COMMUNICATIONS IN NUMERICAL METHODS IN ENGINEERING, VOl. 11, 69-72 (1995)

MODIFIED CALCULATION OF THE DISSIPATION TERM IN THERMOPLASTIC FINITE ELEMENT EQUATIONS

LASZLO SZABb* AND IMRE VARGAt Department of Applied Mechanics, Technical University of Budapest, H-11 I I Budapest,

Miiegyetem rkp. 5 , Hungary

SUMMARY In the paper, a simple modification of the internal dissipation term in coupled thermoplastic finite element equations is presented. A modified form of the heat capacity and the thermomechanical coupling matrices are derived. These modifications are based on a decomposition of the internal dissipation term into two parts, which depend on the total strain rate and the rate of temperature change, respectively.

KEY WORDS finite element method; thermoplasticity; internal dissipation

INTRODUCTION

A method for the finite element analysis of coupled thermoplasticity has been presented by Hsu' and applied in metal forming processes.' Based on HSU'S work' a theoretical and numerical investigation of the internal dissipation factor was discussed by SYuialec. In these works, the following governing equations have been derived for the coupled thermoplasticity:

(poCP + +) TT dQ - 1 TBijCij dQ - s DT dQ - 1 QT dQ n n n n

= S kijT,iT,j dQ+ Tqini d(aQ) (2) a an

where C& is the tensor of elastic-plastic moduli, hij is the strain rate tensor, yij is the generalized thermal moduli, T is the temperature, p0Cp is the specific heat, 7 is a specific coefficient, p i j is the modified thermal modulus tensor, D is the internal dissipation factor, kij is the thermal conductivity matrix, Q is the heat generation, qi is the heat flux, ni is the normal to the boundary aQ, V i are the rate of displacement vectors, and f i , p i are the rate of the body force and surface traction vectors, respectively. The superposed dot denotes the material time derivative or rate. The tensors Ci:pkl, yij, Bij and the coefficient + are given in detail in the Appendix.

* Present address: McGill University, Department of Mining and Metallurgical Engineering, 3450 University Street, Montreal, Quebec, Canada H3A 2A7. t On leave from Faculty of Mechanical Engineering, Slovak Technical University, Faculty of Mechanical Engineering 812 31 Bratislava, Slovakia.

CCC 0748-8025/95/010069-04 0 1995 by John Wiley & Sons, Ltd.

Received 8 October 1993 Revised 11 JuQ 1994

70 THERMOPLASTIC FINITE ELEMENT EQUATIONS

In equation (2) the internal dissipation factor can be calculated in the form'

D = v ~ i j i f j (3) where aij is the stress tensor, €fj is the rate of the plastic strain tensor and Y is a variable factor.

Using the finite element discretization scheme as described by Hsu' and S h ~ i a l e c ~ , ~ for (1) and (2), the following finite element equations can be obtained:

(4)

(5 )

where K, is the element mechanical stiffness matrix, MT is the element thermal stiffness matrix, L is the rate of the mechanical load vector, KT is the element conductivity matrix, Q is the element thermal load vector and D is the element dissipation vector. The vectors u(t) and T ( t ) are the respective nodal displacements and temperature in the discretized solid. In addition, the element heat capacity matrix is defined by

KuU(t) + MrT(t) = L(t)

M,U(t) + C T ( t ) = K T T ( ~ ) + Q ( t ) + D

C = 1 b(poCp + r)bT dO n c

and the thermomechanical coupling matrix is given by

where b(x), the interpolation functions, are for the temperature field ( T ( t ) = T(t)b(x)), and the matrices G(x) and h are used in the discretization of the strain field ( E ( x ) = G(x)hu(t)). Here x is the co-ordinate vector, t is the time variable and the superscript T denotes transpose.

In the proposed finite element equation ( 5 ) , the internal dissipation term is computed as an external 'load' vector. In this case, because the internal dissipation is calculated from the plastic strain rate (see equation (3)), which is unknown at the beginning of the time step, the computational algorithm becomes more complicated.

In this paper, a simple modification of the finite element equation (5) is presented. A key step in this development is the derivation of a new expression for the dissipation term. In this expression the internal dissipation is a function of two variables: the rate of temperature change, and the total strain rate, which corresponds to the rate of displacement. Thus, the internal dissipation term can be added to the left side of equation (5). It means that a modified form of the matrices C and Mu must be considered.

MODIFICATION OF MATRICES Mu AND C

The rate of the plastic strain tensor, in the case of the associated flow rule, can be given by

where F(aij, x, T ) = 0 is the yield function. Here, x is the work-hardening parameter. The proportionality factor can be derived from the consistency condition F = 0 as follows:

where crij is the linear thermal expansion coefficient tensor, C : j k [ is the fourth-order isotropic elasticity tensor, and S is given in the Appendix.

L. SZABO AND I . VARGA 71

Substituting (8) into (3) and using (9) for D leads to the following equation:

D = D$)Cij + D'T'j-

where

D$' = V a k l ( C e - l ) k / r s C % i j

Now, by substituting the above-defined D into (2) yields

= S ki jTiTj dQ + 5 Tqini d(aQ) (13) D an

Then, from the above expression, by following the usual discretization procedure, equation (2) can be rewritten as

(14) MUU(t) + eT(t) = KrT( t ) + Q(2)

where the modified matrices eu and M are given by

Mu = - ( i b(p + D(C))TG dQ)h n,

Finally, we conclude that the proposed modification can be easily incorporated into any finite element code. When it is used in thermoelastic-plastic finite element analysis, the effect of the internal dissipation factor may be analysed more accurately.

ACKNOWLEDGEMENT

This research was supported by the OTKA under contract OTKA 5-721. The authors gratefully acknowledge this support.

APPENDIX

In equation (1) the tensors Ciegkl and y;j are considered in the thermoelastic-plastic constitutive equation 1 - 3

(17) u.. 1J - - C'P r j k k k l + ylji'

72 THERMOPLASTIC FINITE ELEMENT EQUATIONS

and

Equation (2) is based on the 'coupled heat conduction equation' (see Hsu, ' p. 259, equation

(22)

(9.14))'

(kT;) , ; + Q + BijT&?j + D - p ~ C p f ' = 0

where k is the thermal conductivity of the solid, h7j is the rate of the elastic strain tensor, and @;j is the thermal modulus tensor. The term flijTh7j in (22) can be expressed in terms of the total strain rate h;j and the rate of temperature change T as

where

and 7 = - TBijCe.-' iJkl y k l

The Bjj , according to Hsu,' are defined by

aa0 a;, @..- - = I J - aT aT

On substituting (17) into the above expression, we may show that B;j = yij, and thus (25) can be rewritten as

(27)

We note that the expression of 7 ((25) or (27)) is not identical to what was given in Hsu' (p. 264, equation (9.44)) and S h i a l e ~ ~ ' ~ (p. 679, equation (58) and p. 400, equation (63)). The derivation of in the above referenced works, in our opinion, was based on a misprinted equation, (see Hsu,' p. 264, equation (3.15)).

7 = - Ty..C'.-' 1J iJkl yk l

REFERENCES 1 . T. R. Hsu, The Finite Element Method in Thermomechanics, Allen & Unwin, Winchester, 1986. 2. A. Sluialec, 'An analysis of thermal effects of coupled thermo-plasticity in metal forming processes',

3. A. Sh'uialec, 'An evaluation of the internal dissipation factor in coupled thermo-plasticity', Znt. J. Commun. Appl. Numer. Methods, 4, 675-685 (1988).

Non-Linear Mech., 4, 395-403 (1990).