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Finite element computation of void growth damage in thermo-viscoplasticity
Int. J. Mech. Sci. Vol. 33, No. 5, pp. 339 350, 1991
0020-7403/91 $3.00 + .00 © 1991 Pnrgamon Press pie
Printed in Great Britain.
FINITE ELEMENT C O M P U T A T I O N OF VOID GROWTH DAMAGE IN THERMO-VISCOPLASTICITY C. LAMY, P. HRYCAJ, J. OUDIN,J. C. GELIN t a n d Y. RAVALARD Laboratoire de G6nie M6canique, CNRS, MECAMAT, Universit6 de Valenciennes et du Hainaut Camhr6sis, 59326 Valenciennes, France; and t Laboratoire de M6canique Appliqu6e, Universit6 de Franche Comt6, 25030 Besanqon, France
(Received 11 January 1989; and in revisedform 10 October 1990) Abstraet--A thermo-viscoplastic damage model based on a visco-plastic potential leads to an expression for a consistent tangent operator for finite element discretization. Implicit schemes are used for strain, void volume fraction and temperature computations which are then available to describe the essential damage evolutions due to microvoids growth in most polycrystalline and ductile materials. A typical numerical test is given for a four node quadrilateral element and a full computation example involves the finite element tensile test model of a standard notched cylindrical specimen.
NOTATION C 0, C 1, C Cv d D fi,f f G I J~ m n ql, q2, q3 s t Ti, T v V~, Vm xi, Yi, x, y eM ~M r/ Z Pl, P aM ¢r am aeq am/a~q fly0
isothermic consistencies, temperature/strain dependent consistency isovolume specific heat macroscopic deviatoric strain rate tensor macroscopic strain rate tensor initial, current void volume fraction variation rate of void volume fraction virtual power functional second order unit tensor second invariant of deviatoric tensor strain rate hardening/softening coefficient strain hardening/softening coefficient material parameters in the visco-plastic potential deviatoric Cauchy stress tensor time variable initial, current matrix temperature velocity field elementary apparent volume of the material, elementary volume of the matrix initial, current particle coordinates equivalent strain in the matrix relative equivalent strain rate in the matrix visco-plasticity multiplicative factor virtual velocity field fraction of plastic work converted into heat initial, current density of the material effective isotropic yield stress of the matrix macroscopic Cauchy stress macroscopic hydrostatic stress Mises effective stress triaxiality ratio isotropic visco-plastic potential
INTRODUCTION T h e m i c r o s t r u c t u r e s o f m o s t d u c t i l e a n d p o l y c r y s t a l l i n e m a t e r i a l s u s e d in m e c h a n i c a l p a r t s a n d w o r k p i e c e s a r e often s t r o n g l y m o d i f i e d by the o c c u r r e n c e o f l o c a l i z e d e l a s t o - p l a s t i c a n d / o r v i s c o - p l a s t i c d e f o r m a t i o n m o d e s in the c o n s t i t u t i n g matrices. T h e r e l a t e d h e t e r o g e n eities of t h e p l a s t i c flows are d u e firstly to the p o l y c r y s t a l l i n e m i c r o s t r u c t u r e s a n d s e c o n d l y to s e c o n d p h a s e particles, i n c l u s i o n s a n d p r e c i p i t a t e s . I n t h e s e z o n e s , the m a t e r i a l is g o i n g to be m o r e o r less d a m a g e d d e p e n d i n g o n the stress a n d s t r a i n paths. T h e u l t i m a t e step in the 339
340
C. LAMYet al.
damage evolution is ductile rupture of the material and this is the result of a complex damage mechanism which mainly follows four typical steps: accommodation of the matrix, nucleation, growth and coalescence of microvoids. This damage mechanism limits the processing ability and diminishes the load carrying capacity. The growing interest in modelling damage occurrences is motivated by the urgent need to improve modern processing and use of polycrystalline materials. The incipient damage begins after an accommodation phase of the material matrix in which high stress and strain gradients appear around second phase particles, inclusions and precipitates [1]. When a function of particles shape factor, equivalent macroscopic Cauchy stress and macroscopic hydrostatic stress reaches a critical local stress value, microvoids nucleate; the critical value depends on the type of incipient damage, either second phase particles and inclusions fractures, matrices fractures, inclusions debonding or decohesions of triple joined grains [2]. The increase of void volume fraction during nucleation has been related to equivalent strain rate in the matrix or to effective yield stress and macroscopic hydrostatic stress [-3]. For very high deformation rates, the rate number of nucleated microvoids depends on a critical nucleation stress and on a critical sensibility factor [4]. The strain required for void nucleation is estimated by a simple experimental method [5]. Further on, when nucleation is almost finished, most of the microvoids grow and the corresponding variation of the void volume fraction can be observed by density [-6], Young's modulus [-7] or microhardness measurements [8]. The modifications of the mechanical properties have been described using state variable damage parameters for isotropic problems and a damage tensor for anisotropic materials [9]. The growth of initially spherical voids and more exact void shapes, void interaction and loss of load carrying capacity depend on the triaxiality of the stress field [ 10]. The void volume fraction has also been considered as an explicit internal parameter of the yield function of the material [11-14] and so modelling of the nucleation and growth of the voids in elasticviscoplastic materials has been developed, for instance, in the case of spherical voids, or of periodically arranged cylindrical voids [15-20]. A mechanistic model in terms of the equivalent deviatoric strain rate tensor has also been used to give the variation rate of the void volume fraction [21]. The ultimate step in damage evolution is the coalescence of microvoids and voids when ductile fracture is achieved. This step is predicted either from the critical dimensions of voids [22], critical dilatancy, critical energy [-23] or an intrinsic limit function [24, 25]. The result is a strong softening of the material and the partial or complete loss of load carrying capacity. As more and more mechanical engineering problems involve visco-plastic behaviour of materials, it becomes important to take into account damage evolution, and therefore to define the available constitutive equations and the related computation procedures. The materials involved in mechanical problems are rarely pore free ones, so the proposed analysis of visco-plastic problems consider microvoids growth as the leading phenomenon in damage evolution. For instance, the initial void volume fraction of a typical 280 mm diameter 4142 steel bar issued from continuous casting is in the range of 2.10 -3 to 2" 10 '* and typical damage leads to 10 2 to 10 -3. For that purpose, the authors propose a new constitutive model based on an isotropic visco-plastic potential available for isotropic materials which already contains an initial volume fraction of voids. The developments are achieved within the non-linear finite element framework and the formulations are implemented in the ASTRID finite element code. Results from typical numerical benchmark and a reference isotropic damage computation, the tensile test of axisymmetric notched cylindrical specimens are presented and stress, strain, void volume fraction and temperature distribution are computed. CONSTITUTIVE MODEL
Yield function and flow rule To take into account nucleation, growth and coalescence of microvoids as a measure of the evolving material damage, the simplest parameter is the current volume fraction f of
Void growth damage in thermo-viscoplasticity
341
microvoids defined by the ratio
f-
Va--Vm Va
in which Va is the elementary apparent volume of the material and I'm is the corresponding elementary volume of the matrix. The void volume fraction is related to the initial volume fraction fi, initial density Pi and current density p at time t by
p/p, = (1 - fi)/(1 - f), the rate of increase of void volume fraction is given by
in which subscripts n, g and c, respectively, correspond to nucleation, growth and coalescence. In most usual polycrystalline materials such as steels at medium and high temperatures, the volume fraction due to void nucleation is rather small compared to the volume fraction of initial voids. So, it can be considered that before the coalescence step
The prediction of the microvoids growth can be obtained by using a global model which involves a visco-plastic potential which is convenient in a finite element framework. The general form is ~'~vp = ~,~vp(O.m,j ~ , 0.M, f ) with o"m the hydrostatic stress, J~ = 1.5 s:s the second invariant of the deviatoric stress tensor s, aM = C (~M)m the isotropic yield stress of the matrix, C the material consistency, ~M the relative equivalent strain rate which is the ratio between equivalent strain rate and the reference strain rate equal to 1 s- 1, m the strain rate coefficient, and f the present void volume fraction. Good predictions have been obtained from the following expression 3 s:s {3 GmX~ fFP - 2 a~ + 2fql c°sh~2 q 2 ~ J - 1 - q3f 2 in which parameters ql, q2 and q3 mainly depend on the distribution of second phase particles and inclusions in the matrix. The normality rule is well suited to most visco-plastic problems and thus, visco-plastic strain rate D is expressed by D=d+
trD 8fF p {3 s 1 //30"m'~ } ~l=}-~-a = p ~ + - - f q! l s i nah ~ 2 qM2 ~ )
with d the deviatoric strain, I the second order unit tensor and ~ the visco-plasticity multiplicative factor. It becomes d = ~3
S
~
and
trD=~
3 fql slnh ~q2
The variations of the matrix volume due to temperature gradients and to elasticity effects are often very small, so mass conservation of the material is expressed by
15 + p t r D = 0 and the rate of void volume fraction when microvoids grow in such a non-dilatant matrix is = (1 - ~ ) t r O and the current void volume fraction is calculated from
f(t) = f + MS 3 3 : 5 - B
fo (
z) dr.
C. LAMYet al.
342
The equivalent strain rate ~M is given by ~M =
DD
=
I[ D
II =
d:d + ~
.
Equivalence between plastic work dissipated in the material and in its plastic matrix leads to a : D = (1
--f)aM~ M.
Thus, the visco-plasticity multiplicative factor ~ is a function of the state variables [C(~M)m+ 13 "
1 -f
~:
S:S ~d + 30"m fql sinh ( ~ q2 O'o.;) 3 ~O' O"M
Variational formulation Let V be the plastic deformed volume of the material, S,, and Sv the parts of its limiting surface on which, respectively, surface tractions and velocities are prescribed. The virtual power theorem leads to the equation
G(v, tl)= fv(S + amI)'VStldV- fs t r / d S = 0 a with r/ a virtual velocity field according to the prescribed conditions on surface Sv VSq = ½(Vq + VVq) is the symmetrized gradient of q, a~ s= ~ d
and
2 aMsinh_ 1 (aMtrD ~ Gm : ~ q 2 ~k3 ~ f ~ l / "
From the Newton-Raphson scheme, the linearized derivative is in the form
Dv G(v, q)Av + G(v, ~) = 0 with the consistent tangent operator + ~-
I : V~t/Av d V.
The solutions S,+l and v.+ 1 at time t,+l are obtained from (i) D~G(v.+I,
~)Au(i)+ 1 -4- (-J(.{i) ~Vn+l,
/7) = 0
in which (i) indicates the ith iteration. Velocities are updated from • (i) ~(i+1) ~n+l ~- ,~(i) ~n+l _~ /~Vn + 1' A fully implicit scheme gives the equivalent strain ~.+1 = /~, + ~,+1 At and the void volume fraction from /n+l : i n "['-/n+ 1 At in which At is the time increment and suffixes n, n + 1 correspond to time t., t,+l.
Influence of strain and temperature The local strain and temperature dependence of the matrix yield stress may be important for the numerical modelling of some problems in which gradients of these two state variables are rather high due to strong heterogeneity in the material deformation. In these cases, the present matrix consistency C is a function which depends on temperature T and on equivalent strain eM,
C = C(T, eM)
Void growth damage in thermo-viscoplasticity
343
and the effective yield stress becomes (7M = C( T, ~M)(~M) m
The rate of the temperature due to plastic deformation is given by ?= ~a:D=
Z
piCv
piCv
(l--f)aM~M
with • the fraction of plastic work converted into heat, Pl the material initial density and Cv the specific heat. The tangent operator is modified by the existence of the derivative of the consistency C with respect to the strain rate tensor: •C _
Z
~C(I_f)C(~M)m+IAt2I
[iCy c~T
63D -Jr"
[c363C M
Z
t3C ( 1 - f ) ( m +
~M ''+- piCv c3T
1)C(~M)"+I]At
d IId[I2"
Temperature is obtained from the fully implicit scheme T.+I = L + ? . + , a t . Temperature values are expected to be close to the exact ones, except in some localized zones where severe thermal boundary conditions are observed. NUMERICAL
EXAMPLES
Two typical examples illustrate the performance and interesting feature of finite element models which take into account the growth of microvoids in materials: first, benchmark traction of a single Q4 element in order to appreciate some numerical performances, second tensile test of a standard ALl0 notched cylinder to expose some effects of void volume growth on the state variables. In the examples, a reference value of 100 N . m m - 2 is chosen for the material consistency, 0.01 and 0.25 for the strain rate hardening coefficients (two reference cases), 0.1 for the strain hardening coefficient, - 3 for the temperature sensitivity coefficient, 10- s and 10- 3 for the initial void volume fraction (two reference cases), and reference parameters for the viscoplastic potential of q, = 1.25, q2 = 1, q3 = (q,)2 = 1.5625 according to Ref. 1-16]. Typical single element traction
Consider the isothermal traction of a four node quadrilateral element (Fig. 1) in which the current particles coordinates x and y at time t are expressed by X = (1 + Off)X i
y = (1 + 20 y~ with 2ty i the prescribed displacement on nodes 3 and 4, xi, y~ the initial particle coordinates. Numerical and analytical calculations are both achieved for displacement factor 2 equal to
y
[
prescribeddisplacement
~\\\\\'~'~
~NX\\\\"¢.
6- x
FIG. 1. Four node quadrilateral isoparametric element: x and y directions boundary conditions, prescribed displacement in the y direction.
C. LAMY et al.
344
TABLE 1. TRACTION OF A FOUR NODE QUADRILATERAL ELEMENT: NUMERICAL AND ANALYTICAL VALUES OF EQUIVALENT STRAIN RATE, STRESS AND VOID VOLUME FRACTION IN A POROUS VISCOPLASTIC MATERIAL a M = |00(F,'M) TM N ' m m - 2 , f = 10 s UNTIL 5 0 0 INCREMENTS OF 0 , 0 0 2 SECONDS (DISPLACEMENT FACTOR /. -- 0.3 S 1)
I
~M
(s)
(s
O~yy (N'mm 21
l)
./ ( x l 0 s)
frill
(N'mm
a)
0.2
0.326783 0.326783
114.177 114.250
57.0874 57.1601
1.56076 1.56118
O.5
0.301193 0.301193
I 14.077 114.085
57.0363 57.0447
2.90856 2.911075
0.8
0.279308 0.279308
I 13.978 113.933
56.9779 56.9755
5.17095 5.16982
1.0
0.266394 0.266395
113.912 113,899
56.9399 56.9378
7.41720 7.41021
B o l d n u m b e r s : n u m e r i c a l v a l u e s . Italic n u m b e r s : a n a l y t i c a l v a l u e s .
0.3 s - 1 until time t is equal to 1 s, the reference visco-plastic yield stress of the matrix being aM = 100(~M) ° ° 1 N ' m m -2 and the initial void volume fraction 10 -5 . Table 1 gives numerical and analytical values of the relative equivalent strain rate, C a u c h y stress ayy, hydrostatic stress a m and void volume f r a c t i o n f The results of the finite element c o m p u t a t i o n agree very well with the analytical ones as shown by the small relative error values A~M
eM ~<3"10-5'
Aayy
--
~<10 3
Gyy
~
AG m
__
<2.10-3
(7 m
'
Af 0_4. <~9'1 f
Tensile test of notched cylindrical specimen The standard tensile test of an AE 10 notched cylinder shown in Fig. 2(a) is now analysed when a porous material is involved. The effective material matrix yield stress value is given by O"M =
100(~M) 0'25
N . m m -2
and the initial void volume fraction is equal to 10 -3. The test conditions are: ram velocity of the test machine 20 m m s t, a specimen reference length from 95 m m at time t = 0 to 102 m m at time t = 0.35 s. Only a quarter of the meridian plane of the specimen is used for the model. The initial mesh is made of 134 Q4 elements with 2 x 2 Gauss integration points [Figs 2(b), (c)]. The prescribed velocities on the Z axis are vr = 0 and on the R axis vz = 0. The c o m p u t a t i o n is achieved in 70 time increments. After the test, the dimensions of the specimens are slightly different when comparing the pore free visco-plastic material to the p o r o u s visco-plastic one. For instance, the reduction of the diameter of the notch is more important for the pore free specimen than for the porous one (Fig. 3). So, from a measure of the notch diameter, an estimate of the void volume fraction is available. Figure 3 also shows the radial stress maps in the notch zones for pore free visco-plastic material and for porous visco-plastic one. It is noticed that the m a x i m u m radial stresses are, respectively, 35.2 N . m m -2 located at coordinates r = 0, z = 0 in the pore free specimen [Fig. 3(a)] and 19.4 N . m m - 2 located at coordinate z = 4.25 m m on the notch surface in the porous one [Fig. 3(b)]. The change in the stress m a p is also significant and from it can be observed some new stress increases and decreases for a p o r o u s material c o m p a r e d to a pore free one in the z-direction and in the notch surface normal direction.
Void growth damage in thermo-viscoplasticity
345
Ca) Scale 1 z
(c) ~
(b) r
r
Scale6
Scale3
FIG. 2. Cylindrical notched AE 10 specimen: (a) dimensions in the testing zone; (b) finite element mesh in a quarter of the meridian plane; (c) zoom in the center cross-section.
.
z
pore f r e e /
=
.
r=8.92 z
~
(N.mm -2)
K1L r
(a)
~
5/,F
(N.
A-2.3
/
B
~
5.2
/
D 20.~
o
""'D \
-9.8
"
A-7.3
Ill
B-1.9
I-
D 8.7
~3z~ !!i:! -
* maximum minimum (b) o
2)
[[~ f I[
~.03 E 27.7 z=0 * 35.2
•
r=8.92 ~ A ~ / C p o r °/ u s
.
Scale5
FIG. 3. Cylindrical notched AE 10 specimen, computations at 0.35 s: (a) radial stress in a pore free specimen; (b) radial stress in a porous specimen.
346
C. LAMYet al.
A n a l o g o u s axial stress m a p s are obtained, the m a x i m u m being l o c a t e d in the n o t c h e d cross-section, 135.21 N . m m - 2 in the vicinity of the z-axis for the p o r e free specimen, 110.84 N . m m - 2 on the free surface for the p o r o u s specimen. In the latter case, a released stress zone is also observed in the vicinity of the z-axis, the m i n i m u m stress value being 4.55 N - r a m -2 at c o o r d i n a t e z -- 7.6 ram. The evolutions of h y d r o s t a t i c stress in the p o r e free m a t e r i a l and in the p o r o u s one are also different: at time t = 0.35 s, the m a x i m u m is l o c a t e d at r = 0, z = 0 for a p o r e free specimen [Fig. 4(a)] but at z = 0 on the notch surface for a p o r o u s one and the m a p is also affected by released stress zones [Fig. 4(b)]. L o o k i n g now for time e v o l u t i o n in the center zone discretized in I, II, III a n d IV Q4 elements [Fig. 2(c)], it is clear that h y d r o s t a t i c stress time derivatives in the I, II, III elements are going from small values 22 N . m m - 2. s ~ as expected in a p o r e free specimen to significant ones 220 N . m m - 2 . s - ~ in p o r o u s specimens (Fig. 5). All a l o n g test time, the m a x i m u m values r e m a i n in element I, that c o r r e s p o n d s to center zone for the pore free specimen, and on the c o n t r a r y they change from element I to element IV for the p o r o u s specimen. The m a x i m u m values of the Mises effective stress at time t = 0.35 s r e m a i n located at c o o r d i n a t e z = 0 on the notch free surface, 132.10 N . m m 2 for pore free m a t e r i a l and 102-41 N . m m 2 for p o r o u s material; however the r g r a d i e n t in the notch cross-section of the p o r o u s specimen reaches 34 N . m m -3 c o m p a r e d to 3 N - r a m -3 for the pore free specimen.
z
r=-8.92
,
z
r=8.92
f,':? / ~B,- /
(N.mm-2) ,
/):-t
D
J r=3.03
ID I / I
IB r:3"161 E 43.0
/*'-.9~~ r
*
51.6
r
(a)
* maximum
(b)
Scale 5
FIG. 4. Cylindrical notched AE 10 specimen, computations at 0.35 s: (a) hydrostatic stress in a pore free specimen; (b) hydrostatic stress in a porous specimen.
o
• 70"
Z
i
!
~
~III
~60"
~, 60
50" __...Iv--
~ ..~
. 70 -"-'II
HI
~
4ff
~
pore free I
I
I
0.1
40
I
I
0.2
I
0.3 time
(a)
50 ...---IV--"--
~
I
I
I
0.1
0.2
(s)
I
I
I
0.3 time (s)
(b)
FIG. 5. Cylindrical notched AE 10 specimen, computations vs time in the center zone: (a) hydrostatic stress in a pore free specimen; (b) hydrostatic stress in a porous specimen.
Void growth damage in thermo-viscoplasticity
1.0.] ~
pore free
1.0.
fF 10"3 porous
o 0.4.
347
I
\ o.4.
"~ o 2
"t~ o.2.
VI
I IH I I~ ~I 0.46 0.56 0.66 equivalent strain
0.36
IV I 0.36
I I1 , 0.46 0.56 0.66 equivalent strain
(a)
(b)
F16. 6. Cylindrical notched AE 10 specimen, triaxiality vs equivalent strain in the center zone: (a) stress triaxiality in a pore free specimen; (b) stress triaxiality in a porous specimen.
Some analyses of the triaxiality ratio O'm/O'eq are given on Fig. 6. For a pore free material, the triaxiality ratio remains almost constant when the equivalent strain increases [Fig. 6(a)] whilst for porous material, triaxiality in elements I, II and III decreases for higher values of equivalent strain [Fig. 6(b)]. The void volume fraction values for the porous material are shown on Fig. 7: either in the z direction or in the notch surface tangential direction, rather regular values map and gradients are obtained [Fig. 7(a)]; volume fraction time derivatives in the center elements I, II, III and IV increase regularly from time t = 0 to time t = 0.35 s, the final derivatives being not very different from 1.2 s- 1 [Fig. 7(b)]. The distribution of equivalent strain is also affected by material porosity: although the isovalue maps look similar for both materials, the maximum values, located twice at the boundary of the notch cross-section, reach 1.006 for the pore free specimen and 1.073 for the porous one. Look now at a thermo-viscoplastic porous material, the yield stress of which being temperature dependent is ~M = 100(eM)°'1 1000
z porous3 f~ 10"
I r=8.92 1 | z=11.32 | 1 0.16
~
(iM)°'25N'mm-2"
/ I //II //Ill
.~
0.12 f
O
C
IV
0.04 ~3 16
l
[Scale5
..
(a)
r
*maximum
0.1 (b)
0.2
0.3 time (s)
FIG. 7. Cylindrical notched AE 10 specimen: computations of void volume fraction: (a) at time 0.35 s; (b) vs time.
348
C. LAMY et al.
z
I
."',,
-\"-./
I
r=-8.85
A o. I
"_77{71
* maximum
Scale 5
FIG. 8. Cylindrical notched AE l0 specimen, computations of void volume fraction at time 0.35 s.
| r=-8.85 I z=10 97
porous Ti= 1100 °C
1112
~
III
1108 ~
~B
-""~ /
•- " ' ~ " ~ D "~'- [ / _ ¢ ' ~ r=3.46 / El Ii -z=O"--
A I101 B 1103 C D E *
1105 1107 1109 Illl
,
r Scale 5 (a)
11 I
,
0.1 * maximum
, 0.2
(b)
,
, 0.3 time
l (s)
FIG, 9. Cylindrical notched AE 10 specimen, computations of temperature: (a) at 0.35 s; (b) vs time.
Computations have been made for a reference steel of initial density Pi = 7800 k g . m - 3 and isovolume specific heat Cv = 580 J . k g - 1. °C- 1. The fraction of plastic work converted into heat is 0.95 and the initial temperature T~ = 1100 °C. The void fraction map (Fig. 8) is similar to that obtained in the previous isothermal analysis. The most important growth of void volume fraction appears in the center of the notch cross-section, from an initial value of 10 -a to 51.10 -3 whereas the maximum increase of temperature is found on the r axis at the notch free surface (Fig. 9). The time evolution in the center zone reveals that the temperature time derivatives rise steadily, the highest derivative being 40°C.s - 1 in element IV [Fig. 9(b)]. CONCLUSION
The growth of microvoids is considered as an important damage parameter for polycrystalline and ductile materials. To allow the secure analysis of visco-plastic structures, a thermo-viscoplastic damage model based on the Gurson yield function is developed within the finite element framework provided by the ASTR1Dcode. The consistent tangent operator for the finite element discretization is defined and the bulk constitutive equations of the material are expressed by exponential functions of strain rate and temperature.
Void growth damage in thermo-viscoplasticity
349
T h e r e p o r t e d n u m e r i c a l tests o n a f o u r n o d e q u a d r i l a t e r a l e l e m e n t s h o w t h a t t h e r e l a t i v e e r r o r s o n t h e strain, stress a n d v o i d v o l u m e f r a c t i o n c o m p u t e d v a l u e s are a l w a y s l o w e r t h a n l 0 - 3 a n d t h a t c o n v e r g e n c e is g o o d . T h e finite e l e m e n t m o d e l o f t h e s t a n d a r d tensile test o f A E l 0 n o t c h e d c y l i n d r i c a l s p e c i m e n s h o w s t h a t h y d r o s t a t i c stress d e c r e a s e s w i t h time, t h e m a x i m u m v a l u e s a r e l o c a t e d in t h e n o t c h e d c r o s s - s e c t i o n , first in the z axis z o n e a n d l a t e r in the n o t c h free surface. T h e m a x i m u m g r o w t h of the m i c r o v o i d s is o b s e r v e d in t h e n o t c h e d c r o s s - s e c t i o n also a n d r e m a i n s the test in the z axis zone. T h e o t h e r state v a r i a b l e s m a p s a n d stress t r i a x i a l i t y a r e also different f r o m the o n e s r e l a t e d to p o r e free m a t e r i a l s . T h e influence of a d i a b a t i c t e m p e r a t u r e i n c r e a s e d u e to v i s c o - p l a s t i c d e f o r m a t i o n was t h e n studied: in the A E l 0 s p e c i m e n , v o i d v o l u m e f r a c t i o n m o d i f i c a t i o n s a n d significant t e m p e r a t u r e g r a d i e n t s are o b s e r v e d w h e n c o n s i d e r i n g a s t a n d a r d steel at f o r g i n g t e m p e r ature. S p e c i m e n profile c h a n g e s are also effective w h e n t a k i n g i n t o a c c o u n t v o i d v o l u m e fraction growth and temperature gradients. I n c o n c l u s i o n , the m o d e l is a b l e to give n e w i n f o r m a t i o n o n w h a t c o u l d h a p p e n w h e n v i s c o - p l a s t i c b e h a v i o r is effective in p o l y c r y s t a l l i n e a n d d u c t i l e m a t e r i a l s : stress t e n s o r m o d i f i c a t i o n s , h e t e r o g e n e i t y of v o i d v o l u m e f r a c t i o n a n d significant t e m p e r a t u r e gradients. The authors would like to thank Nord Pas de Calais Region, C.N.R.S. and the French Ministry of Education for supports.
Acknowledgements
REFERENCES 1. F. MONTHEILLETand F. MoussY, Physique et m~canique de l'endommagement. Editions de Physique (1988). 2. F. LEROY,F. MoussY and F. MONTHEILLET,Interactions inclusion-matrice au cours d'op6rations de formage. Ecole d'Et~ Mat~riaux Mise en Forme Pi~ces Fortunes, GIS Mise en Forme, GRECO Grandes D6formations et Endommagement, Saint Pierre d'O16ron (P. Franciosi and F. Moussy Eds). Presses du CNRS, Paris (1989). 3. E. OI~ATEand E. KLEBER,Plastic flow of void containing metals, applications to axisymmetric sheet forming problems. Proceedings of N U M I F O R M 86, 339 (1986). 4. M. LACOMME, A. FROGER, J. P. ANSART and R. DORMEVAL, Endommagement sous choc de l'alliage d'aluminium AU4G. Congr~s International D Y M A T , Editions de Physique (1988). 5. G. LE Roy, J. D. EMaURY,G. EDWARDSand M. F. ASHBY,A model of ductile fracture based on the nucleation and growth of voids. Acta Metall. 29, 1509 (1981). 6. F. Moussv, Etude de Fendommagement li6 a la pr6sence d'inclusions. Contrat de programme G.I.S. raise en Forme, M R T 83.P.0462, 3, 59 (1987). 7. J. C. GELIN and J. OUDIN, Mesure de l'endommagement des aciers inoxydables aust6nitiques par essais de traction-torsion. Compte rendu du Groupe de R~flexion sur l'Endommagement, Saint Germain en Laye (1983). 8. D. MARQUIS,M6canique de l'endommagement continu. Ecole d'Et~ Mat~riaux Mise en Forme Pi~ces Form~es, GIS Mise en Forme, GRECO Grandes D6formations et Endommagement, Saint Pierre d'O16ron (P. Franciosi and F. Moussy Eds). Presses du CNRS, Paris (1989). 9. M. PREDELEANU,Finite strain plasticity analysis and damage effects in metal forming processes. Computational methods for predicting material processing defects (M. Predeleanu Ed.), p. 295. Elsevier (1987). 10. C. L. HOM, R. M. MCMEEKING, Void growth in elastic-plastic materials. J. Appl. Mech. 56, 309 (1989). 11. A. L. GURSON,Continuum theory of ductile rupture by void nucleation and growth--Part I. Yield criteria and flow rules for porous ductile media. J. Engng Mater. Technol. 99, 2 (1977). 12. A. L. GURSON,Porous rigid-plastic materials containing rigid inclusions--yield function, plastic potential and void nucleation. Advances in Research on the Strength and Fracture of Materials (D.M.R. Taplin Ed.), Pergamon Press, Oxford (1977). 13. K. MOR1 and K. OSAKADA,Analysis of the forming process of sintered power metals by a rigid-plastic finiteelement method. Int. J. Mech. Sci. 29, 229 (1987). 14. A. ROSOCHOWSKIand L. OLEN|K, Damage evolution in mild steel. Int. J. Mech. Sci. 30, 1, 51 (1988). 15. V. TVERGAARD,Ductile fracture by cavity nucleation between larger voids. J. Mech. Phys. Solids 30, 265 (1982). 16. V. TVERGAARD,Material failure by void coalescence in localized shear bands. Int. J. Solids Structures lg, 659 (1982). 17. N. ARAVAS,The analysis of void growth that leads to central bursts during extrusion. J. Mech. Phys. Solids 34, 1, 55 (1986). 18. R. BECKERand A. NEEDLEMAN,O. RICHMONDand V. TVERGAARD,Void growth and failure in notched bars. J. Mech. Phys. Solids 36, 317 (1988). 19. N. ARAVAS,On the numerical integration of a class of pressure dependent plasticity models. Int. J. Numer. Meth. Engng 24, 1395 (1987). 20. V. TVERGAARD,Influence of void nucleation on ductile shear fracture at a free surface. J. Mech. Phys. Solids 30, 399 (1982). MS
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21. K. K. MATFIURand P. R. DAWSON, Damage evolution modeling in bulk forming processes. Computational methods for Predicting Material Processing Defects (M. Predeleanu Ed.), p. 251. Elsevier (1987). 22. H. SEKIGUCHI and K. OSAKADA,Ductile fracture of carbon steel under cold forming conditions. Bull. of J.S.M.E. 24, 534 (1981). 23. M. OYANE,S. SHIMAand T. TABATA,Consideration of basic equations, and their application, in the forming of metal powders and porous metals. J. Mech. Working Tech. l, 325 (1978). 24. J.C. GEL1N, J. OUDIN, Y. RAVALARDand A. MOISAN, An improved finite element method for the analysis of damage and ductile fracture in cold forming processes. Annals of the C.I.R.P. 34/1, 209 (1985). 25. J. OUDIN, G. LACOMaE, Y. RAVALARDand T. LABARTHE-VAQU1ER,D6termination de la forgeabilite des m6taux et alliages. Sixi@me Colloque Mbcanique et Matkriaux de Tarbes, l, 1 41 (1987).
0020-7403/91 $3.00 + .00 © 1991 Pnrgamon Press pie
Printed in Great Britain.
FINITE ELEMENT C O M P U T A T I O N OF VOID GROWTH DAMAGE IN THERMO-VISCOPLASTICITY C. LAMY, P. HRYCAJ, J. OUDIN,J. C. GELIN t a n d Y. RAVALARD Laboratoire de G6nie M6canique, CNRS, MECAMAT, Universit6 de Valenciennes et du Hainaut Camhr6sis, 59326 Valenciennes, France; and t Laboratoire de M6canique Appliqu6e, Universit6 de Franche Comt6, 25030 Besanqon, France
(Received 11 January 1989; and in revisedform 10 October 1990) Abstraet--A thermo-viscoplastic damage model based on a visco-plastic potential leads to an expression for a consistent tangent operator for finite element discretization. Implicit schemes are used for strain, void volume fraction and temperature computations which are then available to describe the essential damage evolutions due to microvoids growth in most polycrystalline and ductile materials. A typical numerical test is given for a four node quadrilateral element and a full computation example involves the finite element tensile test model of a standard notched cylindrical specimen.
NOTATION C 0, C 1, C Cv d D fi,f f G I J~ m n ql, q2, q3 s t Ti, T v V~, Vm xi, Yi, x, y eM ~M r/ Z Pl, P aM ¢r am aeq am/a~q fly0
isothermic consistencies, temperature/strain dependent consistency isovolume specific heat macroscopic deviatoric strain rate tensor macroscopic strain rate tensor initial, current void volume fraction variation rate of void volume fraction virtual power functional second order unit tensor second invariant of deviatoric tensor strain rate hardening/softening coefficient strain hardening/softening coefficient material parameters in the visco-plastic potential deviatoric Cauchy stress tensor time variable initial, current matrix temperature velocity field elementary apparent volume of the material, elementary volume of the matrix initial, current particle coordinates equivalent strain in the matrix relative equivalent strain rate in the matrix visco-plasticity multiplicative factor virtual velocity field fraction of plastic work converted into heat initial, current density of the material effective isotropic yield stress of the matrix macroscopic Cauchy stress macroscopic hydrostatic stress Mises effective stress triaxiality ratio isotropic visco-plastic potential
INTRODUCTION T h e m i c r o s t r u c t u r e s o f m o s t d u c t i l e a n d p o l y c r y s t a l l i n e m a t e r i a l s u s e d in m e c h a n i c a l p a r t s a n d w o r k p i e c e s a r e often s t r o n g l y m o d i f i e d by the o c c u r r e n c e o f l o c a l i z e d e l a s t o - p l a s t i c a n d / o r v i s c o - p l a s t i c d e f o r m a t i o n m o d e s in the c o n s t i t u t i n g matrices. T h e r e l a t e d h e t e r o g e n eities of t h e p l a s t i c flows are d u e firstly to the p o l y c r y s t a l l i n e m i c r o s t r u c t u r e s a n d s e c o n d l y to s e c o n d p h a s e particles, i n c l u s i o n s a n d p r e c i p i t a t e s . I n t h e s e z o n e s , the m a t e r i a l is g o i n g to be m o r e o r less d a m a g e d d e p e n d i n g o n the stress a n d s t r a i n paths. T h e u l t i m a t e step in the 339
340
C. LAMYet al.
damage evolution is ductile rupture of the material and this is the result of a complex damage mechanism which mainly follows four typical steps: accommodation of the matrix, nucleation, growth and coalescence of microvoids. This damage mechanism limits the processing ability and diminishes the load carrying capacity. The growing interest in modelling damage occurrences is motivated by the urgent need to improve modern processing and use of polycrystalline materials. The incipient damage begins after an accommodation phase of the material matrix in which high stress and strain gradients appear around second phase particles, inclusions and precipitates [1]. When a function of particles shape factor, equivalent macroscopic Cauchy stress and macroscopic hydrostatic stress reaches a critical local stress value, microvoids nucleate; the critical value depends on the type of incipient damage, either second phase particles and inclusions fractures, matrices fractures, inclusions debonding or decohesions of triple joined grains [2]. The increase of void volume fraction during nucleation has been related to equivalent strain rate in the matrix or to effective yield stress and macroscopic hydrostatic stress [-3]. For very high deformation rates, the rate number of nucleated microvoids depends on a critical nucleation stress and on a critical sensibility factor [4]. The strain required for void nucleation is estimated by a simple experimental method [5]. Further on, when nucleation is almost finished, most of the microvoids grow and the corresponding variation of the void volume fraction can be observed by density [-6], Young's modulus [-7] or microhardness measurements [8]. The modifications of the mechanical properties have been described using state variable damage parameters for isotropic problems and a damage tensor for anisotropic materials [9]. The growth of initially spherical voids and more exact void shapes, void interaction and loss of load carrying capacity depend on the triaxiality of the stress field [ 10]. The void volume fraction has also been considered as an explicit internal parameter of the yield function of the material [11-14] and so modelling of the nucleation and growth of the voids in elasticviscoplastic materials has been developed, for instance, in the case of spherical voids, or of periodically arranged cylindrical voids [15-20]. A mechanistic model in terms of the equivalent deviatoric strain rate tensor has also been used to give the variation rate of the void volume fraction [21]. The ultimate step in damage evolution is the coalescence of microvoids and voids when ductile fracture is achieved. This step is predicted either from the critical dimensions of voids [22], critical dilatancy, critical energy [-23] or an intrinsic limit function [24, 25]. The result is a strong softening of the material and the partial or complete loss of load carrying capacity. As more and more mechanical engineering problems involve visco-plastic behaviour of materials, it becomes important to take into account damage evolution, and therefore to define the available constitutive equations and the related computation procedures. The materials involved in mechanical problems are rarely pore free ones, so the proposed analysis of visco-plastic problems consider microvoids growth as the leading phenomenon in damage evolution. For instance, the initial void volume fraction of a typical 280 mm diameter 4142 steel bar issued from continuous casting is in the range of 2.10 -3 to 2" 10 '* and typical damage leads to 10 2 to 10 -3. For that purpose, the authors propose a new constitutive model based on an isotropic visco-plastic potential available for isotropic materials which already contains an initial volume fraction of voids. The developments are achieved within the non-linear finite element framework and the formulations are implemented in the ASTRID finite element code. Results from typical numerical benchmark and a reference isotropic damage computation, the tensile test of axisymmetric notched cylindrical specimens are presented and stress, strain, void volume fraction and temperature distribution are computed. CONSTITUTIVE MODEL
Yield function and flow rule To take into account nucleation, growth and coalescence of microvoids as a measure of the evolving material damage, the simplest parameter is the current volume fraction f of
Void growth damage in thermo-viscoplasticity
341
microvoids defined by the ratio
f-
Va--Vm Va
in which Va is the elementary apparent volume of the material and I'm is the corresponding elementary volume of the matrix. The void volume fraction is related to the initial volume fraction fi, initial density Pi and current density p at time t by
p/p, = (1 - fi)/(1 - f), the rate of increase of void volume fraction is given by
in which subscripts n, g and c, respectively, correspond to nucleation, growth and coalescence. In most usual polycrystalline materials such as steels at medium and high temperatures, the volume fraction due to void nucleation is rather small compared to the volume fraction of initial voids. So, it can be considered that before the coalescence step
The prediction of the microvoids growth can be obtained by using a global model which involves a visco-plastic potential which is convenient in a finite element framework. The general form is ~'~vp = ~,~vp(O.m,j ~ , 0.M, f ) with o"m the hydrostatic stress, J~ = 1.5 s:s the second invariant of the deviatoric stress tensor s, aM = C (~M)m the isotropic yield stress of the matrix, C the material consistency, ~M the relative equivalent strain rate which is the ratio between equivalent strain rate and the reference strain rate equal to 1 s- 1, m the strain rate coefficient, and f the present void volume fraction. Good predictions have been obtained from the following expression 3 s:s {3 GmX~ fFP - 2 a~ + 2fql c°sh~2 q 2 ~ J - 1 - q3f 2 in which parameters ql, q2 and q3 mainly depend on the distribution of second phase particles and inclusions in the matrix. The normality rule is well suited to most visco-plastic problems and thus, visco-plastic strain rate D is expressed by D=d+
trD 8fF p {3 s 1 //30"m'~ } ~l=}-~-a = p ~ + - - f q! l s i nah ~ 2 qM2 ~ )
with d the deviatoric strain, I the second order unit tensor and ~ the visco-plasticity multiplicative factor. It becomes d = ~3
S
~
and
trD=~
3 fql slnh ~q2
The variations of the matrix volume due to temperature gradients and to elasticity effects are often very small, so mass conservation of the material is expressed by
15 + p t r D = 0 and the rate of void volume fraction when microvoids grow in such a non-dilatant matrix is = (1 - ~ ) t r O and the current void volume fraction is calculated from
f(t) = f + MS 3 3 : 5 - B
fo (
z) dr.
C. LAMYet al.
342
The equivalent strain rate ~M is given by ~M =
DD
=
I[ D
II =
d:d + ~
.
Equivalence between plastic work dissipated in the material and in its plastic matrix leads to a : D = (1
--f)aM~ M.
Thus, the visco-plasticity multiplicative factor ~ is a function of the state variables [C(~M)m+ 13 "
1 -f
~:
S:S ~d + 30"m fql sinh ( ~ q2 O'o.;) 3 ~O' O"M
Variational formulation Let V be the plastic deformed volume of the material, S,, and Sv the parts of its limiting surface on which, respectively, surface tractions and velocities are prescribed. The virtual power theorem leads to the equation
G(v, tl)= fv(S + amI)'VStldV- fs t r / d S = 0 a with r/ a virtual velocity field according to the prescribed conditions on surface Sv VSq = ½(Vq + VVq) is the symmetrized gradient of q, a~ s= ~ d
and
2 aMsinh_ 1 (aMtrD ~ Gm : ~ q 2 ~k3 ~ f ~ l / "
From the Newton-Raphson scheme, the linearized derivative is in the form
Dv G(v, q)Av + G(v, ~) = 0 with the consistent tangent operator + ~-
I : V~t/Av d V.
The solutions S,+l and v.+ 1 at time t,+l are obtained from (i) D~G(v.+I,
~)Au(i)+ 1 -4- (-J(.{i) ~Vn+l,
/7) = 0
in which (i) indicates the ith iteration. Velocities are updated from • (i) ~(i+1) ~n+l ~- ,~(i) ~n+l _~ /~Vn + 1' A fully implicit scheme gives the equivalent strain ~.+1 = /~, + ~,+1 At and the void volume fraction from /n+l : i n "['-/n+ 1 At in which At is the time increment and suffixes n, n + 1 correspond to time t., t,+l.
Influence of strain and temperature The local strain and temperature dependence of the matrix yield stress may be important for the numerical modelling of some problems in which gradients of these two state variables are rather high due to strong heterogeneity in the material deformation. In these cases, the present matrix consistency C is a function which depends on temperature T and on equivalent strain eM,
C = C(T, eM)
Void growth damage in thermo-viscoplasticity
343
and the effective yield stress becomes (7M = C( T, ~M)(~M) m
The rate of the temperature due to plastic deformation is given by ?= ~a:D=
Z
piCv
piCv
(l--f)aM~M
with • the fraction of plastic work converted into heat, Pl the material initial density and Cv the specific heat. The tangent operator is modified by the existence of the derivative of the consistency C with respect to the strain rate tensor: •C _
Z
~C(I_f)C(~M)m+IAt2I
[iCy c~T
63D -Jr"
[c363C M
Z
t3C ( 1 - f ) ( m +
~M ''+- piCv c3T
1)C(~M)"+I]At
d IId[I2"
Temperature is obtained from the fully implicit scheme T.+I = L + ? . + , a t . Temperature values are expected to be close to the exact ones, except in some localized zones where severe thermal boundary conditions are observed. NUMERICAL
EXAMPLES
Two typical examples illustrate the performance and interesting feature of finite element models which take into account the growth of microvoids in materials: first, benchmark traction of a single Q4 element in order to appreciate some numerical performances, second tensile test of a standard ALl0 notched cylinder to expose some effects of void volume growth on the state variables. In the examples, a reference value of 100 N . m m - 2 is chosen for the material consistency, 0.01 and 0.25 for the strain rate hardening coefficients (two reference cases), 0.1 for the strain hardening coefficient, - 3 for the temperature sensitivity coefficient, 10- s and 10- 3 for the initial void volume fraction (two reference cases), and reference parameters for the viscoplastic potential of q, = 1.25, q2 = 1, q3 = (q,)2 = 1.5625 according to Ref. 1-16]. Typical single element traction
Consider the isothermal traction of a four node quadrilateral element (Fig. 1) in which the current particles coordinates x and y at time t are expressed by X = (1 + Off)X i
y = (1 + 20 y~ with 2ty i the prescribed displacement on nodes 3 and 4, xi, y~ the initial particle coordinates. Numerical and analytical calculations are both achieved for displacement factor 2 equal to
y
[
prescribeddisplacement
~\\\\\'~'~
~NX\\\\"¢.
6- x
FIG. 1. Four node quadrilateral isoparametric element: x and y directions boundary conditions, prescribed displacement in the y direction.
C. LAMY et al.
344
TABLE 1. TRACTION OF A FOUR NODE QUADRILATERAL ELEMENT: NUMERICAL AND ANALYTICAL VALUES OF EQUIVALENT STRAIN RATE, STRESS AND VOID VOLUME FRACTION IN A POROUS VISCOPLASTIC MATERIAL a M = |00(F,'M) TM N ' m m - 2 , f = 10 s UNTIL 5 0 0 INCREMENTS OF 0 , 0 0 2 SECONDS (DISPLACEMENT FACTOR /. -- 0.3 S 1)
I
~M
(s)
(s
O~yy (N'mm 21
l)
./ ( x l 0 s)
frill
(N'mm
a)
0.2
0.326783 0.326783
114.177 114.250
57.0874 57.1601
1.56076 1.56118
O.5
0.301193 0.301193
I 14.077 114.085
57.0363 57.0447
2.90856 2.911075
0.8
0.279308 0.279308
I 13.978 113.933
56.9779 56.9755
5.17095 5.16982
1.0
0.266394 0.266395
113.912 113,899
56.9399 56.9378
7.41720 7.41021
B o l d n u m b e r s : n u m e r i c a l v a l u e s . Italic n u m b e r s : a n a l y t i c a l v a l u e s .
0.3 s - 1 until time t is equal to 1 s, the reference visco-plastic yield stress of the matrix being aM = 100(~M) ° ° 1 N ' m m -2 and the initial void volume fraction 10 -5 . Table 1 gives numerical and analytical values of the relative equivalent strain rate, C a u c h y stress ayy, hydrostatic stress a m and void volume f r a c t i o n f The results of the finite element c o m p u t a t i o n agree very well with the analytical ones as shown by the small relative error values A~M
eM ~<3"10-5'
Aayy
--
~<10 3
Gyy
~
AG m
__
<2.10-3
(7 m
'
Af 0_4. <~9'1 f
Tensile test of notched cylindrical specimen The standard tensile test of an AE 10 notched cylinder shown in Fig. 2(a) is now analysed when a porous material is involved. The effective material matrix yield stress value is given by O"M =
100(~M) 0'25
N . m m -2
and the initial void volume fraction is equal to 10 -3. The test conditions are: ram velocity of the test machine 20 m m s t, a specimen reference length from 95 m m at time t = 0 to 102 m m at time t = 0.35 s. Only a quarter of the meridian plane of the specimen is used for the model. The initial mesh is made of 134 Q4 elements with 2 x 2 Gauss integration points [Figs 2(b), (c)]. The prescribed velocities on the Z axis are vr = 0 and on the R axis vz = 0. The c o m p u t a t i o n is achieved in 70 time increments. After the test, the dimensions of the specimens are slightly different when comparing the pore free visco-plastic material to the p o r o u s visco-plastic one. For instance, the reduction of the diameter of the notch is more important for the pore free specimen than for the porous one (Fig. 3). So, from a measure of the notch diameter, an estimate of the void volume fraction is available. Figure 3 also shows the radial stress maps in the notch zones for pore free visco-plastic material and for porous visco-plastic one. It is noticed that the m a x i m u m radial stresses are, respectively, 35.2 N . m m -2 located at coordinates r = 0, z = 0 in the pore free specimen [Fig. 3(a)] and 19.4 N . m m - 2 located at coordinate z = 4.25 m m on the notch surface in the porous one [Fig. 3(b)]. The change in the stress m a p is also significant and from it can be observed some new stress increases and decreases for a p o r o u s material c o m p a r e d to a pore free one in the z-direction and in the notch surface normal direction.
Void growth damage in thermo-viscoplasticity
345
Ca) Scale 1 z
(c) ~
(b) r
r
Scale6
Scale3
FIG. 2. Cylindrical notched AE 10 specimen: (a) dimensions in the testing zone; (b) finite element mesh in a quarter of the meridian plane; (c) zoom in the center cross-section.
.
z
pore f r e e /
=
.
r=8.92 z
~
(N.mm -2)
K1L r
(a)
~
5/,F
(N.
A-2.3
/
B
~
5.2
/
D 20.~
o
""'D \
-9.8
"
A-7.3
Ill
B-1.9
I-
D 8.7
~3z~ !!i:! -
* maximum minimum (b) o
2)
[[~ f I[
~.03 E 27.7 z=0 * 35.2
•
r=8.92 ~ A ~ / C p o r °/ u s
.
Scale5
FIG. 3. Cylindrical notched AE 10 specimen, computations at 0.35 s: (a) radial stress in a pore free specimen; (b) radial stress in a porous specimen.
346
C. LAMYet al.
A n a l o g o u s axial stress m a p s are obtained, the m a x i m u m being l o c a t e d in the n o t c h e d cross-section, 135.21 N . m m - 2 in the vicinity of the z-axis for the p o r e free specimen, 110.84 N . m m - 2 on the free surface for the p o r o u s specimen. In the latter case, a released stress zone is also observed in the vicinity of the z-axis, the m i n i m u m stress value being 4.55 N - r a m -2 at c o o r d i n a t e z -- 7.6 ram. The evolutions of h y d r o s t a t i c stress in the p o r e free m a t e r i a l and in the p o r o u s one are also different: at time t = 0.35 s, the m a x i m u m is l o c a t e d at r = 0, z = 0 for a p o r e free specimen [Fig. 4(a)] but at z = 0 on the notch surface for a p o r o u s one and the m a p is also affected by released stress zones [Fig. 4(b)]. L o o k i n g now for time e v o l u t i o n in the center zone discretized in I, II, III a n d IV Q4 elements [Fig. 2(c)], it is clear that h y d r o s t a t i c stress time derivatives in the I, II, III elements are going from small values 22 N . m m - 2. s ~ as expected in a p o r e free specimen to significant ones 220 N . m m - 2 . s - ~ in p o r o u s specimens (Fig. 5). All a l o n g test time, the m a x i m u m values r e m a i n in element I, that c o r r e s p o n d s to center zone for the pore free specimen, and on the c o n t r a r y they change from element I to element IV for the p o r o u s specimen. The m a x i m u m values of the Mises effective stress at time t = 0.35 s r e m a i n located at c o o r d i n a t e z = 0 on the notch free surface, 132.10 N . m m 2 for pore free m a t e r i a l and 102-41 N . m m 2 for p o r o u s material; however the r g r a d i e n t in the notch cross-section of the p o r o u s specimen reaches 34 N . m m -3 c o m p a r e d to 3 N - r a m -3 for the pore free specimen.
z
r=-8.92
,
z
r=8.92
f,':? / ~B,- /
(N.mm-2) ,
/):-t
D
J r=3.03
ID I / I
IB r:3"161 E 43.0
/*'-.9~~ r
*
51.6
r
(a)
* maximum
(b)
Scale 5
FIG. 4. Cylindrical notched AE 10 specimen, computations at 0.35 s: (a) hydrostatic stress in a pore free specimen; (b) hydrostatic stress in a porous specimen.
o
• 70"
Z
i
!
~
~III
~60"
~, 60
50" __...Iv--
~ ..~
. 70 -"-'II
HI
~
4ff
~
pore free I
I
I
0.1
40
I
I
0.2
I
0.3 time
(a)
50 ...---IV--"--
~
I
I
I
0.1
0.2
(s)
I
I
I
0.3 time (s)
(b)
FIG. 5. Cylindrical notched AE 10 specimen, computations vs time in the center zone: (a) hydrostatic stress in a pore free specimen; (b) hydrostatic stress in a porous specimen.
Void growth damage in thermo-viscoplasticity
1.0.] ~
pore free
1.0.
fF 10"3 porous
o 0.4.
347
I
\ o.4.
"~ o 2
"t~ o.2.
VI
I IH I I~ ~I 0.46 0.56 0.66 equivalent strain
0.36
IV I 0.36
I I1 , 0.46 0.56 0.66 equivalent strain
(a)
(b)
F16. 6. Cylindrical notched AE 10 specimen, triaxiality vs equivalent strain in the center zone: (a) stress triaxiality in a pore free specimen; (b) stress triaxiality in a porous specimen.
Some analyses of the triaxiality ratio O'm/O'eq are given on Fig. 6. For a pore free material, the triaxiality ratio remains almost constant when the equivalent strain increases [Fig. 6(a)] whilst for porous material, triaxiality in elements I, II and III decreases for higher values of equivalent strain [Fig. 6(b)]. The void volume fraction values for the porous material are shown on Fig. 7: either in the z direction or in the notch surface tangential direction, rather regular values map and gradients are obtained [Fig. 7(a)]; volume fraction time derivatives in the center elements I, II, III and IV increase regularly from time t = 0 to time t = 0.35 s, the final derivatives being not very different from 1.2 s- 1 [Fig. 7(b)]. The distribution of equivalent strain is also affected by material porosity: although the isovalue maps look similar for both materials, the maximum values, located twice at the boundary of the notch cross-section, reach 1.006 for the pore free specimen and 1.073 for the porous one. Look now at a thermo-viscoplastic porous material, the yield stress of which being temperature dependent is ~M = 100(eM)°'1 1000
z porous3 f~ 10"
I r=8.92 1 | z=11.32 | 1 0.16
~
(iM)°'25N'mm-2"
/ I //II //Ill
.~
0.12 f
O
C
IV
0.04 ~3 16
l
[Scale5
..
(a)
r
*maximum
0.1 (b)
0.2
0.3 time (s)
FIG. 7. Cylindrical notched AE 10 specimen: computations of void volume fraction: (a) at time 0.35 s; (b) vs time.
348
C. LAMY et al.
z
I
."',,
-\"-./
I
r=-8.85
A o. I
"_77{71
* maximum
Scale 5
FIG. 8. Cylindrical notched AE l0 specimen, computations of void volume fraction at time 0.35 s.
| r=-8.85 I z=10 97
porous Ti= 1100 °C
1112
~
III
1108 ~
~B
-""~ /
•- " ' ~ " ~ D "~'- [ / _ ¢ ' ~ r=3.46 / El Ii -z=O"--
A I101 B 1103 C D E *
1105 1107 1109 Illl
,
r Scale 5 (a)
11 I
,
0.1 * maximum
, 0.2
(b)
,
, 0.3 time
l (s)
FIG, 9. Cylindrical notched AE 10 specimen, computations of temperature: (a) at 0.35 s; (b) vs time.
Computations have been made for a reference steel of initial density Pi = 7800 k g . m - 3 and isovolume specific heat Cv = 580 J . k g - 1. °C- 1. The fraction of plastic work converted into heat is 0.95 and the initial temperature T~ = 1100 °C. The void fraction map (Fig. 8) is similar to that obtained in the previous isothermal analysis. The most important growth of void volume fraction appears in the center of the notch cross-section, from an initial value of 10 -a to 51.10 -3 whereas the maximum increase of temperature is found on the r axis at the notch free surface (Fig. 9). The time evolution in the center zone reveals that the temperature time derivatives rise steadily, the highest derivative being 40°C.s - 1 in element IV [Fig. 9(b)]. CONCLUSION
The growth of microvoids is considered as an important damage parameter for polycrystalline and ductile materials. To allow the secure analysis of visco-plastic structures, a thermo-viscoplastic damage model based on the Gurson yield function is developed within the finite element framework provided by the ASTR1Dcode. The consistent tangent operator for the finite element discretization is defined and the bulk constitutive equations of the material are expressed by exponential functions of strain rate and temperature.
Void growth damage in thermo-viscoplasticity
349
T h e r e p o r t e d n u m e r i c a l tests o n a f o u r n o d e q u a d r i l a t e r a l e l e m e n t s h o w t h a t t h e r e l a t i v e e r r o r s o n t h e strain, stress a n d v o i d v o l u m e f r a c t i o n c o m p u t e d v a l u e s are a l w a y s l o w e r t h a n l 0 - 3 a n d t h a t c o n v e r g e n c e is g o o d . T h e finite e l e m e n t m o d e l o f t h e s t a n d a r d tensile test o f A E l 0 n o t c h e d c y l i n d r i c a l s p e c i m e n s h o w s t h a t h y d r o s t a t i c stress d e c r e a s e s w i t h time, t h e m a x i m u m v a l u e s a r e l o c a t e d in t h e n o t c h e d c r o s s - s e c t i o n , first in the z axis z o n e a n d l a t e r in the n o t c h free surface. T h e m a x i m u m g r o w t h of the m i c r o v o i d s is o b s e r v e d in t h e n o t c h e d c r o s s - s e c t i o n also a n d r e m a i n s the test in the z axis zone. T h e o t h e r state v a r i a b l e s m a p s a n d stress t r i a x i a l i t y a r e also different f r o m the o n e s r e l a t e d to p o r e free m a t e r i a l s . T h e influence of a d i a b a t i c t e m p e r a t u r e i n c r e a s e d u e to v i s c o - p l a s t i c d e f o r m a t i o n was t h e n studied: in the A E l 0 s p e c i m e n , v o i d v o l u m e f r a c t i o n m o d i f i c a t i o n s a n d significant t e m p e r a t u r e g r a d i e n t s are o b s e r v e d w h e n c o n s i d e r i n g a s t a n d a r d steel at f o r g i n g t e m p e r ature. S p e c i m e n profile c h a n g e s are also effective w h e n t a k i n g i n t o a c c o u n t v o i d v o l u m e fraction growth and temperature gradients. I n c o n c l u s i o n , the m o d e l is a b l e to give n e w i n f o r m a t i o n o n w h a t c o u l d h a p p e n w h e n v i s c o - p l a s t i c b e h a v i o r is effective in p o l y c r y s t a l l i n e a n d d u c t i l e m a t e r i a l s : stress t e n s o r m o d i f i c a t i o n s , h e t e r o g e n e i t y of v o i d v o l u m e f r a c t i o n a n d significant t e m p e r a t u r e gradients. The authors would like to thank Nord Pas de Calais Region, C.N.R.S. and the French Ministry of Education for supports.
Acknowledgements
REFERENCES 1. F. MONTHEILLETand F. MoussY, Physique et m~canique de l'endommagement. Editions de Physique (1988). 2. F. LEROY,F. MoussY and F. MONTHEILLET,Interactions inclusion-matrice au cours d'op6rations de formage. Ecole d'Et~ Mat~riaux Mise en Forme Pi~ces Fortunes, GIS Mise en Forme, GRECO Grandes D6formations et Endommagement, Saint Pierre d'O16ron (P. Franciosi and F. Moussy Eds). Presses du CNRS, Paris (1989). 3. E. OI~ATEand E. KLEBER,Plastic flow of void containing metals, applications to axisymmetric sheet forming problems. Proceedings of N U M I F O R M 86, 339 (1986). 4. M. LACOMME, A. FROGER, J. P. ANSART and R. DORMEVAL, Endommagement sous choc de l'alliage d'aluminium AU4G. Congr~s International D Y M A T , Editions de Physique (1988). 5. G. LE Roy, J. D. EMaURY,G. EDWARDSand M. F. ASHBY,A model of ductile fracture based on the nucleation and growth of voids. Acta Metall. 29, 1509 (1981). 6. F. Moussv, Etude de Fendommagement li6 a la pr6sence d'inclusions. Contrat de programme G.I.S. raise en Forme, M R T 83.P.0462, 3, 59 (1987). 7. J. C. GELIN and J. OUDIN, Mesure de l'endommagement des aciers inoxydables aust6nitiques par essais de traction-torsion. Compte rendu du Groupe de R~flexion sur l'Endommagement, Saint Germain en Laye (1983). 8. D. MARQUIS,M6canique de l'endommagement continu. Ecole d'Et~ Mat~riaux Mise en Forme Pi~ces Form~es, GIS Mise en Forme, GRECO Grandes D6formations et Endommagement, Saint Pierre d'O16ron (P. Franciosi and F. Moussy Eds). Presses du CNRS, Paris (1989). 9. M. PREDELEANU,Finite strain plasticity analysis and damage effects in metal forming processes. Computational methods for predicting material processing defects (M. Predeleanu Ed.), p. 295. Elsevier (1987). 10. C. L. HOM, R. M. MCMEEKING, Void growth in elastic-plastic materials. J. Appl. Mech. 56, 309 (1989). 11. A. L. GURSON,Continuum theory of ductile rupture by void nucleation and growth--Part I. Yield criteria and flow rules for porous ductile media. J. Engng Mater. Technol. 99, 2 (1977). 12. A. L. GURSON,Porous rigid-plastic materials containing rigid inclusions--yield function, plastic potential and void nucleation. Advances in Research on the Strength and Fracture of Materials (D.M.R. Taplin Ed.), Pergamon Press, Oxford (1977). 13. K. MOR1 and K. OSAKADA,Analysis of the forming process of sintered power metals by a rigid-plastic finiteelement method. Int. J. Mech. Sci. 29, 229 (1987). 14. A. ROSOCHOWSKIand L. OLEN|K, Damage evolution in mild steel. Int. J. Mech. Sci. 30, 1, 51 (1988). 15. V. TVERGAARD,Ductile fracture by cavity nucleation between larger voids. J. Mech. Phys. Solids 30, 265 (1982). 16. V. TVERGAARD,Material failure by void coalescence in localized shear bands. Int. J. Solids Structures lg, 659 (1982). 17. N. ARAVAS,The analysis of void growth that leads to central bursts during extrusion. J. Mech. Phys. Solids 34, 1, 55 (1986). 18. R. BECKERand A. NEEDLEMAN,O. RICHMONDand V. TVERGAARD,Void growth and failure in notched bars. J. Mech. Phys. Solids 36, 317 (1988). 19. N. ARAVAS,On the numerical integration of a class of pressure dependent plasticity models. Int. J. Numer. Meth. Engng 24, 1395 (1987). 20. V. TVERGAARD,Influence of void nucleation on ductile shear fracture at a free surface. J. Mech. Phys. Solids 30, 399 (1982). MS
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