Stochastic equations of rigid-thermo-viscoplasticity in metal forming process

International Journal of Engineering Science 40 (2002) 367±383

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Stochastic equations of rigid-thermo-viscoplasticity in metal forming process Maksym Grzywi nski, Andrzej Søu_zalec

*

Technical University of Czestochowa, 42-200 Czestochowa, Poland Received 13 April 2001; accepted 20 June 2001

Abstract Stochastic rigid-thermo-viscoplastic metal forming process is described and discussed. The theoretical formulation for stochastic equations in rigid-thermo-viscoplasticity is presented. It is based on the combination of the second-order perturbation technique and second-moment analysis. The principle allows incorporation of system uncertainties into ®nite-element equations. Probabilistic distributions for temperatures and strains taking into account random initial and boundary conditions are given. Example of stochastic analysis in shell nosing process is shown. Ó 2002 Elsevier Science Ltd. All rights reserved.

1. Introduction Problems of structural mechanics and material processing are considered usually by deterministic theories. But many of these problems have stochastic character. Stochastic analysis in structural mechanics and material engineering is an active area of research. Researches in this ®eld combine numerical and analytical solutions with stochastic one and undertake the problem of random deformation of structures which can be solved with applications of the probabilistic approach. The literature on probabilistic methods in mechanics is considerable see [1±4], for instance. It can be divided into two parts: i.e. using statistical and nonstatistical approaches. Non-statistical methods include stochastic ®nite-element method, which are discussed in [5±7]. Non-statistical probabilistic methods are analysed in [8,9]. The above references describe linear [1±5,7] and nonlinear [6,8,9] problems. Thermo-plastic processes in metal forming are usually described by a rigid-thermo-viscoplastic theory. For this theory the numerical solutions for equations describing stochastic coupled

*

Corresponding author. Tel.: +48-34-3250-920; fax: +48-34-3250-920. E-mail address: [email protected] (A. Søu_zalec).

0020-7225/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 2 2 5 ( 0 1 ) 0 0 0 8 0 - 5

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thermo-mechanical metal forming processes are not undertaken in the literature. In [16] Tomita analysed the in¯uence in parameters in the constitutive equation for rigid plastic body, but he did not undertake the problem of stochastic equations for rigid-plasticity. Stochastic approach to coupled-thermo-mechanical metal forming problems is important not only because of random material parameters, but particularly because of boundary problems appearing in these processes. Contact problems die-workpiece have exceptional random character and lead to determine the boundary forces in the contact die-workpiece considering random character of friction between them. If we combine the above-mentioned conditions with stochastic analysis many important problems appear to be solved by de®ning a proper stochastic formulation for metal forming process taking into account boundary conditions. The method presented is based on the combination of the second-order perturbation technique and second-moment analysis. The principle allows incorporation of system uncertainties into ®nite-element equations. The matrix formulation of rigid-thermo-viscoplasticity is standard and is based on the Galerkin method: such a formulation in turn uses the standard ®nite-element analysis [10,11]. 2. Equations of random thermal process 2.1. Matrix thermal equations Thermal deterministic equations for the problem considered are placed in Appendix A. Now we consider matrix equations of heat ¯ow where the heat capacity matrix C, the thermal conductivity matrix K, the thermal load vector F, and the column vector T of nodal temperature Ti are functions of the discretized random variable vector b ˆ b…x†, where x is the coordinate vector _ C…b†T…b; t† ‡ K…b†T…b; t† ˆ F…b; t†:

…1†

The random function b…x† is approximated using shape functions Ni …x† by b…x† ˆ

q X

Ni …x†bi ˆ Nb;

…2†

iˆ1

where bi are nodal values of b…x†, that is the values of b at xi , i ˆ 1; . . . ; q. The mean value of b denoted by E…b† is expressed as E…b† ˆ

q X

Ni E…bi †

…3†

iˆ1

and the variance by V …b† ˆ a2 E…b†2 ; where a is the coecient of variation.

…4†

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369

All the random functions are expanded about the mean value E…b† via a Taylor series and only up to second-order terms are retained. For any small parameter c we have q X

T…b; t† ˆ E…T…t†† ‡ c

iˆ1

q 1 X E…T;bi …t††Dbi ‡ c2 E…T;bi bj …t††Dbi Dbj ; 2 i;jˆ1

…5†

where Dbi represents the ®rst-order variation of bi about E…bi † and for any function g E…g…x†† ˆ g…x; E…b††; E…g;b1 † ˆ

og ; ob1

E…g;b1 b2 † ˆ

…6†

o2 g : ob1 ob2

In a similar manner we can express C…b†, K…b† and F…b; t† as C…b† ˆ E…C† ‡ c

q X iˆ1

K…b† ˆ E…K† ‡ c

q X iˆ1

F…b; t† ˆ E…F† ‡ c

q 1 2X E…C;bi †Dbi ‡ c E…C;bi bj †Dbi Dbj ; 2 i;jˆ1

…7†

q 1 X E…K;bi †Dbi ‡ c2 E…K;bi bj †Dbi Dbj ; 2 i;jˆ1

…8†

q X iˆ1

q 1 X E…F;bi …t††Dbi ‡ c2 E…F;bi bj †Dbi Dbj : 2 i;jˆ1

…9†

Substitution of Eqs. (5) and (7)±(9) into Eq. (1) and collecting terms of order 1; c and c2 yields the following equations for E…T…t††, E…T;bi …t†† and E…T;bi bj …t†† zeroth order: _ E…C†E…T…t†† ‡ E…K†E…T…t†† ˆ E…F…t††;

…10†

®rst order E…C†E…T_ ;bi …t†† ‡ E…K†E…T;bi …t†† ˆ E…F1;bi …E…T†; t††;

…11†

where E…F1;bi …E…T†; t†† ˆ E…F;bi …t††

_ …E…C;bi †E…T…t†† ‡ E…K;bi †E…T…t†††;

…12†

second order ^ 2 …t† ˆ F ^ 2 …E…T†; E…T;b †; t†; ^_ 2 …t† ‡ E…K†T E…C†T i

…13†

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^2 ˆ F

q  X 1

  E…F;bi bj …t†† Cov…bi ; bj †

2 q  X

i;jˆ1

1 1 _ ‡ E…K;bi bj †E…T…t†† E…C;bi bj †E…T…t†† 2 2 i;jˆ1   _ ‡ E…C;bi †E…T;bj …t†† ‡ E…K;bi †E…T;bj …t†† Cov…bi ; bj †

…14†

and ^ 2 …t† ˆ 1 T 2

q X

E…T;bi bj † Cov…bi ; bj †;

…15†

 1=2 Cov…bi ; bj † ˆ V …b…xi ††V …b…xj †† R…b…xi †; b…xj ††

…16†

i;jˆ1

and R…b…xi †; b…xj †† is the autocorrelation. 2.2. Random matrix equation of thermo-viscoplasticity We consider Eqs. (B.7) and (B.12) in which matrices K…l† and the vectors Q; q_ and p are functions of the discretized random variable b ˆ b…x† and the yield stress is a function of temperature T Then we have _ p…b† ˆ Q…b†; ‡ KT…p†  Kl …T…b†; b†q…b†

…17†

_ K…p† q…b† ˆ 0:

…18†

For any function g ˆ g…T …b†; b† we have g ˆE…g† ‡ c

q X

E…g;bi ‡ g;T T;bi †Dbi

iˆ1

q 1 X ‡ c2 E…g;bi bj ‡ g;Tbj T;bi ‡ g;T T;bi bj ‡ g;bi T_ T;bj ‡ g;TT T;bi T;bj †Dbi Dbj : 2 i;jˆ1

…19†

Assuming g ˆ g1 …T …b†† ‡ q2 …b† and g1 …T …b†† is a linear function T we get g ˆ E…g† ‡ c

q X iˆ1

q
1 X E…g;bi ‡ g;T T;bi †Dbi ‡ c2 E g;bi bj ‡ g;T T;bi bj Dbi Dbj : 2 i;jˆ1

…20†

All the random functions are expanded about the mean value E…b† via a Taylor series and only up to second order terms are retained. For any small parameter c we have

M. Grzywinski, A. Søu_zalec / International Journal of Engineering Science 40 (2002) 367±383

_ t† ˆ E…q…t†† _ q…b; ‡c

q X iˆ1

 p…b† ˆ E… p…b†† ‡ c

q X iˆ1

q 1 X E…q_ ;bi …t†Dbi † ‡ c2 E…q_ ;bi bj …t††Dbi Dbj ; 2 i;jˆ1

q 1 X E… p;bi †Dbi ‡ c2 E…p;bi bj …t††Dbi Dbj : 2 i;jˆ1

371

…21†

…22†

In a similar manner we can express Q…b† as Q…b† ˆ E…Q† ‡ c

q X iˆ1

2 1 X E…Q;bi †Dbi ‡ c2 E…Q;bi bj †Dbi Dbj ; 2 i;jˆ1

Kl …T …b†; b† ˆ E…Kl † ‡ c

q X iˆ1

q

1 X E Kl;bi ‡ Kl;T T;bi Dbi ‡ c2 E Kl;bi bj ‡ Kl;T T;bi bj Dbi bj : 2 i;jˆ1

…23†

…24†

Substitution of Eqs. (21), (22) and (23), (24) into Eqs. (17) and (18) and collecting terms of order 1; c and c2 yields the following equations for: _ E…q; _ bi †; E…q; _ bi bj † and E… E…q†; p†; E… p;bi †; E…p;bi bj †; zero order _ ‡ E…KT…p† †E… E…K…l† †E…q† p† ˆ E…Q†;

…25†


_ ˆ 0; E K…p† E…q†

…26†

®rst order p;bi † ˆ E…Q1;bi †; E…K…l† †E…q_ ;bi † ‡ E…KT…p† †E…

…27†

_ bi † ˆ 0; E…K…p† †E…q;

…28†

where E…Q1;bi † ˆ E…Q;bi †

_ …E…Kl;bi ‡ K…l†T T;bi †E…q††;

…29†

second order ^ zˆ2 ; ^ E…K…l† †^ q_ zˆ2 ‡ E…KT…p† † pzˆ2 ˆ Q

…30†

E…K…p† †^ q_ 2 ˆ 0;

…31†

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where

   q  X 1 E Q;bi bj Cov…bi ; bj † 2 i;jˆ1 (" ! )  q X 1 _ ‡ E…K…l†;bi †E…q; _ bi † Cov…bi ; bj † ; E K…l†;bi bj ‡ K…l†;T T;bi bj E…q† 2 i;jˆ1

^ zˆ2 ˆ Q

1 ^_ 2 ˆ q 2 ^ p2 ˆ

q X

…32†

E…q;_ bi bj † Cov…bi ; bj †;

…33†

q 1X E… p;bi bj † Cov…bi ; bj †: 2 i;jˆ1

…34†

i;jˆ1

The de®nition for the expectation and autocovariance of the velocity matrix q_ are given by Z ‡1 _ _ q…b†f…b† db …35† E…q† ˆ 1

and i

Z

j

Cov…q ; q † ˆ

‡1 1

…qi

E…qi †…qj

E…qj †††f…b† db;

where f…b† is the joint probability density function. The second-order estimate of the mean value of q_ is obtained from Eq. (35) to give ( ) q 1 X _ bi bj † Cov…bi ; bj † : _ ˆ q…E…b†† _ E…q; E…q† ‡ 2 i;jˆ1 The matrix E…K…l† † is expressed as Z 2l…b†BT B dX: E…K…l† † ˆ

…36†

…37†

…38†

X

The element strain-rate vector e_ is of the form _ e_ ˆ Bq: Then 1 _ ‡ _ ˆ BE…q† E…e† 2 ( Cov…e_e ; e_f † ˆ

(

q X i;jˆ1

q X

) _ bi bj Cov…bi ; bj † ; Bq;

…39†

i;jˆ1

) T

_ ebi †…Bf q_ fbj † Cov…bi ; bj † : …Be q;

…40†

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2.3. Stochastic frictional forces In stochastic analysis we assume that the friction factor and shear yield stress of the material are random, so the shear friction stress has also the random character. Consider the equation for friction factor. We can express m…b† and ky …b† as m…b† ˆ E…m† ‡ c

q X iˆ1

ky …b† ˆ E…ky † ‡ c

q X iˆ1

s…b† ˆ E…s† ‡ c

2
1 X E…m;bi †Dbi ‡ c2 E m;bi bj Dbi bj ; 2 i;jˆ1

q X iˆ1

…41†

2

1 X E ky;bi Dbi ‡ c2 E ky;bi bj Dbi Dbj ; 2 i;jˆ1

…42†

2
1 X E…s;bi †Dbi ‡ c2 E s;bi bj Dbi Dbj ; 2 i;jˆ1

…43†

so we have the following equations: zero order E…s† ˆ E…m†E…ky †;

…44†

®rst order
E…s1;bi † ˆ E…m†E ky;bi ;

…45†

where E…s1;bi † ˆ E…s;bi †

E…m;bi †E…s†;

…46†

second order s^2 ˆ E…m†k^y2 ;

…47†

where s^2 ˆ

q  X 1 i;jˆ1

2




E s;bi bj Cov…bi ; bj †



q  X 1 i;jˆ1

2




E m;bi bj E…ky † ‡ E…m;bi †E ky;bi




 Cov…bi ; bj † ; …48†

1 k^y2 ˆ 2

q X i;jˆ1


E ky;bi bj Cov…bi ; bj †:

…49†

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3. Example As an example we consider the simulation of hot nosing process. Shell nosing refers to the process of forming an give nose at the end of a tubular part by pressing the tube into a suitable formed die (Fig. 1). The ®nite-element simulations were carried out using the mesh system and dimensions shown in Fig. 2. In the analysis 9-node isoparametric elements were used. The validity of the assumed mesh system has been tested by introducing more dense meshes. It did not follow changes in results obtained more than 0.01%. So the other meshes used have been omitted in presentation. The following random data are adopted: Expectations of mechanical parameters E…m† ˆ 0:02: E…n…E…T ††† ˆ 0:1743  10 9 T 3 ‡ 0:1026  10 5 T 2 0:1917  10 2 T ‡ 1:3, E…c…E…T ††† ˆ 0:8682  10 8 T 3 ‡ 0:5285  10 4 T 2 0:107 ‡ 72:28, E…r0 …E…T ††† ˆ 7:2588  10 8 T 3 ‡ 4:4912  10 4 T 2 0:938T ‡ 664:6; cross-correlation functions l…nq ; nr † ˆ exp‰ abs…xi xj †=nn Š, l…cq ; cr † ˆ exp‰ abs…xi xj †=nc Š, l…rq0 ; rr0 † ˆ exp‰ abs…xi xj †=nr Š, l…mq ; mr † ˆ exp‰ abs…xi xj †=nm Š with correlation lengths nn ˆ nc ˆ nr ˆ nm ˆ 1, and coecient of variation aqn ˆ aqc ˆ aqr ˆ aqm ˆ 0:14. Expectations of thermal parameters: thermal conductivity E…K† ˆ 1:70, heat capacity E…c† ˆ 1000, density E…q† ˆ 2400, surface ®lm conductance E…h† ˆ 20,

Fig. 1. Scheme of shell nosing.

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375

Fig. 2. Mesh system for ®nite-element simulation of shell nosing.

cross-correlation functions: l…k q ; k r † ˆ exp‰ abs…zi zj †=nk Š, l…cq ; cr † ˆ exp‰ abs…zi zj †=nc Š, l…qq ; qr † ˆ exp‰ abs…zi zj †=nq Š, l…hq ; hr † ˆ exp‰ abs…zi zj †=nh Š, coecient of variations nk ˆ nc ˆ nq ˆ nh ˆ 1, correlation lengths ak ˆ ac ˆ aq ˆ ah ˆ 0:14. The assumed temperature distribution of shell prior to nosing is shown in Fig. 3. Table 1 presents the expectations of temperatures for the given initial temperatures and assuming nosing speed of 41 mm/s and die penetration 139 mm. In Fig. 4 expectation of temperatures are shown. Tables 2 and 3 show expectations of strain components at two di€erent penetration distances. Figs. 5 and 6 show expected values of strains at two penetration of 139 and 220 mm, respectively. 4. Concluding remarks The study presented show that thermo-mechanical analysis of metal forming with spatially random parameters can be e€ectively carried out using the stochastic ®nite-element technique.

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Fig. 3. Temperature distribution of preform to nosing. Table 1 Expectations of temperatures after nosing Distance from nose tip (mm) 2

Temperature (°C) 10 S.D.

50

100

150

200

6.57 1.61

6.73 1.65

6.51 1.59

6.45 1.53

Fig. 4. Expected values of temperature at die penetration 220 mm. Table 2 Expectations of strains eh ; ez ; er at penetration 139 mm Distance from nose tip (mm)

25

50

75

100

125

Strain eh …%† S.D. Strain ez (%) S.D. Strain er (%) S.D.

)21.1 5.21 8.4 2.13 8.5 2.16

)18.5 4.63 6.9 1.71 7.2 1.74

)11.2 2.72 4.7 1.17 5.1 1.19

)7.4 1.87 2.2 0.57 2.7 0.61

)1.2 0.32 0.7 0.18 1.2 0.31

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377

Table 3 Expectations of strains eh ; ez ; er at penetration 220 mm Distance from nose tip (mm)

50

100

150

200

Strain eh (%) S.D. Strain ez (%) S.D. Strain er (%) S.D.

)44.2 11.09 18.3 4.58 19.1 4.73

)27.4 6.79 7.6 1.87 8.4 2.15

)14.6 3.62 4.2 1.07 4.9 1.21

)4.2 1.08 0.4 0.11 0.9 0.23

Fig. 5. Expected values of strain at penetrations of 139 mm.

Fig. 6. Expected values of strains at penetrations of 220 mm.

Metal forming processes are characterized in many parameters having random values, so stochastic analysis seems to be particularly important in these situations. Random temperature and deformation ®elds in¯uence on ®nal mechanical properties of workpieces.

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The numerical algorithms developed are likely to be accurate and cost-e€ective for complex large-scale systems. Appendix A. Thermal deterministic equations The heat ¯ow equation in it deterministic form is given by kr2 T ‡ qs ˆ c
q

oT ; ot

…A:1†

where T is the temperature, r is the nabel operator, which in the case of axisymmetrical problem in cylindrical coordinate system …r; z† takes the form  rˆ

 o2 1 o o2 ‡ ‡ ; or2 r or oz2

…A:2†

where c is the heat capacity of the material, q is the mass density, k is the thermal conductivity, and t is the ®ne variable, qs is the heat ¯ux generated in the material qs ˆ vsij eij ;

…A:3†

where v is the coecient taking the values from the interval [0.9,1] see [18]. The boundary conditions describe: · heat ¯ux between the die and the workpiece qp ˆ hp …Td

Tw †

…A:4†

where hp is the coecient of heat exchange between the die and the workpiece, Td is the die temperature and Tw is the workpiece temperature; · heat ¯ux from friction forces qt ˆ jf j
…vR †;

…A:5†

where f is the friction force, and vR is the relative velocity between the die and the workpiece; · heat ¯ux taken to the surrounding q0 ˆ qc ‡ qr ;

…A:6†

where qc is the convective heat ¯ux qc ˆ hc …T

T1 †;

…A:7†

where hc is the surface ®lm conductance, and T1 is the surrounding temperature, qr is the radiation heat ¯ux

M. Grzywinski, A. Søu_zalec / International Journal of Engineering Science 40 (2002) 367±383

qr ˆ re…T 4

4 T1 †;

379

…A:8†

where r is the Stefan±Bolzman constant and e is the emissivity. Application of the ®nite-element method to heat ¯ow Eq. (A.1) leads to matrix equation CT_ ‡ KT ˆ F…t†;

…A:9†

where C is the heat capacity matrix, K is the thermal conductivity matrix, F is the thermal load vector, T is the vector of nodal temperatures. The form of matrices C and K and the vector F are standard and can be found in general textbook on ®nite-element analysis. Appendix B. Rigid thermo-viscoplasticity equations We shall brie¯y present in this section the set of equations typical of the ¯ow formulation. Using notation which has already been standardised in the literature [10,11,13], continuum equations describing the rigid thermo-viscoplastic ¯ow are as follows: rij ˆ sij

pdij ;

sij ˆ 2le_ij ; 1 e_ij ˆ …ti;j ‡ tj;i †; 2 e_kk ˆ 0; lˆ e_ ˆ

…B:1†

r0 ‡ …e_ =c†1=n ; 3e_ q 2 e_ e_ ; 3 ij ij

where l is the nonlinear viscosity, r0 the dependent on temperature static tensile yield limit, sij the stress deviator, p the pressure (assumed positive in compression) and c, n parameters of the model dependent on temperature. The weak form of the equilibrium equation reads Z Z Z ^ ^tv d…oX†; fv dX ‡ …B:2† re dX ˆ X

X

oXt

where ^f are body forces and ^t surface tractions. We now introduce the mixed ®nite-element expansion (the standard matrix notation is used) ti ˆ Nq_ i ; _ e_ ˆ Bq;  p ˆ B p;

i ˆ 1; 2; . . . ; I; …B:3†

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where N and B are the shape functions for the velocity and pressure and B is the standard strain rate-velocity matrix while ^

^

q_ ˆ fq_ a g ˆ fq_ 11 ; . . . ; q_ 1I ; q_ 21 ; . . . ; q_ 2I ; q_ N1 ; . . . ; q_ NI g

a ˆ 1; 2; . . . ; N ;

and a ˆ 1; 2; . . . ; N

 p ˆ fpag;

are the nodal velocity and pressure vectors, respectively, with N and N denoting the respective numbers of kinematic and pressure degrees of freedom, I the number of velocity components (i.e. I ˆ 2 or I ˆ 3) and N^ the number of nodes for velocity discretisation (hence N ˆ I
N^ ). The vector q_ is split in vectors q_ i each consisting of subsequent nodal velocity components along the _ we obtain from Eq. (B.2) `i'-axis. By splitting Q, ^f and ^t in the same way as q, Z X

BT r dX ˆ Q; Z

Qi ˆ

X

N^f i dX ‡

…B:4† Z

N^ti d…oX† ‡ Fi ;

oXt

…B:5†

in which the superscript `T' indicates transposition and F is the vector of concentrated loads applied at the nodes. By substituting Eq. (B.1) into it we get Z X

 2lBT B dX q_

Z X

  dX  B T mB p ˆ Q;

…B:6†

i.e., K…l† q_ ‡ KT…p†  pˆQ

…B:7†

with Z K…l† ˆ KT…p†

Z X

lk0 dX ˆ

Z ˆ

X

X

 dX BT mB

2lBT B dX;

…B:8† …B:9†

the row vector m ˆ ‰1 1 1 0 0 0Š converting the total strain into the volumetric component and k0 being dependent only on the geometry. The equilibrium equation (B.2) has to be solved together with the incompressibility condition (B.1)4 written in the weak form as

Z X

M. Grzywinski, A. Søu_zalec / International Journal of Engineering Science 40 (2002) 367±383

 kk dX ˆ 0 Be

which after discretisation reads Z  T T  B m B dX q_ ˆ 0 X

381

…B:10†

…B:11†

or, more compactly K…p† q_ ˆ 0:

…B:12†

Appendix C. Frictional forces

Boundary tractions imposed in metal forming problems are mainly frictional forces. The modeling of these forces is traditionally made in two ways: by means of a friction coecient and by means of a friction factor [17]. A directly applied friction coecient leads to non-symmetric matrices and is not commonly used in bulk deformation metal forming. A friction factor is de®ned by the equation s ˆ mky ;

…C:1†

where s-shear friction stress, ky is the shear yield stress of the material and m is the friction factor. In considering the boundary conditions a major problems is faced when a point of reverse relative velocity between workpiece and die exists, and its position is not known a priori. This happens, for instance, in ring compression with high friction. The orientation of the friction force changes at that point, so an abrupt jump in its value shows up. In such a case we can use the equation for the friction force     2 vR 1 tan ; …C:2† f ˆ mky P ajvD j where m is the friction factor …0 6 m 6 1†, ky is the shear yield stress, vR is the magnitude of relative velocity between die and workpiece, a is a constant several orders of magnitude less than the die velocity, and jvD j is the magnitude of die velocity. The introduction of frictional force requires linearization in the corresponding surface integrals. The derivations produce very cumbersome expressions, which can be found in [12]. There are two other methods referred to in the literature for taking into account frictional forces which explore the fact that a nonlinear system in which only essential boundary conditions (velocities) are much better conditioned than one with suppressible boundary conditions (forces).

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The ®rst one, used by Zienkiewicz, Jakin and Onate [11], places a very thin layer of elements in the boundary, with the die side nodes ®xed and having material behavior such that the yield stress is a certain fraction ± ``coecient of friction'' ± of the hydrostatic pressure of the deforming material in contact with the layer. The reasoning behind this approach is that the layer is so thin that all of its deformation would be shear, thus the shear strength transmitted would be the shear yield strength of the material and the deviatoric normal stresses are negligible when compared to the hydrostatic stress. In the type of problems considered in this work, a check on their values indicates that such things do not happen frequently, so this approach did not work. A second approach, used by Hartley et al. [14,15], considers a thin layer of elements too, but its behavior is such that the sti€ness is the sti€ness of the material adjacent, multiplied by a function of the ``friction factor'' m bˆ

m 1

m

:

…C:3†

When m ˆ 0, no resistance to relative motion between workpiece and die exists, and when m ˆ 1 the workpiece sticks to the die. However, experiments have shown that only for high temperatures does m ˆ 1 correspond to a sticking condition; in addition, if for instance m is made 0.5, the behavior of the whole body is similar to a little extended body, sticking to the die, which is quite di€erent from the sliding usually observed. References [1] E. Vanmarcke, Random Fields, Analysis and Synthesis, second ed., MIT Press, Cambridge, MA, 1984. [2] S. Ang, W.H. Tang, Probability Concepts in Engineering, Planning and Design, vol. I, Basic Principles, Wiley, New York, 1975. [3] F. Ma, Approximate analysis of a class of linear stochastic systems with colored noise, Int. J. Eng. Sci. 24 (1986) 19±34. [4] H.J. Larson, in: Probabilistic Models in Engineering Science, vols. 1&2, Wiley, New York, 1979. [5] H. Contreras, The stochastic ®nite-element method, Comput. Struct. 12 (1980) 341±348. [6] M.A. Lawrence, A basic random variable approach to stochastic structural analysis, Ph.D. Thesis, Department of Civil Engineering, University of Illinois at Urbana-Champaign, Urbana-Champaign, Illinois, 1986. [7] S. Nakagiri, T. Hisada, K. Toshimitsu, Stochastic time-history analysis of structural vibration with uncertain damping, ASME PVP 93 (1984) 109±120. [8] K. Liu, T. Belytschko, A. Mani, Probabilistic ®nite elements for nonlinear structural dynamics, Comput. Meth. Appl. Mech. Eng. 56 (1986) 61±81. [9] K. Liu, T. Belytschko, A. Mani, Random ®eld ®nite elements, Int. J. Numer. Meth. Eng. 23 (1986) 1831±1845. [10] O.C. Zienkiewicz, The Finite Element Method, McGraw-Hill, New York, 1977. [11] O.C. Zienkiewicz, P.C. Jain, E. Onate, Flow of solids during forming and extrusion: some aspects of numerical solutions, Int. J. Solids Struct. 24 (1978) 15±38. [12] W.R.D. Wilson, Friction and lubrication in bulk metal-forming process, J. Appl. Metalworking 1 (1979) 7±14. [13] C.C. Chen, S. Kobayashi, Plastic ®nite-element analysis of ring compression, applications of numerical methods to forming processes, ASME AMD 28 (1978) 163. [14] P. Hartley, C.E.N. Sturgess, G.W. Rowe, Friction in ®nite-element analysis of metalforming process, Int. J. Mech. Sci. 21 (1979) 301±307. [15] P. Hartley, C.E.N. Sturgess, G.W. Rowe, A prediction of the in¯uence of friction in the ring test by the ®niteelement method, in: 7th NAMRC, Ann Arbor Science, Ann Arbor, MI, 1979, pp. 151±158.

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[16] Y. Tomita, A rigidplastic ®nite-element perturbation method for the e€ective prediction of the in¯uence of the change in parameters in the constitutive equation upon deformation behaviour, Int. J. Mech. Sci. 24 (1982) 711± 717. [17] M. Gierzy nska, Friction, Wear and Lubrication in Metal Forming Processes, PWN, Warsaw, 1983 (in Polish). [18] A. Søu_zalec, Introduction to Nonlinear Thermomechanics, Springer, Berlin, 1992.