ij,
the components of the deviatoric stress and the strain rate tensors and a n = aii/3,
are
~,, = ~ii
are the hydrostatic pressure and the volume dilatation strain rate. Consider the following problem: Find the velocity u, stress tr, equivalent strain g, damage D and wear w fields, satisfying -
the following equations and relations: - equation of equilibrium aij,j=O
in
f l x ( 0 , T),
(2)
incompressibility condition
-
iv = 0
in
(3)
f~ x (0, T),
- strain rate - velocity relations 1
(4)
gij = ~(ui,j + uj,~), yield criterion and flow rule
-
F(alj,e,g,D)
3~ ~j = ~ s ~ j ,
- d2 - (1 - D)2 a~,(& g) = O,
(5)
equivalent strain evolution equation
-
g(x,t) =
f0 t g ( x , r ) d r
{o,
in
f l x (0, T),
(6)
in
flx(O,T),
(7)
F3 x (0, T),
(8)
damage evolution equation
-
D(x, t) =
D ~ (el-~0) (~-~o) '
e e [0, eo] e e leo, el]
Oct -
wear evolution equation ¢b(x,t) -- k,~latch(u)l lUr - r(w)urR[
on
284
T. ANGELOV - boundary conditions
oz, n i = 0
on
orT = 0 ,
if if
such -
F ~ U F 2 U F 4 U F s x ( 0 , T), uN =0
on
u N = tb
and
[~rv(u)[ < rf(eh, Dh,u),
than
lO'r(U)i=Tf(eh,Dh, u), that
(9)
F6 x ( 0 , T),
than
ur--r(w)urR=--A~rr(u
(10)
u r - r(w)urR = O, ~ const. ) on
A>0,
F 3 x ( 0 , T),
(11)
initial conditions
~(x,0) = 0 ,
w(x,0) = 0,
D(x,0) = 0 .
(12)
Here 6ij is Kronecker symbol; n = {hi} is the unit normal vector outward to r; uN, u r and crN, o' r are the normal and tangential components of the velocity and the stress vector; urR is the rolling velocity; the shear strength limit for the material of F3, according to Coulomb-Siebel friction law, is defined by: rf(gh, Dh, u) = min {#f[~rNh(U)[, m j,(1 -- Dh)rp(gh, gh(u))}:
(13)
p j ( x ) and rnf(x) are the friction coefficient and factor; k~(x) is the wear constant; r0 is the initial and r(w) = (r0 - w)/r0 is the relative roll radius; De,, 6 [0, 1) is the critical damage value; e0 and c1 are the strain thresholds for initiation of damage and rupture; rp(eh, gh) =
~Tp(gh,2h)/V/3
is the shear yield limit for the material of F3. The subindex
h denotes that the corresponding quantities are appropriately mollified on F3. The strain and strain rate dependent uniaxial yield limit cry(c, ~5) is assumed monotonically increasing, almost everywhere differentiable function on both variables, with bounded derivatives and such that:
7]2 --~
~
~ 711,
where 711 and 7]2 are positive constants. usually accepted
V~, CE
[0, oo),
(14)
The following empirical yield limit expression is
~(~,~) =~o(l+c,~F~(~) ~ for ~e [~1,~], ~e [0,~1,
O'p(2,~q)
for C. e [0,~~1] U [~2, 00),
q = 1,2,
(15)
where cl > 0, (0 < I < 3), are material constants, qq(2) = 7]q(g2) for 2 _> 22 and 711 = ql(g2), 7]~ = q2(0), ¢1 is the equivalent strain rate at static loading, g2 and g2 are the equivalent strain and strain rate at fracture.
ROLLING WITH DAMAGE AND WEAR
285
Variational formulation
Let V = {v:
v • (Ul(~'/)) k,
v N = 0 on FaUF6},
U°(12) -= L2(F~),
L2(F3),
be the spaces of admissible velocities, equivalent strains, damage and wear. The velocities are shifted by a function z(t) • (H(~)) k, such that zN(t ) = tb(t) on F3 and zN(t ) = 0 on F6, and for a sake of simplicity, the same notations are used. Consider the following penalty variational formulation associated with problem (2)-(12): - Find u(t) • V, ~(t), D(t) • H°(~) and w(t) • L2(F~), satisfying for all t • [0, T]
a(t, u; u, v - u) + j(t, u; v) - j(t, u; u) >_O,
/o
=
•
f D~dx=~Y(g)~dx,
(16)
(17)
V~ • " ° ( a )
fr wCdr=~ W(w,u)¢dr, 3
Vv•V
VCeL:(rz)
(18) (19)
3
and the initial conditions (12). Here the following notations are used: [
a ( t , w ; u , v ) = Jn
(1
-
j(t,u;v)
D)%(e(t),g(w)) ~(w)
[2gij(u)~ij(v) + dg~(u)~.(v)] dx,
f
l_ rJ(gh'Dh'U)lVr - r(w)u~,Ldr,
(20)
(21)
,2[' 3
are the virtual power of the actual stresses and friction forces, d > 0 is a penalty constant,
E(t), Y(g) and W(w, u) are the functions in the right-hand side of equations (6)- (8). Let the friction and wear coefficients be sufficiently small constants [1, 2, 6] and let
~s(x), max), k~(x) e Loo(r3), u.,~ • (Loo(F3))*.
(22)
Then under the propositions and assumptions made, the following result holds. Theorem: The problem (16) - (19) has a unique solution, such that u • Loo(0, T;V),
~, D • Loo(O,T;g°(~)),
w • Loo(O,T;L2(F3)).
(23)
The proof of this theorem is based on an auxiliary problem with time delay in the equivalent strain, damage and wear terms. Let r/ > 0 be a small constant and set in (16) gn(t - r]), Dn(t - r]), wn(t - T/), assumed equal to zero for (t - 7/) < 0. Then the auxiliary problem is:
286
T. ANGELOV - Find un(t) C V, g,(t), Dn(t) e H°([~) and w,(t) E L2(F3), satisfying for all t C [0, 7']
a(t-q,u,;un,v-u,)+)(t-q,u~;v)-j(t-q,u,7;u,)>_O,
VvE V
(24)
fa,% Tdx = fatf,,(t)'yda'.
V? E H°(f~)
(25)
fa D,, ~ d.r = ~a Y(gn)~ dx'
V~ E H°(a)
(26)
dl
and the initial conditions e,7(-q) = 0, Dn(-q) = 0, w,~(-q) = 0. Following [1,2,6], it can be shown that for any fixed q, this problem has a unique solution. It can be shown further that the obtained sequence of q-solutions, when q -~ 0, converges weakly-, to a solution of (16) - (19) and this solution is unique. This scheme of proof generates an algorithm, which gives as a final result the steady-state solution of (16) - (19) and the time T, when it is reached.
Acknowledgements
This work is supported by Bulgarian Science Fund with the grant MM447/94.
References
[1] Angelov, T., A secant-modulus method for a rigid-plastic rolling problem,
Int. J. Nonlinear Mech., 30, 169-178 (1995) [2] Angelov, T., Baltov, A., Nedev, A., Existence and uniqueness of the solution of a rigid-plastic rolling problem, Int. J. Eng. Sci., 33, 1251-1261 (1995) [3] Baltov, A., Angelov, T., Variational approach of a problem for the non-steady rolling process, Mech. Res. Comm., 22,555-560 (1995) [4] Lemaitre, J., Chaboche, J.-L., Mechanics of solid materials, Cambridge Univ. t:ress, Cambridge (1990). [5] Stromberg, N., Johansson, L., Klarbring, A., Derivation and analysis of a generalised standard model for contact, friction and wear, Int. J. Solids Struct., 33, 1817-1836 (1996). [6] Andrews, K.T., Klarbring, A., Shillor, M., Wright, S., A dynamic thermoviscoelastic contact problem with friction and wear, Int. ,I. En9. Sci., 35. 1291-1309 (1997).
Copyright © 1999 Elsevier Science Ltd Printed in the USA. All rights reserved 0093-6413/99IS-see front matter
Pergamon PII S0093-6413(99)00025-7
ON A ROLLING PROBLEM WITH DAMAGE AND WEAR
T. A. Angelov Institute of Mechanics, Bulgarian Academy of Sciences, "Acad.G.Bonchev" str., bl.4, Sofia 1113, Bulgaria
(Received 17 March 1998; accepted for print 9 March 1999)
Introduction
The late 1970s and early 1980s were noted with significant achievements in the field of mechanical modelling and computer simulation of rolling processes. The flow theory of plasticity with rigid-plastic(viscoplastic), incompressible or slightly compressible material model was found to describe quite well the mechanical behaviour of the workpiece at a hot rolling process. Steady-state, transient and thermomechanically coupled problems were stated, taking also into account different material effects as rate sensitivity and hardening. A constant Coulomb or Siebel friction law for the roll-workpiece interface was usually accepted. The computational experiments are usually based on the finite element method applied to the virtual power principle, combined with either a penalty function or a Lagrange multiplier method. Different iterative computational procedures are proposed and used for solving the obtained systems of strongly nonlinear equations. The increasing requirements for quality of the rolling products states the future research trends namely in developing adequate material damage, wear and friction models. Corresponding problems have to be stated and studied, for which efficient numerical methods have to be developed.
281
282
T. ANGELOV
Recently, an isothermal steady-state rolling problem with a rigid-plastic, rate-sensitive at a constant strain, slightly compressible material model and a velocity dependent Coulomb or Siebel friction law, was stated and studied in [1, 2]. Existence and uniqueness results were obtained and the convergence of an iterative numerical method was proved. A basically new approach for treating transient problems, taking into account strain hardening, was presented in [3]. in this work following [1 - 3], a non-steady hot-strip rolling problem for incompressible materials, taking into account strata hardening, damage and wear effects, is stated. The material damage is introduced by a Lemaitre type ductile damage model [4]. The roll wear, interpreted as a change of the radius, due to a loss of roll material, or a gain of workpiece material, is introduced by Archard wear law [5, 6]. A variational formulation is given and an existence and uniqueness theorem is stated. The main steps for a computational algorithm are outlined.
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, oc) [:3]. The workpiece is supposed isotropM rigidplastic, incompressible metallic body occupying the domain ~ C Rk(k = 2, 3) (Fig.l), with 6 sufficiently regular boundary P = Urn= l[ • .,.
F1 U P2 U F4 U F5
is the contact boundary and F6 is the boundary of symmetry.
q v,
gl
FIG. 1 Workpiece geometry at rolling
is assumed tractions free, r3
ROLLING WITH DAMAGE AND WEAR
283
The points of (~ = Q U P are identified by their cartesian coordinates x = { x i } , and the standard indicial notation is used. T h e superposed dot denotes partial time derivative. Let u ( x , t ) = {ui(x,t)}, ~r(x,t) = { a i j ( x , t ) } ,
k ( x , t ) = {&j(x,t)}, (1 < i , j <_ k), D ( x , t ) and
w(x, t) denote the velocity vector, stress and strain rate tensors, damage and wear variables respectively. A notation as e.g. u(t) = u ( . , t ) will be also used. Let
2. o
~ -~- ~Eij6ij ,
@ .--__~ S i j S i j '
be the equivalent stress and strain rate, where sij = oij
(l) eij = eij -- ~ i j / 3 ,
-- O'H~ij,
the components of the deviatoric stress and the strain rate tensors and a n = aii/3,
are
~,, = ~ii
are the hydrostatic pressure and the volume dilatation strain rate. Consider the following problem: Find the velocity u, stress tr, equivalent strain g, damage D and wear w fields, satisfying -
the following equations and relations: - equation of equilibrium aij,j=O
in
f l x ( 0 , T),
(2)
incompressibility condition
-
iv = 0
in
(3)
f~ x (0, T),
- strain rate - velocity relations 1
(4)
gij = ~(ui,j + uj,~), yield criterion and flow rule
-
F(alj,e,g,D)
3~ ~j = ~ s ~ j ,
- d2 - (1 - D)2 a~,(& g) = O,
(5)
equivalent strain evolution equation
-
g(x,t) =
f0 t g ( x , r ) d r
{o,
in
f l x (0, T),
(6)
in
flx(O,T),
(7)
F3 x (0, T),
(8)
damage evolution equation
-
D(x, t) =
D ~ (el-~0) (~-~o) '
e e [0, eo] e e leo, el]
Oct -
wear evolution equation ¢b(x,t) -- k,~latch(u)l lUr - r(w)urR[
on
284
T. ANGELOV - boundary conditions
oz, n i = 0
on
orT = 0 ,
if if
such -
F ~ U F 2 U F 4 U F s x ( 0 , T), uN =0
on
u N = tb
and
[~rv(u)[ < rf(eh, Dh,u),
than
lO'r(U)i=Tf(eh,Dh, u), that
(9)
F6 x ( 0 , T),
than
ur--r(w)urR=--A~rr(u
(10)
u r - r(w)urR = O, ~ const. ) on
A>0,
F 3 x ( 0 , T),
(11)
initial conditions
~(x,0) = 0 ,
w(x,0) = 0,
D(x,0) = 0 .
(12)
Here 6ij is Kronecker symbol; n = {hi} is the unit normal vector outward to r; uN, u r and crN, o' r are the normal and tangential components of the velocity and the stress vector; urR is the rolling velocity; the shear strength limit for the material of F3, according to Coulomb-Siebel friction law, is defined by: rf(gh, Dh, u) = min {#f[~rNh(U)[, m j,(1 -- Dh)rp(gh, gh(u))}:
(13)
p j ( x ) and rnf(x) are the friction coefficient and factor; k~(x) is the wear constant; r0 is the initial and r(w) = (r0 - w)/r0 is the relative roll radius; De,, 6 [0, 1) is the critical damage value; e0 and c1 are the strain thresholds for initiation of damage and rupture; rp(eh, gh) =
~Tp(gh,2h)/V/3
is the shear yield limit for the material of F3. The subindex
h denotes that the corresponding quantities are appropriately mollified on F3. The strain and strain rate dependent uniaxial yield limit cry(c, ~5) is assumed monotonically increasing, almost everywhere differentiable function on both variables, with bounded derivatives and such that:
7]2 --~
~
~ 711,
where 711 and 7]2 are positive constants. usually accepted
V~, CE
[0, oo),
(14)
The following empirical yield limit expression is
~(~,~) =~o(l+c,~F~(~) ~ for ~e [~1,~], ~e [0,~1,
O'p(2,~q)
for C. e [0,~~1] U [~2, 00),
q = 1,2,
(15)
where cl > 0, (0 < I < 3), are material constants, qq(2) = 7]q(g2) for 2 _> 22 and 711 = ql(g2), 7]~ = q2(0), ¢1 is the equivalent strain rate at static loading, g2 and g2 are the equivalent strain and strain rate at fracture.
ROLLING WITH DAMAGE AND WEAR
285
Variational formulation
Let V = {v:
v • (Ul(~'/)) k,
v N = 0 on FaUF6},
U°(12) -= L2(F~),
L2(F3),
be the spaces of admissible velocities, equivalent strains, damage and wear. The velocities are shifted by a function z(t) • (H(~)) k, such that zN(t ) = tb(t) on F3 and zN(t ) = 0 on F6, and for a sake of simplicity, the same notations are used. Consider the following penalty variational formulation associated with problem (2)-(12): - Find u(t) • V, ~(t), D(t) • H°(~) and w(t) • L2(F~), satisfying for all t • [0, T]
a(t, u; u, v - u) + j(t, u; v) - j(t, u; u) >_O,
/o
=
•
f D~dx=~Y(g)~dx,
(16)
(17)
V~ • " ° ( a )
fr wCdr=~ W(w,u)¢dr, 3
Vv•V
VCeL:(rz)
(18) (19)
3
and the initial conditions (12). Here the following notations are used: [
a ( t , w ; u , v ) = Jn
(1
-
j(t,u;v)
D)%(e(t),g(w)) ~(w)
[2gij(u)~ij(v) + dg~(u)~.(v)] dx,
f
l_ rJ(gh'Dh'U)lVr - r(w)u~,Ldr,
(20)
(21)
,2[' 3
are the virtual power of the actual stresses and friction forces, d > 0 is a penalty constant,
E(t), Y(g) and W(w, u) are the functions in the right-hand side of equations (6)- (8). Let the friction and wear coefficients be sufficiently small constants [1, 2, 6] and let
~s(x), max), k~(x) e Loo(r3), u.,~ • (Loo(F3))*.
(22)
Then under the propositions and assumptions made, the following result holds. Theorem: The problem (16) - (19) has a unique solution, such that u • Loo(0, T;V),
~, D • Loo(O,T;g°(~)),
w • Loo(O,T;L2(F3)).
(23)
The proof of this theorem is based on an auxiliary problem with time delay in the equivalent strain, damage and wear terms. Let r/ > 0 be a small constant and set in (16) gn(t - r]), Dn(t - r]), wn(t - T/), assumed equal to zero for (t - 7/) < 0. Then the auxiliary problem is:
286
T. ANGELOV - Find un(t) C V, g,(t), Dn(t) e H°([~) and w,(t) E L2(F3), satisfying for all t C [0, 7']
a(t-q,u,;un,v-u,)+)(t-q,u~;v)-j(t-q,u,7;u,)>_O,
VvE V
(24)
fa,% Tdx = fatf,,(t)'yda'.
V? E H°(f~)
(25)
fa D,, ~ d.r = ~a Y(gn)~ dx'
V~ E H°(a)
(26)
dl
and the initial conditions e,7(-q) = 0, Dn(-q) = 0, w,~(-q) = 0. Following [1,2,6], it can be shown that for any fixed q, this problem has a unique solution. It can be shown further that the obtained sequence of q-solutions, when q -~ 0, converges weakly-, to a solution of (16) - (19) and this solution is unique. This scheme of proof generates an algorithm, which gives as a final result the steady-state solution of (16) - (19) and the time T, when it is reached.
Acknowledgements
This work is supported by Bulgarian Science Fund with the grant MM447/94.
References
[1] Angelov, T., A secant-modulus method for a rigid-plastic rolling problem,
Int. J. Nonlinear Mech., 30, 169-178 (1995) [2] Angelov, T., Baltov, A., Nedev, A., Existence and uniqueness of the solution of a rigid-plastic rolling problem, Int. J. Eng. Sci., 33, 1251-1261 (1995) [3] Baltov, A., Angelov, T., Variational approach of a problem for the non-steady rolling process, Mech. Res. Comm., 22,555-560 (1995) [4] Lemaitre, J., Chaboche, J.-L., Mechanics of solid materials, Cambridge Univ. t:ress, Cambridge (1990). [5] Stromberg, N., Johansson, L., Klarbring, A., Derivation and analysis of a generalised standard model for contact, friction and wear, Int. J. Solids Struct., 33, 1817-1836 (1996). [6] Andrews, K.T., Klarbring, A., Shillor, M., Wright, S., A dynamic thermoviscoelastic contact problem with friction and wear, Int. ,I. En9. Sci., 35. 1291-1309 (1997).