Contribution to Numerical Treatment of Forming Processes under Consideration of Plastic Compressibility

ZAMM � Z. Angew. Math. Mech. 80 (2000) 4, 233±±244

Betten, J.; Schumacher, R.

Contribution to Numerical Treatment of Forming Processesunder Consideration of Plastic Compressibility

F�ur starrplastisches, kompressibles Materialverhalten unter Ber�ucksichtigung eines unterschiedlichen Zug- und Druckver-haltens werden die Stoffgleichungen aufgestellt und in die proze�beschreibenden Grundgleichungen nach den Methodender Gleitlinien und der Finiten Elemente eingebracht. L�osungsalgorithmen, mit denen FE- und GL-Verfahren kombiniertangewendet werden k�onnen, und L�osungen f�ur station�are und instation�are Prozesse werden vorgestellt.

Constitutive equations describing rigid-plastic compressible material behaviour are derived taking into account astrength-differential effect. Basic systems of equations according to the slip-line and finite element methods are intro-duced. Solution algorithms combining the FE and SL-methods and solutions for stationary and unsteady processes arepresented.

MSC (1991): 73E05, 73E50, 73G05, 73V05, 73V20

1. Introduction

For optimal component and tool construction an understanding of mechanical behaviour is especially important. Nu-merical simulations using the Finite Element Method (FEM) or the Slip-Line Method (SLM) are increasingly replacingcost-intensive experiments. At the same time the design and construction phases can be optimised through supportingcalculations and the whole process can be directly controlled through real time simulations. Modular CAD/CAEpackages as for example IDEAS and PRO-ENGINEER guarantee a `loss free' data transfer between construction,calculation, and manufacture and thus help cutting down the developmental period. More important however than themanipulation of complex software packages is a fundamental understanding of the describing basic equations, so thatthe correct conclusions for constructive changes can be drawn when evaluating numerical or theoretical material limita-tions. Also the often favoured FEM does not necessarily yield the best and fastest results. Hence element stressescalculated as secondary quantities from a global equilibrium condition in areas with large displacement or velocitygradients can have large errors. Finite Difference Methods (FDM), as for example characteristics methods or slip-linemethods, which have receded into the background because of their less systematic programmability, can in individualcases yield better results globally as well as locally. Under simplified assumptions analytical solutions can also succeed[10], although only for an initial rough calculation. The situation in fluid mechanics is different. Here the FDM and theFinite Volume Methods (FVM) have stood their ground because of better convergence behaviour compared with theFEM. The FVM in particular have been widely used in commercial CFD (Computational Fluid Dynamics) programssuch as FLUENT or STAR-CD. Currently combinations of CFD- and structural mechanics FE-programs which allowfor interaction with respect to temperature, heat transfer coefficient, etc. receive much attention. Combinations ofFDM and FEM seem also sensible, with the aim of balancing the disadvantages of one method with the advantages ofthe other.

2. Constitutive equations

Permanent deformations can be defined through the theory of the plastic potential F � F �sij� which for the specialisotropic case is characterized by the scalar function [3, 5]

F � F �J1; J2; J3� and F � F �J1; J02; J

03� ; resp: ; �2:1�

where J1; J2; J3 are the invariants of the Cauchy stress tensor sij and J 02; J03 the invariants of the stress deviator s0ij.

A relationship between the static and the kinematic variables can be found through the flow rule (normal rule)

deij � @F �spq�@sij

dl and _eij � @F �spq�@sij

_l; resp:; �2:2�

where the strain rate tensor _eij is described by

_eij � 1

2

@ui@xj� @uj@xi

� ��2:3�

(ui are the coordinates of the velocity vector). With sufficiently small displacement gradients and velocity gradienttensor corresponds approximately to the material derivative of the classical strain tensor [5]. Eq. (2.2) is called the

Betten, J.; Schumacher, R.: Numerical Treatment of Forming Processes 233

associate flow rule, where the gradient points into the direction of the plastic deformation and is orthogonal to theequipotential surface (yield surface). Nonassociative material behaviour can be observed for instance with differentgranular materials (for example soils). Here the flow characteristics are pressure dependent because of interior friction,although the incompressability condition has to be satisfied in order for the invariant J1 not to enter the plasticpotential.

For parabolic and elliptic flow conditions [4] we use the approach

F � AJ 02 � BJ21 � CJ1 with A � A�r� ; B � B�r� ; C � C�r; sFZ ; sFD� ; �2:4�

where r = total density/density of the basic and matrix material, resp., sFZ and sFD are the tension and pressure flowlimit, resp., and the free variable C controls a strength-differential effect (SD-effect) [4]. For the approach (2.4) thematerial equations (2.2) become

deij ÿ 1

3dij 1ÿ A

6B

� �devol � A sij � C

6Bdij

� �dl �2:5a�

with

devol � 3�6BsM � C� dl; sM � J1=3: �2:5b; c�Included is the proportionality factor dl which can be determined from the one-dimensional equivalent state [5]:

dl � deV

C � 23 sV �A� 3B� ; �2:6�

That is, C, depending on r, causes an expansion and a displacement of the yield surface on the J1-axis (in analogywith the isotropic and kinematic hardening).

Through the equivalent stress [5]

sV1; 2� ÿ 3C

4�A� 3B� ��

3

A� 3BAJ 02 �BJ2

1 �C

2J1

� �� 9

16

C

A� 3B

� �2s

�2:7�

we obtain the transmission conditions at the boundary at the start of flow:

limsij! sFZ ;ÿsFD

sV1; 2�sij� � sFZ and ÿ sFD : �2:8a; b�

From the yield condition F � DY 2 we determine through the one-dimensional tension pressure test [3]:

C � �B�A=3� �sFD ÿ sFZ � ; D � A=3�B ; Y � ��sFZsFDp

: �2:9a; b; c�For the case C � 0 we obtain an elliptic flow region, whose centre coincides with the origin of the basic stress space.For sFD > sFZ we observe a shift into the pressure region, where for the special case B � 0, sFD > sFZ the ellipsoidturns into a paraboloid which is open towards the pressure region. A yield cylinder according to Mises can be obtainedwith B � 0 and sFD � sFZ . From the validity of the plastic Poisson ratios of isotropic materials [4] �0 � pn � 0:5�where by means of

pn � sV �Aÿ 6B� ÿ �A� 6B� �sFD ÿ sFZ ��A� 3B� �2sn � sFD ÿ sFZ �

�2:10�

it follows that

1 � SD � 2Aÿ 3B

A� 3Bwith SD � sFD

sFZ: �2:11�

According to [4] the following restrictions are imposed on the free variables in (2.4), taking into account the convexitycondition:

1 � SD � 2 ; 0 � B � �2ÿ SD�=�3�1� SD�� for A � 3 : �2:12a; b�For this basic representation of the free variables it remains to say that the level of the yield surface locally rises abovethe Mises-cylinder, i.e., under certain stress conditions the porous material has greater strength. To what extent thisfact is physically meaningful for all sinter materials has to be investigated more closely experimentally in the individualcase [7]. If the ellipsoid yield condition is not allowed to rise above the Mises-cylinder anywhere and hence its maxi-mum is smaller than or equal to the Mises level of the yield surface, the following restrictions for D have to apply:

D � A

3SDÿ 1

4B

C

Y

� �2

for B > 0: �2:13�

234 ZAMM � Z. Angew. Math. Mech. 80 (2000) 4

For the equivalent strain-increment it follows from snden � sijdeji that

deV � ÿC deVol

12BsV�

��2B� 2

3A� C

sV

� �deij deijA

�Aÿ 6B

18ABde2

Vol

� �� C deVol

12BsV

� �2s

: �2:14�

3. Modified FE-approach according to Markov

For rigid-plastic material laws the method of the upper bound leads to a suitable FE formulation. For the assumedcompressible characteristics of the material a modified Markov principle [14] is applied:

P � �V

sF _en dV ��SF

tut dS ÿ�ST

piui dS ! stat: ; sF � Y ; �3:1�

with outer normal and shear forces applied separately to ST and SF [14] taking into account the Siebel friction law

t � mY =��3p

0 � m � 1 : �3:2�Assuming a piecewise differentiable velocity field (3.1) can be minimized with respect to the velocities:�

V

sFd

��2B� 2

3A� C

sF

� �_eij _eijA�Aÿ 6B

18AB_e2Vol

� �� C _eVol

12BsF

� �2s24 35 dV

��V

sFC

12BYd _eVol

� �dV ÿ

�SF

t dut dS ��ST

pi dui dS : �3:3�

Areas without plastic flow are considered to be quasi-rigid with a lower bound for the equivalent strain rate. Thevolume integrals are obtained through a numerical integration method with Gauss points, where the right hand side ofthe eq. (3.3) depends on the volumetric strain rate.

The modified variational principle can also be used for the simulation of incompressible processes with a relativedensity nearly 1, for example r = 0.99. The mean stress in the FE calculations is known for a compressible materiallaw, so that even with approximately incompressible material behaviour all stresses, and not just the stress deviators,can be determined directly from the applied material law. This means that a Lagrange parameter function, as is oftenadded to the functional as a secondary condition to observe the incompressibility and which, through the mean princi-ple stress, increases the number of degrees of freedom in the element, is not required.

4. Basic equations for the Slip-Line Method

For making the basic system the following definitions are holding in case of isotropy:

2_e12

_eI ÿ _eII� sin 2q � s12

q; �4:1�

I, II signalize main directions, III peripheral direction in case of axisymmetry, q: main direction of the I/II-plane, withthe maximal shearing stress or half the principal stress difference, resp., in the �x1; x2�-plane:

q � sI ÿ sII

2�4:2�

and the friction angle known from soil mechanics [6, 11]

w � arcsin ÿ @q@p

� �with p � ÿsI � sII

2: �4:3�

Using the Lode parameters

Ls � m � 2sIII ÿ sII ÿ sI

sI ÿ sII; Le � 2_eIII ÿ _eII ÿ _eI

_eI ÿ _eII�4:4a; b;�

the invariants can be formulated as follows:

J1 � 3p� mq ; J 02 � q2�I � m2=3� ; J 03 �2

3q3m

m2

9ÿ I

� �: �4:5�

For plastic potentials of the form F � F �J1; J02� the material law represented suitably for the Slip-Line Method can be

derived from

dF � 0�) @F

@J 02

@J 02@q

@q

@p� @F

@J1

@J1

@q

@q

@p� @F

@J1

@J1

@p� 0


as

@q

@p� 3�@F=@J1� _l

�@F=@J 02� �sI ÿ sII� _l� m��@F=@J1� � �@F=@J 02� s0III� _l� ÿ sinw ; �4:6�

such that for the dilatation angle being identical with the friction angle it holds [14]

sin w � ÿ _eI � _eII � _eIII

_eI ÿ _eII � m _eIII: �4:7�

In case of isotropy without effect of J 03 on statics and kinematics the Lode parameters (4.4a, b) are equal: m � LE [5].For the radius of the Mohr circle q the yield condition F � DY 2 yields the conditions q � q�p;r� in case of plane

strain � _eIII � 0� and q � q�p;r; m� in case of axisymmetrical deformation � _eIII � u2=r�, resp.:

q ��DY 2

Aÿ 9B

3B�A p� C

6B

� �2

�C2

4B

1

Aÿ 1

9B

� �" #vuut ; �4:8a�

q � 3=2

3A� m2�A� 3B�� ÿm�C ÿ 6Bp� � 1��

3p

��4DY 2 ÿ C2=B� �3A� m2�A� 3B�� AB�3� m2� �C ÿ 6Bp�2

q� �: �4:8b�

With the aid of the basic system and the equilibrium conditions sji;j � 0 the systems of partial differential equa-tions arising from that can be investigated with respect to their characteristics.

Plane deformationIt follows for the kinematics

�cos 2q� sin w� @u1

@x1� �cos 2qÿ sin w� @u2

@x1� 0 ; �4:9a�

2 sin 2q@u2

@x1ÿ �cos 2qÿ sin w� @u1

@x2ÿ �cos 2qÿ sin w� @u2

@x1� 0 �4:9b�

and for the statics

ÿ �1� cos 2q sin w� @p

@x1ÿ 2q sin 2q

@q

@x1ÿ @p

@x2sin 2q sin w� 2q sin 2q

@q

@x2

� ÿ cos 2q@q

@r

@r

@x1ÿ sin 2q

@q

@r

@r

@x2� R1 ; �4:10a�

ÿ �1ÿ cos 2q sin w� @p

@x2� 2q sin 2q

@q

@x2ÿ @p

@x1sin 2q sin w� 2q cos 2q

@q

@x1

� cos 2q@q

@r

@r

@x2ÿ sin 2q

@q

@r

@r

@x1� R2 : �4:10b�

If the systems of equations are solved separately and the right-hand sides of the statics R1; R2 can be presumedto be known because of numerical integration then we have systems of quasilinear partial differential equations asfollows:

Li �Pnj�1

aij@gj@x1�Pn

j�1

bij@gj@x2ÿRi � 0 ; i � 1; 2; j � 1; 2 �?� ; �4:11�

aij; bij; Ri : functions of the dependent variables gj ;

xi : independent variables ;

if gj � gj�xi� are solutions of (4.11) and if x2 � x2�x1� is the equation of a curve with given dependent variables gj.The coefficient determinant of (4.9a, b), (4.10a, b) (determinant of direction) necessarily has to vanish if derivativesperpendicularly to a certain curve (characteristic) cannot be given uniquely:

ÿ1ÿ cos 2q sin w ÿ sin 2q sin w ÿ2q sin 2q 2q sin 2q

ÿ sin 2q sin w ÿ1� cos 2q sin w 2q cos 2q 2q sin 2q

dx1 dx2 0 0

0 0 dx1 dx2

��

�� 0 : �4:12�

The determinant of direction classifies partial differential equations into hyperbolic, parabolic, and elliptic equations[12, 14].


The systems of differential equations (4.9a, b) and (4.10a, b) always are hyperbolic [6] for

jwj < p=2 �4:13�with the nonorthogonal characteristic directions [6, 11, 12] (Fig. 1)

dx2

dx1

��1; 2

� tan�qa; b� with qa;b � q� �p=4ÿ w=2� ; �4:14�

i.e., they are of hyperbolic type over the whole envelope face (Fig. 2) [14].Stresses outside of the hyperbolic area with

jpjY� A

3�B

� � ��D

AB�A� 12B�

s; C � 0 ; �4:15�

characterizing the parabolic limit by coinciding characteristics or characterizing the elliptic area cannot be investigatedby the SLM [9].

Using (4.14) the geometrical relations

@xi@xj� sin�q� p=4� w=2� cos�q� p=4ÿ w=2�ÿ cos�q� p=4� w=2� sin�q� p=4ÿ w=2��

; �4:16a�

@xi@xj� sin�q� p=4ÿ w=2� ÿ cos�q� p=4ÿ w=2�

cos�q� p=4� w=2� sin�q� p=4� w=2��

1

cos w�4:16b�

can be derived from a local coordinate system associated with the characteristics at each grid point. Then the xi areinterpreted as arc lengths and not as coordinates [2], with nonexisting equality of the mixed derivatives:

@2xp@xi @xj

6� @2xp@xj @xi

: �4:17�

A sufficient condition for the existence of characteristics is given by

ÿ1ÿ cos 2q sin w ÿ sin 2q sin w ÿ2q sin 2q R1

ÿ sin 2q sin w ÿ1� cos 2q sin w 2q cos 2q R2

dx1 dx2 0 dp

0 0 dx1 dq

��

�� 0 �4:18�


Fig. 1. Characteristic directions

Fig. 2. Representation of the yield con-dition in the form q�p�

(compatibility condition) [13]. The compatibility conditions or a transformation of the partial differential equations tocharacteristics lead to the characteristic systems of kinematics:

cos w@na

@xa

ÿ �nbÿna sin w� @qa

@xa

� 0 ; �4:19a�

cos w@nb

@xb

ÿ �naÿnb sin w� @qb

@xb

� 0 �4:19b�

and of statics [6]:

cos w@p

@xa

� 2q@q

@xa

� @q

@r

@r

@xb; �4:20a�

cos w@p

@xb

ÿ 2q@q

@xb

� @q

@r

@r

@xa ; xa;b? xb;a : contravariant arc lengths : �4:20b�

In streamlines the continuity condition can be transformed into

@�ln r�@h

� @�ln w�@h

2 sin w

cos 2fÿ sinw; �4:21�

i.e., for stationary processes the density can be calculated as dependent variable without being necessary to developfinite differences of the velocity of changing the volume (strain rate). For f � 0 the case of a diagonal flow (a bundle ofprincipal lines is identical with the flow lines, w: velocity of the flow lines) arises [8], i.e., it appears a homogeneousdeformation having minimal forming forces.

Axisymmetrical deformationFor the statics it follows

ÿ @p@z�1� cos 2q sin w� ÿ 2q sin 2q

@q

@zÿ @p@r

sin 2q sin w� 2q sin 2q@q

@r

� ÿ q

rsin 2qÿ @q

@r

@r

@z� @q

@m

@m

@z

� �cos 2q� @q

@r

@r

@z� @q

@m

@m

@z

� �sin 2q ; �4:22a�

ÿ @p@r�1ÿ cos 2q sin w� � 2q sin 2q

@q

@rÿ @p@z

sin 2q sin w� 2q cos 2q@q

@z

� ÿ q

r�cos 2q� m� ÿ @q

@r

@r

@z� @q

@m

@m

@z

� �sin 2q� @q

@r

@r

@r� @q

@m

@m

@r

� �cos 2q : �4:22b�

If the quantities r; m are known by a suitable numerical method you will find the characteristic directions from(4.14) with the characteristic system for p; q:

cos w dp� 2q dq

� dxa

q

r�ÿm sin�qÿ p=4ÿ w=2� � sin�q� p=4� w=2�� @q

@mÿ @m

@xa

sin w� @m

@xb

� �1

cos w

�� @q

@rÿ @r

@xa

sin w� @r

@xb

� �1

cos w

�; �4:23a�

cos w dpÿ 2q dq

� dxb

q

r�m cos�qÿ p=4� w=2� � cos�q� p=4ÿ w=2�� @q

@mÿ @m

@xa

� @m

@xb

sin w

� �1

cos w

�� @q@r

ÿ @r

@xa

� @r

@xb

sin w

� �1

cos w

�; �4:23b�

and for the kinematics:

@u1

@xa

sin�q� p=4� w=2� ÿ @u2

@xa

cos�q� p=4� w=2� � ÿu2

2r�1� m sinw� ; �4:24a�

@u1

@xb

cos�q� p=4ÿ w=2� ÿ @u2

@xb

sin�q� p=4ÿ w=2� � ÿ u2

2r�1� m sinw� �4:24b�

with, contrarily to the case of plane deformation, non-vanishing strain rate in the direction of the characteristics:

_eaa � _ebb � u2

2r�1� m sin w� : �4:25�


Moreover one can derive two equations of the angular velocity w � 12 �@u2=@x1 ÿ @u1=@x2�:

wa � ÿ @u1

@xa

sin�q� p=4� w=2� ÿ @u2

@xa

cos�q� p=4� w=2� � u2

r

1

m� h

� �cos w ; �4:26a�

wb � @u1

@xb

sin�q� p=4ÿ w=2� ÿ @u2

@xb

cos�q� p=4ÿ w=2� ÿ u2

r

1

m� h

� �cos w �4:26b�

with

h�m; r; p� � 1

m

1

3

@F

@J 02mq ÿ @F

@J1

2

3

@F

@J 02mq � @F

@J1

; �4:26c�

which in a numerical integration procedure can be used as equations for controlling the estimates [1].

5. Integration method

For an integration of the basic equations it is necessary to define the plastic zone through a characteristic grid structure,where with the aid of the Massau grid construction the regions of influence can be treated according to Fig. 3 [11, 13].


Fig. 3. Regions of influence for the Cauchy, Riemann, and mixed intial value problem

Fig. 4. Integration diagram for a collocation method with plane deformation

Through a suitable estimate of free variables and through decoupling, the global elliptic system can be decom-posed into mutually dependent hyperbolic partial systems. The partial systems can then be solved successively, wherethe last solved partial system, according to the choice of variables, either yields new values (controls) for the initialestimates (method A) or is overdetermined and yields two values for the unknown variable, whose difference is mini-mized iteratively (method B) [1].

As well as the common estimates towards the solution of the differential equation system, estimates for theinitial values are usually required, which are then improved through suitable iterative methods [14] under observationof boundary conditions.

5.1 Plane deformation

Collocation method (method A)Through a polynomial approach for the density the static equations are determined and characteristic and can besolved iteratively at the grid point (point iteration) in conjunction with the geometric equations. If the velocity field isknown the new density can be calculated using the streamlines. The difference between the actual and calculatedboundary conditions, and the difference between the given densities and the densities calculated from the streamlinesconstitute the controls. These are minimized in a grid iteration and hence the free variables, consisting of the coeffi-cients of the polynomial approach and the initial estimates, are improved. The integration procedure is completedwhen the sum of the squares of the controls is small enough and satisfies the condition of positive specific plastic rate


Fig. 5. Velocity field, w-distribution (rad), and distribution of density

everywhere [14]. Since this is an approximate method we use more controls than variables, that is the solution isapproached in the mean.

Difference method (method A, B)The so-called `normal' or `stable difference method' [1] is applied for example to plane incompressible deformation ofhardening materials. With given angles q (free variables) the differential equations of the geometry, the kinematic andthe static can be worked out successively. Since the static is overdetermined, the difference Dp � pa ÿ pb withp � �pa � pb�=2 results from the arclengths (method B). For compressible material laws with variable density the fric-tion angle w is also part of the free variables. The controls for the friction angles estimated at the outset are thedifferences Dw � wÿ �ÿ arcsin�@q=@p�� (method A). This way it is again possible to solve geometry, kinematics, andstatics successively. The balance free variables/controls is usually even. For higher accuracy computation time andmemory requirement are considerably higher than for the collocation method.

5.2 Axisymmetrical deformation

Collocation method (method A, B)Through the polynomial approaches for the density and for the Lode parameter (4.4b) the static equations togetherwith the geometric equations in p, q are defined and characteristic. Accordingly the velocity field is calculated through(4.25a, b). Control equations for the initially estimated Lode parameter (through the coefficient of the polynomial)constitute the differences at the gridpoints Dw � wa ÿ wb (method B). For the density the controls follow from thedensities newly calculated through the streamlines (method A).

Mixed FE/SL methodq, w are given free variables and are used for the calculation of the geometry. Then the quantities r, m, which havebeen determined more accurately, are transferred from the FE calculation into the SL grid. To ensure the convergenceof the integration a bilinear polynomial is assigned to the transferred values in the element via the nodal values, sothat a continuous, but not continuously differentiable data surface is produced at the element boundaries. Now it ispossible to solve the characteristic system of the static, where iterations are required at the gridpoint because of thenonlinear dependency on p. Finally with known q the kinematic equations yield the velocity field. The controls aremade up in the same way as for the difference method with plane deformation.

6. Applications

Forward extrusionIn case of plane transformation the collocation method shall be tested exemplarily for extrusion. There the tangential-ity of inlet and outlet results in a nearly diagonal flow with a nearly homogeneous distribution of density at the outlet(Fig. 5a, b, c). The Figs. 6a, b show that the initial density controls the mean inlet pressure. In Fig. 6b the calculationhad to be abandoned at the value r0 � 0:8 of the initial density since the net of characteristics was displaced too muchand there arose areas with a discretization being too coarse.

Pipe extrusionIn Fig. 8b an application of the mixed method in case of axisymmetrical deformation is presented [14]. Considering thefact that qSL is directly derived from the flow condition and qFE is derived from the flow rule in connection with theplastic potential we can observe good correspondence of distribution and domain of data range (Fig. 8a, b).


Fig. 6. Mean inlet pressure


Fig. 7. Global quantities

Fig. 8. Distribution q=Y according to FEM/SLM with A � 1; B � �1ÿ r�2


Fig. 9. Relative decrease of the inner symmetry radius with SD-influence for r0 � 0:8; 0:9

Fig. 10. Velocity field and distribution of density for reduction of 20% and 40%,Ra :Ri :H � 6 :3 :2; sFZ � 65 N=mm2; A � 2� r2; B � 1ÿ r2

Fig. 7a, b presents a comparison of the global data of both the methods. It is seen that in spite of the differentmethods there is a maximal deviation of 4% for the mean inlet pressure and around 3% for the outlet velocity.

Ring upsetA frequently used example of unsteady deformation is the axisymmetrical ring upset (Fig. 10) which is especially suita-ble because of sensitive reactions of the material flow to changes in the contact zone. Depending on the friction coeffi-cient the material at the inner side of the ring bulges out inwards or outwards.

A higher SD-quotient reduces the possibility of changes of the volume, therefore for higher SD-values analo-gously to higher initial densities the material in case of large and small friction coefficients bulges out more outwardsand more inwards, resp. (Fig. 9a, b). In any case the needed force increases.

7. Summary

Considering the rigid-plastic behaviour of porous material the basic equations have been derived according to theMethods of Finite Elements and of Slip-Lines. A modified FE-approach according to Markov shows an influence of thechange of volume to the ªright-hand sideº. According to the Slip-Line Method a numerical integration of quasilinearpartial differential equations can be realized by decoupling into partial systems and prescribing unknown values. Theapplicability of the methods could be proved at some forming processes.

References

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Aachen 1995. VDI-Fortschrittsberichte, Reihe 20, Nr. 189, 1996.

Received June 23, 1997, revised September 10, 1998, accepted November 23, 1998

Addresses: Univ.-Prof. Dr.-Ing. J. Betten, Technical University Aachen, Department of Mathematical Models in Materials Science,D-52056 Aachen, Templergraben 55, Germany; Dr.-Ing. R. Schumacher, Dr. Schrick GmbH, Dreherstra�e 5, D-42899Remscheid, Germany


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Contribution to Numerical Treatment of Forming Processes under Consideration of Plastic Compressibility