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A finite elastoplastic flow theory for porous media
Internattonttl Journal o f Plastiot.v, Vol. 4, pp. 301-316, 1988 Printed in the U.S.A.
0749-6419/88 $3.00 + .00 Copyright L 1988 Pergamon Press pie
A FINITE ELASTOPLASTIC FLOW THEORY FOR POROUS MEDIA
Y.K. LEE General Motors Research Laboratories
Abstract-A yield criterion and basic flow equations are developed for finite elastoplastic defor-
mation of porous materials. The yield criterion is derived using a specific micromechanical model of noninteracting spherical pores, with the matrix material being of the yon Mises type. The criterion satisfies the convexity requirement of plasticity theory and accounts for the effect of hydrostatic stress. The theoretical prediction of yield stress is in good agreement with published experimental data. The field equations for finite elastoplastic deformation of the porous media are obtained by generalizing those for plastically incompressible materials. The equations are derived in the form of a combined initial- and boundary-value problem whose spatial domain is the current configuration of the deformed body. They are distinguished by their quasi-linear nature. The theoretical model has two important features: 1. It shows an exclusive dependence of the yield surface of the porous medium on the properties of the constituent matrix material. 2. It shows that there is a porosity effect that itself leads to material instability. Since the theoretical formulation displays the role of dilatancy in controlling the instability of plastic flow, it is anticipated that the theory will lead to a better understanding of the material failure in ductile fracture.
I. INTRODUCTION
Although the advantages of powder metallurgy (P/M) forging have long been appreciated, the feasibility and cost effectiveness of this process have been recognized only recently [HALL& MOCA~KI, 1985]. Of all industries, automakers have been among the first to lead the way to utilize this process [Ktmr,Al~i, 1985]. Forged P / M parts serve in many areas of automobiles, such as transmissions, chassis, accessory mechanisms, and even engines. Basically, P / M forging consists of making a preform by compacting the powder, followed by a sintering operation, then forging the preform into a finished part. As sintered, the preform usually contains about 10 to 25°70 porosity. The purpose of the forging step in the P / M forging process is to reduce the porosity by proper metal flow and die filling, in order to produce a part having full density in critical areas. By proper preform and loading designs, one can have more efficient use of material and greater improvement in mechanical properties over conventional P / M parts. However, the study of deformation of porous materiais has been hampered by the complex behavior due to porosity changes in the process. Because the preform contains porosity, its deformation behavior is considerably different from that of conventional wrought metals. In addition, the deformation analysis for the porous medium cannot be represented by conventional plasticity models, such as that due to yon Mises, which do not predict the dependence of yielding on porosity. 301
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Y.K. LEE
The motivation of this work stems from the need for a criterion on workability during the deformation process. To start with, the current study aims to develop improved flow equations for modeling the elastoplastic deformation of porous materials. Earlier theories for describing the deformation behavior of porous material can be catalogued into two extremes. One is based on an assumption that the constituent matrix material is rigid, perfectly plastic [ K U H N & DOWNEY, 1971; GREEN, 1972; OYANE et al., 1973]. The results were derived from the discussion of physical behavior merged with a mechanics of materials approach. Their analyses were limited to simple stress states and not connected to the development of general constitutive models. The other theory is based on a spherical model [CARROLL& HOLT, 1972a,1972b; BUTCHERet al., 1974] for which the entire void containing material is idealized and modeled by a single hollow sphere. The model is restricted to hydrostatic loading. Lately, more complicated theories [GuRsON, 1977; TVERGAAD, 1982] assuming a rigid perfectly plastic matrix in a continuum model and accounting for both void nucleation and growth have been proposed. However, the choice of the criterion for void nucleation, whether plastic-strain control as used by Gurson and Tvergaad or stress control as proposed by ARGON et al. [1975a,1975b], is unsettled due to the statistical nature of the problem [THoMsoN & HANCOCK, 1984]. Apart from the aforementioned works, SPITZIG et al. [1975,1976] have extensively investigated the effect of hydrostatic pressure on the deformation behavior of certain types of steel. They have suggested a yield criterion that has a linear dependence of the second stress invariant on the mean stress but retains the assumption of constant density. Consequently, the measured values for the ratio of volume expansion to axial strain are less than one-fifteenth of that predicted by the normality flow rule [DRucKEa, 1951] in all cases. The discrepancy may be partly due to the generation of new dislocations and vacancies that are associated with permanent bulk-density changes. Their work surely shows the evidence of pressure dependence and dilatancy of plastic flow for fully dense metals due to the existence of microvoids even in a scale of several hundred angstroms. Immediately, the question raised here is: Should we consider metal as a fully dense material? This would depend on the scale of size of the element to which the theory applies. Unfortunately, it is very difficult to collocate different phenomenological origins (e.g., the micro and the macro events of material deformation) into a single theory. In spite of that, their theory does not predict the effect of elastic volume expansion occurring during deformation. In this report, we propose a yield criterion for porous materials, neglecting the effect of void nucleation. In addition, we also establish the field equations to characterize the behavior of the porous material undergoing quasistatic, isothermal, finite deformation. The analyses are based on an isotropic continuum model of void growth, in which the material ductility depends solely on the volume fraction of the void but not on the void size and spacing (i.e., no anisotropy is introduced due to void growth and the voids are regarded as isolated and noninteracting). Of course, this representation entails a few limitations, but these are common to other analyses [Race a TRACEY, 1969; BERG, 1970; RUDNICKI & RICE, 1975]. Based o n the argument of BISHOP& HILL [1951]*, Berg [1970] remarked that the normality flow rule also applies to porous aggregates if the matrix *Bishop and Hill derived the normality flow rule from Schmid's law for single crystals. They showed that the principle of m a x i m u m plastic resistance holds not only when the strain is due to slip on a single crystallographic system, but also for single crystals when strain occurs simultaneously on a number of systems and for whole aggregates of single crystals as well.
Finite elastoplasticflow theory
303
material is modeled locally by the same rule. This is relevant in the context developed in this article. The procedure to develop constitutive and equilibrium equations follows OstAs & SWEDLOW'S [1974] work* for the rate formulation for finite deformation of elastoplastic solids, in which the descriptions of continuum deformation and loading are developed in a flexed reference frame and the governing equations of the finite deformation problem are distinguished by their quasilinear nature. The hypothesis of DRucram [1951] as restated by Osias and Swedlow was used here as a foundation for developing flow equations. Nonetheless, it proves to he most powerful and provides the information we require. In addition to Drucker's hypothesis, we also assume the existence of a load function and the additivity of elastic and plastic strain increments.
II. YIELDCRITERION II. 1 Generalform The determination of an apparentt yield criterion for porous materials is troublesome because of its dependency on porosity, which changes during deformation. The essential macroscopic characteristics of an apparent yield criterion must reflect volume changes and the dependence of the yield surface on the hydrostatic stress. In addition, the problem is complicated by how the apparent yield criterion relates to that of the constituent matrix material through a suitable definition of yielding. To establish such a relationship, we postulate several basic assumptions: 1. The porous material is modeled as an isotropic continuum. 2. The elastic response of porous material is considered to be linear up to the yield point. 3. The matrix material is assumed to satisfy the normality flow rule based on the yon Mises yield condition. 4. The densification begins on matrix yielding. 5. The elastic modulus of the porous medium, E, can be predicted by the rule of mixtures upper bound [PAUL, 1960] (i.e., E = peru, because there is no restriction to the matrix deformation imposed by the voids, where Em is the elastic modulus of the matrix material at full density and p is the relative density of the porous medium). In the last assumption, the relative density is defined as the ratio of the density of the porous aggregate to that of matrix material and is an alternative expression for porosity. T h e value of o is bounded by 0 < p _< 1. The elastic deformation energy for the porous medium can be written as 1
U = ~ ave o
(I)
*Their work is limited to plastically imcompressible materials. tThe adjective "apparent" refers to average values of stress that represent the behavior of the total voided aggregate.
304
Y.K. LEE
where o 0 and eo are, respectively, the components of stress and strain tensors in a Cartesian coordinate system.* For uniaxial tension or compression in the xl direction, I
1 oi1
U= 2 O r l l e l l - 2 E
(2)
Suppose S~ is the corresponding tensile stress in the matrix. Then we can write 1 S?I 1 all 2 p E,,, - 2 E "
(3)
At yielding (i.e., when aIr reaches the apparent yield stress ay of the porous material, which is equivalent to saying that L'~I reaches the yield stress ro of the matrix material), we have
p
E,,
-
"
E
(4)
Equation (4) gives a clear definition of the yielding of porous materials. Making use o f E = pE,,, we have 1
ro = - . o~.
p
(5)
For subsequent yielding, the actual stress state in the matrix material can be represented by the equivalent flow stress r~q. Therefore, we write 1
r~q = - O~q P
(6)
where Creqis the apparent equivalent stress in the porous medium. It should be noted that subscripts eq are not indices. OYANE et al. [1973] reached the same expression as shown by eqn (6) using a unit-cell model based on the Levy-Mises flow rule [HrLL, 1950] which does not take the elastic strains into account. Consequently, no definition o f matrix yielding is given in their work. The relationship represented by eqn (6) is similar to CARROLL& HOLT'S [1972a] result in which they suggested that the relation between the compaction pressure in the porous aggregate, P, and the average pressure in t h e matrix material, P,,,, in a compaction process should differ by a factor p (i.e., P m = P / p ) . Therefore, eqn (6) applies to hydrostatic compression as well. The implication of eqn (6) is that, given the relative density o f porous medium and the yield stress o f constituent matrix, one can calculate the apparent yield stress of the porous medium at that density. Then the question is: W h a t should a suitable form be for ~eq? T o preserve the yield criterion at the fully dense condition as used in conventional *For simplicity, we use a Cartesian coordinate system here. However, the yield stress has the same value in any coordinate system.
Finite elastoplastic flow theory
305
plasticity [Fut~G, 1965a], a general form of the apparent yield criterion without the Bauschinger effect [MCCLIt~TOCKR, ARGOTS, 1966] is proposed as
where lj is the first invariant of the stress tensor and J2 is the second invariant of the stress deviator. The criterion represented by eqn (7) satisfies the convexity requirement of plasticity theory [Y~G, 1980] and produces the class of hardening behavior accounting for the effect of hydrostatic stress. At first glance, frepresents the degree of influence of the hydrostatic stress component on the onset of yielding of the porous material. In reality, f constitutes the effect of hydrostatic stress on the porosity of the material and can be explicitly expressed as a function of relative density. This can easily be seen from the next section. II.2 Determination of the function f To determine the function f, let us examine a pressure-density relationship in hydrostatic compaction loading, which is based on a specific micromechanical model of noninteracting spherical pores proposed by C~*dmLL • HOLT [1972b]. As they point out, there are three phases in a compaction process for an elastoplastic porous material. 1. An elastic phase during which any porosity change is recoverable as the load releases. 2. A transitional elastic-plastic phase during which the elastic-plastic interface propagates from the inner boundary around the pore to the outer boundary. 3. A fully plastic phase in which the pore collapses. Based on a spherical model in the context of elastic-perfect plasticity, Carroll and Holt obtained an approximate static pore-collapse relation by neglecting the change in porosity during the first two phases (i.e., assuming that the porosity remains unchanged until the pressure reaches a critical value and then follows plastic collapse for increasing loading) in which the pressure is shown as*
P= ~ Yln
(8)
where Y is the yield stress of the constituent matrix material and is equivalent to the deviatoric stress,t which is an induced shear stress when the void is subject to hydrostatic compression and is responsible for void collapse. Therefore, it includes a mechanism by which the dilatations would decrease and eventually cease as a result of shearing motion. If the expression (8) is used as a yield criterion in compaction, eqn (7) can then be written as *This result is integrated from the equilibrium equation which has no restriction on the magnitude of deformation. tln this case, therefore, it is intuitiveto specify the yield criterionin terms of octahedral shear stress [N~, 1937].
306
Y.K. LEE
(9) Note that the relation I1 = - 3 P has been used and Y is replaced by r~q. Finally, eqns (6) and (9) give
In eqn (10), therefore, the larger the relative density, the larger the value o f f becomes. When f = = (i.e., p = l), the yield surface represented by eqn (7) reduces to a cylinder that coincides with the yon Mises yield surface (i.e., O=q = req with Zeq = ~ ) . In this case, req is the octahedral shear stress. Therefore, a=q denotes the apparent octahedral shear stress for a porous medium. We shall replace treq and req by ao,. and to,., respectively, when we derive the flow equations later. To test the validity of eqn (7), we have computed the yield stress at different densities by using the experimental data for sintered copper obtained by StoMA ~, OYANE [1976] and then compared with their theoretical predictions. Shima and Oyane have proposed a yield criterion similar to eqns (6) and (7), but used p" instead o f p for the denominator in eqn (6) together with a different value of f derived from a curve-fitting technique that requires numerous test data. They concluded that n = 2.5 is the best fit for the data they presented. However, as shown in Fig. l, the distinction in accuracy between n = 2 and n = 2.5 is not obvious. For n = l, the distinction between eqn (7) and the theory o f Shima and Oyane lies on the different choice o f compaction model. Taking the yield stress of solid copper (p = 1) equal to 54 MPa as inferred from their data for the case o f n = l, we calculated the yield stress for p = 0.9, 0.8, and 0.7. The
60 •
so ~
]
Compression
o T~.~io°
Q.
~
8,-.q
'e'~ - - .
~
"0 ou
>-
20
'°o I 1.0
o
="
prese ~t
theory
I \% n=2.0 • n=2.5
O.8
O.6
Relative
Density
II
0.4
Fig. !. Relationshipsbetweenyield stress and relative density.
Finite elastoplastic flow theory
307
result is then shown by the solid line in Fig. 1. Good agreement with the published experimental data is a p p a r e n t at l e a s t when p > 0.8.
III. ELASTOPLASTICFLOW THEORY III. 1 Reference frame The total deformation is described in a fixed reference frame by the mapping*
x i = x i ( X J, t)
(11)
with det(xi, J) ~: 0. In eqn (11), x ~are s p a t i a l coordinates of material panicles comprising domain B at time t > to which were located in Bo at to. Choosing the spatial, or Eulerian, viewpoint, we can obtain a velocity field v i = vi(x j, t) and compute the velocity gradient
oi;j = diy + wij
(12)
where d~j is the symmetric deformation rate tensor and wij is the skew-symmetric spin tensor. They are given as 1 Wij = ~ (Ui; j -- Uj;i)
l
d,-~.= ~ (v,-;j + vj.),
(13)
III.2 Load function Treating the porous material as a continuous medium and considering its density change during deformation process, we take the loading function to be of the form?
F = ~(Ii,J2)
-
K(W(P)(P))
(14)
where F = 0, ~ > 0 during loading F = 0, ~ = 0 during neutral loading F = 0, ~ < 0 during unloading *Tensor notation follows O s ~ s ~ SWEDLOW'Swork [1974]. Covariant (contravarian0 character is denoted by subscript (superscript) indices; repeated indices in subscript-superscript pairs indicate summation over the range 1,2,3; a comma denotes partial and a semicolon covariant differentiation; x i and X "~are coordinates in a single, fixed, orthogonal curvilinear system ya, gij is the metric tensor o f y i ; 8~ is the Kronecker delta; and the overdot denotes the time rate of change (i.e., material derivative). tAssuming that such a function exists is tantamount to having said that the effect of the stress tensor on the yield funmion can be measured in terms of a scalar and, moreover, that the scalar is a measure of the degree of plastic straining. To the extent that one can determine the behavior experimentally, this is unobjectionable for most metallic alloys.
308
Y . K . LEE
and where KI~ = Jz =
apparent yield function o f porous material, strain-hardening function, a~, the first invariant of the stress tensor, oj, ! ~.j~,, q(~J the second invariant of the stress deviator, _, Sj - aj - ~a~.gj, and gji --- mixed metric tensor. In eqn (14), it is clear that the yield function dependence on the current stress state is restricted to the invariants of the stresses (i.e., I1 and J,_). The strain-hardening function is a scalar and depends solely on the prior plastic strain energy density W `v~, which is also an implicit function of the relative density of the porous medium. In a stress space, the apparent yield function is a measure of the vector from the origin to the point describing the stress field at a point in the body. If this measure is less than the scalar ~, no flow ensues. The plastic flow will alter x so that the stress vector will never pierce the apparent yield surface. To explain the scalar K, we need to use the assumption made in a previous s e c t i o n - t h a t the void reduction or enlargement may occur only when the stress state at the surrounding matrix material reaches its yield criterion. The implication of this statement is consistent with the fact that the elastic part of strain is infinitesimal and reversible. Since a unit volume o f a porous aggregate with a relative density n consists o f the matrix material of volume p, the rate of the plastic work ~VU,> done per unit volume o f the porous aggregate may be expressed as
W'"'(o )
=
(15)
=
where req is the equivalent flow stress of the matrix material, d ~ ') is the corresponding plastic strain rate also referring to the matrix material, a u is the component o f the Cauchy stress tensor, and d~ p) is the plastic component of the deformation rate. Note that both (p) and eq are not indices. When the porous aggregate reaches the fully dense condition, p becomes unity and eqn (15) reduces to the original form as used by Ostas SWEDI.OW [ 1974]. I I 1.3 D r u c k e r ' s h y p o t h e s i s
The relationship between the plastic strains and the yield function is given by the normality flow rule derived from DRtrC~ER'S [1951 ] postulate. For finite plastic deformation, the statement of Drucker's hypothesis is written as [OsIAs s, SwEDtOW, 1974] 6iJd~ n) > 0
(16)
where 6 e is the Jaumann stress rate given by biJ ~___biJ @ aim W mj -- tyJmWim
(17)
" Oau Ou = Ot
(18)
and Oou v k +
~x k
"
Finite elastoplastic flow theory
309
The Jaumann rate in eqn (16) is used to provide an objective measure of the change in stress viewed from a frame rotating with the material, as follows from the work of PRAGER [1961].* It was shown by OSIAS • SWEDtOW [1974], independent of any rotation, that eqn (16) requires that the work done by the external agency in producing d~ p~ is nonnegative. Therefore, the convexity of yield surface in the loading space, the normality of plastic strain increment, and the linearity for the relation between plastic strain rate and stress rate implied by DRtrCKER'S hypothesis for the infinitesimal case were well preserved in the formulation of finite elastoplastic flow. III.4 Flow equations The flow rule associated with eqn (14) can be derived as
d~ v~ = Nij~,tOtl
(19)
with Nijkl "~ ( l / K ' ) ( 6ijd~kl/(arsorS), O,nn ~" O~/otlmn = ~lg,nn + ~2S,~.gkn, ~t = 0 # / 0 I ~ ,
42 --O~/OJ2, and ~, = dK/dW. The value of x' can be determined by experiment as in a fully dense condition [SwEDtOW 1968]. Unfortunately, a deforming process usually involves nonhomogeneous deformation and the strain-hardening function of the porous materials depends on density• As a result, it may require many tests and interpolations in order to obtain a meaningful value of x'. This becomes a very big drawback for the development of a theory. To overcome this drawback, an appropriate expression for determining the value of x' needs to be established. Based on the functional dependence between req and d~p~ implied in eqn (15) for the matrix material, together with the quasistatic nature of the process, we can define the equivalent plastic shear modulus of the matrix material, /'/'eq
(20)
in which, e t~,~ = f,d~p~. Note that the equivalent plastic shear modulus depends solely on the equivalent stress (i.e.,/zJqp~ = #JqU~(req)), and can be derived from uniaxial tensile test data. Combining eqns (15) and (20) then gives ~'eq =
2lzJqp~o~Jd~.p~ p'Teq
(21)
The rate change of yield function can be written, from eqn (6), as -----ibl"eq + pi'eq. *A further discussion, also by Prager, on the choice of stress rates appears in PRAGER [1962].
(22)
310
Y . K . LEE
Substituting for req and ~'~q from eqns (15) and (21), respectively, gives 9- (p)
oiJdi}p ' -
dJ~ + - - r e q
(23)
where dn, - - p / o is the total dilation rate. Equation (23) gives the values for K'. The following important conclusions can be drawn from eqn (23): 1. The plastic flow of the porous material is determined by its porosity and the strainhardening character of the constituent matrix material. 2. The value of K', which determines the growth of the apparent yield surface, can be completely specified by conducting a single experiment on the constituent matrix material at full density, from which the equivalent plastic shear modulus is measured. 3. The kinematic part of pressure dependent yielding, represented by the first term of the right-hand side of eqn (23), derives not only from the effect of associated plastic dilatancy due to void growth (or contraction) but also from the basic flow mechanisms controlled by elastic constants. 4. During deformation the strain-hardening of the matrix material steadily grows. The role of the kinematic part in the process is different. During compression the porosity decreases (i.e., o > 0), constituting a strengthening factor. During extension the material becomes less dense (i.e., p < 0), and its porosity grows, the kinematic part in this case being a strain-softening factor. Combining eqns (19) and (23) yields the plastic flow rule in the following form
d(p) ij = [ 1/(2~t~qp) 7"eq
~dn~,loijO~, ) - - 6 k k ~rs 0 rs
(24)
The total strain rate comprises both elastic and plastic portions and we have so far dealt with the latter. Deriving from GREEN & NAGHDI'S [1965] work that the elastic strain component is defined as the difference between total and permanent deformation, OStAS [1972] has shown that the objective relation between the elastic deformation rate and the Jaumann stress rate is ditje) = M i j k l 6 kl
(25)
provided that the elastic strain is infinitesimal. In (25), M~jkl is defined as
Mijkt ~ ~
(gikgjt + gilgjk) -- 1 +"~ gijglct
(26)
in which gu is the metric tensor. The shear modulus # and Poisson's ratio v are defined in the elastic range as in classical linear isotropic elasticity [Tiuosrmr~Ko ~, GOODmR, 1970]. However, the shear modulus differs from that of the fully dense state and depends on the density of the porous aggregate. For consistency, we use/~ = P~m, where t~m is the elastic shear modulus of the constituent matrix material. Combining eqns (19) and (26) (i.e., using dij = d~e) + d~ p)) we have the total flow rule
21~d6 = BiyktOkl
(27)
Finite elastoplastic flow theory
311
with 1
v
Bsj~,l =- ~ (g,,gjl + g~t&k) -
1 +------~gijgkt + 13#PijdPkt
(28)
where 3 is defined as
I_ [ \
req
de(qP)/JdPrs a's"
(29)
The inverse form of eqn (29) can be written as
0ii = Diiktdkt
(30)
with
. ] D ijkl m I~ (gikgjl _{_ gilgjk) "4- 2____L__, (1 2~') gUgkt -
- 21Xq~abdPcdgaigb'ig ckgdl
(31)
1/f3 4- ~mndPrsgmrg ns
Note that utility of the last expression awaits a definite form of the apparent yield function ¢. the essential information needed for the material representation is the slope of the stress-plastic strain curve, as a function of stress. Just precisely which stress is needed depends on how we choose to present ¢. If we choose ¢ = aoc as represented by eqn (7), then 1
.
1
(32)
and
[
/3=3 2#/~ ro¢
~o~ ]J~r.a"s"
(33)
In eqn (33), the factor 3 is introduced because the octahedral variables are used in place of the equivalent quantities in eqn (29). Similar to eqn (20), the octahedral shear modulus of the matrix material/~P) is defined as 2 / ~ p) = ~o¢/.i,~p) and the octahedral plastic shear strain 3,~p) is found by integrating the rate of octahedral shear strain d~p~ with respect to time (i.e., ~o~fl' --ftd~U)). Under these definitions, the octahedral plastic shear modulus can be related to the equivalent plastic modulus by #~P~ = 3~qpJ. III.5 Stress rate equilibrium In equilibrium, it requires that the total force acting on a body must vanish:
fs
tidS = 0
(34)
312
Y . K . LEE
where t ~ are surface tractions along boundary S, and requires that the time rate of change of this integral also vanishes. As a result, the rate equilibrium equations are written as o• ; u/ -
J a:~ki v:~ = O.
(35)
O f course, eqn (35) includes no body force. The time rate of traction i ~ is found by directly differentiating the familiar Cauchy's condition [Fu~G, 1965b] on the boundary of the deforming body. The result is i' = ( o './+ o~/ n/ nx t,..~) - o': vi/~ ) n / .
(36)
Other than the traction boundary conditions specified by eqn (36), one may specify velocity boundary conditions as well. Apart from boundary conditions, initial conditions are required to complete a problem specification since both the constitutive and the equilibrium equations are given in terms o f rates. 111.6 F i e l d e q u a t i o n s The route to assemble the field equations is straightforward and is given by OsIas SWEDLOW [1974]. With eqns (13) and (17) inserted into eqn (30) and then proceeding to equations of equilibrium (36), we can arrive at coupled, second-order differential equation in terms of velocities. For convenient reference, the field equations are: ( ~ ) [ 8 ,i( 6 , ,/, g ,k - z' k / ,i ][o ......t,;l]:i + [Dijk%):l]:i + okpufpl = O. v , , g_/~,) - ,~,~, v ..... , g /i - 8,,6,,,g (37) Presuming knowledge of the stress field, these differential equations are intrinsically linear* at an instant of time. They provide a quasilinear model for the entire deformation process. One should note that the flow rate in the form shown loses meaning as K' --, 0 (i.e., no hardening response for the porous aggregate). However, both the inverse and the differential equations (37) go smoothly to the limit and are tractable. In a case when the plastic shear modulus o f the constituent matrix tZ~qp~ --, ~ (i.e., /3 = 0 and elastic loading only), the plastic flow D ijk~ in (37) reduces to the constant linear elastic form. The velocity equilibrium equations may then be immediately integrated with respect to time yielding the Navier displacement equations o f classical linear elasticity. This implies that the elastic state exists simultaneously for both the constituent matrix and the porous aggregate. Therefore, it justifies the second assumption proposed for the yield criterion in a previous section. IV. D I S C U S S I O N
For the small-strain deformation o f pore-free materials, it is known that as long as the elastic modulus is finite and the plastic behavior is characterized by work harden*Formally, the equations are said to be quasilinear in that nonlinearities appear in the coefficients of derivatives rather than the derivatives per se.
Finite elastoplastic flow theory
313
ing (i.e, g~qP) > 0), the field equations are elliptic [SwEDtOW, 1968]. In the event of nonwork hardening, however, the field equations can lose ellipticity and lead to an hyperbolic problem statement in the yield zone, depending on the specific problem and its degree of excitation [LEE ~ SWEDLOW, 1980; 1984a]. The occurrence of shear banding is then a consequence of the hyperbolicity of the field equations. In the context of finite deformation, the field equation is complicated by the terms involving stressvelocity gradient coupling in eqn (37), which results from the distinction between deformed and undeformed coordinates. The criterion for shear banding derived for small-strain deformation may not apply to finite deformation, and the effect of large strain in the deformation needs to be studied. In viewing the quasilinear nature of the field equations, the procedure to find the conditions for the shear bands to occur may follow either SWEDLOW[1968] for special cases in two-dimensional or Guo ~ SWEDLOW [1983] for general three-dimensional problems. Further work is needed to settle fully the mathematical details and we examine these matters in a subsequent paper. We now turn our attention to the physical interpretation of material or flow instability for porous materials. In viewing eqn (24), we observe that, for finite values of plastic shear modulus (i.e, #~ff) :- 0), there is a possibility that d~.p~ and #,t will not be uniquely related if dn, is nonnegative. The ratio of dilatation to distortion alone is involved in the instability. Regardless of the origins of the deformation mechanism, either shear or dilation, the material instability occurs when the strain-induced hardening reflected by a positive slope along the stress-strain curve of the matrix material in an increment of deformation is cancelled out by an accompanying strain-induced softening due to geometrical changes or changes in porosity. The onset of such instability can then be found by setting eqn (23) equal to zero. As a result, we have 7"eq
=
(38)
For physical reasons, eqn (38) must represent a maximum in dn,, beyond which instability grows under falling load. For the plastic flow to be stable, it requires r~q >
dn-"-~n
(39) "/'eq •
As long as the slope of the stress-strain curve for the porous materials remains positive (i.e., x' > 0), the inequality in eqn (39) will be observed. As a result, the dilatational region will grow by encroaching on neighboring material and propagate in a stable manner. As strain accumulates, the value of i',q/d,,,, is lowered so that softening cancels the hardening and leads to material instability. In some cases, depending on the stress states, the formation of shear bands can be a supplement to void growth, and the matrix material with a high hardening rate will act to delay initiation of shear bands but at the expense of void growth. We leave this issue to the subsequent paper. Regardless of the deformation mechanisms, there is a porosity effect that itself leads to material instability. At one extreme, the material condition alone is sufficient to initiate material instability if the matrix material behaves elastic-perfectly plastically (i.e., when/~,~) = 0). For example, the material instability may occur immediately at yield without void growth in a simple shear case because the fact that ann = 0 leads to K" = 0. The above discussion makes clear the importance of a high hardening rate relative to the flow stress
314
Y.K. LEE
for resisting the shear localization. In viewing eqn (24) again, as long as the value of /~P~ is positive, a proper choice of processing conditions during deformation, making dnn < 0 at all times, will prevent material failure from occurring. The inclusion of dilatancy and compressibility into the constitutive relations provides an insight about the effect of porosity on material instability. Without porosity, the material instability may not occur during the deformation when the matrix material is characterized as strain hardening, because the deforming material has to trace its own stress-strain curve. V. FINAL REMARK
We have posed a generalized yon Mises yield criterion based on the first and second stress invariants. The model incorporates both elastic and plastic dilatancy and hydrostatic stress dependence of yield criterion in a form intended to model the deformation behavior of porous materials. The degree of accuracy of the theoretical prediction on yield stress at different levels of porosity was encouraging. Following the development of yield criterion, we have also posed the finite elastoplastic deformation of porous medium in the form of an initial- and boundary-value problem whose spatial domain is the current configuration of the deformed body. Since the derivation of field equations is generalized from OSIAS & SWEDLOW'S [1974] work, the deformation process is governed by quasilinear equations whose solution requires a sequence of spatial integrations of the instantaneously linear equations. One of the most important features of this constitutive model is its capability of showing the important effect of dilatation on the initiation of material instability. This could lead to a better understanding of material failure in ductile fracture. Problem solving will usually entail numerical procedures. The issue then becomes identification of an appropriate method that will not be restricted by a particular material response (i.e., nonwork hardening). Recently, LEE 8, SWEDLOW[1980,1984a,b] have shown the means to present shear bands, in the context of a small strain formulation, as they grow from incipient yield to plastic collapse, together with a continuum field of elastoplastic deformation. Currently, this new computational ability is being generalized for the theory developed here. Acknowledgements-The author wishes to express his sincere appreciation to Professor J.L. Swedlow of the Department of Mechanical Engineering at Carnegie-Mellon University, for his conceptual counsel and fruitful discussion during the course of this work. He is also indebted to his colleagues Drs. D.A. Caulk and M.R. Barone, for their comments on various aspects of this endeavor. REFERENCES 1937 1950 1951
Nho,~, A., "Plastic Behavior of Metal in the Strain-Hardening Range. Part 1," J. Appl. Phys., 8, 205. HILL,R., The Mathematical Theory of Plasticity, p. 38, Clarendon Press, Oxford. BISHOP,J.F.W. and HILL, R., "A Theory of the Plastic Distortion of a Polyscrytalline Aggregate under Combined Stresses," Phil. Mag.., 42, 414. 1951 DXOCKER, D.C., "A More Fundamental Approach to Plastic Stress-Strain Relations," Proc. First U.S. Natl. Congr. Appl. Mech., p. 487. 1960 PAUL,B., "Prediction of Elastic Constants of Multiphase Materials," Trans. AIME, 2t9, 36. 1961 PRAGEI~,W., "An Elementary Discussion of Definitions of Stress Rate," Quart. AppL Math., 18, 403. 1962 PV.AOER,W., "On Higher Rates of Stress and Deformation," J. Mech. Phys. Solids, !0, 133. 1965ab FtmG, Y.C., Foundations of Solids Mechanics, a: Chapter 6, b: Chapter 3, Prentice-Hall, Englewood Cliffs, NJ. 1965 G ~ E s , A.E. and N^Gn~I, P.M., "A General Theory of an Elastic-Plastic Continuum," Arch. Rat. Mech. Anal., 18, 251.
Finite elastoplastic flow theory
1966
315
McCtINTOCK F.A. and ARGON,A.S., Mechanical Behavior of Materials, Sec. 5.8, Addison-Wesley, Reading, MA. 1968 SwEDLOW, J.L., "Character of the Equations of Elasto-Plastic Flow in Three Independent Variables," Int. J. Nonlin. Mech., 3, 325. 1969 RICE, J.R. and Tt~tCEY, D.M., "On the Ductile Enlargement of Voids in Triaxial Stress Fields," J. Mech. Phys. Solids, 17, 201. BERG, C.A., "Plastic Dilatation and Void Interaction," in Inelastic Behavior of Solids, eds. M.F. Kan1970 ninen, W.F. Adler, A.R. Rosenfield, and R.I. Jaffee, p. 171, McGraw-Hill, New York. TIMOSHE,~KO, S.P. and GOODtER, J.N., Theory of Elasticity, McGraw-Hill, New York. 1970 KUHN, H.A. and DOWNEr, C.L., "Deformation Characteristics and Plasticity Theory of Sintered 1971 Powder Materials," Int. J. Powder Metall., 7, 15. 1972a CARROLL, M.M. and Hotr, A.C., "Suggested Modification of the P-c~ Model for Porous Materials," J. Appl. Phys., 43, 759. 1972b CARROLL, M.M. and HOLT, A.C., "Static and Dynamic Pore-Collapse Relations for Ductile Porous Materials," J. Appl. Phys., 43, 1626. GREEts, R.J., "A Plasticity Theory for Porous Solids," Int. J. Mech. Sci., 14, 215. 1972 1972 OSIAS, J.R., "Finite Deformation of Elasto-Plastic Solids: The Example of Necking in Flat Tensile Bars," Ph.D. Thesis, Report SM-72-22, Dept. of Mech. Engrg. CIT, Carnegie-Mellon University, Pittsburgh, PA. OY~a,~l~,M., SHIMA,S. and Ko~o, Y., "Theory of Plasticity for Porous Metals," Bull. JPN Soc. Mech. 1973 Engrs., 16, 1254. BUTCHER, B.M., CARROLL,M.M. and HotT, A.C., "Shock-Wave Compaction of Porous Aluminum," 1974 J. Appl. Phys., 45, 3864. 1974 OstAs, J.R. and SWEDtOW, J.L., "Finite Elasto-Plastic Deformation-l, Theory and Numerical Examples," Int. J. Solids & Structs, 10, 321. 1975a ARGON, A.S., Ira, J. and N~FDtEraAN, A., "Distribution of Plastic Strain and Negative Pressure in Necked Steel and Copper Bars," Metall. Trans., 6A, 815. 1975b ARGO~, A.S., I~, J. and SArOGLU, R., "Cavity Formation from Inclusions in Ductile Fracture," Metall. Trans., 6A, 825. 1975 RUDNICXt, J.W. and RtcE, J.R., "Conditions for the Localization of Deformation in PressureSensitive Dilatant Materials," J. Mech. Phys. Solids, 23, 371. 1975 SPITZIG, W.A., SOBER, R.J. and RICHraOND,O., "Pressure Dependence of Yielding and Associated Volume Expansion in Tempered Martensite," Acta Met., 23, 885. 1976 SHtraA, S. and OrAN~, M., "Plasticity Theory for Porous Metals," Int. J. Mech. Sci., 18, 285. 1976 SPITZta, W.A., SOBER, R.J. and RICHMOND,O., "The Effect of Hydrostatic Pressure on the Deformation Behavior of Maraging and HY-80 Steels and its Implication for Plasticity Theory," Metall. Trans., 7A, 1703. 1977 GuxsoN, A.L., "Continuum Theory of Ductile Rupture by Void Nucleation and Growth: Part l Yield Criterion and Flow Rules for Porous Ductile Media," Trans. ASME J. Engng. Mater. & Technol., 99, 2. 1980 LE~:, Y.K. and SWEDtOW, J.L., "Numerical Analysis of Elliptic-Hyperbolic Plastic Flow in Torsion," J. Appl. Mech., 47, 283. 1980 Y~G, W.H., "A Generalized yon Mises Criterion for Yield and Fracture," J. Appl. Mech., 47, 297. 1982 TVERG.O,D, V., "On Localization in Ductile Materials Containing Spherical Voids," Int. J. Fract., 18, 237. 1983 Guo, R.M. and SWEDLOW,J.L., "On the Characterie Equations of Elasto-Plastic Flow," Int. J. Nonlin. Mech., 18, 321. 1984a L~:t:, Y.K. and SW~:DLOW,J.L., "Computation of Shear Bands Using A New Variational Approach, I: Theory and Elementary Examples," Int. J. Num. Meth. Engng., 20, 409. 1984b LEE, Y.K. and SW'EDLOW,J.L., "Computation of Shear Bands Using A New Variational Approach, I1: Use of Weight Function in a Crack Problem," Int. J. Num. Meth. Engng., 20, 423. 1984 THOMSON, R.D. and H.~d,rcoc~, I.W., "Ductile Failure by Void Nucleation, Growth, and Coalescence," Int. J. Fract., 26, 99. 1985 HALt, D.W. and Moc,~csEI, S., "Update on P/M Automotive Application," Int. J. Powder Metall. & Powder Technol., 21, 79. 1985 Ktn.gxxN1, K.M., " P / M Forging Moves into Volume Production," Math. Des. $7(14), 74.
Engineering Mechanics Department General Motors Research Laboratories Warren, MI 48090-9055, U.S.A.
(Received 2 May 1987)
316
Y . K . LyE
NOMENCLATURE elastic modulus of the porous medium
E f m
elastic modulus of the matrix material = shear modulus of the porous medium
btm
~--- shear
modulus of the matrix material
Poisson's ratio (p) (p) # e q ,/'to'.:
p
equivalent and octahedral plastic shear moduli of the matrix material = relative density of the porous medium
F
= load function
4'
= apparent yield function of the porous material
=
apparent strain hardening function
K
U
= elastic deformation energy for the porous medium
TO, Y
----- yield
stress of the matrix material
apparent yield stress of the porous material equivalent flow stress of the matrix material
Teq
d~ )
= rate of equivalent plastic strain in the matrix material
Toc
octahedral shear stress in the matrix material
dot p )
rate of octahedral plastic shear strain
Oeq
=
apparent equivalent stress in the porous medium apparent octahedral shear stress in the porous medium
O'oc
11
= first invariant of the stress tensor
"/2 f
= function
P
= compaction pressure in the porous aggregate
= second invariant of the stress deviator
P,n
= average pressure in the matrix material
t
= time
S
= surface
nj
= unit normal to surface
X i
= spatial coordinate
Xj
-- material coordinate
Ui
=
contravariant components of the velocity field
Ii
=
contravariant components of the traction
d,j, d~ e ~, ditjp' = components of the deformation rate tensor, elastic part, plastic part Wij
=
components of the spin tensor
wtP)
=
plastic work components of Cauchy's stress tensor
oiJ
components of the strain tensor
6ij
=
-'Pll
~--- axial
sj go O0
stress in the matrix
-- components of stress deviator = metric tensor = components of J a u m a n n stress rate tensor Kronecker delta
0749-6419/88 $3.00 + .00 Copyright L 1988 Pergamon Press pie
A FINITE ELASTOPLASTIC FLOW THEORY FOR POROUS MEDIA
Y.K. LEE General Motors Research Laboratories
Abstract-A yield criterion and basic flow equations are developed for finite elastoplastic defor-
mation of porous materials. The yield criterion is derived using a specific micromechanical model of noninteracting spherical pores, with the matrix material being of the yon Mises type. The criterion satisfies the convexity requirement of plasticity theory and accounts for the effect of hydrostatic stress. The theoretical prediction of yield stress is in good agreement with published experimental data. The field equations for finite elastoplastic deformation of the porous media are obtained by generalizing those for plastically incompressible materials. The equations are derived in the form of a combined initial- and boundary-value problem whose spatial domain is the current configuration of the deformed body. They are distinguished by their quasi-linear nature. The theoretical model has two important features: 1. It shows an exclusive dependence of the yield surface of the porous medium on the properties of the constituent matrix material. 2. It shows that there is a porosity effect that itself leads to material instability. Since the theoretical formulation displays the role of dilatancy in controlling the instability of plastic flow, it is anticipated that the theory will lead to a better understanding of the material failure in ductile fracture.
I. INTRODUCTION
Although the advantages of powder metallurgy (P/M) forging have long been appreciated, the feasibility and cost effectiveness of this process have been recognized only recently [HALL& MOCA~KI, 1985]. Of all industries, automakers have been among the first to lead the way to utilize this process [Ktmr,Al~i, 1985]. Forged P / M parts serve in many areas of automobiles, such as transmissions, chassis, accessory mechanisms, and even engines. Basically, P / M forging consists of making a preform by compacting the powder, followed by a sintering operation, then forging the preform into a finished part. As sintered, the preform usually contains about 10 to 25°70 porosity. The purpose of the forging step in the P / M forging process is to reduce the porosity by proper metal flow and die filling, in order to produce a part having full density in critical areas. By proper preform and loading designs, one can have more efficient use of material and greater improvement in mechanical properties over conventional P / M parts. However, the study of deformation of porous materiais has been hampered by the complex behavior due to porosity changes in the process. Because the preform contains porosity, its deformation behavior is considerably different from that of conventional wrought metals. In addition, the deformation analysis for the porous medium cannot be represented by conventional plasticity models, such as that due to yon Mises, which do not predict the dependence of yielding on porosity. 301
302
Y.K. LEE
The motivation of this work stems from the need for a criterion on workability during the deformation process. To start with, the current study aims to develop improved flow equations for modeling the elastoplastic deformation of porous materials. Earlier theories for describing the deformation behavior of porous material can be catalogued into two extremes. One is based on an assumption that the constituent matrix material is rigid, perfectly plastic [ K U H N & DOWNEY, 1971; GREEN, 1972; OYANE et al., 1973]. The results were derived from the discussion of physical behavior merged with a mechanics of materials approach. Their analyses were limited to simple stress states and not connected to the development of general constitutive models. The other theory is based on a spherical model [CARROLL& HOLT, 1972a,1972b; BUTCHERet al., 1974] for which the entire void containing material is idealized and modeled by a single hollow sphere. The model is restricted to hydrostatic loading. Lately, more complicated theories [GuRsON, 1977; TVERGAAD, 1982] assuming a rigid perfectly plastic matrix in a continuum model and accounting for both void nucleation and growth have been proposed. However, the choice of the criterion for void nucleation, whether plastic-strain control as used by Gurson and Tvergaad or stress control as proposed by ARGON et al. [1975a,1975b], is unsettled due to the statistical nature of the problem [THoMsoN & HANCOCK, 1984]. Apart from the aforementioned works, SPITZIG et al. [1975,1976] have extensively investigated the effect of hydrostatic pressure on the deformation behavior of certain types of steel. They have suggested a yield criterion that has a linear dependence of the second stress invariant on the mean stress but retains the assumption of constant density. Consequently, the measured values for the ratio of volume expansion to axial strain are less than one-fifteenth of that predicted by the normality flow rule [DRucKEa, 1951] in all cases. The discrepancy may be partly due to the generation of new dislocations and vacancies that are associated with permanent bulk-density changes. Their work surely shows the evidence of pressure dependence and dilatancy of plastic flow for fully dense metals due to the existence of microvoids even in a scale of several hundred angstroms. Immediately, the question raised here is: Should we consider metal as a fully dense material? This would depend on the scale of size of the element to which the theory applies. Unfortunately, it is very difficult to collocate different phenomenological origins (e.g., the micro and the macro events of material deformation) into a single theory. In spite of that, their theory does not predict the effect of elastic volume expansion occurring during deformation. In this report, we propose a yield criterion for porous materials, neglecting the effect of void nucleation. In addition, we also establish the field equations to characterize the behavior of the porous material undergoing quasistatic, isothermal, finite deformation. The analyses are based on an isotropic continuum model of void growth, in which the material ductility depends solely on the volume fraction of the void but not on the void size and spacing (i.e., no anisotropy is introduced due to void growth and the voids are regarded as isolated and noninteracting). Of course, this representation entails a few limitations, but these are common to other analyses [Race a TRACEY, 1969; BERG, 1970; RUDNICKI & RICE, 1975]. Based o n the argument of BISHOP& HILL [1951]*, Berg [1970] remarked that the normality flow rule also applies to porous aggregates if the matrix *Bishop and Hill derived the normality flow rule from Schmid's law for single crystals. They showed that the principle of m a x i m u m plastic resistance holds not only when the strain is due to slip on a single crystallographic system, but also for single crystals when strain occurs simultaneously on a number of systems and for whole aggregates of single crystals as well.
Finite elastoplasticflow theory
303
material is modeled locally by the same rule. This is relevant in the context developed in this article. The procedure to develop constitutive and equilibrium equations follows OstAs & SWEDLOW'S [1974] work* for the rate formulation for finite deformation of elastoplastic solids, in which the descriptions of continuum deformation and loading are developed in a flexed reference frame and the governing equations of the finite deformation problem are distinguished by their quasilinear nature. The hypothesis of DRucram [1951] as restated by Osias and Swedlow was used here as a foundation for developing flow equations. Nonetheless, it proves to he most powerful and provides the information we require. In addition to Drucker's hypothesis, we also assume the existence of a load function and the additivity of elastic and plastic strain increments.
II. YIELDCRITERION II. 1 Generalform The determination of an apparentt yield criterion for porous materials is troublesome because of its dependency on porosity, which changes during deformation. The essential macroscopic characteristics of an apparent yield criterion must reflect volume changes and the dependence of the yield surface on the hydrostatic stress. In addition, the problem is complicated by how the apparent yield criterion relates to that of the constituent matrix material through a suitable definition of yielding. To establish such a relationship, we postulate several basic assumptions: 1. The porous material is modeled as an isotropic continuum. 2. The elastic response of porous material is considered to be linear up to the yield point. 3. The matrix material is assumed to satisfy the normality flow rule based on the yon Mises yield condition. 4. The densification begins on matrix yielding. 5. The elastic modulus of the porous medium, E, can be predicted by the rule of mixtures upper bound [PAUL, 1960] (i.e., E = peru, because there is no restriction to the matrix deformation imposed by the voids, where Em is the elastic modulus of the matrix material at full density and p is the relative density of the porous medium). In the last assumption, the relative density is defined as the ratio of the density of the porous aggregate to that of matrix material and is an alternative expression for porosity. T h e value of o is bounded by 0 < p _< 1. The elastic deformation energy for the porous medium can be written as 1
U = ~ ave o
(I)
*Their work is limited to plastically imcompressible materials. tThe adjective "apparent" refers to average values of stress that represent the behavior of the total voided aggregate.
304
Y.K. LEE
where o 0 and eo are, respectively, the components of stress and strain tensors in a Cartesian coordinate system.* For uniaxial tension or compression in the xl direction, I
1 oi1
U= 2 O r l l e l l - 2 E
(2)
Suppose S~ is the corresponding tensile stress in the matrix. Then we can write 1 S?I 1 all 2 p E,,, - 2 E "
(3)
At yielding (i.e., when aIr reaches the apparent yield stress ay of the porous material, which is equivalent to saying that L'~I reaches the yield stress ro of the matrix material), we have
p
E,,
-
"
E
(4)
Equation (4) gives a clear definition of the yielding of porous materials. Making use o f E = pE,,, we have 1
ro = - . o~.
p
(5)
For subsequent yielding, the actual stress state in the matrix material can be represented by the equivalent flow stress r~q. Therefore, we write 1
r~q = - O~q P
(6)
where Creqis the apparent equivalent stress in the porous medium. It should be noted that subscripts eq are not indices. OYANE et al. [1973] reached the same expression as shown by eqn (6) using a unit-cell model based on the Levy-Mises flow rule [HrLL, 1950] which does not take the elastic strains into account. Consequently, no definition o f matrix yielding is given in their work. The relationship represented by eqn (6) is similar to CARROLL& HOLT'S [1972a] result in which they suggested that the relation between the compaction pressure in the porous aggregate, P, and the average pressure in t h e matrix material, P,,,, in a compaction process should differ by a factor p (i.e., P m = P / p ) . Therefore, eqn (6) applies to hydrostatic compression as well. The implication of eqn (6) is that, given the relative density o f porous medium and the yield stress o f constituent matrix, one can calculate the apparent yield stress of the porous medium at that density. Then the question is: W h a t should a suitable form be for ~eq? T o preserve the yield criterion at the fully dense condition as used in conventional *For simplicity, we use a Cartesian coordinate system here. However, the yield stress has the same value in any coordinate system.
Finite elastoplastic flow theory
305
plasticity [Fut~G, 1965a], a general form of the apparent yield criterion without the Bauschinger effect [MCCLIt~TOCKR, ARGOTS, 1966] is proposed as
where lj is the first invariant of the stress tensor and J2 is the second invariant of the stress deviator. The criterion represented by eqn (7) satisfies the convexity requirement of plasticity theory [Y~G, 1980] and produces the class of hardening behavior accounting for the effect of hydrostatic stress. At first glance, frepresents the degree of influence of the hydrostatic stress component on the onset of yielding of the porous material. In reality, f constitutes the effect of hydrostatic stress on the porosity of the material and can be explicitly expressed as a function of relative density. This can easily be seen from the next section. II.2 Determination of the function f To determine the function f, let us examine a pressure-density relationship in hydrostatic compaction loading, which is based on a specific micromechanical model of noninteracting spherical pores proposed by C~*dmLL • HOLT [1972b]. As they point out, there are three phases in a compaction process for an elastoplastic porous material. 1. An elastic phase during which any porosity change is recoverable as the load releases. 2. A transitional elastic-plastic phase during which the elastic-plastic interface propagates from the inner boundary around the pore to the outer boundary. 3. A fully plastic phase in which the pore collapses. Based on a spherical model in the context of elastic-perfect plasticity, Carroll and Holt obtained an approximate static pore-collapse relation by neglecting the change in porosity during the first two phases (i.e., assuming that the porosity remains unchanged until the pressure reaches a critical value and then follows plastic collapse for increasing loading) in which the pressure is shown as*
P= ~ Yln
(8)
where Y is the yield stress of the constituent matrix material and is equivalent to the deviatoric stress,t which is an induced shear stress when the void is subject to hydrostatic compression and is responsible for void collapse. Therefore, it includes a mechanism by which the dilatations would decrease and eventually cease as a result of shearing motion. If the expression (8) is used as a yield criterion in compaction, eqn (7) can then be written as *This result is integrated from the equilibrium equation which has no restriction on the magnitude of deformation. tln this case, therefore, it is intuitiveto specify the yield criterionin terms of octahedral shear stress [N~, 1937].
306
Y.K. LEE
(9) Note that the relation I1 = - 3 P has been used and Y is replaced by r~q. Finally, eqns (6) and (9) give
In eqn (10), therefore, the larger the relative density, the larger the value o f f becomes. When f = = (i.e., p = l), the yield surface represented by eqn (7) reduces to a cylinder that coincides with the yon Mises yield surface (i.e., O=q = req with Zeq = ~ ) . In this case, req is the octahedral shear stress. Therefore, a=q denotes the apparent octahedral shear stress for a porous medium. We shall replace treq and req by ao,. and to,., respectively, when we derive the flow equations later. To test the validity of eqn (7), we have computed the yield stress at different densities by using the experimental data for sintered copper obtained by StoMA ~, OYANE [1976] and then compared with their theoretical predictions. Shima and Oyane have proposed a yield criterion similar to eqns (6) and (7), but used p" instead o f p for the denominator in eqn (6) together with a different value of f derived from a curve-fitting technique that requires numerous test data. They concluded that n = 2.5 is the best fit for the data they presented. However, as shown in Fig. l, the distinction in accuracy between n = 2 and n = 2.5 is not obvious. For n = l, the distinction between eqn (7) and the theory o f Shima and Oyane lies on the different choice o f compaction model. Taking the yield stress of solid copper (p = 1) equal to 54 MPa as inferred from their data for the case o f n = l, we calculated the yield stress for p = 0.9, 0.8, and 0.7. The
60 •
so ~
]
Compression
o T~.~io°
Q.
~
8,-.q
'e'~ - - .
~
"0 ou
>-
20
'°o I 1.0
o
="
prese ~t
theory
I \% n=2.0 • n=2.5
O.8
O.6
Relative
Density
II
0.4
Fig. !. Relationshipsbetweenyield stress and relative density.
Finite elastoplastic flow theory
307
result is then shown by the solid line in Fig. 1. Good agreement with the published experimental data is a p p a r e n t at l e a s t when p > 0.8.
III. ELASTOPLASTICFLOW THEORY III. 1 Reference frame The total deformation is described in a fixed reference frame by the mapping*
x i = x i ( X J, t)
(11)
with det(xi, J) ~: 0. In eqn (11), x ~are s p a t i a l coordinates of material panicles comprising domain B at time t > to which were located in Bo at to. Choosing the spatial, or Eulerian, viewpoint, we can obtain a velocity field v i = vi(x j, t) and compute the velocity gradient
oi;j = diy + wij
(12)
where d~j is the symmetric deformation rate tensor and wij is the skew-symmetric spin tensor. They are given as 1 Wij = ~ (Ui; j -- Uj;i)
l
d,-~.= ~ (v,-;j + vj.),
(13)
III.2 Load function Treating the porous material as a continuous medium and considering its density change during deformation process, we take the loading function to be of the form?
F = ~(Ii,J2)
-
K(W(P)(P))
(14)
where F = 0, ~ > 0 during loading F = 0, ~ = 0 during neutral loading F = 0, ~ < 0 during unloading *Tensor notation follows O s ~ s ~ SWEDLOW'Swork [1974]. Covariant (contravarian0 character is denoted by subscript (superscript) indices; repeated indices in subscript-superscript pairs indicate summation over the range 1,2,3; a comma denotes partial and a semicolon covariant differentiation; x i and X "~are coordinates in a single, fixed, orthogonal curvilinear system ya, gij is the metric tensor o f y i ; 8~ is the Kronecker delta; and the overdot denotes the time rate of change (i.e., material derivative). tAssuming that such a function exists is tantamount to having said that the effect of the stress tensor on the yield funmion can be measured in terms of a scalar and, moreover, that the scalar is a measure of the degree of plastic straining. To the extent that one can determine the behavior experimentally, this is unobjectionable for most metallic alloys.
308
Y . K . LEE
and where KI~ = Jz =
apparent yield function o f porous material, strain-hardening function, a~, the first invariant of the stress tensor, oj, ! ~.j~,, q(~J the second invariant of the stress deviator, _, Sj - aj - ~a~.gj, and gji --- mixed metric tensor. In eqn (14), it is clear that the yield function dependence on the current stress state is restricted to the invariants of the stresses (i.e., I1 and J,_). The strain-hardening function is a scalar and depends solely on the prior plastic strain energy density W `v~, which is also an implicit function of the relative density of the porous medium. In a stress space, the apparent yield function is a measure of the vector from the origin to the point describing the stress field at a point in the body. If this measure is less than the scalar ~, no flow ensues. The plastic flow will alter x so that the stress vector will never pierce the apparent yield surface. To explain the scalar K, we need to use the assumption made in a previous s e c t i o n - t h a t the void reduction or enlargement may occur only when the stress state at the surrounding matrix material reaches its yield criterion. The implication of this statement is consistent with the fact that the elastic part of strain is infinitesimal and reversible. Since a unit volume o f a porous aggregate with a relative density n consists o f the matrix material of volume p, the rate of the plastic work ~VU,> done per unit volume o f the porous aggregate may be expressed as
W'"'(o )
=
(15)
=
where req is the equivalent flow stress of the matrix material, d ~ ') is the corresponding plastic strain rate also referring to the matrix material, a u is the component o f the Cauchy stress tensor, and d~ p) is the plastic component of the deformation rate. Note that both (p) and eq are not indices. When the porous aggregate reaches the fully dense condition, p becomes unity and eqn (15) reduces to the original form as used by Ostas SWEDI.OW [ 1974]. I I 1.3 D r u c k e r ' s h y p o t h e s i s
The relationship between the plastic strains and the yield function is given by the normality flow rule derived from DRtrC~ER'S [1951 ] postulate. For finite plastic deformation, the statement of Drucker's hypothesis is written as [OsIAs s, SwEDtOW, 1974] 6iJd~ n) > 0
(16)
where 6 e is the Jaumann stress rate given by biJ ~___biJ @ aim W mj -- tyJmWim
(17)
" Oau Ou = Ot
(18)
and Oou v k +
~x k
"
Finite elastoplastic flow theory
309
The Jaumann rate in eqn (16) is used to provide an objective measure of the change in stress viewed from a frame rotating with the material, as follows from the work of PRAGER [1961].* It was shown by OSIAS • SWEDtOW [1974], independent of any rotation, that eqn (16) requires that the work done by the external agency in producing d~ p~ is nonnegative. Therefore, the convexity of yield surface in the loading space, the normality of plastic strain increment, and the linearity for the relation between plastic strain rate and stress rate implied by DRtrCKER'S hypothesis for the infinitesimal case were well preserved in the formulation of finite elastoplastic flow. III.4 Flow equations The flow rule associated with eqn (14) can be derived as
d~ v~ = Nij~,tOtl
(19)
with Nijkl "~ ( l / K ' ) ( 6ijd~kl/(arsorS), O,nn ~" O~/otlmn = ~lg,nn + ~2S,~.gkn, ~t = 0 # / 0 I ~ ,
42 --O~/OJ2, and ~, = dK/dW
(20)
in which, e t~,~ = f,d~p~. Note that the equivalent plastic shear modulus depends solely on the equivalent stress (i.e.,/zJqp~ = #JqU~(req)), and can be derived from uniaxial tensile test data. Combining eqns (15) and (20) then gives ~'eq =
2lzJqp~o~Jd~.p~ p'Teq
(21)
The rate change of yield function can be written, from eqn (6), as -----ibl"eq + pi'eq. *A further discussion, also by Prager, on the choice of stress rates appears in PRAGER [1962].
(22)
310
Y . K . LEE
Substituting for req and ~'~q from eqns (15) and (21), respectively, gives 9- (p)
oiJdi}p ' -
dJ~ + - - r e q
(23)
where dn, - - p / o is the total dilation rate. Equation (23) gives the values for K'. The following important conclusions can be drawn from eqn (23): 1. The plastic flow of the porous material is determined by its porosity and the strainhardening character of the constituent matrix material. 2. The value of K', which determines the growth of the apparent yield surface, can be completely specified by conducting a single experiment on the constituent matrix material at full density, from which the equivalent plastic shear modulus is measured. 3. The kinematic part of pressure dependent yielding, represented by the first term of the right-hand side of eqn (23), derives not only from the effect of associated plastic dilatancy due to void growth (or contraction) but also from the basic flow mechanisms controlled by elastic constants. 4. During deformation the strain-hardening of the matrix material steadily grows. The role of the kinematic part in the process is different. During compression the porosity decreases (i.e., o > 0), constituting a strengthening factor. During extension the material becomes less dense (i.e., p < 0), and its porosity grows, the kinematic part in this case being a strain-softening factor. Combining eqns (19) and (23) yields the plastic flow rule in the following form
d(p) ij = [ 1/(2~t~qp) 7"eq
~dn~,loijO~, ) - - 6 k k ~rs 0 rs
(24)
The total strain rate comprises both elastic and plastic portions and we have so far dealt with the latter. Deriving from GREEN & NAGHDI'S [1965] work that the elastic strain component is defined as the difference between total and permanent deformation, OStAS [1972] has shown that the objective relation between the elastic deformation rate and the Jaumann stress rate is ditje) = M i j k l 6 kl
(25)
provided that the elastic strain is infinitesimal. In (25), M~jkl is defined as
Mijkt ~ ~
(gikgjt + gilgjk) -- 1 +"~ gijglct
(26)
in which gu is the metric tensor. The shear modulus # and Poisson's ratio v are defined in the elastic range as in classical linear isotropic elasticity [Tiuosrmr~Ko ~, GOODmR, 1970]. However, the shear modulus differs from that of the fully dense state and depends on the density of the porous aggregate. For consistency, we use/~ = P~m, where t~m is the elastic shear modulus of the constituent matrix material. Combining eqns (19) and (26) (i.e., using dij = d~e) + d~ p)) we have the total flow rule
21~d6 = BiyktOkl
(27)
Finite elastoplastic flow theory
311
with 1
v
Bsj~,l =- ~ (g,,gjl + g~t&k) -
1 +------~gijgkt + 13#PijdPkt
(28)
where 3 is defined as
I_ [ \
req
de(qP)/JdPrs a's"
(29)
The inverse form of eqn (29) can be written as
0ii = Diiktdkt
(30)
with
. ] D ijkl m I~ (gikgjl _{_ gilgjk) "4- 2____L__, (1 2~') gUgkt -
- 21Xq~abdPcdgaigb'ig ckgdl
(31)
1/f3 4- ~mndPrsgmrg ns
Note that utility of the last expression awaits a definite form of the apparent yield function ¢. the essential information needed for the material representation is the slope of the stress-plastic strain curve, as a function of stress. Just precisely which stress is needed depends on how we choose to present ¢. If we choose ¢ = aoc as represented by eqn (7), then 1
.
1
(32)
and
[
/3=3 2#/~ ro¢
~o~ ]J~r.a"s"
(33)
In eqn (33), the factor 3 is introduced because the octahedral variables are used in place of the equivalent quantities in eqn (29). Similar to eqn (20), the octahedral shear modulus of the matrix material/~P) is defined as 2 / ~ p) = ~o¢/.i,~p) and the octahedral plastic shear strain 3,~p) is found by integrating the rate of octahedral shear strain d~p~ with respect to time (i.e., ~o~fl' --ftd~U)). Under these definitions, the octahedral plastic shear modulus can be related to the equivalent plastic modulus by #~P~ = 3~qpJ. III.5 Stress rate equilibrium In equilibrium, it requires that the total force acting on a body must vanish:
fs
tidS = 0
(34)
312
Y . K . LEE
where t ~ are surface tractions along boundary S, and requires that the time rate of change of this integral also vanishes. As a result, the rate equilibrium equations are written as o• ; u/ -
J a:~ki v:~ = O.
(35)
O f course, eqn (35) includes no body force. The time rate of traction i ~ is found by directly differentiating the familiar Cauchy's condition [Fu~G, 1965b] on the boundary of the deforming body. The result is i' = ( o './+ o~/ n/ nx t,..~) - o': vi/~ ) n / .
(36)
Other than the traction boundary conditions specified by eqn (36), one may specify velocity boundary conditions as well. Apart from boundary conditions, initial conditions are required to complete a problem specification since both the constitutive and the equilibrium equations are given in terms o f rates. 111.6 F i e l d e q u a t i o n s The route to assemble the field equations is straightforward and is given by OsIas SWEDLOW [1974]. With eqns (13) and (17) inserted into eqn (30) and then proceeding to equations of equilibrium (36), we can arrive at coupled, second-order differential equation in terms of velocities. For convenient reference, the field equations are: ( ~ ) [ 8 ,i( 6 , ,/, g ,k - z' k / ,i ][o ......t,;l]:i + [Dijk%):l]:i + okpufpl = O. v , , g_/~,) - ,~,~, v ..... , g /i - 8,,6,,,g (37) Presuming knowledge of the stress field, these differential equations are intrinsically linear* at an instant of time. They provide a quasilinear model for the entire deformation process. One should note that the flow rate in the form shown loses meaning as K' --, 0 (i.e., no hardening response for the porous aggregate). However, both the inverse and the differential equations (37) go smoothly to the limit and are tractable. In a case when the plastic shear modulus o f the constituent matrix tZ~qp~ --, ~ (i.e., /3 = 0 and elastic loading only), the plastic flow D ijk~ in (37) reduces to the constant linear elastic form. The velocity equilibrium equations may then be immediately integrated with respect to time yielding the Navier displacement equations o f classical linear elasticity. This implies that the elastic state exists simultaneously for both the constituent matrix and the porous aggregate. Therefore, it justifies the second assumption proposed for the yield criterion in a previous section. IV. D I S C U S S I O N
For the small-strain deformation o f pore-free materials, it is known that as long as the elastic modulus is finite and the plastic behavior is characterized by work harden*Formally, the equations are said to be quasilinear in that nonlinearities appear in the coefficients of derivatives rather than the derivatives per se.
Finite elastoplastic flow theory
313
ing (i.e, g~qP) > 0), the field equations are elliptic [SwEDtOW, 1968]. In the event of nonwork hardening, however, the field equations can lose ellipticity and lead to an hyperbolic problem statement in the yield zone, depending on the specific problem and its degree of excitation [LEE ~ SWEDLOW, 1980; 1984a]. The occurrence of shear banding is then a consequence of the hyperbolicity of the field equations. In the context of finite deformation, the field equation is complicated by the terms involving stressvelocity gradient coupling in eqn (37), which results from the distinction between deformed and undeformed coordinates. The criterion for shear banding derived for small-strain deformation may not apply to finite deformation, and the effect of large strain in the deformation needs to be studied. In viewing the quasilinear nature of the field equations, the procedure to find the conditions for the shear bands to occur may follow either SWEDLOW[1968] for special cases in two-dimensional or Guo ~ SWEDLOW [1983] for general three-dimensional problems. Further work is needed to settle fully the mathematical details and we examine these matters in a subsequent paper. We now turn our attention to the physical interpretation of material or flow instability for porous materials. In viewing eqn (24), we observe that, for finite values of plastic shear modulus (i.e, #~ff) :- 0), there is a possibility that d~.p~ and #,t will not be uniquely related if dn, is nonnegative. The ratio of dilatation to distortion alone is involved in the instability. Regardless of the origins of the deformation mechanism, either shear or dilation, the material instability occurs when the strain-induced hardening reflected by a positive slope along the stress-strain curve of the matrix material in an increment of deformation is cancelled out by an accompanying strain-induced softening due to geometrical changes or changes in porosity. The onset of such instability can then be found by setting eqn (23) equal to zero. As a result, we have 7"eq
=
(38)
For physical reasons, eqn (38) must represent a maximum in dn,, beyond which instability grows under falling load. For the plastic flow to be stable, it requires r~q >
dn-"-~n
(39) "/'eq •
As long as the slope of the stress-strain curve for the porous materials remains positive (i.e., x' > 0), the inequality in eqn (39) will be observed. As a result, the dilatational region will grow by encroaching on neighboring material and propagate in a stable manner. As strain accumulates, the value of i',q/d,,,, is lowered so that softening cancels the hardening and leads to material instability. In some cases, depending on the stress states, the formation of shear bands can be a supplement to void growth, and the matrix material with a high hardening rate will act to delay initiation of shear bands but at the expense of void growth. We leave this issue to the subsequent paper. Regardless of the deformation mechanisms, there is a porosity effect that itself leads to material instability. At one extreme, the material condition alone is sufficient to initiate material instability if the matrix material behaves elastic-perfectly plastically (i.e., when/~,~) = 0). For example, the material instability may occur immediately at yield without void growth in a simple shear case because the fact that ann = 0 leads to K" = 0. The above discussion makes clear the importance of a high hardening rate relative to the flow stress
314
Y.K. LEE
for resisting the shear localization. In viewing eqn (24) again, as long as the value of /~P~ is positive, a proper choice of processing conditions during deformation, making dnn < 0 at all times, will prevent material failure from occurring. The inclusion of dilatancy and compressibility into the constitutive relations provides an insight about the effect of porosity on material instability. Without porosity, the material instability may not occur during the deformation when the matrix material is characterized as strain hardening, because the deforming material has to trace its own stress-strain curve. V. FINAL REMARK
We have posed a generalized yon Mises yield criterion based on the first and second stress invariants. The model incorporates both elastic and plastic dilatancy and hydrostatic stress dependence of yield criterion in a form intended to model the deformation behavior of porous materials. The degree of accuracy of the theoretical prediction on yield stress at different levels of porosity was encouraging. Following the development of yield criterion, we have also posed the finite elastoplastic deformation of porous medium in the form of an initial- and boundary-value problem whose spatial domain is the current configuration of the deformed body. Since the derivation of field equations is generalized from OSIAS & SWEDLOW'S [1974] work, the deformation process is governed by quasilinear equations whose solution requires a sequence of spatial integrations of the instantaneously linear equations. One of the most important features of this constitutive model is its capability of showing the important effect of dilatation on the initiation of material instability. This could lead to a better understanding of material failure in ductile fracture. Problem solving will usually entail numerical procedures. The issue then becomes identification of an appropriate method that will not be restricted by a particular material response (i.e., nonwork hardening). Recently, LEE 8, SWEDLOW[1980,1984a,b] have shown the means to present shear bands, in the context of a small strain formulation, as they grow from incipient yield to plastic collapse, together with a continuum field of elastoplastic deformation. Currently, this new computational ability is being generalized for the theory developed here. Acknowledgements-The author wishes to express his sincere appreciation to Professor J.L. Swedlow of the Department of Mechanical Engineering at Carnegie-Mellon University, for his conceptual counsel and fruitful discussion during the course of this work. He is also indebted to his colleagues Drs. D.A. Caulk and M.R. Barone, for their comments on various aspects of this endeavor. REFERENCES 1937 1950 1951
Nho,~, A., "Plastic Behavior of Metal in the Strain-Hardening Range. Part 1," J. Appl. Phys., 8, 205. HILL,R., The Mathematical Theory of Plasticity, p. 38, Clarendon Press, Oxford. BISHOP,J.F.W. and HILL, R., "A Theory of the Plastic Distortion of a Polyscrytalline Aggregate under Combined Stresses," Phil. Mag.., 42, 414. 1951 DXOCKER, D.C., "A More Fundamental Approach to Plastic Stress-Strain Relations," Proc. First U.S. Natl. Congr. Appl. Mech., p. 487. 1960 PAUL,B., "Prediction of Elastic Constants of Multiphase Materials," Trans. AIME, 2t9, 36. 1961 PRAGEI~,W., "An Elementary Discussion of Definitions of Stress Rate," Quart. AppL Math., 18, 403. 1962 PV.AOER,W., "On Higher Rates of Stress and Deformation," J. Mech. Phys. Solids, !0, 133. 1965ab FtmG, Y.C., Foundations of Solids Mechanics, a: Chapter 6, b: Chapter 3, Prentice-Hall, Englewood Cliffs, NJ. 1965 G ~ E s , A.E. and N^Gn~I, P.M., "A General Theory of an Elastic-Plastic Continuum," Arch. Rat. Mech. Anal., 18, 251.
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1966
315
McCtINTOCK F.A. and ARGON,A.S., Mechanical Behavior of Materials, Sec. 5.8, Addison-Wesley, Reading, MA. 1968 SwEDLOW, J.L., "Character of the Equations of Elasto-Plastic Flow in Three Independent Variables," Int. J. Nonlin. Mech., 3, 325. 1969 RICE, J.R. and Tt~tCEY, D.M., "On the Ductile Enlargement of Voids in Triaxial Stress Fields," J. Mech. Phys. Solids, 17, 201. BERG, C.A., "Plastic Dilatation and Void Interaction," in Inelastic Behavior of Solids, eds. M.F. Kan1970 ninen, W.F. Adler, A.R. Rosenfield, and R.I. Jaffee, p. 171, McGraw-Hill, New York. TIMOSHE,~KO, S.P. and GOODtER, J.N., Theory of Elasticity, McGraw-Hill, New York. 1970 KUHN, H.A. and DOWNEr, C.L., "Deformation Characteristics and Plasticity Theory of Sintered 1971 Powder Materials," Int. J. Powder Metall., 7, 15. 1972a CARROLL, M.M. and Hotr, A.C., "Suggested Modification of the P-c~ Model for Porous Materials," J. Appl. Phys., 43, 759. 1972b CARROLL, M.M. and HOLT, A.C., "Static and Dynamic Pore-Collapse Relations for Ductile Porous Materials," J. Appl. Phys., 43, 1626. GREEts, R.J., "A Plasticity Theory for Porous Solids," Int. J. Mech. Sci., 14, 215. 1972 1972 OSIAS, J.R., "Finite Deformation of Elasto-Plastic Solids: The Example of Necking in Flat Tensile Bars," Ph.D. Thesis, Report SM-72-22, Dept. of Mech. Engrg. CIT, Carnegie-Mellon University, Pittsburgh, PA. OY~a,~l~,M., SHIMA,S. and Ko~o, Y., "Theory of Plasticity for Porous Metals," Bull. JPN Soc. Mech. 1973 Engrs., 16, 1254. BUTCHER, B.M., CARROLL,M.M. and HotT, A.C., "Shock-Wave Compaction of Porous Aluminum," 1974 J. Appl. Phys., 45, 3864. 1974 OstAs, J.R. and SWEDtOW, J.L., "Finite Elasto-Plastic Deformation-l, Theory and Numerical Examples," Int. J. Solids & Structs, 10, 321. 1975a ARGON, A.S., Ira, J. and N~FDtEraAN, A., "Distribution of Plastic Strain and Negative Pressure in Necked Steel and Copper Bars," Metall. Trans., 6A, 815. 1975b ARGO~, A.S., I~, J. and SArOGLU, R., "Cavity Formation from Inclusions in Ductile Fracture," Metall. Trans., 6A, 825. 1975 RUDNICXt, J.W. and RtcE, J.R., "Conditions for the Localization of Deformation in PressureSensitive Dilatant Materials," J. Mech. Phys. Solids, 23, 371. 1975 SPITZIG, W.A., SOBER, R.J. and RICHraOND,O., "Pressure Dependence of Yielding and Associated Volume Expansion in Tempered Martensite," Acta Met., 23, 885. 1976 SHtraA, S. and OrAN~, M., "Plasticity Theory for Porous Metals," Int. J. Mech. Sci., 18, 285. 1976 SPITZta, W.A., SOBER, R.J. and RICHMOND,O., "The Effect of Hydrostatic Pressure on the Deformation Behavior of Maraging and HY-80 Steels and its Implication for Plasticity Theory," Metall. Trans., 7A, 1703. 1977 GuxsoN, A.L., "Continuum Theory of Ductile Rupture by Void Nucleation and Growth: Part l Yield Criterion and Flow Rules for Porous Ductile Media," Trans. ASME J. Engng. Mater. & Technol., 99, 2. 1980 LE~:, Y.K. and SWEDtOW, J.L., "Numerical Analysis of Elliptic-Hyperbolic Plastic Flow in Torsion," J. Appl. Mech., 47, 283. 1980 Y~G, W.H., "A Generalized yon Mises Criterion for Yield and Fracture," J. Appl. Mech., 47, 297. 1982 TVERG.O,D, V., "On Localization in Ductile Materials Containing Spherical Voids," Int. J. Fract., 18, 237. 1983 Guo, R.M. and SWEDLOW,J.L., "On the Characterie Equations of Elasto-Plastic Flow," Int. J. Nonlin. Mech., 18, 321. 1984a L~:t:, Y.K. and SW~:DLOW,J.L., "Computation of Shear Bands Using A New Variational Approach, I: Theory and Elementary Examples," Int. J. Num. Meth. Engng., 20, 409. 1984b LEE, Y.K. and SW'EDLOW,J.L., "Computation of Shear Bands Using A New Variational Approach, I1: Use of Weight Function in a Crack Problem," Int. J. Num. Meth. Engng., 20, 423. 1984 THOMSON, R.D. and H.~d,rcoc~, I.W., "Ductile Failure by Void Nucleation, Growth, and Coalescence," Int. J. Fract., 26, 99. 1985 HALt, D.W. and Moc,~csEI, S., "Update on P/M Automotive Application," Int. J. Powder Metall. & Powder Technol., 21, 79. 1985 Ktn.gxxN1, K.M., " P / M Forging Moves into Volume Production," Math. Des. $7(14), 74.
Engineering Mechanics Department General Motors Research Laboratories Warren, MI 48090-9055, U.S.A.
(Received 2 May 1987)
316
Y . K . LyE
NOMENCLATURE elastic modulus of the porous medium
E f m
elastic modulus of the matrix material = shear modulus of the porous medium
btm
~--- shear
modulus of the matrix material
Poisson's ratio (p) (p) # e q ,/'to'.:
p
equivalent and octahedral plastic shear moduli of the matrix material = relative density of the porous medium
F
= load function
4'
= apparent yield function of the porous material
=
apparent strain hardening function
K
U
= elastic deformation energy for the porous medium
TO, Y
----- yield
stress of the matrix material
apparent yield stress of the porous material equivalent flow stress of the matrix material
Teq
d~ )
= rate of equivalent plastic strain in the matrix material
Toc
octahedral shear stress in the matrix material
dot p )
rate of octahedral plastic shear strain
Oeq
=
apparent equivalent stress in the porous medium apparent octahedral shear stress in the porous medium
O'oc
11
= first invariant of the stress tensor
"/2 f
= function
P
= compaction pressure in the porous aggregate
= second invariant of the stress deviator
P,n
= average pressure in the matrix material
t
= time
S
= surface
nj
= unit normal to surface
X i
= spatial coordinate
Xj
-- material coordinate
Ui
=
contravariant components of the velocity field
Ii
=
contravariant components of the traction
d,j, d~ e ~, ditjp' = components of the deformation rate tensor, elastic part, plastic part Wij
=
components of the spin tensor
wtP)
=
plastic work components of Cauchy's stress tensor
oiJ
components of the strain tensor
6ij
=
-'Pll
~--- axial
sj go O0
stress in the matrix
-- components of stress deviator = metric tensor = components of J a u m a n n stress rate tensor Kronecker delta