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Axially symmetrical problems of the micropolar theory of plasticity
Inr .I Enp,(: SC! Vol. 23, No. 7. PP. 709-715. Pnnted in Great Britam
0020-7225/U $3.00 t .oO 0 1985 Pergamon Press Ltd.
1985
AXIALLY SYMMETRICAL PROBLEMS OF THE MICROPOLAR THEORY OF PLASTICITY
University
M. WAGROWSKA and 2. OLESIAK of Warsaw. Institute of Mechanics, Warsaw,
Poland
Abstract-We consider an elastic-plastic continuum with free rotations, and two partial yield conditions, namely shear yield and rotational yield. As a special case we discuss axially symmetric plane strain. The system of equations for plastic region has been reduced to a single differential equation. From the approximate solution we have obtained the plastic region and stress distributions.
INTRODUCTION THIS PAPER is concerned with axially symmetric problems of plane strain theory of micropolar plasticity. The theory of micropolar plasticity constitutes a generalization of both the classical theory of plasticity [2] and the moment stress theory. Here a fundamental problem arises, namely the number of yield conditions, and their form. For continua with constrained rotations the yield conditions have been discussed by S. Bodaszewski [I], M. Mi$cu [5], and A. Sawczuk [lo], and in a more general case of nonhomogeneous continuum by J. A. Kiinig and W. Olszak [3]. H. Lippmann [4] has formulated the theory of micropolar plasticity with free rotations of Cosserats’ type. In [9] we discuss a possible number of yield conditions and we introduce the notion of partial yielding, namely shear yielding and rotational one. The total yielding means the conjunction of both the partial ones. Since we consider an elastic-plastic continuum the constitutive equations have to be known in all four possible regions: elastic, shear plastic (SY), rotationally plastic (RY), and totally plastic. It is of interest when and how much the micropolarity of the continuum affects the extent of plastic regions and the distribution of stresses. In the case of plane strain axially symmetric region we arrive at a system of 12 differential and algebraic equations. This system can be reduced to a single, complicated ordinary differential equation. The equation, by reasonable simplifying assumption, can be reduced to Riccati’s equation (nonhomogeneous). As an example we considered a hollow cylinder subjected to an internal pressure and a distribution of body moments. For a certain set of numerical values we found the new extent of plastic zone, and the change in the distribution of stresses. 1. YIELD
We depart 4u=1
CONDITIONS
from the expression
AND
CONSTITUTIVE
for the strain
EQUATIONS
energy density
1 u(,)u(lj)
CL
+
;
a(
-
p(2P:
3h) uliukk 1 1 + ; mw(iJ) + ; P(ij)P(iJ)
-
r(27
P (1.1) + 3p) llkkPLIl
where u(~), u(~) denote the symmetric and skew-symmetric parts of the force stress tensor, similarly for the moment stress tensor pij = P(U) + P(C), II, X, (Y, y, /3, and E denote the material constants of a micropolar elastic body, sij, and mij are the components of the force stress and moment stress tensor deviators, respectively. We assume that the voluminal part of the strain energy density (or tr (au)) does not affect the yielding. We define (compare the discussion in [9]) the criterion of shear yield (SY)
709
M. WAGROWSKA
710
Up, =
and 2. OLESlAK
k; A U,, < L&
(1.2)
and the criterion of rotational yielding (RT) (1.3) The total yielding is the conjunction
of both the partial yieldings (1.4)
where
The criteria result from the invariant considerations, and are invariant with respect to any rigid body motion. Thus for the bodies with internal free rotations we may distinguish the following regions: (a) elastic, (b) shear plastic, rotationally elastic, (c) rotationally plastic, strain elastic, and (d) totally plastic. In Case (a) the constitutive equations of the micropolar elasticity with free rotations hold true (see, e.g. 161). In Case (b) we have the following constituti~e relations: 1 + P, 2r’(iJ,
=
-
I s(Ij) >
P
2r(ij)
=
i
S{ij),
where I”&are the components of the strain tensor yii deviator, K~denote the torsion tensor components, Pi is nondimensional constant for shear plasticity region. For Case (c) we obtain
2q
1 i-P2
= -
Y
where V2 is nondimensional In Case (d) we have
1 fP* L
wt + --
(l + +u12)p pkk& 38) ’ U’
&ij) - r(2r
(1.7)
constant for rotational plasticity region.
(1.8)
2. CONSTITUTIVE
EQUATIONS
FOR
AXIAL
SYMMETRY
Within the elastic region and the plane strain state these are the following nonvanishing components of stress tensor:
Axially symmetrical problems of the micropolar theory of plasticity 6, = C& + v&J
711
+ x ; %,
~*,=,(,,-~.)-*(~~,~+~~~)+2~~=.
(2.1)
In addition, we have x zi-= (Y - #z,r,
%z = (7 + W,,,.
The yield conditions reduce to the form, respectively 1 1 (SY) - (u& + trf + cr:_-- u,,cr,, - aeecr, - (JZZIJ~T) + - (us + @BP)2= k;, 6r 8~ WY) & Ifir + fiz,12 + ;
(I+: - EL,)~ = J%.
(2.2)
(2.3)
In a region of shear yielding we have the constitutive equations IL, = (7’ i- +bz,
Kzr= 0
3YV = y
(20, -
3y,, = 9
(2a, - CT,-
Uefj -
while in a region of rotational yielding
U,)
+
&
(U,
+
Ueee +
Uzz),
Uer)
+
&-g
(U,
+
Ueff +
Uz;),
(2.4)
712
M. W&ZROWSKA
and 2. OLESIAK
(2.5) In this case, the compatibility
equations reduce to the form
(2.6)
3. We
SPECIAL
CASES
shall discuss and compare the solutions of two problems, namely: (a) cylinder of a micropolar material with free rotations with internal pressure -p applied at r = a, ti&) = -p, (bf the same cylinder with internal pressure -p and body moments Y = (0, 0, Y=), where Y, = H In b r
A micropolar cylinder under tractions only does not know 17, 81 that its material is a micropolar one. The difference becomes evident when the body moments have been applied. The solution to problem (a) within the frames of elasticity reduces to Lam& solution. In the case of an elastic-plastic solid, proceeding exactly like in the case of the classical theory of plasticity, we find the extent of the plastic region and the value of the critical pressure for which the zones of plasticity start developing [2]. For the sake of convenience we introduce two new constants 3 and 4 resembling Young’s modulus and Poisson’s ratio of the classical elasticity. We do so in order to obtain certain estimations on the value of the material constants. We have
P = p(211 + 3X) = 2/.&(1-t Y), p+x
2V =: h PfA’
J(2y+3Qy(t+[) r+P
2[ =P
here we denote
3
u+P’
The solution to (b) for an elastic case takes the following form [7, 81:
Axially symmetrical problems of the micropolar theory of plasticity
713
where, taking into account the boundary conditions
we obtain
1 2cx bA2 = 5 Bz + (II + +2
2pae2B2 = -
v A
=
3
A 1 I,(+)
[.A,it(&
pa2bz 2pb2
-
a2’
+
B,
&(dQ
+ B, K,(&]
B3
1 ,
-I-
- f +
7 in ha) + FIn hb)
Pa2 = 2(fi i- A)@ -
a2) -
(3.3)
In the plastic zone we have to solve the following system of equations ~equilib~um equations, yield conditions, compatibility equations, and constitutive equations):
(3.4)
714
M. WAGROWSKA
and Z. OLESIAK
The preceding system of 12 differential and algebraic equations can be reduced to the following system of four equations: 2~~ + ram,, + prz,r + i
CL,
+
H In
! =0,
In turn the system of eqns (3.5) can be reduced to the single differential equation of the following form: 8k;j.L r’
Pc.rr
-+p,,, +f
j.LFtc +
4 D = clr; r3 > [ y+c + f
+ i ru,., + !yL - 7 - rH 4cu
)I312
(r2p(rz,r- rp,: - lr*H + 2D)2
,
(3.6)
where C and D denote the constants of integration. For o,,, P dnY,reqn (3.6) can be linearized to the following equation: = 0, (3.7)
where
The substitutions C = 2k&&,
a-’ = 4B,,
and s\l(y + c)B, = i(A, + B,r2), enable us to reduce eqn (3.7) to the following form: d*Jn + 3s2(y + t)B, - 3A, -_ d&t ds2 s[s2(y + t)B: - A,] ds +
and consequently
JN.
4A l DB: - NB, s2(y + t)& - A,]4
B A, B + B,[s’(y
+ t)Bf - A]
= 0,
(3.8)
to Riccati’s equation z, + 3.s2(y + e)& - 3Ar ; + -2 +1=0, s(.s’B:(y + e) - /i] -
(3.9)
Axially symmetrical problems of the micropolar theory of plasticity
715
Figure I
where
An approximate solution for (urr - (~~~1 4 ICT@ + ~~1, and uti + uor = const = A0 is shown in Fig. 1. We have assumed the following data: a = 0.3, b = 1.0, P = 0.67 * 106, y = 8.33. lo*, 7 = 2.2~ 10m2, A0 = -2.7. lo-‘, p = 2.1 *lo’, H = 10m7. Dotted lines denote the distribution of stresses for an elastic case, dashed lines for an elastic-plastic body without body moments. We see from the solution shown in Fig. 1 that even a small additional field of body moments acting on a micropolar body changes the extent of the shear plastic zone. Here we have shown the influence of micropolar properties on the extent of shear plastic region. In [ 1 I] an example is given that a shear yielding may exist, at least theoretically, exerted by body moments only. REFERENCES [I] S. BODASZEWSKI, 0 niesymetrycznym stanie napiecia i jego zastosowaniu w mechanice osrodkow ciaglych. Arch. Mech. Stos. 5, 3, 351-396 (1953). [2] L. M. KACHANOV, Foundufions ofthe theory of plasticify, North-Holland, Amsterdam (197 1). [3] J. A. KONIG and W. OLSZAK, The yield criterion in the general case of nonhomogeneous stress and deformation fields. in Topics in Applied Continuum Mechanics, (Edited by J. L. Zeman and F. Ziegler), 58-70. Springer-Verlag, New York 1974. [4] H. LIPPMANN, Eine Cosserat Theorie des plastischen Fliessens. Acta Mechanics 8, 255-284 (1969). [5] M. MISICU, On a theory of asymmetric plastic and viscoplastic solids. Met. Appt. 9, 477-495 (1964). [6] W. NOWACKI, Theory ofmicropolar elasticity. 25, Springer-Verlag, Udine (1970). [7] Z. OLESIAK and M. WAGROWSKA, Zagadnienia mikropolarnie sprezystej rmy grubosciennej. M.T. i S. 14, 493-503 (1976). [8] Z. OLESIAK and M. WAGROWSKA, Micropolar, elastic, thick-walled tube. Bull. de lilcad. Polon. Sci. (ser. IV) XXIV, I89- I96 (1976). [9] Z. OLESIAK and M. WAGROWSKA, On Shear and rotational yield, submitted for publication in Arch. Mech.
[lo] A. SAWCZUK, On yielding of Cosserat continua. Arch. Mech. Stos. 19, 471-480 (1967). [I l] M. WAGROWSKA, Osiowo symetryczne zagadnienia mikropolarnej teorii plastycznoSci, submitted for publication in M.T. i S. (Received 2 March 1984)
0020-7225/U $3.00 t .oO 0 1985 Pergamon Press Ltd.
1985
AXIALLY SYMMETRICAL PROBLEMS OF THE MICROPOLAR THEORY OF PLASTICITY
University
M. WAGROWSKA and 2. OLESIAK of Warsaw. Institute of Mechanics, Warsaw,
Poland
Abstract-We consider an elastic-plastic continuum with free rotations, and two partial yield conditions, namely shear yield and rotational yield. As a special case we discuss axially symmetric plane strain. The system of equations for plastic region has been reduced to a single differential equation. From the approximate solution we have obtained the plastic region and stress distributions.
INTRODUCTION THIS PAPER is concerned with axially symmetric problems of plane strain theory of micropolar plasticity. The theory of micropolar plasticity constitutes a generalization of both the classical theory of plasticity [2] and the moment stress theory. Here a fundamental problem arises, namely the number of yield conditions, and their form. For continua with constrained rotations the yield conditions have been discussed by S. Bodaszewski [I], M. Mi$cu [5], and A. Sawczuk [lo], and in a more general case of nonhomogeneous continuum by J. A. Kiinig and W. Olszak [3]. H. Lippmann [4] has formulated the theory of micropolar plasticity with free rotations of Cosserats’ type. In [9] we discuss a possible number of yield conditions and we introduce the notion of partial yielding, namely shear yielding and rotational one. The total yielding means the conjunction of both the partial ones. Since we consider an elastic-plastic continuum the constitutive equations have to be known in all four possible regions: elastic, shear plastic (SY), rotationally plastic (RY), and totally plastic. It is of interest when and how much the micropolarity of the continuum affects the extent of plastic regions and the distribution of stresses. In the case of plane strain axially symmetric region we arrive at a system of 12 differential and algebraic equations. This system can be reduced to a single, complicated ordinary differential equation. The equation, by reasonable simplifying assumption, can be reduced to Riccati’s equation (nonhomogeneous). As an example we considered a hollow cylinder subjected to an internal pressure and a distribution of body moments. For a certain set of numerical values we found the new extent of plastic zone, and the change in the distribution of stresses. 1. YIELD
We depart 4u=1
CONDITIONS
from the expression
AND
CONSTITUTIVE
for the strain
EQUATIONS
energy density
1 u(,)u(lj)
CL
+
;
a(
-
p(2P:
3h) uliukk 1 1 + ; mw(iJ) + ; P(ij)P(iJ)
-
r(27
P (1.1) + 3p) llkkPLIl
where u(~), u(~) denote the symmetric and skew-symmetric parts of the force stress tensor, similarly for the moment stress tensor pij = P(U) + P(C), II, X, (Y, y, /3, and E denote the material constants of a micropolar elastic body, sij, and mij are the components of the force stress and moment stress tensor deviators, respectively. We assume that the voluminal part of the strain energy density (or tr (au)) does not affect the yielding. We define (compare the discussion in [9]) the criterion of shear yield (SY)
709
M. WAGROWSKA
710
Up, =
and 2. OLESlAK
k; A U,, < L&
(1.2)
and the criterion of rotational yielding (RT) (1.3) The total yielding is the conjunction
of both the partial yieldings (1.4)
where
The criteria result from the invariant considerations, and are invariant with respect to any rigid body motion. Thus for the bodies with internal free rotations we may distinguish the following regions: (a) elastic, (b) shear plastic, rotationally elastic, (c) rotationally plastic, strain elastic, and (d) totally plastic. In Case (a) the constitutive equations of the micropolar elasticity with free rotations hold true (see, e.g. 161). In Case (b) we have the following constituti~e relations: 1 + P, 2r’(iJ,
=
-
I s(Ij) >
P
2r(ij)
=
i
S{ij),
where I”&are the components of the strain tensor yii deviator, K~denote the torsion tensor components, Pi is nondimensional constant for shear plasticity region. For Case (c) we obtain
2q
1 i-P2
= -
Y
where V2 is nondimensional In Case (d) we have
1 fP* L
wt + --
(l + +u12)p pkk& 38) ’ U’
&ij) - r(2r
(1.7)
constant for rotational plasticity region.
(1.8)
2. CONSTITUTIVE
EQUATIONS
FOR
AXIAL
SYMMETRY
Within the elastic region and the plane strain state these are the following nonvanishing components of stress tensor:
Axially symmetrical problems of the micropolar theory of plasticity 6, = C& + v&J
711
+ x ; %,
~*,=,(,,-~.)-*(~~,~+~~~)+2~~=.
(2.1)
In addition, we have x zi-= (Y - #z,r,
%z = (7 + W,,,.
The yield conditions reduce to the form, respectively 1 1 (SY) - (u& + trf + cr:_-- u,,cr,, - aeecr, - (JZZIJ~T) + - (us + @BP)2= k;, 6r 8~ WY) & Ifir + fiz,12 + ;
(I+: - EL,)~ = J%.
(2.2)
(2.3)
In a region of shear yielding we have the constitutive equations IL, = (7’ i- +bz,
Kzr= 0
3YV = y
(20, -
3y,, = 9
(2a, - CT,-
Uefj -
while in a region of rotational yielding
U,)
+
&
(U,
+
Ueee +
Uzz),
Uer)
+
&-g
(U,
+
Ueff +
Uz;),
(2.4)
712
M. W&ZROWSKA
and 2. OLESIAK
(2.5) In this case, the compatibility
equations reduce to the form
(2.6)
3. We
SPECIAL
CASES
shall discuss and compare the solutions of two problems, namely: (a) cylinder of a micropolar material with free rotations with internal pressure -p applied at r = a, ti&) = -p, (bf the same cylinder with internal pressure -p and body moments Y = (0, 0, Y=), where Y, = H In b r
A micropolar cylinder under tractions only does not know 17, 81 that its material is a micropolar one. The difference becomes evident when the body moments have been applied. The solution to problem (a) within the frames of elasticity reduces to Lam& solution. In the case of an elastic-plastic solid, proceeding exactly like in the case of the classical theory of plasticity, we find the extent of the plastic region and the value of the critical pressure for which the zones of plasticity start developing [2]. For the sake of convenience we introduce two new constants 3 and 4 resembling Young’s modulus and Poisson’s ratio of the classical elasticity. We do so in order to obtain certain estimations on the value of the material constants. We have
P = p(211 + 3X) = 2/.&(1-t Y), p+x
2V =: h PfA’
J(2y+3Qy(t+[) r+P
2[ =P
here we denote
3
u+P’
The solution to (b) for an elastic case takes the following form [7, 81:
Axially symmetrical problems of the micropolar theory of plasticity
713
where, taking into account the boundary conditions
we obtain
1 2cx bA2 = 5 Bz + (II + +2
2pae2B2 = -
v A
=
3
A 1 I,(+)
[.A,it(&
pa2bz 2pb2
-
a2’
+
B,
&(dQ
+ B, K,(&]
B3
1 ,
-I-
- f +
7 in ha) + FIn hb)
Pa2 = 2(fi i- A)@ -
a2) -
(3.3)
In the plastic zone we have to solve the following system of equations ~equilib~um equations, yield conditions, compatibility equations, and constitutive equations):
(3.4)
714
M. WAGROWSKA
and Z. OLESIAK
The preceding system of 12 differential and algebraic equations can be reduced to the following system of four equations: 2~~ + ram,, + prz,r + i
CL,
+
H In
! =0,
In turn the system of eqns (3.5) can be reduced to the single differential equation of the following form: 8k;j.L r’
Pc.rr
-+p,,, +f
j.LFtc +
4 D = clr; r3 > [ y+c + f
+ i ru,., + !yL - 7 - rH 4cu
)I312
(r2p(rz,r- rp,: - lr*H + 2D)2
,
(3.6)
where C and D denote the constants of integration. For o,,, P dnY,reqn (3.6) can be linearized to the following equation: = 0, (3.7)
where
The substitutions C = 2k&&,
a-’ = 4B,,
and s\l(y + c)B, = i(A, + B,r2), enable us to reduce eqn (3.7) to the following form: d*Jn + 3s2(y + t)B, - 3A, -_ d&t ds2 s[s2(y + t)B: - A,] ds +
and consequently
JN.
4A l DB: - NB, s2(y + t)& - A,]4
B A, B + B,[s’(y
+ t)Bf - A]
= 0,
(3.8)
to Riccati’s equation z, + 3.s2(y + e)& - 3Ar ; + -2 +1=0, s(.s’B:(y + e) - /i] -
(3.9)
Axially symmetrical problems of the micropolar theory of plasticity
715
Figure I
where
An approximate solution for (urr - (~~~1 4 ICT@ + ~~1, and uti + uor = const = A0 is shown in Fig. 1. We have assumed the following data: a = 0.3, b = 1.0, P = 0.67 * 106, y = 8.33. lo*, 7 = 2.2~ 10m2, A0 = -2.7. lo-‘, p = 2.1 *lo’, H = 10m7. Dotted lines denote the distribution of stresses for an elastic case, dashed lines for an elastic-plastic body without body moments. We see from the solution shown in Fig. 1 that even a small additional field of body moments acting on a micropolar body changes the extent of the shear plastic zone. Here we have shown the influence of micropolar properties on the extent of shear plastic region. In [ 1 I] an example is given that a shear yielding may exist, at least theoretically, exerted by body moments only. REFERENCES [I] S. BODASZEWSKI, 0 niesymetrycznym stanie napiecia i jego zastosowaniu w mechanice osrodkow ciaglych. Arch. Mech. Stos. 5, 3, 351-396 (1953). [2] L. M. KACHANOV, Foundufions ofthe theory of plasticify, North-Holland, Amsterdam (197 1). [3] J. A. KONIG and W. OLSZAK, The yield criterion in the general case of nonhomogeneous stress and deformation fields. in Topics in Applied Continuum Mechanics, (Edited by J. L. Zeman and F. Ziegler), 58-70. Springer-Verlag, New York 1974. [4] H. LIPPMANN, Eine Cosserat Theorie des plastischen Fliessens. Acta Mechanics 8, 255-284 (1969). [5] M. MISICU, On a theory of asymmetric plastic and viscoplastic solids. Met. Appt. 9, 477-495 (1964). [6] W. NOWACKI, Theory ofmicropolar elasticity. 25, Springer-Verlag, Udine (1970). [7] Z. OLESIAK and M. WAGROWSKA, Zagadnienia mikropolarnie sprezystej rmy grubosciennej. M.T. i S. 14, 493-503 (1976). [8] Z. OLESIAK and M. WAGROWSKA, Micropolar, elastic, thick-walled tube. Bull. de lilcad. Polon. Sci. (ser. IV) XXIV, I89- I96 (1976). [9] Z. OLESIAK and M. WAGROWSKA, On Shear and rotational yield, submitted for publication in Arch. Mech.
[lo] A. SAWCZUK, On yielding of Cosserat continua. Arch. Mech. Stos. 19, 471-480 (1967). [I l] M. WAGROWSKA, Osiowo symetryczne zagadnienia mikropolarnej teorii plastycznoSci, submitted for publication in M.T. i S. (Received 2 March 1984)