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The effect of porosity and micro-defects on plasticallydeformed porous materials

A.G. Mamalisa,*, G.L. Petrossianb, D.E. Manolakosa

aDepartment of Mechanical Engineering, National Technical University of Athens, 42, 28th October Ave., 106 82 Athens, GreecebDepartment of Mechanics and Machine Science, State Engineering University of Armenia, Yerevan, Armenia

Received 30 April 1998

Abstract

A modi®ed plasticity theory for sintered powder materials, taking into account the real complex state of porosity and micro-defects

(cracks, non-metallic inclusions, etc.) is proposed. Equations for plastic ¯ow were developed and methods for calculating the true

mechanical properties are suggested. The yield criterion derived, in conjuction with simple uniaxial compression tests conducted on

specimens of the porous material, may evaluate the real state of the material porosity related to micro-defects which may constitute the

sources for the initiation and propagation of cracks, even in high compressive stress ®elds such as those developed in most manufacturing

processes, as well as the formation of new pores during plastic deformation. # 1999 Elsevier Science S.A. All rights reserved.

Keywords: Porosity; Micro-defects; Plastically deformed porous materials

1. Introduction

Static and dynamic powder compaction have been

combined with plastic deformation processes for fabricat-

ing sound engineering products [1]. Sintered porous

materials in the form of three-dimensional plates and

axisymmetric billets are employed as the initial materials

for subsequent rolling, forging, extrusion, wire-drawing,

etc., operations. Of great importance for the manufacturing

engineer are, therefore, theoretical approaches for analys-

ing and predicting the working load and the deformation

characteristics and the density of the plastically deformed

components.

Various methods of analysis for metal working processes

are on hand, based on the classical plasticity theory for

solids, where volume constancy is assumed for the material

undergoing deformation; this assumption applies in effect to

pore-free materials. However, during the deformation of

porous materials the volume does not remain constant. Even

a very small amount of porosity affects the mechanical

properties and contributes to failure.

Yield criteria and stress±strain relations for plastically

deformed porous metals were proposed initially by Kuhn

and Downey [2] and Green [3]; simple stress states, such as

uniaxial compression and tension and plane-strain compres-

sion were utilised. Basic equations of plasticity theory of

such deformed bodies have been also suggested by Shima

and Oyane [4] Petrossian [5] and others [4,6±11]. A com-

prehensive account of the methods for determining the

mechanical properties of common and composite porous

materials undergoing plane strain and/or axisymmetric plas-

tic deformation is given in [1]. Note, however, that, in these

cases, the real state of the material porosity related to micro-

defects is not taken into account; such micro-defects may

constitute the sources for the initiation and propagation of

cracks, even in high compressive stress ®elds such as those

developed in solid hydrostatic extrusion [1] and the forma-

tion of new pores during deformation.

In the present paper, a modi®ed plasticity theory for

sintered powder materials, taking into account the real

complex state of porosity and micro-defects (cracks, non-

metallic inclusions, etc.), is proposed. Equations for plastic

¯ow are developed and methods for calculating the true

mechanical properties are suggested. Experimental valida-

tion of the proposed theory for various loading conditions is

discussed.

Journal of Materials Processing Technology 96 (1999) 117±123

*Corresponding author. Tel.: +30-1772-3688; fax: +30-1772-3689

E-mail address: [email protected] (A.G. Mamalis)

0924-0136/99/$ ± see front matter # 1999 Elsevier Science S.A. All rights reserved.

PII: S 0 9 2 4 - 0 1 3 6 ( 9 9 ) 0 0 2 6 5 - 4

2. Fundamentals of plasticity theory for porousmaterials

2.1. Plastic flow

The modelling of the consolidation of porous materials

needs a yield function that takes into consideration the

change in volume of the porous body during compaction

and it must include the in¯uence of the hydrostatic stress

component and satisfy the symmetry and convexity condi-

tions required for the development of plasticity theory for

porous materials. Under these requirements, it seems rea-

sonable for the plastic ¯ow of sintered materials to depend

mainly on the linear invariant of the stress tensor

I1��1��2��3�3�m and the quadratic invariant of the

deviator stress tensor J2�(1/6)[(�1ÿ�2)2�(�2ÿ�3)2�(�3ÿ�1)2]. The relevant yield surface must be smooth,

convex and close, and has to approximate the Huber±Mises

cylinder as the relative density of the porous body

approaches unity. In the present analysis, in view of the

assumptions made above, the following generalized yield

criterion for compressible materials (metal powder com-

pacts and sintered metals) is proposed:

F��ij� � AJ2 � BI21 ÿ �Y2

0 � 0; (la)

where A is a constant equal to 3, such as to suit in von Mises

criterion; B a parameter equal to �m representing the degree

of in¯uence of the hydrostatic stress component, �m;

��2n�1 is a parameter equal to the ratio (YR/Y0)2; YR the

yield stress of the porous material; Y0 the yield stress of solid

(pore-free) matrix material; �, � are parameters that are

functions of relative density � (or equivalently of porosity f);

and m, n are constants that are determined experimentally,

also dependent upon porosity.

Thus, the yield criterion may be represented in terms of

the principal stress components by:

F��ij� � 12��1 ÿ �2�2 � ��2 ÿ �3�2 � ��3 ÿ �1�2h i� 9�m�2

m ÿ �2n�1Y20 � 0: (1b)

Considering the problem of yield and plastic ¯ow in a

more general manner and using function F as a plastic

potential for porous materials, the relevant stress±incremen-

tal strain relationships can be derived by partially differ-

entiating this function with respect to �1, �2 and �3, i.e.:

d"1 � d�@F

@�1

� d��2�1 ÿ �2 ÿ �3 � 6�m�m�; (2a)

d"2 � d�@F

@�2

� d��2�2 ÿ �3 ÿ �1 � 6�m�m�; (2b)

d"3 � d�@F

@�3

� d��2�3 ÿ �1 ÿ �2 � 6�m�m�; (2c)

where d� is a non-negative constant of proportionality.

If �eq denotes the equivalent stress referring to the matrix

material, it may be expressed by the relationship:

�eq � 1

�n�0:5

1

2��1 ÿ �2�2 � ��2 ÿ �3�2 � ��3 ÿ �1�2h i�

� 9�m�2m

�0:5

:

Combining Eqs. (2), the following expressions are

obtained:

�1 ÿ �2 � d"1 ÿ d"2

3d�; (4a)

�2 ÿ �3 � d"2 ÿ d"3

3d�; (4b)

�3 ÿ �1 � d"3 ÿ d"1

3d�; (4c)

�m � d"1 � d"2 � d"3

18�md�(4d)

and by replacement into Eq. (3) the equivalent stress can be

expressed in terms of the principal strain increments by:

�eq � 1

2d� � �n�0:5

� 2

9�d"1 ÿ d"2�2 � �d"2 ÿ d"3�2 � �d"3 ÿ d"1�2h i�

��d"m�2�m

)0:5

; (5)

where d"m�(d"1�d"2�d"3)/3.

If d"eq denotes the equivalent strain increment referred to

the matrix, then the plastic work done per unit volume of an

element of the porous body is expressed in terms of the stress

and strain increment components as:

dW � �1d"1 � �2d"2 � �3d"3 � � � �eqd"eq: (6)

By substituting Eq. (2) into Eq. (6) and re-arranging, the

following expression for constant d� is obtained:

d� � � � d"eq

2�2n�1�eq

: (7)

Combining Eqs. (5) and (7), the equivalent strain incre-

ment, d"eq, can be derived from:

d"eq � �n�0:5

�

�(

2

9�d"1 ÿ d"2�2 � �d"2 ÿ d"3�2 � �d"3 ÿ d"1�2h i

��d"m�2�m

)0:5

(8)

whilst substituting Eq. (7) into Eqs. (2), the following

relationship for incremental strains in terms of equivalent

stress and equivalent strain increments are obtained:

d"1 � � � d"eq

2�2n�1�eq

�2�1 ÿ �2 ÿ �3 � 6�m�m�; (9a)

118 A.G. Mamalis et al. / Journal of Materials Processing Technology 96 (1999) 117±123

d"2 � � � d"eq

2�2n�1�eq

�2�2 ÿ �3 ÿ �1 � 6�m�m�; (9b)

d"3 � � � d"eq

2�2n�1�eq

�2�3 ÿ �1 ÿ �2 � 6�m�m�: (9c)

From Eqs. (9), the volumetric strain increment, d"v, may

be predicted as

d"v � d"1 � d"2 � d"3 � 9� � �m�md"eq

�2n�1�eq

: (10a)

However, taking also into account that ��1ÿf, the volu-

metric strain increment may be related to the increments of

relative density, d�, and of porosity, df, by:

d"v � ÿ d�

�� df

1ÿ f: (10b)

Combining Eqs. (10) and re-arranging:

df � 9�1ÿ f �2�m�md"eq

�2n�1�eq

: (11)

2.2. Determination of porosity functions: effect of the

initial porosity

Porosity functions � and � both depend upon many

technological factors such as the compaction process

applied, the size and shape of the powders, the dimensions

of the sintered sample obtained, etc. A variety of similar

porosity functions have been proposed by various theoretical

treatments, but the choice of proper magnitudes is dif®cult

and often aimless. A typical empirical formulation regarding

porosity functions � and � has been developed in [8] and

used in the present analysis:

(i) in the range 0�f�0.25:

� � 0:15f 0:6; (12a)

� � 1ÿ 1:8f 0:9: (12b)

(ii) in the range 0.25�f�0.60:

� � 0:0185� 0:183f ; (12c)

� � 0:73ÿ f : (12d)

In case of the simple loading of porous materials under

uniform stress state, it was observed that the function of

porosity � takes always a constant value �0 depending only

on the initial porosity, f0 and furthermore, the stress com-

ponents at each point of the porous body increase propor-

tionally, i.e.:

�1 � � � ��1; (13a)

�2 � � � ��2; (13b)

�3 � � � ��3; (13c)

where � is a proportionality parameter and ��1, ��2 and ��3 are

constant values of stress components, respectively.

Under this concept, the formulation derived above

reduces to:

d"1 � � � d"eq

2�n�0:5�eq

�2��1 ÿ ��2 ÿ ��3 � 6�m0 ��m�; (14a)

d"2 � � � d"eq

2�n�0:5�eq

�2��2 ÿ ��3 ÿ ��1 � 6�m0 ��m�; (14b)

d"3 � � � d"eq

2�n�0:5�eq

�2��3 ÿ ��1 ÿ ��2 � 6�m0 ��m�; (14c)

where

��eq �1

2��1 ÿ ��2�2 � ��2 ÿ ��3�2 � ��3 ÿ ��1�2�

��9�m

0 ��m�2o0:5

: (15)

Integrating Eqs. (14), gives, respectively:

"1 � 1

2��eq

�2��1 ÿ ��2 ÿ ��3 � 6�m0 ��m�Z� � d"eq

�n�0:5; (16a)

"2 � 1

2��eq

�2��2 ÿ ��3 ÿ ��1 � 6�m0 ��m�Z� � d"eq

�n�0:5; (16b)

"3 � 1

2��eq

�2��3 ÿ ��1 ÿ ��2 � 6�m0 ��m�Z� � d"eq

�n�0:5: (16c)

The equivalent strain, "eq, may be predicted from Eq. (8)

under the condition ��0:

"eq � �n�0:5

�

� 2

9��"1ÿ "2�2 � �"2ÿ "3�2 � �"3ÿ "1�2� � "2

m

�m0

� �0:5

:

(17)

Substituting from Eqs. (16) into Eq. (17) and simplifying:Z� � d"eq

�n�0:5� � � "eq

�n�0:5: (18)

Therefore, substituting Eq. (18) into Eqs. (16), the for-

mulation regarding the strain components reduces to the

following general form, respectively:

"1 � � � "eq

2�2n�1�eq

�2�1 ÿ �2 ÿ �3 � 6�m0 �m�; (19a)

"2 � � � "eq

2�2n�1�eq

�2�2 ÿ �3 ÿ �1 � 6�m0 �m�; (19b)

"3 � � � "eq

2�2n�1�eq

�2�3 ÿ �1 ÿ �2 � 6�m0 �m�: (19c)

Recalling Eq. (11) for current porosity, f, under the con-

ditions introduced above and integrating, gives:Zff0

df

�1ÿ f �2 �9�m

0 �m"eq

�2n�1�eq

;

or upon simplifying:

f � 1ÿ 1

1ÿ f0

� 9�m0 �m"eq

�2n�1�eq

� �ÿ1

: (20)

A.G. Mamalis et al. / Journal of Materials Processing Technology 96 (1999) 117±123 119

3. Modified plasticity theory: effect of real complexstate of porosity and microdefects

The real state of the material porosity related to micro-

defects (pores and metallic inclusions) may constitute the

sources for the initiation and propagation of cracks inside the

porous body, even in high compressive stress ®elds, such as

those developed in typical mechanical processes, resulting

in the formation of new pores (micro-defects) during defor-

mation. The so-obtained crack porosities, fc affect greatly

the amount of plastic deformation imposed and dominate the

fracture phenomena occurring in sintered or compacted

porous materials. To analyse the conditions governing the

associated fracture mechanism, the following simpli®ca-

tions may be introduced: (i) the encountered new crack

porosities result in changes only in Eqs. (11) and (20); (ii)

new porosities depend only on the initial porosity, f0; and

(iii) cracking initiation and propagation depend on the

stress±strain state and the degree of deformation imposed

on the material during a manufacturing process.

Based on these assumptions, it is reasonable to express the

crack porosity, fc, as a function of the initial porosity, f0, and

of a strain, "c, typical for each manufacturing process, by:

fc � Kcf0"2c ; (21)

where Kc is a constant determined experimentally and

related to the stress±strain state of the material during its

plastic deformation and fracture.

Therefore, the total current porosity, f, of the porous body

will be the sum of porosity due only to existing pure pores,

fp, and of porosity due to micro-defects developed, fc, i.e.:

f � fp � fc:

where substituting for fp and fc from Eqs. (20) and (21),

respectively:

f � 1ÿ 1

1ÿ f0� 9�m

0 �m"eq

�2n�1�eq

� �ÿ1

�Kcf0"2c ; (22)

whilst by using Eq. (11) and differentiating Eq. (21), the

corresponding porosity increment df is obtained:

df � 9�1ÿ f �2�m�md"eq

�2n�1�eq

� 2Kcf0"c d"c: (23)

4. Theoretical and experimental determination of thestress±strain and porosity±strain characteristics ofengineering sintered materials

4.1. Effect of initial porosity

For sintered materials, the strainR

d"eq, integrated over a

known strain path, provides a measure of the plastic defor-

mation in the matrix material. Therefore, in order to predict

the various parameters related to the present theoretical

approach and check the validity of the formulation proposed

above and simultaneously to gain useful information on the

mechanical properties of a porous body, it is simply needed

to construct a curve of the form �eq � g�R d"eq�. This can be

achieved very easily by conducting a uniaxial compression

test. The stress±strain state for a sintered material under

uniaxial compression, described by �1�ÿ�, �2��3�0 and

d"2�d"3, can be converted to the stress±strain state for the

matrix material by substitution into Eqs. (5) and (8) as:

�eq � �1� �m�0:5�

�n�0:5; (24a)

d"eq � ÿ �n�0:5

��1� �m�0:5 � d"c; (24b)

whilst the state of porosity, by replacement into Eq. (11), is

expressed by:

df � ÿ 3�1ÿ f ��m d"c

1� �m; (25)

where ��P/A0 is the axial compressives stress; P the

compressive load; A0 the cross-sectional area of the speci-

men tested; "c�ln(h/h0) the axial compressive strain; h0 the

initial height of the specimen; and h is the current height of

the specimen.

Conducting an incremental compression test in c loading

steps and applying a variational formulation for the predic-

tion of the current porosity and strain level:

fi � fiÿ1 ��fi; (26a)Zd"eq � �

c

i�1�"eqi

: (26b)

In this manner the following step-by-step calculation

procedure may be applied for a complete mechanical char-

acterisation of porous materials through a simple uniaxial

compression test.

Step 1. At least two cylindrical samples of different initial

porosity, say f01 and f02, respectively, with initial height h0

and diameter d0, have to be subjected to incremental com-

pression test under identical loading conditions and in 3±5

loading stages. After each loading stage, the current height

h, current diameter d, current porosity f, axial compressive

stress � and axial compressive strain "c are properly mea-

sured and recorded. Thereafter, experimental curves f±"c and

�±"c are constructed easily.

Step 2. For constant strain increment �"c and initial

porosity f0, the parameter �0 is calculated from Eqs. (12)

and then, by combining Eqs. (25) and (26a), a family of

theoretical f±"c curves for various values of exponent m are

predicted.

Step 3. The obtained theoretical curves f±"c are compared

with the corresponding experimental graph. From the coin-

cidence of the experimental curve with one of the curves

predicted theoretically, the best-®t value of exponent m for

the material tested is obtained.

Step 4. From two experimental curves f±"c of the same

material (i.e. two different testing samples), current porosity


values f are recorded per loading stage and a pair (�, �)

for each specimen is evaluated by Eqs. (12), respectively.

Then, equivalent stress �eq may be predicted and recorded

for various values of exponent n for both of the test

samples.

Step 5. Equivalent strain increments (�"eq1, �"eq2) and

total equivalent strains (R

d"eq1;R

d"eq2) loading stage are

calculated by using Eqs. (24b) and (26b), respectively.

Step 6. Combining the results obtained in Steps 4 and 5,

a family of curves �eq±R

d"eq is constructed for each

specimen. Since the matrix material for both of the

test samples requires the coincidence of curves seq±Rd"eq, a pair of curves from the above-constructed families

that satis®es this requirement gives the proper value for

exponent n.

4.2. Effect of complex state of porosity and microdefects

A similar procedure may be followed to predict the

conditions for cracking initiation and propagation inside

the powdered body. Combining Eqs. (23) and (25), the

critical porosity increment during a uniaxial compression

process may be written in the form:

dfc � ÿ 3�m�1ÿ f � d"c

1� �m� 2Kc1f0"c � d"c (27)

and taking also into consideration that initially porosity f

decreases as compression proceeds and that after the occur-

rence of cracking starts to increase, the limiting (critical)

value of porosity is expected when dfc�0. Under this con-

dition, Eq. (27) yields:

Kc1 � 3�mmin�1ÿ fmin�

2f0�1� �mmin�"�c

; (28)

where substituting into Eq. (27):

dfc � 3�m

min�1ÿ fmin�"c

�1� �mmin�"�c

ÿ �m�1ÿ f �1� �m

� �d"c; (29)

where fmin and �min are the critical values of porosity f and

parameter �, respectively, when critical deformation "�coccurs; see also Figs. 1±3.

Under uniaxial compression conditions, Eq. (22) reduces

to:

fc � 1ÿ �1ÿ f0��1� �m0 �

1� �m0 ÿ 3�m

0 "c

� Kcf0"2c (30)

from which, under the condition dfc�0:

Kc2 � 3�m0 �1ÿ f0��1� �m

0 �2f0"�c�1� �m

0 ÿ 3�m0 "�c�2

(31)

and substituting into Eq. (30) yields:

fc � 1ÿ �1ÿ f0��1� �m0 �

� 1

1� �m0 ÿ 3�m

0 "c

ÿ 3�m0 "

2c

2"�c�1� �m0 ÿ 3�m

0 "�c�2

" #:

Implementing Eqs. (29) and (30) or Eqs. (31) and (32)

with experimental curve f±"c the critical values of para-

meters m and Kc when cracking is starting inside the porous

body may be predicted.

Fig. 1. Variation of porosity with compressive strain. Determination of

constant m. Curves 1 and 2: experimental curves for specimens 1 and 2,

respectively; Curves 3±9: theoretical curves corresponding to m�0.4, 0.5,

0.6, 0.7, 0.8, 0.9 and 1.0, respectively.

Fig. 2. Determination of constant n. Curves 1 and 2: stress±strain curves

as obtained from compression tests for specimens 1 and 2, respectively;

Curve 3: constitutive flow curve �eq±R

d"eq for sintered brass.

Fig. 3. A typical graph showing the variation of porosity with critical

strain for evaluating the micro-defects constants.


4.3. Experimental validation

All parameters included in the proposed yield criterion

concerning the mechanical characteristics and governing the

limiting conditions for cracking initiation inside a porous

body, are predicted by carrying out simple uniaxial com-

pression tests on two sintered powder compacted samples of

the same material but with different initial porosity.

The specimens tested were produced by cold compacting

brass 59-1 powder, with a particle size ranging between 0.25

and 0.63 mm, to the required green density and sintering

them in a hydrogen environment at 750±7808C for 1.5 h.

The sintered preforms were machined properly to cylind-

rical samples with initial dimensions of 10 mm diameter and

12 mm height, measuring, in this manner, initial porosities

f01�0.17 and f02�0.09, respectively.

Both of the sintered samples were subjected to incre-

mental uniaxial compression conducted at room temperature

between the ¯at platens on an Instron testing machine at a

constant initial strain rate of 0.005 sÿ1. The ram displace-

ment was controlled to produce loading steps such as to

attain constant strain increments of 0.025. A MoS2 paste

lubricant was used to achieve near-homogeneous deforma-

tion. The specimen density was also measured and recorded

after each loading step.

Experimental results pertaining to the variation of poros-

ity f and compressive stress �with strain level " are shown as

graphs f±" and �±" in Figs. 1 and 2, respectively, for both of

the specimens tested. Implementation of the theoretical step-

by-step treatment described above gave the following results

for the various porosity parameters: (i) m�0.75 and n�0.5,

without consideration of micro-defects; and m�0.4 and

n�0.5, when the effect of micro-defects is examined; whilst

typical values of critical strain, "c and porosities fp, fc and f

are recorded in Table 1.

5. Discussion and conclusions

A new yield criterion for porous materials is proposed

from a phenomenological point of view, capable, apart from

predicting the mechanical properties of the porous body, of

considering also, in a simple way, the effect of micro-defects

on the conditions for cracking development inside the

material under severe plastic deformation.

The validity of the formulation pertaining to the plasticity

of porous metals is achieved by determining experimentally

all of the parameters incorporating the yield criterion as

functions of porosity.

A step-by-step procedure has been developed in combi-

nation with simple uniaxial compression tests on sintered

samples of the material examined. Application of the

method to brass 59-1 sintered specimens leads to the

following concluding remarks; see also Figs. 1±3 and

Table 1.

1. Consideration of new pores (i.e. micro-defects) results

in change to the porosity distribution inside the porous

body, however, in any case, the quantity of micro-

defects depends on the initial porosity of the material.

2. Cracking initiation and propagation inside the porous

body depends on the existing stress±strain state in the

vicinity of the micro-defect and the plastic deformation

imposed on the material.

3. At critical strain, "c, the porosity of pores, fp, decreases

(from 0.17 to 0.045 under "c�0.2 in the example studied)

and the porosity of cracks, fc, increases (from 0 to 0.07

under "c�0.2), whilst the total current porosity, f, finally

decreases (from 0.17 to 0.115 under "c), see Figs. 1±3

and Table 1.

The deformation mechanism of a porous material arising

from the above-described results seems to follow the

sequence: opening and closing of pores and cracks; devel-

opment of new pores (micro-defects); and so on.

6. Nomenclature

A constant

B constant

F plastic potential

f current porosity

fc porosity due to micro-defects (cracks)

f0 initial porosity

fp porosity due only to pores, when micro-defects

are developed

h height of the specimen

h0 initial height of the specimen

I1 linear invariant of stress tensor

J2 quadratic invariant of deviator stress tensor

Kc coefficient of micro-defects

m constant of the material, depending on the

porosity

n constant of material depending on the porosity

Y0 yield stress of the matrix material

YR yield stress of the porous material

Greek letters

� parameter, function of porosity



"1,"2,"3 principal strain components

Table 1

Values of porosities fp, fc, f for different plastic deformations in uniaxial

compression

"c 0.050 0.100 0.150 0.200

fp 0.140 0.110 0.080 0.045

fc 0.004 0.018 0.040 0.070

f 0.144 0.128 0.120 0.115


"c typical critical strain for a manufacturing

process

"eq equivalent strain

"m mean strain

"v volumetric strain

� proportionality constant

� relative density

�1, �2, "3 principal stress components

�m hydrostatic stress component

�eq equivalent stress

References

[1] A.G. Mamalis, A. Kandeil, M.C. de Malherbe, W. Johnson, Defects

in cold-hydrostatically extruded aluminium, iron and nickel-base

powder products, J. Mech. Working Technol. 4 (1980) 327.

[2] H.A. Kuhn, C.L. Downey, Deformation characteristics and

plasticity theory of sintered powder materials, Int. J. Powder Met.

7 (1971) 15.

[3] R.J. Green, A plasticity theory for porous solids, Int. J. Mech. Sci. 14

(1972) 215.

[4] S. Shima, M. Oyane, Plasticity theory for porous metals, Int. J. Mech.

Sci. 18 (1976) 285.

[5] G.L. Petrossian, Plastic deformation of powder materials, Metallurgy,

Moscow, 1988.

[6] M. Oyane, S. Shima, T. Tabata, Consideration of basic equations, and

their application, in the forming of metal powders and porous metals,

J. Mech. Working Technol. 1 (1978) 325.

[7] S.M. Doraivelu, H.L. Gegel, J.S. Gunasekera, J.C. Malas, J.T.

Morgan, J.F. Thomas Jr., A new yield function for compressible P/M

materials, Int. J. Mech. Sci. 26 (1984) 527.

[8] Y.K. Lee, A finite elastoplastic flow theory for porous media, Int. J.

Plasticity 4 (1988) 301.

[9] B-B. Hwang, S. Kobayashi, Deformation characterization of

powdered metals in compaction, Int. J. Mach. Tools Manufact. 30

(1990) 309.

[10] A. de Matos Dias, L. Schaeffer, Theoretical and experimental

analysis of plasticity equations for porous metals, Int. J. Mach. Tools

Manufact. 33 (1993) 577.

[11] H.S. Kim, C.W. Won, B.S. Chun, Plastic deformation of porous

metals with an initial inhomogeneous density distribution, J. Mater.

Process. Technol. 74 (1998) 213.


Journal of Materials Processing Technology 172 (2006) 277–282

Determination of initial compression stresses when extrudingbimetallic tubes with porous internal layer

A.G. Mamalisa,∗, G.L. Petrosyanb, D.E. Manolakosa, H.G. Petrosyanb

a Department of Mechanical Engineering, National Technical University of Athens, Athens, Greeceb Department of Mechanics and Machine Science, State Engineering University of Armenia, Yerevan, Armenia

Received 15 March 2004; received in revised form 15 March 2004; accepted 27 September 2005

Abstract

This paper presents an analytical approach of the extrusion process applied to biomaterial tubes made of two prebonded metallic materials. Theouter component of the bimetallic is a pore-free layer, whilst the internal one is a porous sintered tube.

A generalized “slab method” of analysis, properly modified to describe the plastic behaviour of a porous-sintered material, developed incombination with the theory of thin shells, in order to investigate the initial stage of plastic deformation in the cylindrical chamber of the extrusiondie. Numerical solution of the formulation suggested is conducted for components made of sintered and pore-free steel, respectively, and the effectof porosity on the stress–strain state of the bimetal is examined, allowing, in this manner, application of the method in relevant practical problemsin the field of bimetallics.© 2005 Elsevier B.V. All rights reserved.

Keywords: Bimetallics; Extrusion; Plasticity of porous materials

1. Introduction

Various manufacturing branches of modern technology in thefield of filters and heat-transfer elements widely use porous andbimetallic laminated-porous tubes (thermal tubes), produced bydifferent methods of powder metallurgy[1].

Tube extrusion and tube drawing without mandrel have beenstudied extensively for dense materials, in the frame of improv-ing mechanical properties and external surface quality of thedeformed workpiece[3]. Similarly, extensive activities seem tobe devoted to the design of bimetallics (rods and tubes) extrusion,made also of dense materials[4–9]. However, the processing ofporous tubular samples, which possess great practical and theo-retical interest, has attracted much less effort in the internationalliterature.

The present contribution is limited to investigate the initialstage of plastic deformation in the cylindrical chamber of theextrusion die of biomaterial tubes, composed by an outer pore-free component and an internal porous sintered one. An extendedformulation of the “slab method” of analysis is produced, taking

∗ Corresponding author.E-mail address: [email protected] (A.G. Mamalis).

into consideration a convenient yield function – developed bythe authors – which incorporates the stress field and porosity asthe main variables[1,2]. Numerical solution of the problem wasconducted for components made of same material – sintered andpore-free steel, respectively – and the effect of porosity on thestress–strain state of the bimetal is extensively examined.

2. Theoretical

Consider the case of a bimetallic tube, consisting of a fullycompacted external layer and a porous-sintered internal one,subjected to compression in the cylindrical part of an extru-sion die through the initial deformation stage of the process, seeFig. 1(a). The metallic layers are interconnected through inter-ference fit, allowing no clearance at their contact surface.

If r1 is the radius at the contact surface, under the above-described circumstances, it may be written for the incrementsof plastic deformation in the axial and circumferential direction,respectively:

dεm1 = dεm2 (axial direction) (1a)

dεθ1 = dεθ2 = 0 (circumferential direction) (1b)

0924-0136/$ – see front matter © 2005 Elsevier B.V. All rights reserved.doi:10.1016/j.jmatprotec.2005.09.018

278 A.G. Mamalis et al. / Journal of Materials Processing Technology 172 (2006) 277–282

Fig. 1. (a) Extrusion of a bimetallic tube; (b) stress state of the external solid tube; (c) stress state of the internal porous tube; (d) equilibrium ofa strip element ofexternal tube.

For establishing the regularities of the plastic deformationof the bimetal, the plasticity theory of real porous materials(PTRPM) is used[1]. According to this procedure, the concep-tions of equivalent stressσeqand equivalent increment of plasticdeformationdεeq are combined with the mass preservation lawand the flow law associated with the plasticity conditions, pro-viding, in this way, the following basic equations of PTRPM

[5]:

σeq = 1

βn+0.5 (σ2i + 9αmσ2

0)0.5

(2)

dεeq = βn+0.5

ρ

[(dεi)

2 + (dε0)2

αm

]0.5

(3)

A.G. Mamalis et al. / Journal of Materials Processing Technology 172 (2006) 277–282 279

dεij = 3ρ dεeq

2β2n+1σeq[σij − (1 − 2αm)δijσ0] (4)

df = 9(1− f )2αmσ0dεeq

β2n+1σeq(5)

whereσi is the intensity of stresses,σ0 the mean (hydrostatic)stress, dεi the intensity of the increments of plastic deformations,dε0 the mean strain increment,δij Kronecker’s symbol,f the realporosity,m, n parameters determined experimentally andα,βporosity functions evaluated by the following simple expressions[6]:

α = f and β = 1 − f (6)

2.1. Analysis of external layer

According to the principles of the theory of thin shells andtaking into account that the tangential stressesµpr are very small,the stress state of the outer layer – designated hereinafter by suf-fix 1 – may be expressed in terms of principal stress componentsas

σ1 = σr1 = −pr1, σ2 = σθ1, σ3 = σm1 (7)

whereσr, σθ, σm are the radial, circumferential and meridionalstress components, respectively, andpr1 the contact pressurebetween the layers, seeFig. 1(b).

On the basis of the PTRPM analysis, a system of equa-tions is obtained, which allows determination of the necessarystress–strain state of a porous material (internal layer) during itscompression in the cylindrical chamber of the extrusion tooling.Application of the formulation established may be also appliedto describe the same conditions for compact materials (externallayer) by assumingf = 0,α = 0, andβ = 1. Therefore, the follow-ing known equations for pore-free materials are produced:

σi = {12[(σ1 − σ2)2 + (σ2 − σ3)2 + (σ3 − σ1)2]}0.5

(8)

dεi = {29[(dε1 − dε2)2 + (dε2 − dε3)2 + (dε3 − dε1)2]}0.5

(9)

dεij = 3dεi

2σi

(σij − δijσ0) (10)

Replacing from Eq.(7) into Eq.(8) yields:

σi = {12[(pr1 + σθ1)2 + (σθ1 − σm1)2 + (σm1 + pr1)2]}0.5

with solution:

σθ1 = 12(σm1 − pr1 ± Q1) (11)

where

Q1 = [4σ2i − 3(σm1 + pr1)2]

0.5(12)

or taking into account the axial compression of the layer, i.e.σm1 =− pz1:

σθ1 = 12(−pz1 − pr1 − Q1) (13)

Substituting into Eq.(10) for the components of the incre-ment of plastic deformation and considering for the mean stress

for compact materials the relationσ0 = (σ1 +σ2 +σ3)/3, we havefor the increments of plastic deformations in the axial and cir-cumferential directions, respectively:

dεm1 = dεi

2σi

(2σm1 − σθ1 + pr1) (14a)

dεθ1 = dεi

2σi

(2σθ1 − σm1 + pr1) (14b)

or combining Eqs.(13) and (14):

dεm1 = dεi

4σi

[3(σm1 + pr1) + Q1] (15a)

dεθ1 = − dεi

2σi

Q1 (15b)

2.2. Analysis of internal layer

The stress state of the sintered-porous layer (internal tube,seeFig. 1c) – designated hereinafter by suffix 2 – is expressedin terms of principal stress components as

σ1 ≈ 0, σ2 = σθ2, σ3 = σm2 (16)

Substituting into the yield criterion(2) and following thesame procedure as for external tube, we obtain:

σeq ={

12[σ2

θ2 + (σθ2 − σm2)2 + σ2m2] + αm(σθ2 + σm2)2

}0.5

βn+0.5

(17)

and solving forσθ2:

σθ2 = (1 − 2αm)σm2 ± Q2

2(1+ αm)(18)

where

Q2 = [4β2n+1(1 + αm)σ2eq − 3(1+ 4αm)σ2

m2]0.5

(19)

or taking into account the axial compression of the layer, i.e.σm2 =− pz2:

σθ2 = −(1 − 2αm)pz2 − Q2

2(1+ αm)(20)

Substituting to the constitutive Eq.(4) and taking also intoaccount the stress conditions(16) for the internal layer, it isobtained for the deformation increments:

dεm2 = ρ dεeq

4β2n+1(1 + αm)σeq[3(1 + 4αm)σm2 + (1 − 2αm)Q2]

(21a)

dεθ2 = − ρ dεeq

2β2n+1σeqQ2 (21b)

Since the deformation increments for both layers, as predictedfrom Eqs.(15a), (15b), (21a) and (21b), must fulfill the contactconditions(1), we obtain successively:


dεm1 = dεm2 or combining Eqs.(15a) and (21a)

ρ dεeq

β2n+1(1 + αm)σeq[3(1 + 4αm)σm2 + (1 − 2αm)Q2

= dεi

σi

[3(σm1 + pr1) + Q1] (22)

dεθ1 = dεθ2 = 0 or taking into account Eqs.(15b) and (21b)areobtained:

Q1 = Q2 = 0 (23)

On the foundation of conditions(23)and taking into accountEqs.(12) and (19), yields:

σm1 = −pr1 − 2σi√3

(24)

σm2 = −2σeq

(β2n+1(1 + αm)

3(1+ 4αm)

)0.5

(25)

Substituting the values(23) into Eq.(22), yields:

ρ dεeq

β2n+1(1 + αm)σeq(1 + 4αm)σm2 = dεi

σi

(σm1 + pr1) (26)

From which is obtained:

dεi

dεeq= ρ(1 + 4αm)σiσm2

β2n+1(1 + αm)σeq(σm1 + pr1)(27)

By combining Eqs.(4) and (10), similar expressions for theradial deformation increment of each layer may be produced,i.e.:

dεr1 = − dεi

2σi

(2pr1 + σm1 + σθ1) (28a)

dεr2 = − ρ dεeq

2β2n+1σeq(1 − 2αm)(σm2 + σθ2) (28b)

From the consideration of equilibrium conditions for aninfinitesimal element of an axisymmetric shell, the followinggeneral expression is valid:

σm

rm+ σθ

rθ= −pv − pr

δ(29)

In terms of the internal tube characteristicspr = pr2 = 0:

σm2

rm2+ σθ2

rθ2= −pv2

δ2(30)

whererm is the radius of curvature of a meridional section,rθ

the radius of curvature of a cross-section perpendicular to themeridional arc,δ the shell thickness andpv is the intensity ofthe normal distributed load.

For the cylindrical internal layer is:rm2 =∞, rθ2 = r1 andpv2 = pr1 and, therefore, Eq.(30), taking also into account Eq.(20), reduces to the form:

pr1 = − δ2

2r1(1 + αm)(1 − 2αm)σm2 (31)

In terms of the external tube characteristics,rm = rm1 =∞,rθ = rθ1 = r1, δ= δ1, σθ =σθ1, pv = pv1, pr = pr1, and, therefore,from Eq.(29), taking also into account Eq.(13)are obtained:

pν1 = pr1 + δ1

2r1(pz1 + pr1) (32)

Considering the equilibrium of a ring element of the externallayer as shown inFig. 1(d) gives the following relation for theaxial stressσm1:

σm1 = σm10 + µpr1z

r1(33)

whereσm10 is an initial value of axial stress,µ the friction coeffi-cient in the contact area with the die,z the distance of the currentsection from the punch andδ1 is the thickness of external layer.

3. Results and discussion

The stress–strain curves of both – internal and external mate-rials – under uniaxial compression are needed for numericallyanalyzing the initial stage of plastic deformation. The relevantequations may be approximated by the following expressions,respectively:

External tube : σi = σy1 + A1

(∫dεi

)b1

(34a)

Internal tube : σeq = σy2 + A2

(∫dεeq

)b2

(34b)

whereσy1, σy2 are the yield stresses of dense materials andA1,A2, b1 andb2 material constants.

A sintered steel with chemical composition: 4% Ni, 1.5% Cuand 0.5% Mo was selected as fabrication material of internaltube, whilst a similar pore-free material was considered for theexternal layer. For this material, the following mechanical prop-erties were experimentally predicted[10]: σy1 =σy2 = 630 MPa,A1 = A2 = 750 MPa andb1 = b2 = 0.286.

Numerical results were obtained, assuming different initialporosity of internal layer, i.e.f0 = 0.05, 0.1, 0.15, 0.2, 0.25, 0.3,and porosity parametersm = 1.1 andn = 1.4.

To facilitate the numerical procedure, it is convenient toexpress the system of equations governing the problem in adimensionless form. For this purpose, the following dimension-less quantities are introduced:

σeq = σeq

σy

, σi = σi

σy

, σm = σm

σy

,

pr1 = pr1

σy

, σθ = σθ

σy

, pν1 = pν1

σy

resulting in the above stress–strain Eq.(34) to be written in theform:

σi = 1 + 1.19

(∫dεi

)0.286

(35a)

σeq = 1 + 1.19

(∫dεeq

)0.286

(35b)


Table 1Numerical results

No. f α β σm2 σθ2 pr1 σm1 σθ1 pv d = dεi/dεeq

1 0 0 1.00 1.155 0.578 0.039 1.194 0.620 0.101 1.0002 0.05 0.05 0.95 0.994 0.444 0.029 1.184 0.610 0.090 1.1823 0.10 0.10 0.90 0.856 0.334 0.022 1.177 0.600 0.082 1.4084 0.15 0.15 0.85 0.736 0.246 0.017 1.172 0.595 0.077 1.6795 0.20 0.20 0.80 0.63 0.178 0.012 1.167 0.590 0.071 1.9986 0.25 0.25 0.75 0.539 0.125 0.009 1.164 0.587 0.068 2.3997 0.30 0.30 0.70 0.459 0.161 0.006 1.161 0.580 0.064 2.892

Fig. 2. Variation of meridional stress with porosityf (1: σm1, 2: σm2).

Fig. 3. Variation of contact pressure with porosityf (1: pv1, 2: pr1).

Numerical results are tabulated inTable 1and representedgraphically inFigs. 2–5. From these graphs, the following inves-tigations on the initial plastic deformation of the bimetallic tubemay be drawn:

• Axial stressesσm1, σm2 and hoop stressesσθ1, σθ2 decreasewith increasing porosity f of internal layer. It must be notedthat the stress state in the axial and tangential directions seems

Fig. 4. Variation of circumferential stresses with porosityf (1: σθ1, 2: σθ2).

Fig. 5. Variation of ratiodεi/dεeq with porosityf.

to be more severe for internal tube as compared to that ofexternal layer.

• Contact pressures between layers of bimetal, ¯pr1 and betweenexternal layer and container, ¯pv1 also decrease with increasingporosityf of internal layer, but their values are very low.

• The external pore-free layer of bimetal seems to sustain muchhigher plastic deformation than the porous internal one, whichincreases with increasing porosity of the internal material.

4. Conclusions

A generalized “slab method” of analysis, properly modifiedto describe the plastic behaviour of a porous-sintered material,developed in combination with the theory of thin shells, in orderto investigate the initial stage of plastic deformation in the cylin-drical chamber of the extrusion die. The effect of porosity on thestress–strain state of the bimetal was examined, allowing, in thismanner, application of the method in relevant practical problemsin the field of bimetallics.

Numerical results were conducted for bimetallics of exten-sively tested materials for various levels of initial porosity ofinternal layer and graphs showing the influence of porosityon process parameters of valuable practical interest were con-structed.

References

[1] G.L. Petrossian, Plastic Deformation of Powder Materials, Metallurgy,Moscow, 1988.

[2] A.G. Mamalis, G.L. Petrossian, D.E. Manolakos, The effect of porosityand micro-defects on plastically deformed porous materials, J. Mater.Process. Technol. 96 (1999) 117.

[3] W. Johnson, P.B. Mellor, Engineering Plasticity, Van Nostrand ReinholdCo., London, 1973.


[4] J.L. Alcaraz, J.M. Martınez-Esnaola, J. Gil-Sevillano, An analyticalapproach to the stress field in the extrusion of bimetallic tubes, Int.J. Solids Struct. 33 (1996) 2075.

[5] J.L. Alcaraz, J. Gil-Sevillano, J.M. Martınez-Esnaola, A fracture condi-tion of the upper bound method for the extrusion of bimetallic tubes, J.Mater. Process. Technol. 61 (1996) 265.

[6] J.L. Alcaraz, J. Gil-Sevillano, Safety maps in bimetallic extrusions, J.Mater. Process. Technol. 60 (1996) 133.

[7] J.L. Alcaraz, J. Gil-Sevillano, An analysis of the extrusion of bimetallictubes by numerical simulation, Int. J. Mech. Sci. 38 (1996) 157.

[8] N.R. Chitkara, A. Aleem, Extrusion of axi-symmetric bi-metallictubes from solid circular billets: application of a generalized upperbound analysis and some experiments, Int. J. Mech. Sci. 43 (2001)2833.

[9] N.R. Chitkara, A. Aleem, Extrusion of axi-symmetric bi-metallic tubes:some experiments using hollow billets and application of a generalizedslab method of analysis, Int. J. Mech. Sci. 43 (2001) 2857.

[10] G.L. Petrossian, P. Beiss, P. Hambardzuyan, H. Badalyan, Peculiaritiesof deformation diagram construction of porous sintered samples fromsteel powder, Collect. Mater. Annu. Sci. Conf. SEUA 1 (2001) 209.

Journal of Materials Processing Technology 172 (2006) 243–248

Mathematical modelling of plastic deformation processes ofbimetallic tubes with porous-internal layer in conical dies

A.G. Mamalisa,∗, G.L. Petrosyanb,D.E. Manolakosa, A.F. Hambardzumyanb

a Department of Mechanical Engineering, National Technical University of Athens, Greeceb Department of Mechanics and Machine Science, State Engineering University of Armenia, Yerevan, Armenia

Received 6 July 2004; received in revised form 7 July 2005; accepted 27 September 2005

Abstract

The present paper deals with the mathematical analysis of the extrusion of bimetallic tubes, made of porous materials. The Plasticity Theoryof Real Porous Materials [G.L. Petrossian, Plastic Deformation of Powder Materials, Metallurgy, Moscow, 1988] is used for the establishmentof real regularities of bimetal’s plastic deformation and development of traditional technological schemes of its treatment. An algorithm and acomputer program were composed to integrate the derived system of differential equations. The model created allows revealing the influenceof initial porosity, applied loading, level of deformation, materials hardening and other technological parameters on the stress-strain state of theprocess. On the basis of numerical results obtained, graphs between components of stress-strain state of bimetal and porosity of internal layer aredrawn, which allow solving important technological problems.© 2005 Elsevier B.V. All rights reserved.

Keywords: Bimetallic tube; Extrusion; Porous material

1. Introduction

It is well known, that filtering and heat conducting porous-laminated bimetallic tubes are widely used in modern engineer-ing. Because of the technological peculiarities of powder tubesforming, the sintered tubes generally need an additional treat-ment: pressing (extruding) through the conical dies, which givesopportunity to reach given sizes.

Technological processes of tube extrusion and drawing with-out mandrel are very productive techniques giving improvedexternal surface quality and mechanical properties[2]. So,except the alteration of overall sizes the value of initial poros-ity changes, hardening of material and changing of mechanicalcharacteristics of samples must be taken into account duringforming.

There are some papers giving solutions on the extrusion ofbimetallic tubes and rods[3,4]. However, on the treatment of theextrusion of porous-monometallic and bimetallic tubes there areonly strictly limited quantity of solutions[1]. It is obvious that

∗ Corresponding author.E-mail address: [email protected] (A.G. Mamalis).

elaboration of adequate mathematical model on this topic willallow to research and reveal real regularities of samples exploita-tion and mechanical properties alteration during deformationprocess and to control the technological process purposefully.

2. Theoretical

Consider the extrusion of a bimetallic tube through a conicaldie, as shown inFig. 1. The tube consists of two layers, a free-pore external layer and a porous-sintered internal one, connectedthrough an interference fit, allowing no clearance at their contactsurface.

For establishing the regularities of the plastic deformation,the Plasticity Theory of Real Porous Materials (PTRPM)[1,5]is applied. According to this methodology, the conceptions ofequivalent stressσeq and equivalent strain incrementdεeq arecombined along with the mass preservation law and the flow-law associated with the plasticity conditions, providing, in thismanner, the following basic equations of the flow-theory of realporous materials[5]:

σeq = 1

βn+0.5 (σ2i + 9αmσ2

0)0.5

(1)

0924-0136/$ – see front matter © 2005 Elsevier B.V. All rights reserved.doi:10.1016/j.jmatprotec.2005.09.017


Fig. 1. Extrusion of a bimetallic tube.

dεeq = βn+0.5

ρ

(dε2

i + dε20

αm

)0.5

(2)

dεij = 3ρdεeq

2β2n+1σeq[σij − (1 − 2αm)δijσ0] (3)

df = 9αm(1 − f )2σ0dεeq

β2n+1σeq(4)

whereσi is the stress-intensity,σij components of stress-tensor,σ0 = δijσij/3 the mean stress, dεi the intensity of strain incre-ments, dεij components of strain tensor, dε0 = δijdεij/3 the meanstrain increment andδij Kronecker’s symbol.

The variation of porosityf is expressed by introducing simpleporosity functionsα andβ, given in Ref.[6] as:

α = f, β = 1 − f (5)

when f = 0 (α = 0, β = 1), the above formulation describes thebehaviour of pore-free-materials[7]. In the following, it isassumed that the equivalent stressσeq is a function of integralof equivalent strain increment

∫dεeq, which does not depend on

the stress-state type and the porosity of material, i.e.

σeq = F

(∫dεeq

)(6)

2.1. Analysis of internal layer

To theoretically investigate the technological parameters,which govern the forming mechanism of a bimetallic tubethrough a conical die, it is necessary to establish the stress-strainstate of both layers, taking simultaneously in to account theircontact conditions. With this respect, a theoretical model hasbeen developed[1,8], combining the principles of the classicalslab method and the theory of thin-shells.

According to this procedure and ignoring the effect of bend-ing moments, the equilibrium equations of a tube (shell) element,

Fig. 2. Stress-state of internal porous-tube.

see alsoFigs. 2 and 3, are written as:

d

dr1(σm2r1δ2) − σθ2δ2 + pm2r1

sinφ= 0 (7)

σm2

ρm+ σθ2

ρθ

= −pv2

δ2(8)

wherer is the radius of the circular cross-section,ρm the radiusof curvature of a meridional section,ρθ the radius of curvaturein a direction perpendicular to the meridional arc,δ2 the wallthickness of internal tube,φ the semi-apical angle of the die,σm2andσθ2 the meridional and circumferential stresses, respectively,pm2, p�2 = pr1 the vertical pressure in the meridional and normaldirections, respectively.

For a conical shell, it may be assumed thatρm =∞,ρθ = r1/cosφ and the strain increments in the circumferential(dεθ) and normal (dεv) directions are related with the incrementsof tube radius and wall thickness by the following expressions,respectively[8]

dεθ2 = dr1

r1, dεv2 = dδ2

δ2(9)

By introducing appropriate non-dimensional quantities, the fol-lowing equations, governing the plastic deformation of porous-

Fig. 3. Equilibrium of a shell element of internal tube.


layer when extruded through a conical die, are obtained:

dσm2

dr1= k[(1 − 2αm

2 )σm2 − Q2]

2r1(1 + αm2 )

− σm2[3(1 − 2αm2 )σm2 + (1 + 4αm

2 )Q2]

2r1(1 + αm2 )Q2

(10)

k = 1 + µ cotφ (11)

dδ2

dr1= δ2(3σm2 − Q2)(1 − 2αm

2 )

2r1(1 + αm2 )Q2

(12)

df2

dr1= −3αm

2 (1 − f )2(3σm2 − Q2)

r1(1 + αm2 )Q2

(13)

dεeq2

dr1= −2β2n+1

2 σeq2

ρ2r1Q2(14)

σθ2 = (1 − 2αm2 )σm2 − Q2

2(1+ αm2 )

(15)

dεm2 = 3ρ2dεeq

2β2n+1σeq

[σm2 − (1 − 2αm)(σm2 + σθ2)

3

](16)

dεθ2 = 3ρ2dεeq

2β2n+1σeq

[σθ2 − (1 − 2αm)(σm2 + σθ2)

3

](17)

Q2 =√

4β2n+12 (1 + αm

2 )σ2eq − 3(1+ 4αm

2 )σ2m2 (18)

pr1 = pv2 = − σθ2δ2cosφ

r1

δ20

r10(19)

pm2 = µpv2 (20)

wherer10 and δ20 are the initial values of tube initial radiusand wall thickness, respectively,µ is the friction coefficientat the layers interface,σm2 = σm2/σy2, σθ2 = σθ2/σy2, σeq2 =σeq2/σy2, Q2 = Q2/σy2, pv2 = pv2/σy2, pm2 = pm2/σy2, r1 =r1/r10, δ2 = δ2/δ20 andσy2 is the yield stress of internal layer’smaterial.

2.2. Analysis of external layer

External tube is treated as a thin-walled shell subjected toexternal (pv1) and internal (pr1) pressures (seeFig. 4). From theequilibrium of an infinitesimal shell element it is obtained:

σm1

ρm+ σθ1

ρθ

= −pv1 − pr1

δ1−(

1

ρm+ 1

ρθ

)pv1 + pr1

2(21)

For a conical die it is assumedρm =∞ andρθ = r1/cosφ. Takingalso into account, thatpm1 =µpv1 and combining Eqs.(7) and(21), yields:

r1dσm1

dr1+ σm1 + r1

δ1

dδ1

dr1σm1 − c1σθ1 + c2r1pr1

δ1= 0 (22)

where

c1 = 1 + 2µr1cotφ

2r1 + δ1cosφandc2 = µ(2r1 − δ1cosφ)

(2r1 + δ1cosφ)sinφ(23)

Fig. 4. Stress-state of external solid tube.

The variation of the applied meridional stress (forming pressure)over the length of the conical part may be easily drawn from Eq.(22)

dσm1

dr1= 1

r1

[c1σθ1 − c2r1pr1

δ1− σm1

(1 + r1

δ1

dδ1

dr1

)](24)

Using the validσ–dε relations for porous materials[5], we find

dεθ1 = 3

2

dεi

σi

(σθ1 − σ01) dεv1 = 3

2

dεi

σi

(−pr1 − σ01) (25)

where dεθ1, dεv1 the plastic strain increments in the circumfer-ential and normal directions, dεi the intensity of plastic deforma-tion increments,σi the stress-intensity,σ01 = (σm1 +σθ1−pr1)/3the mean stress.

Substituting forσ01 in Eq.(25)we obtain:

dεθ1 = 3

2

dεi

σi

(2σθ1 − σm1 + pr1)

dεv = 3

2

dεi

σi

(−2pr1 − σθ1 − σm1) (26)

or forming the ratio dεv/dεθ1 and taking also into account Eq.(9), finally yields:

−2pr1 − σθ1 − σm1

2σθ1 − σm1 + pr1= dδ1

δ1

r1

dr1(27)

Applying expression(27), it is possible to evaluate the variationof external wall thickness:

dδ1

dr1= −δ1(2pr1 + σθ1 + σm1)

r1(2σθ1 − σm1 + pr1)(28)

Equating the first equations of(9) and(26)gives:

dεi

2σi

(2σθ1 − σm1 + pr1) = dr1

r1


or finally

dεi

dr1= 2σi

r1(2σθ1 − σm1 + pr1)(29)

The exerted pressure on the tube by the die may be determinedby using Eq.(22), i.e.

σθ1

r1cosφ = −pv1 − pr1

δ1− cosφ

r1

pv1 + pr1

2(30)

Similarly, expression(30) after mathematical simplificationsgives:

2δ1σθ1cosφ = −pv1(2r1 + δ1cosφ) + pr1(2r1 − δ1cosφ)

or finally

pv1 = pr1(2r1 − δ1cosφ) − 2δ1σθ1cosφ

2r1 + δ1cosφ(31)

The yield condition for pore-free-materials[6], consideringsimultaneously thatσ1 ≡ σm1, σ2 ≡ σ�1, σ3 ≡ −pr1, is writtenin the form:

σi = 1√2

√(σm1 − σθ1)2 + (σθ1 + pr1)2 + (−pr1 − σm1)2

which reduces to the following square equation:

σ2θ1 + (pr1 − σm1)σθ1 + σ2

m1 + p2r1 − σ2

i + pr1σm1 = 0

with a discriminant

D = 4σ2i − 3p2

r1 − 3σ2m1 − 6pr1σm1

Considering the quantity

Q1 =√

D =√

4σ2i − 3p2

r1 − 3σ2m1 − 6pr1σm1 (32)

the circumferential stress is found equal to:σθ1 = (σm−pr1 ±Q1)/2.

The sign ofQ1 should be chosen from the condition of|σθ1|increase during the extrusion process, i.e.

σθ1 = σm − pr1 − Q1

2(33)

Eqs. (23), (24), (26), (28), (29), (31–33) may be expressedin dimensionless form by introducing the following notations:σm1 = σm1/σy1, σθ1 = σθ1/σy1, σi = σi/σy1, Q1 = Q1/σy1,δ1 = δ1/δ10, dδ1 = d(δ1/δ10), pv1 = pv1/σy1, whereσy1 is theyield limit of external layer’s material.

So, the following set of equations, describing the plasticdeformation of external layer is obtained:

dσm1 = dr1

r1

[c1σθ1 − c2r1pr1

δ1

1

k1− σm1

(1 + r1

δ1

dδ1

dr1

)](34)

c1 = 1 + 2µr1cotφ

2r1 + δ1k1cosφc2 = µ(2r1 − δ1k1cosφ)

(2r1 + δ1k1cosφ)sinφ(35)

dδ1

dr1= − δ1(2pr1 + σθ1 + σm1)

r1(2σθ1 − σm1 + pr1)(36)

dεi

dr1= 2σi

r(2σθ1 − σm1 + pr1)(37)

pv1 = pr1(2r1 − δ1k1cosφ) − 2δ1σθ1k1cosφ

2r1 + δ1k1cosφ(38)

σθ1 = σm1 − pr1 − Q1

2(39)

Q1 =√

4σ2i − 3σ2

m1 − 3p2r1 − 6pr1σm1 (40)

dεm1 = dεi

2σi

(2σm1 − σθ1 + pr1) (41)

dεθ1 = dεi

2σi

(2σθ1 − σm1 + pr1) (42)

where k1 = δ10/r10—geometrical coefficient of external thin-walled layer of bimetallic tube.

3. Results and Discussion

A computer program, based on the programming package“MATLAB”, was composed to numerically solve the set of equa-tions governing the extrusion of a bimetallic porous-tube througha conical die. The flow-chart of the procedure has been tabulatedin Fig. 5.

A sintered steel with chemical composition: 4% Ni, 1.5% Cuand 0.5% Mo was selected as fabrication material of internaltube, whilst a similar pore-free-material was considered for theexternal layer. Uniaxial compression of steel samples with dif-ferent initial porosity has been analyzed in Ref.[6]. Values ofporosity parametersm = 1.1 andn = 1.4 were also estimated andthe stress-strain curve for the porous material may be approxi-mated by the following expression:

σeq2 = σy2 + B2

(∫dεeq2

)b2

where σy2 = 630 MPa,B2 = 750 MPa,b2 = 0.286. Data of thesame material were also used for the external pore-free-layer(σy1 =σy2, B1 = B2, b1 = b2). Dimensionless stress-strain dia-grams for internal and external layers may be written in theform:

σeq = 1 + 1.19

(∫dεeq

)0.286

(43a)

σi = 1 + 1.19

(∫dεi

)0.286

(43b)

Numerical integration of obtained differential equations wasconducted under the following initial data:µ = 0.1, ϕ = 14◦,k1 = 0.1, k2 = 0.667,∆r1 = −0.02, f20 = 10%. In Ref.[8] thecomponents of bimetallic tube’s stress-strain state in the cylin-drical part of the extrusion die were calculated for non-hardeningmaterial. Using these data as initial conditions when tube entersthe conical part of the die, we findσm10 = −1.203, σm20 =−0.882, pr10 = 0.0227, dεi0/dεeq20= 1.408 and in case of


Fig. 5. Flow-chart of numerical integration procedure.

�εeq20= 0.01, ∆εi = 0.014, �εm10=−0.012, ��r10 = 0.012,�εθ10 = 0,�εθ20 = 0,�εm20=−0.012,�εr20 = 0.0078.

Fig. 6shows the stress-strain diagrams for both external andinternal materials. Solid line corresponds to the external pore-free-layer and dashed line to the internal porous-layer. It maybe drawn that external pore-free-tube deforms more intensivelythan the internal porous-layer.

Stress-state curves of bimetallic tube are presented inFig. 7.Analysis takes place until both meridional stresses approachzero. Diagrams of meridional stresses applied show that extin-guishing negativeσm2 develops, increasing in a lower rate com-pared with meridional componentσm1. Circumferential stresses

Fig. 6. Variation of equivalent strain (—— pore-free-layer, – – – porous-layer).

σθ1 andσθ2 are highly compressive. Initially, they decrease upto a minimum value and thereafter they slightly increase.

Since, all stresses are compressive, the porosity of internallayer decreases (i.e. voids close), seeFig. 8. It must be notedthat we may influence the variation of porosity by consideringdifferent initial parameters.

Strain components are described inFig. 9. Bimetallic tubeincreases in length (

∫dεm > 0), its diameter decreases (

∫dεθ <

0). This leads to the conclusion that wall thickness increases, seeFig. 10. From the same figure it can be seen that increasing of

Fig. 7. Variation of stress-components (—— pore-free-layer, – – – porous-layer).


Fig. 8. Variation of porosity.

Fig. 9. Variation of strain components (—— pore-free-layer, – – – porous-layer).

Fig. 10. Variation of tube wall (—— pore-free-layer, – – – porous-layer).

external tube’s wall thickness is more intensive than this one forinternal layer.

4. Conclusions

The theoretical analysis of bimetallics extrusion was car-ried out based on PTRPM principles. The model created allowsrevealing the influence of initial porosity, applied external, levelof materials’ hardening and other technological parameters onthe stress-strain state of the process.

The particularities of extrusion process and regularities ofstress-strain state components’ variation are revealed and com-parison between bimetallic tube’s internal and external lay-ers’ stress-strain state is realised. Regularities of stress-strainstate characteristics’ distribution show that the model createddescribes adequately real deformation process of bimetallic tubein a conical die. On the basis of numerical results obtained, thediagrams of stress-strain state components variations over the dielength were constructed, which give opportunity to solve techni-cally important technological problems. Calculations show, thatexternal pore-free-layer is under more “hard” stress-strain con-ditions than internal porous-layer and can serve as a protectivelayer for internal one.

References

[1] G.L. Petrossian, Plastic deformation of powder materials, Metallurgy,Moscow, 1988.

[2] W. Johnson, P.B. Mellor, Engineering Plasticity, Van Nostrand ReinholdCompany, London, 1973.

[3] J.L. Alcaraz, J.M. Martınez-Esnaola, J. Gil-Sevillano, An analyticalapproach to the stress-field in the extrusion of bimetallic tubes, Int. J.Solids Struct. 33 (1996) 2075.

[4] N.R. Chitkara, A. Aleem, Extrusion of axisymmetric bimetallic tubesfrom solid circular billets: application of a generalized upper bound anal-ysis and some experiments, Int. J. Mech. Sci. 43 (2001) 2833.

[5] A.G. Mamalis, G.L. Petrossian, D.E. Manolakos, The effect of porosityand micro-defects on plastically deformed porous materials, J. Mater.Process. Technol. 96 (1999) 117.

[6] G.L. Petrosyan, P. Beiss, A.F. Hambardzumyan, H.A. Badalyan, Peculiar-ities of deformation diagram construction of porous-sintered samples fromsteel powder. Collection of materials, in: Annual Scientific Conference ofSEUA, vol. 1, 2001, pp. 209–210.

[7] N.N. Malinin, The Applied Theory of Plasticity and Creep, Moscow,1975.

[8] A.G. Mamalis, G.L. Petrosyan, D.E. Manolakos, H.G. Petrosyan, Deter-mination of initial compression stresses in case of extruding bimetallictubes with porous-internal layer, J. Mater. Process. Technol. (in press).

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