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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2008; 74:1329–1343 Published online 9 October 2007 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.2214 Thermoplasticity with diffusion in welding problems Andrzej Sluzalec , Technical University of Czestochowa, 42-200 Czestochowa, ul. Dabrowskiego 69, Poland SUMMARY A thermomechanical model for the analysis of thermoplasticity with a diffusion term is presented. This model is defined in infinitesimal strain thermoplasticity framework. Thermal hardening is coupled with a diffusion process. The example presents thermodiffusion process in titanium friction welding parts. The migration of hydrogen in the process of thermoplastic deformation is described. Copyright 2007 John Wiley & Sons, Ltd. Received 19 February 2007; Revised 5 September 2007; Accepted 7 September 2007 KEY WORDS: thermomechanical process; thermoplasticity; thermodiffusion; friction welding; titanium alloys 1. INTRODUCTION The problems of coupling between thermoplasticity and chemical diffusion appear in some tech- nological problems in aerospace and nuclear industry. The purpose of this paper is to discuss the coupled problems of thermoplasticity and diffusion in a general way and show some numerical solutions in the example of friction welding of titanium alloys that are widely used in the industry. The paper is divided in two sections. In the first one, a theory of thermoplasticity with a diffusion term is discussed. In the second one, an illustration of the problem in the example of friction welding is presented. Suppose that the thermomechanical state of each material element is uniquely defined by the values of a finite set of state variables. Such a phenomenological theory is, of course, restricted to a limited class of materials on the one hand and to processes running not too far from thermodynam- ical equilibrium on the other hand. A thermodynamical process starts in the initial state of the body, which is characterized by the initial configuration and the initial thermodynamical state of each material element. The initial state is described by the prescribed thermodynamical boundary condi- tions, the prescribed body forces and energy sources acting inside and on the surface of the body. Correspondence to: Andrzej Sluzalec, Technical University of Czestochowa, 42-200 Czestochowa, ul. Dabrowskiego 69, Poland. E-mail: [email protected], [email protected] Copyright 2007 John Wiley & Sons, Ltd.

Thermoplasticity with diffusion in welding problems

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Page 1: Thermoplasticity with diffusion in welding problems

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2008; 74:1329–1343Published online 9 October 2007 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.2214

Thermoplasticity with diffusion in welding problems

Andrzej Sluzalec∗,†

Technical University of Czestochowa, 42-200 Czestochowa, ul. Dabrowskiego 69, Poland

SUMMARY

A thermomechanical model for the analysis of thermoplasticity with a diffusion term is presented. Thismodel is defined in infinitesimal strain thermoplasticity framework. Thermal hardening is coupled with adiffusion process. The example presents thermodiffusion process in titanium friction welding parts. Themigration of hydrogen in the process of thermoplastic deformation is described. Copyright q 2007 JohnWiley & Sons, Ltd.

Received 19 February 2007; Revised 5 September 2007; Accepted 7 September 2007

KEY WORDS: thermomechanical process; thermoplasticity; thermodiffusion; friction welding; titaniumalloys

1. INTRODUCTION

The problems of coupling between thermoplasticity and chemical diffusion appear in some tech-nological problems in aerospace and nuclear industry. The purpose of this paper is to discuss thecoupled problems of thermoplasticity and diffusion in a general way and show some numericalsolutions in the example of friction welding of titanium alloys that are widely used in the industry.

The paper is divided in two sections. In the first one, a theory of thermoplasticity with a diffusionterm is discussed. In the second one, an illustration of the problem in the example of frictionwelding is presented.

Suppose that the thermomechanical state of each material element is uniquely defined by thevalues of a finite set of state variables. Such a phenomenological theory is, of course, restricted to alimited class of materials on the one hand and to processes running not too far from thermodynam-ical equilibrium on the other hand. A thermodynamical process starts in the initial state of the body,which is characterized by the initial configuration and the initial thermodynamical state of eachmaterial element. The initial state is described by the prescribed thermodynamical boundary condi-tions, the prescribed body forces and energy sources acting inside and on the surface of the body.

∗Correspondence to: Andrzej Sluzalec, Technical University of Czestochowa, 42-200 Czestochowa, ul. Dabrowskiego69, Poland.

†E-mail: [email protected], [email protected]

Copyright q 2007 John Wiley & Sons, Ltd.

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1330 A. SLUZALEC

The process is governed by the field equations (balance equations) and the constitutive law of thematerial. We focus our attention on the constitutive law that governs the local thermomechanicalprocess within the thermodynamical state space. In this paper the thermoplasticity with thermod-iffusion appearing in friction welding process of titanium and its alloys is considered. Such anexample is very convenient for the analysis, because thermodiffusion in friction-welded joints maybe analyzed as a one-dimensional case.

Titanium and its alloys may experience a progressive embrittlement leading to catastrophicfailure. Such embrittlement is a consequence of hydrogen migration and the accumulation of brittletitanium hydrides in the presence of a sharp alloy composition gradient near a weld fusion line(for instance, see [1]). Theoretical considerations are conducted to determine the combined effectson thermodiffusion and thermoplasticity considering gradients in interstitial concentration, solventcomposition, stress and temperature. The example of thermodiffusion process in titanium frictionwelding parts is presented, where the diffusion process is coupled with thermoplastic deformation.It should be noted that there exist several papers on thermoplasticity without a diffusion term.They present models that are similar, but they consider only thermoplastic part of the problem[2–5]. Thermoplasticity with diffusion and its application in welding have not been considered inliterature. There is no specific experimental verification of the model carried out by the author.This problem is left open.

2. THE ENERGY EQUATION

Consider any domain � consisting of a base material and a diffusing material. The first law ofthermodynamics together with the kinetic energy theorem yields the relationship expressing theinternal energy variation de during the time interval dt :

de

dt+edivv=−div(ecu)+rde

dt+r−divq (1)

where r is the stress tensor, e is the strain tensor, q is the heat flow vector, r represents the volumerate density of the heat provided to � by possible external heat sources, ec is the internal chemicalpotential energy, v is the material velocity and u is the diffusion velocity. Let sc be the entropy ofthe diffusing material. The total entropy s reads

d

dt

∫�s d�=

∫�

[�s�t

+div(sv)+div(scu)

]d� (2)

The volume integral must be non-negative for any subsystem �. Hence, we obtain

ds

dt+s divv+div(scu)+div

qT

− r

T�0 (3)

where T is the absolute temperature.Multiplying Equation (3) by d�, the above equality becomes

d

dt(s d�)+div(scu)d�+

(div

qT

− r

T

)d��0 (4)

which expresses the second law for the elementary material system d�.

Copyright q 2007 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2008; 74:1329–1343DOI: 10.1002/nme

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THERMOPLASTICITY WITH DIFFUSION IN WELDING PROBLEMS 1331

Let �c be the free enthalpy of the chemical potential defined by

�c=ec−T sc (5)

where sc is the entropy of chemical potential.Now, let � be the free volume energy defined by

�=e−T s (6)

Using (4) we have

rde

dt−s

dT

dt− d�

dt−�divv−scugradT −div(�cu)− q

T·gradT�0 (7)

and using the identity

div(�cu)=u·grad�c+�c divu (8)

we obtain

rde

dt−�c divu−s

dT

dt− d�

dt−�divv−u[grad�c+sc gradT ]− q

T·gradT�0 (9)

Equation (9) is the fundamental inequality extended to thermodiffusion phenomena.

3. IDENTIFICATION OF DISSIPATIONS

The left-hand side of inequality (9) is the dissipation per unit initial volume d� and will be denotedby �.

The second law requires the dissipation � and the associated internal entropy production �/Tto be non-negative:

�=�1+�2+�3�0 (10)

where

�1=rdedt

−�c divu−sdT

dt− d�

dt−�divv (11)

is the intrinsic volume dissipation described in small deformation theory:

�2 = − qT

·gradT (12)

�3 = −u[grad�c+sc gradT ] (13)

where �2 is the thermal dissipation associated with heat conduction and dissipation associatedwith mass transfer.

If we define �m =�1+�3, then the energy equation (1) can be rewritten as

T

[ds

dt+s divv+div(scu)

]=r−divq+�m (14)

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1332 A. SLUZALEC

4. EQUATIONS OF STATE

Consider free energy as a function of variables T , �i j , C , j, where T denotes the temperature,C the concentration and �i j the strain components. The hardening state is characterized by theinternal variable j. These variables constitute a set of state variables which characterize the stateof the system. If we apply the Helmholtz hypothesis that it is always possible to find a set ofvariables which is normal with respect to absolute temperature, i.e. in the real evolutions of theelementary system, variations of a particular state can occur independently of the variations ofthe other variables of the set. The free energy volume density � will depend locally on the statevariables, but not on their rates or spatial gradients. Thus, it depends neither on gradT nor on u.Letting gradT =u=0, the non-negativeness of intrinsic dissipation �1 is derived independently ofthe non-negativeness of total dissipation �:

�1=rdedt

+�cdC

dt−s

dT

dt− d�

dt�0 (15)

The above inequality results from the second law.The non-negativeness of the intrinsic dissipation (15) yields(

r− ��

�e

)de

dt+

(�c−

��

�C

)dC

dt− ��

�j· �j�t

−(s− ��

�T

)dT

dt�0 (16)

where

��

�j· djdt

= ��

�j· j (17)

Then the non-negativeness of the intrinsic dissipation yields

s=−��

�T, r= ��

�e, �c=− ��

�C(18)

The above equations yield the symmetry relationships:

�s��i j

=−��i j�T

,�s�C

=−��c

�T,

��i j��kl

= ��kl��i j

,��c

��i j=−��i j

�C(19)

The thermodynamic states of a thermoplastic material are characterized by external variables T ,�i j , C and internal variables �pi j , plastic strain, and j.

We have

�=�(T,�i j ,C,�pi j ,j) (20)

on account of the local state postulate. The variables appearing in the bracket constitute a setof state variables which characterize the state of an open system, whether at equilibrium or inevolution.

As stated above, the state equations read

s=−��

�T, r= ��

�e, �c=− ��

�C, sc=−��c

�T(21)

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THERMOPLASTICITY WITH DIFFUSION IN WELDING PROBLEMS 1333

The above equations are based on the normality of external variables T , �i j and C and with regardto the whole set of state variables, there are always actual evolutions for which each of theseexternal variables varies independently of the other.

The expression of free energy � with respect to variables T , �i j , C and �pi j can be presented as

� = r0(e−ep)−s0T + 1

2(e−ep)C(e−ep)−CB(e−ep)

−�A(e−ep)− 1

2

c�T0

�2−T scC− 1

2

M

T0C2+U (j) (22)

where �=T −T0, U (j) is the frozen energy due to hardening and c� is the volume heat capacity.Then, the thermoelastic state equations read

r = r0+C(e−ep)−BC−A�

s = s0+Csc+A(e−ep)+ MC

T0+ c��

T0

(23)

The energy U (j) that will be identified as the frozen energy due to hardening is assumed to beindependent of the external state variables e and C .

The thermodynamic force to be associated with the rate of the hardening variables is thethermodynamic hardening force n defined by

n=−�U�j

(24)

The above equation can be inverted in the form

j=−�U∗

�n(25)

where U∗(n) is the Legendre–Fenchel transform of U (j):

U∗ =−nj−U (26)

Therefore, the intrinsic dissipation �1 is

�1=�p− dU

dt=�p+n· dj

dt(27)

where �p is the plastic work rate defined by

�p=rdep

dt(28)

The absence of thermal hardening implies that the hardening force n is used to representthe evolution of the elasticity domain. To give an account of thermodiffusion hardening effects,

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1334 A. SLUZALEC

the free energy � is now expressed in the form:

�=�(�,�,C,ep,j)=C(�c−�sc)+�(e−ep,C,�)+U (j,�,C) (29)

By (24) and (21)

n=−��

�j, s=−��

��(30)

where the hardening force n depends on temperature variation �=T −T0 through the function U .If the temperature variation � and concentration variation C are assumed to be small, the functionU (j,�,C) can be expressed in the form:

U (j,�,C)=U (j)−�S f1(j)−CS f2(j) (31)

The function U (j) can be identified as the frozen-free energy, since it is the free energy recoveredafter restoring the initial temperature T0 and stress r0. Hence, obtain

n = −�U�j

+��S f1

�j+C

�S f2

�j

s = Csc− ��

��− ��

�C+S f1(j)+S f2(j)

(32)

In the loading space {r}, the elasticity domain is defined by the inequality

f = f (r,n)�0 (33)

Then, even if no plastic loading occurs and consequently the hardening variables j retain its presentvalue, and due to a temperature variation d� and concentration variation dC , the hardening forcen may change and hence the elasticity domain as well.

This phenomenon corresponds to a hardening which will be called the thermodiffusion hard-ening.

In the case of a zero frozen-free energy U (j)=0, only this kind of a hardening occurs. It shouldbe noted that an inverse temperature variation −d� and concentration variation −dC restoresthe previous elasticity domain. The second relationship (32) shows that after restoring the initialtemperature T0, stress r0 and concentration C , there is an unrecovered change in entropy, yieldingthe frozen entropy terms S f1(j) and S f2(j).

The flow rule can be then expressed as

(dep,dj)=d�h(r,n), d��0 with

{d��0 if f =0 and d f =0

d�=0 if f <0 or d f <0(34)

where h(r,n) represents the set of directions for the increments (dep,dj).The increment dj of the hardening variable is expressed in the form

dj=d�hj(r,n) (35)

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THERMOPLASTICITY WITH DIFFUSION IN WELDING PROBLEMS 1335

where hj represents the direction actually realized among permissible directions for dj, and theplastic multiplier d� and the hardening modulus H are expressed in forms:

d�= dj f

H, H =−� f

�n· �n�j

· djd�

= � f

�n· �

2U

�j2·hj(r,n) (36)

The expression for dj f is then given as

dj f =−� f

�rdr+ � f

�n· �S

f1

�jd�+ � f

�n· �S

f2

�jdC (37)

In order to express the loading function f in terms of �, C and j rather than of n, substituteexpression (32) of n to Equation (33).

If d f =0, then Equations (34) and (36) give

f (r,�,C,j+dj)= f (r,�,C,j)+ � f

�j·dj= f (r,�,C,j)−H d� (38)

Consider an initial loading state defined by the stress r, temperature � and concentration C , suchthat the plastic criterion is satisfied f (r,�,C,j)=0. If plastic loading occurs, d f =0, the hardeningvariable j undergoes the infinitesimal change dj. By Equation (38) and since H d� is positive forpositive hardening, the initial loading point (r) satisfies f (r,�,C,j+dj)<0. In the loading space{r}, it lies inside the new elasticity domain associated with the new hardening state defined byj+dj and with temperature T and concentration C recorded before the plastic loading occurred.Softening is defined by a negative hardening modulus H , and plastic loading occurs for negativevalues of dj f . Then after plastic loading, the initial loading point lies outside the new elasticitydomain. Ideal plasticity is defined by a zero hardening modulus H , while the plastic multiplier d�remains undetermined during plastic loading.

5. DISCRETIZATION

Assuming the incremental nature of the equations, the time is discretized by

tn =n ·�t (39)

where n is the time step and t is the time variable. Let

Sn−1=(rn−1,wn−1,�n−1,Cn−1,�n−1,sn−1) (40)

be the known solution to the problem at time step n−1, where w is the displacement field.The solution Sn at time tn is expressed as

Sn = Sn−1+�n S where �n S=(�nr,�nw,�n�,�nC,�n�,�ns) (41)

The problem discretized with respect to time satisfies the discretized momentum equation

div�nr=0 (42)

the compatibility equation

2�ne=grad�nw+Tgrad�nw (43)

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1336 A. SLUZALEC

and the discretized boundary conditions

�nrn = �nt∗ on ��t

�nw = �nw∗ on ��u(44)

In thermoplasticity the discretized state equation can be expressed as

�nr=C0(�ne−�nep−�ne

C−�ne�) (45)

For a hardening material, the state equation between hardening variable j and hardening force nis added to the above equations

�nj=−H ·�nn (46)

The flow rule for an ideal plastic material is discretized in an implicit way:

�nep=��

� f

�r(rn−1+�nr), ���0 if f (rn−1+�nr)=0 (47)

In plasticity an iterative scheme needs to be applied because the problem of determining thefinite increment solution �n S=(�nr,�nw,�n�,�nC,�n�,�ns) is non-linear. In time-discretizedproblems, the solution �n S is obtained by successive iterations within the same time step. �n S isdetermined as the limit of a series of function �n,k S as the number k of iterations tends to infinity.The iterative field equations are

div�n,k r= 0 (48)

2�n,ke= grad�n,ke=grad�n,kw+Tgrad�n,kw (49)

The constitutive equation is

�n,kr=C0(�n,ke−�n,k−1ep−�n,k−1e

�−�n,k−1eC) (50)

with initial conditions

�n,k−1ep with �n,0e

p=0 (51)

The finite stress �n,kr and the plastic finite increments �n,kep are determined and are combinedthrough the discretized constitutive equation (50). The flow rule for the ideal standard plasticmaterial is

�n,kep=��

� f

�r(rn−1+�n,kr), ���0 if f (rn−1+�n,kr)=0 (52)

A variational formulation of the linear problem is needed to apply the iterative method. Rewritethe discretized equation (50) in the form

�n,kr=�W ∗

n,k

�(�n,ke)(53)

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THERMOPLASTICITY WITH DIFFUSION IN WELDING PROBLEMS 1337

where W ∗n,k(�n,ke) is the finite increment potential defined by

W ∗n,k(�n,ke) = 1

2 (�n,ke−�n,k−1ep−�n,k−1e

C−�n,k−1e�)

×C0(�n,ke−�n,k−1ep−�n,k−1e

C−�n,k−1e�) (54)

The variational approach can be applied to the problem. The solution �n,kw is the unique onewhich, among all solutions kinematically and statically admissible, minimizes with respect to�n,kw the functional. Jn,k(�n,kw) is defined by

Jn,k(�n,kw)=∫

�W ∗

n,k(�n,ke)d�−∫

��t

�nt�n,kwd(��) (55)

The solution (�n,kw) satisfies∫�

�e[C0(�n,ke−�n,k−1ep−�n,k−1e

C−�n,k−1e�)]d�−

∫��t

�nt�wd(��)=0 (56)

for any variation �e kinematically admissible with zero boundary conditions imposed on ��n .Rewrite the discretized state equations (50) in the form

rn,k = rn−1+�n,kr (57)

�n,kr=C(�n,ke−�n,k−1ep−�n,k−1e

C−�n,k−1e�) (58)

The inversion of the relationship (58) is

�n,ke=C−1 ·�n,kr+�n,k−1ep+�n,k−1e

C+�n,k−1e� (59)

At the first iteration (i.e. for k=1) �n,k−1=0ep is equal to zero, and the field rn,1 is equal torn−1+C ·�n,1e.

Using Equation (53) we have

�n,kep=��

� f (rn,k)

�rn,k, f (rn,k)�0, ���0, �� f (rn,k)=0 (60)

The discretized state equations (58) and (60) give

rn,k−rn,k−1=C ·�n,ke−C ·�n,k−1e (61)

The thermal equations in its discretized form are

Tn

[sn−sn−1

�t+sn divvn+div((sc)nun)

]=rn−divqn+(�m)n (62)

sn =s0+Cn(sc)n+A(e−(ep)n)+ MnCn

T0+ (ce)n�n

T0(63)

where the subscript n denotes the time step.

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1338 A. SLUZALEC

6. SOLUTION PROCEDURE

Consider the virtual work equation for a finite element assemblage for a thermoplastic materialmodel at time t+�t (step n+1). The mechanical equations are of the form∫

�BTLrn+1 d� =Rn+1 (64)

rn+1 =Cn+1(en+1−epn+1−e�n+1−eCn+1) (65)

epn = �nDrn (66)

e�n+1 =A(�n+1−�R)d (67)

eCn+1 =B(Cn+1−CR)d (68)

en+1 =BUn+1 (69)

where B is the total strain–displacement transformation matrix, Un+1 is the nodal point displace-ment vector, Rn+1 is nodal point external load vector, D is the deviatoric stress operator matrix,dT is [1,1,1,0,0,0] and �n is given by Equations (36) and (37). Substituting Equations (65) and(69) into Equation (64) we obtain

KUn+1=Rn+1+∫

�BTCn+1(e

p+e�+eC)d� (70)

where

K=∫

�BTCn+1Bd� (71)

is the so-called elastic stiffness matrix.The thermodiffusion matrix equations are of the form

Ln−1Tn+Kn−1Tn+Rn−1Cn−1+En−1Cn−1+Fn−1=0 (72)

where

L=∫

cHHT d�

K=∫

gradHk gradHT d�

R=∫

MNNT d�+∫

1

TscNNT d�

E=∫

1

�sc divvNNT d�

F=∫

Hqv d�+∫

H�m d�−∫

Hqnd(��)+∫

1

�div(scu)d�+

∫1

�A(e−ep)divvHd�

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THERMOPLASTICITY WITH DIFFUSION IN WELDING PROBLEMS 1339

T is the vector of nodal temperatures and C is the vector of nodal concentrations and H and Nare shape functions for temperature and concentrations, respectively.

The following simple procedure is applied to solve the problem described. First, the mechanicalequations are solved for the values of T and C for step n−1. After solving mechanical equations,thermodiffusion equations are solved for obtained mechanical field variables r and e for time step n.Thermodiffusion equations are solved for T for time step n with assumed C for step n−1. Theintegration is done by a Newton–Raphson method of the whole system of equations.

7. NUMERICAL EXAMPLE

Experimental tests indicate that effects of thermodiffusion in friction welding joints are significantand this phenomenon should be properly modeled. The titanium and its alloys may experience aprogressive embrittlement leading to catastrophic failure. Such embrittlement is a consequence of

Figure 1. Friction welding process.

Table I. Thermal properties of BT6.

Temperature (◦C) 20 300 500 700 900 1100 1300 1600

Thermal conductivity k (W/mK) 46 52 60 74 70 67 67 67Specific heat c (J/kgK) 600 580 570 560 550 540 530 520

Table II. Thermal properties of CP.

Temperature (◦C) 20 300 500 700 900 1100 1300 1600

Thermal conductivity k (W/mK) 262 273 280 291 302 310 320 324Specific heat c (J/kgK) 1760 1750 1740 1730 1720 1710 1710 1700

Table III. Mechanical properties of BT6.

Temperature (◦C) 20 300 500 900 1100 1300 1600

Elastic modulus E (GPa) 200 183 148 148 148 148 148Poisson ratio � 0.25 0.26 0.32 0.32 0.32 0.32 0.32Flow stress (MPa) 1010 790 580 460 195 95 80

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1340 A. SLUZALEC

Table IV. Mechanical properties of CP.

Temperature (◦C) 20 300 500 900 1100 1300 1600

Elastic modulus E (GPa) 200 183 148 148 148 148 148Poisson ratio � 0.25 0.26 0.32 0.32 0.32 0.32 0.32Flow stress (MPa) 1010 790 580 460 195 95 80

Figure 2. Temperature field on the element surface for time 3 s.

hydrogen migration and the accumulation of brittle titanium hydrides in the presence of a sharpalloy composition gradient near a weld fusion line [1, 6].

The numerical and experimental studies of friction welding without diffusion are considered inthe author’s works [7, 8]. The simulations of thermo-elastic–plastic analysis with thermodiffusionare carried out on the basis of finite element code THERMO-PLAST [9] which has been modifiedto take into account the thermodiffusion effects. Let us consider a titanium alloy BT6 which has

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THERMOPLASTICITY WITH DIFFUSION IN WELDING PROBLEMS 1341

Figure 3. Temperature distribution along the cross section: (A) at the place of contact; (B) at the distance4mm; and (C) at the distance 10mm.

Figure 4. The radial stress distribution for t=2s (deformation phase).

Figure 5. The axial stress distribution for t=2s (deformation phase).

6% aluminum and 4.5% vanadium which are simulated to be welded with z commercially puretitanium (CP). A principle of friction welding process is shown in Figure 1. It is assumed that thevariable C introduced in the above sections is the concentration of hydrogen.

It is assumed third that at the place of abutment the heat source is given by the expressionq=∫

�A�r d�, where � is the axial pressure (40N/m2), is the coefficient of friction (0.5),

is the angular speed 1460 rpm, r is the radius and �A is the surface upon which the heat rate acts.The assumed thermal and mechanical parameters are listed in Tables I–IV. The following material

Copyright q 2007 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2008; 74:1329–1343DOI: 10.1002/nme

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1342 A. SLUZALEC

Figure 6. The tangential stress distribution for t=2s (deformation phase).

Figure 7. Initial hydrogen distribution in the welding parts for time t=0s (heating phase).

Figure 8. Hydrogen distribution in the time of deformation process.

densities have been assumed in the simulation: �BT6=7500kg/m3 and �CP=1000kg/m3. Theother material parameters assumed for simulation are A=12×10−6 1/N, B=12×10−10 1/% andM=6×10−4. The initial hydrogen distribution in BT6 rod is assumed to be 100 ppm and in theCP rod is 5 ppm.

In modeling the friction welding process, two phases of the process are considered: first theheating phase 3 s and then the deformation phase 2 s. It is assumed in the heating phase that twoworkpieces subjected to welding undergo thermal loading without any external mechanical loading.

Copyright q 2007 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2008; 74:1329–1343DOI: 10.1002/nme

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THERMOPLASTICITY WITH DIFFUSION IN WELDING PROBLEMS 1343

The hydrogen diffusion is neglected in this phase of process. In the deformation phase, thermo-elastic–plastic equations with a diffusion term are solved. The temperature field in workpiecesfor the time 3 s is given in Figures 2 and 3. The distribution of stresses in workpieces (CP) indeformation phase for time 2 s is given in Figures 4–6. The assumed initial hydrogen distributionfor time 0 s is presented in Figure 7, and Figure 8 shows the changes of hydrogen concentrationin the deformation process.

8. CONCLUDING REMARKS

During friction welding, temperature, stress, strain and their variations govern welding parametersand knowledge of them helps to determine optimum welding parameters as well as improve thedesign and manufacture of welding machines [8, 10].

The variation in temperature and deformation during inertia welding is systematically investi-gated and analyzed.

For many technically important welds appearing in civil and mechanical structures, models forthe determination of hydrogen distribution and its effects should be analyzed. Combinations ofhydrogen, stress and compositions history suggest complicated situation of hydrogen migration.Determinations of such combinations are under way in this research.

The novelty of the recent investigation presented in this paper is the study of coupled problemsof thermoplasticity and diffusion. These problems are discussed in this paper in a general way.The numerical results are shown in the example of friction welding of the titanium alloys that arewidely used in the industry.

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6. Shewmon PG. Diffusion in Solids. McGraw-Hill: New York, 1963.7. Sluzalec A. Theory of Metal Forming Plasticity, Classical and Advanced Topics. Springer: Berlin, 2004.8. Sluzalec A. Thermal effects in friction welding. International Journal of Mechanical Sciences 1990; 32(6):

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University of Czestochowa, 1996.10. Sluzalec A. Theory of Thermomechanical Processes in Welding. Springer: Berlin, 2005.

Copyright q 2007 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2008; 74:1329–1343DOI: 10.1002/nme