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Statistically informed dynamics of void growth in rate dependent materials
International Journal of Impact Engineering 36 (2009) 1242–1249
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Statistically informed dynamics of void growth in rate dependent materials T.W. Wright a, b, K.T. Ramesh b, * a b
U.S. Army Research Laboratory, 4600 Deer Creek Loop, Aberdeen Proving Ground, MD 21005, USA Mechanical Engineering, Johns Hopkins University, Baltimore, MD 21218, USA
a r t i c l e i n f o
a b s t r a c t
Article history: Received 27 August 2008 Received in revised form 1 May 2009 Accepted 17 May 2009 Available online 27 May 2009
An inherently rate dependent material model is used to model the nucleation and growth of voids in metals, leading to spall fracture. Intrinsic material rate dependence introduces a third time scale in addition to those introduced by rates of nucleation and growth. Material rate dependence does increase the spall strength of metals, but it is not nearly as important as local inertia in doing so. Ó 2009 Elsevier Ltd. All rights reserved.
Keywords: Spall Voids Nucleation Growth Local inertia Rate dependence
1. Introduction The rapid cooperative growth of voids within materials is a classic failure mechanism, and it is particularly important under impact loading. Impact events typically produce large amplitude compressive stress waves, which reflect from free surfaces. The interaction of a reflected wave with other waves can generate nearly hydrostatic tensile stress at high loading rates, resulting in rapid nucleation, growth and interaction of voids. The result often is internal rupture through the process known as spallation. In previous work the authors have developed a statistically informed model for the cooperative nucleation, growth and early interaction of voids in a plastically deforming metal [1]. That work focused on rate independent materials, but most metals are well known to be rate-sensitive at the high strain rates associated with impact. Spall data from many uniaxial strain experiments performed on many different materials have been represented in the form m _ sspall ¼ AðV=V (see Antoun et al., [2]), where A and m are 0Þ _ material constants, and V=V 0 is the maximum volumetric strain rate at the spall plane as estimated from data collected at a free surface. Ref. [2] also provides an excellent introduction to the vast literature on spall fracture, which we could not begin to summarize
* Corresponding author. Department of Mechanical Engineering, Johns Hopkins University, Baltimore, MD 21218, USA. Tel: þ1 410 516 7735; fax: þ1 410 516 7254. E-mail address: [email protected] (K.T. Ramesh). 0734-743X/$ – see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijimpeng.2009.05.007
here. However, the apparent rate sensitivity, m, as used above for spall and determined experimentally, is generally much higher than that known from other experiments designed to produce high-rate, homogeneous straining in a small volume of material. Examples of such tests are the split Hopkinson pressure or torsion bar (due to Kolsky) and the pressure/shear experiment (due to Clifton). For a brief discussion of these and other techniques, see Ramesh [3]. The object of this paper is to examine the role of true material rate dependence in contributing to spall strength. As in [1] previously, we postulate that spall strength is not in itself a fundamental property of a material, but rather it is the result of a deformation process that is fully controlled by the bulk homogeneous properties of the intact matrix material, including material rate dependence. In most ductile metals the process underlying spall is void nucleation and growth as the material responds to rapid expansion, e.g., see Meyers [4]. Voids formed under dynamic conditions often seem rapidly to become spherical in shape and to stay approximately spherical during expansion, as in Fig. 1. The figure also shows the preference for voids to form on or near grain boundaries, but voids can also form on other disruptions of the fundamental material lattice or within the bulk lattice if the local mean stress is strong enough. Spall strength can be much greater than the threshold for void nucleation, which is itself often four or more times larger than the ordinary tensile yield strength. From the continuum point of view nucleation represents a bifurcation in the response to hydrostatic
T.W. Wright, K.T. Ramesh / International Journal of Impact Engineering 36 (2009) 1242–1249
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because the homogenized quantities are work conjugate, within an error term, provided the kinetic energy relative to the c.g. is retained:
Sij Dij ¼
sij dij
þ
K_ rel
1 Vtotal
#
x_ i Dij xj
Sext
ðsik Sik Þnk ds
Fig. 1. Voids at grain boundaries in Cu. Dynamically formed voids tend to be spherical even when close to another void [4]. Reproduced with permission.
tension. Ball [5] demonstrated the existence of hydrostatic bifurcation for nonlinear elastic materials, and Huang et al. [6] obtained an explicit formula for the bifurcation stress for elastic/plastic materials. Wu et al. [7] later showed that the calculated value for many materials corresponded roughly to the lowest level of experimentally observed spall stress in those same materials. We will refer to the bifurcation stress as a critical stress. It is not unusual for spall strength to be 10 times the static tensile yield stress, which means that the tensile pressure at spallation can be nearly an order of magnitude larger than any measure of deviatoric stress. We demonstrate that when local inertia around growing voids, together with intrinsic bulk material rate dependence, is taken into account, spall data can be accurately estimated over several decades of the rate of volumetric expansion, and that in all cases examined local inertia around the voids accounts for a greater proportion of the increase than does intrinsic material rate dependence. Thus, inertia is the main contributor to the apparently higher rate sensitivity in spall. 2. Background For clarity we repeat the chief elements of the theory as presented in [1]. Following Molinari and Mercier [8], but using a position vector x relative to the center of mass of a material volume element rather than relative to a fixed frame, we define the macroscopic or homogenized strain rate and stress in terms of average fields within a representative volume element or RVE as follows: 1 # x_ ði njÞ ds ¼ Vtotal
1 Dij ðx;tÞhVtotal
Sext 1 Sij ðx;tÞhVtotal
# xðj tiÞ ds ¼ Sext
Z
dij dv þ
Vsol 1 Vtotal
X
#
!
x_ ði njÞ ds
voids Svoid
! o € sij þ rxðj xiÞ dv
Z n
(1)
Vsolid
In Eq. (1) x is the vectorial position of the center of gravity (c.g.) of the RVE, x is the position vector relative to the c.g., Vtotal and Sext are the total volume and exterior material surface respectively of the RVE, n is the exterior normal on Sext or on the voids. Tractions on the exterior surface and the stress and strain rate within the RVE are given by ti ¼ sij nj ; sij ; and dij, respectively. Thus the homogenized strain rate and stress are defined completely by particle velocities relative to the c.g. and by tractions, both taken only on the exterior surface of the RVE, and by the geometrical properties of the RVE itself. The right hand sides of Eq. (1) follow exactly from the definitions, given sufficient smoothness of the fields and surfaces. As shown by the following identity, the definitions can be useful
(2)
where CC$DD indicates the volumetric average of the quantity within the double brackets. Again the expression is exact given sufficient smoothness. However, its usefulness does require that the last term is sufficiently small, which we assume in this paper. With the porosity 4 defined as the void volume divided by the total volume in an RVE of total mass m, the macroscopic density of the RVE may be defined as rhm=Vtotal . We also define the average density of just the solid material in the RVE, taken by itself, as b r hm=Vsolid , and therefore 1 4 ¼ Vsolid =Vtotal ¼ r=br . Macroscopic conservation of mass is given by r_ =r þ vx_ i =vxi ¼ 0. With total rate of volumetric expansion given by e_ ¼ vx_ i =vxi, conservar_ =br 4_ =ð1 4Þ þ e_ ¼ 0. From the tion of mass demands that b r it seems reasonable to associate the first term with definition of b elastic expansion in the macroscopic porous material. Elastic dilation of the matrix material will increase the size of voids in approximately the same degree as the matrix itself, but have little if any effect on the porosity (void volume per unit total volume). It follows then that change in porosity should be associated with plastic deformation around voids, and that therefore, it is an essentially incompressible process within the matrix material. With these ideas in mind, and with the macroscopic mean stress given by p ¼ ð1=3ÞSii, a constitutive law for the elastic rate of _ r_ =br ¼ p=kð 4; pÞ, and the macroscopic expansion may be written b balance of mass becomes
p_ kð4; pÞ
þ
4_ ¼ e_ 14
(3)
The instantaneous bulk modulus for the homogenized RVE is k, which may depend on the porosity and the mean stress. Eq. (3) shows that volumetric expansion will force a competition between elastic expansion of the porous macroscopic material and more essential changes in porosity. The macroscopic mean stress driving the RVE is p, and for the homogenized bulk modulus we use
14 1 þ ð3k=4mÞ4
k ¼ k
(4)
where k; m are the standard bulk and shear moduli, respectively, for the matrix material. The terms in the square brackets modify the elastic bulk modulus to account for porosity according to a selfconsistent elasticity argument cited by Christensen [9]. The homogenized bulk modulus is predicted to decrease with increasing porosity, which would be expected, but the quantitative accuracy of these terms is not known. In the essential part of our calculations porosity remains much less than a few percent, so the variation of the bulk modulus is unlikely to be a critical issue. The simplest model of a process for increasing porosity is radial expansion of an incompressible, thick walled, hollow sphere, with inner radius a and outer radius b, which is driven by radial tractions only on its outer surface. The surfaces of the voids will be traction free. Incompressibility is captured by the expression r 3 a3 ¼ R3 A3 where upper case denotes initial values, lower case denotes current values, and a ¼ aðA; tÞ and r ¼ rðR; tÞ, respectively. The _ 2, particle velocity and acceleration may be expressed as r_ ¼ a2 a=r 2 2 2 2 2 2 5 _ _ =r . Radial integration of the þ 2aa_ =r 2ða aÞ and r_ ¼ a a=r equation of motion through the wall of the hollow sphere results in the equation
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raa€ 1
Zb a 3 a 1 a4 2 2 2se _ ¼ p se ðbÞ þr a dr 2 þ r b 2 b 2 b4 3
(5)
a
p 2=3se ðbÞ.
where srr ðbÞ ¼ In our case Eq. (5) will be interpreted to represent void growth in a material that is embedded in a porous material whose macroscopic porosity 4 is exactly the same as the local porosity, a3 =b3 , thus a=b ¼ 41=3 . Previously in [1] we had omitted the term ð2=3Þse ðbÞ on the right hand side of Eq. (5) on the grounds that it is much less than p, but here for consistency in developing the growth equation, we leave it in place. As reported later in Section 5, actual calculations show that it has an insignificant effect. Eq. (5) is essentially the equation of motion given by Knowles and Jakub [10] for an elastic incompressible material, but here it is interpreted as applying to an elastic/plastic, incompressible material where the effective tensile stress in spherical expansion is
pffiffiffiffiffiffiffi
se h 3J2 ¼ jsrr sqq j:
(6)
In expansion srr sqq < 0. The right hand side of Eq. (5) represents the difference between the driving macroscopic mean stress, p, and the total resistive effect of an elastic/plastic deviatoric stress, Rb Ihð2=3Þse ðbÞ þ a ð2se =rÞdr. We will require that a_ 0, thus eliminating any consideration of reverse plastic flow (collapsing rather than expanding voids). To ensure this requirement, we impose the joint conditions that if ða_ ¼ 0ÞWðp I 0Þ, then the r.h.s. of Eq. (5) is set equal to zero so that a_ ¼ 0 as well, and void growth stops completely; otherwise the r.h.s. is equal to p I. Thus, if a_ > 0, then the r.h.s. is as written in Eq. (5), even if p I is negative. But if a_ ¼ 0, then the r.h.s. as written in Eq. (5) is not applied until p I becomes positive. With these conditions a_ can never be negative. Finally, as a surrogate for microstructure, which necessarily introduces many defects and stress concentrations into an otherwise perfect crystalline lattice, we assume that there are many sites where a void might nucleate and that there is a corresponding distribution of critical stresses for nucleation. Once voids have nucleated, the total void volume density per unit of solid material in the RVE we denote as y. The contribution from all voids with size ac , which in the absence of porosity have nucleation stress pc , are weighted by a distribution function, gðpc p0c Þ, where p0c is the lowest critical stress for any void in the material. The increment of volume contributed by all voids of radius ac is 4=3pa3c dn, where R dn ¼ Ngðpc Þdpc and dn ¼ N. Here N is the number of potential nucleation sites per unit volume of matrix material. In this article we use only gðpc Þdpc ¼ 2CxDexpðx2 Þdx, where x ¼ ðpc p0c Þ=a. This is a Weibull distribution with modulus 2, stress scale a, and lower limit p0c . Using the relationships between void volume density and porosity, ð1 þ yÞð1 4Þ ¼ 1, and their rates y_ ¼ 4_ =ð1 4Þ2 we arrive at an equation for the rate of growth of porosity
ZN 4_ ¼ 4pNð1 4Þ a2c a_ c gðpc Þdpc 2
or it may also depend on a measure of effective strain rate, Ac , as explained below. Note that the critical stress, pc , which is the variable of integration in Eq. (7), is also used in determining the total deviatoric resistance in Eq. (5), as shown below. Before continuing our discussion of the right hand side of Eq. (5) and introducing a rate model, it is necessary to examine the assumed kinematics of the thick walled sphere further. The spatial representation of the strain rate tensor in spherical coordinates for this case is diagonal with entries ðvr_ =vr; r_ =r; r_ =rÞ. Because the matrix material is assumed to be incompressible, the sum of the three rates must vanish, so the first term equals 2r_ =r. Consequently the total effective deviatoric strain rate that is work conjugate to the effective stress above is given by
3_ T h
rffiffiffiffiffiffiffiffiffiffiffiffiffiffi r_ 2 a3 a_ dij dij ¼ 2 ¼ 2 3 r 3 r a
(8)
The right hand side of Eq. (8) may also be written as 3 A3 þ a3 Þ so the total strain at a fixed value of R may be _ 2a2 a=ðR found by integration to be
3T ¼
2 R3 A3 þ a3 2 r3 ¼ ln ln 3 3 r 3 a 3 þ A3 R3
(9)
Assuming that initial void sizes are small enough to be ignored, we set A ¼ 0. At a large enough distance from the void the total strain will be less than the elastic yield strain, 3Y ¼ sY =E, and the transition will occur at approximately a3 =rp3 ¼ 33Y =2. If As0, first yield will occur at the void surface when ða AÞ=Az1=23Y , but as the void grows the elastic/plastic transition will occur at approximately rp3 za3 ð1 A3 =a3 Þð2=33Y Þ, which will rapidly approach the first approximation for A ¼ 0. 3. Rate model The main difference between this and the previous work is the construction of the deviatoric resistance, which now is modulated by the rate sensitive nature of the material. For present purposes it is sufficient to model plastic strain rate within each hollow sphere by the simple expression 3_ p ¼ 3_ 0 Cjse =sY j1=m 1D, where the strain rate sensitivity is small, m 1, 3_ 0 ¼ Oð104 Þ s1 is a characteristic constant, and sY is the static yield stress in tension. In general sY will evolve with the history of deformation and temperature at each point in the material, but for purposes of this paper, we assume that work hardening has saturated before rapid expansion begins, and that sY is constant. Recalling that the total strain rate within an incompressible hollow _ 3, and that the elastic plastic sphere is given by 3_ T ¼ 2r_ =r ¼ 2a2 a=r 3 3 boundary is located at rp =a z2=33Y , it is shown in Appendix A that the effective stress in the elastic and plastic regions, respectively, is
selas e =sY z2x=33Y ; (7)
p0c
Eqs. (3), (5), and (7) have the same overall structure as that established previously for rate independent materials [1]. Eq. (5) will be applied to every void with internal radius, ac , effective external radius, bc ¼ ac 41=3 , and critical or nucleation stress, pc . Here the cumulative porosity 4 from all voids in the RVE has been used to generate an effective outer radius for each hollow sphere. As before, the integrated momentum equation is driven by the excess of mean stress over the total deviatoric resistance in a thick-walled R a 41=3 ð2se =rÞdr, but hollow sphere, Ið4; Ac ; pc Þh2=3se ðac 41=3 Þ þ acc now the equivalent stress, se , may be due only to elastic deformation
x ¼ a3c =r 3 33Y =2 m 1 þ A R xR m splas s = wð1 þ A xÞ þ c Y e 1 þ AR xR eð1 þ AR xR Þ3_ 0 ðt tR Þ=3Y m ð1 þ AR xR Þm !ð1 þ Ac xÞm ;
t[tR
x ¼ a3c =r 3 > 33Y =2
(10)
where Ac ¼ 2a_ c ðtÞ=3_ 0 ac ðtÞ, AR ¼ 2a_ c ðtR Þ=3_ 0 ac ðtR Þ, and tR is the reference time at which plasticity first begins at particles with initial plas radius R. The expression for se has been found from an asymptotic boundary layer analysis, described in Appendix A. Note that the full asymptotic expression is continuous at t ¼ tR . However, we will use
T.W. Wright, K.T. Ramesh / International Journal of Impact Engineering 36 (2009) 1242–1249
only the limiting expression, which is not continuous at rp . This procedure will tend to overestimate deviatoric resistance slightly by transitioning too rapidly to full viscoplastic flow. After taking account of both elastic and viscoplastic responses, we find that the total resistance in Eq. (5) for each void may now be expressed as:
2 e s ðbc Þþ 3
¼
Zbc ac
2se dr r
8 ( ) Z1 > > > 2s m 1 > ð1þAc xÞ x dx ; > Y 1þ > > <3
4 33Y =2
33Y =2
( ) > Z1 > > 2 > m m 1 > > sY ð1þAc 4Þ þ ð1þAc xÞ x dx ; 4 >33Y =2 > :3
homogeneous material, the number p0c is a calibrated value for the actual lowest value for void nucleation in a real material. We would expect that j1 p0c =p0 j 1 in most cases. Thus the range of pc represents all the possible values at which a void could nucleate at various sites in a material, and the distribution function gðpc p0c Þ represents the probability of void nucleation at pc . The resulting set of Eqs. (3), (5), and (7) thus provides a means for representing an inhomogeneous material with microstructure. Voids nucleate over a range of stresses, each grows according to its own dynamics, but all voids contribute to the total porosity, and the driving tensile pressure, the total porosity, and the growth of all voids, are fully coupled dynamically. Nondimensionalization of Eqs. (3), (5), and (7) brings out other features. Our approach in this paper is slightly different from that used in [1]. We choose the following:
~ ¼ ea= _ X ¼ p=sY ; a
4
(11) where m is now only the material strain rate sensitivity in the matrix material. The lower limit in the integral corresponds to the plastically deforming points most distant from the cavity and the upper limit to the surface of the cavity. The integrand in Eq. (11) is positive and decreases monotonically as x increases within the range of integration. Therefore, even though the normalized strain rate, Ac x, is highest near the surface of the cavity (the upper limit of the integral), it is not those points that contribute the most to the total deviatoric resistance. Rather, it is the points that are closest to the elastic plastic boundary (the lower limit) that contribute the most because they represent a larger volume of matrix material. Consequently we expect that strong interaction between voids may occur either as their outer plastic zones collide or certainly as the voids coalesce to become larger voids. We do not attempt to model void collisions explicitly, and as we shall see, estimation of spall strength does not appear to depend on it, although subsequent fracture and fragmentation certainly do. There are two obvious limits for the integrals in Eq. (11) that can be evaluated explicitly, namely, the rate independent limit ðm/0 or A/0Þ and the high rate limit ðA4[1Þ. These are
Z1 4
8 < ln41 ; dx ð1 þ AxÞm / : m 1 x A m ð1 4m Þ;
as A/0
(12)
for A4[1
hð4; Ac Þ ¼
( ) Z1 2 e 2 s ðbc Þ þ ðse =xÞdx : 3 3
(13)
4
We now represent the total deviatoric resistance in Eq. (5) for any void in the distribution as
Ið4; Ac ; pc Þ ¼ pc hð4; Ac Þ;
p0c pc < N
pffiffiffiffiffiffiffiffiffiffi sY =r; 3 ¼ sY =k
_ t0 Þ; x ¼ pc p0c s ¼ eðt
.
a
which reduces the three equations to:
3k 4 þ 4s ¼ 1 4 4m
a 3 1 p0 ~2s ¼ X ~a ~ss þ x þ c hð4;AÞ 1 41=3 a 241=3 þ 4 4=3 a sY sY 2 2 ZN 4s ¼ 4pk3 ð1 4Þ2 a~2 a~s gðxÞdx (15)
3Xs 1 þ
0
pffiffiffiffiffiffiffiffiffiffi ~s =3_ 0 a ~ and k ¼ N 1=3 e= _ sY =r. Writing the In Eq. (15) A ¼ 2e_ a equations in this form reveals several physical parameters, the most important of which are 3 ¼ Oð102 Þ and k ¼ Oð103 104 Þ. Initial conditions are: Xð0Þ ¼ p0c =sY ; 4ð0Þ ¼ 0; ac ð0Þ ¼ a_ c ð0Þ ¼ 0. 4. Results and comparison to rate independent case Figs. 2–5 summarize the major differences between our previous rate independent results for Cu and the current results with material rate dependence included. The properties we have used for polycrystalline Cu are:
Cu : r ¼ 8930 kg=m3 ; k ¼ 140 GPa ; m ¼ 47:5 GPa ;
These two cases become identical as m/0, but if Ax ¼ Oð1Þ somewhere within the limits of integration, then for ms0, neither approximation is valid over the whole range. However, the rate independent limit is a lower bound for all cases so in calculations we use the maximum of the two limiting cases. When 4 ¼ 0, the rate independent value of Eq. (11) reduces to 2=3sY ð1 þ ln 2=33Y Þ, which we denote as p0 and use to normalize the total resistance,
p1 0
1245
(14)
Whereas the number called p0 above is the calculated estimate of the tensile pressure for spherical bifurcation at which a first void can appear spontaneously at some point in a nominally
E ¼ 128 GPa ; sY ¼ 330 MPa ; N ¼ 1015 m3 ; p0c ¼ 0:4 GPa ; a ¼ 1:0 GPa Fig. 2 shows examples of evolving macroscopic mean stress and porosity for rate independent and rate dependent materials with total volumetric rates of expansion, e_ ¼ 106 s1 . The first nucleation of voids begins at 0.4 GPa, but because growth begins relatively slowly (initially each void radius grows as the 3/2 power of the time since nucleation), porosity is extremely small until about 10 ns for the rate independent material, and about 18 ns for the rate dependent one. During this early period, the mean stress increase is nearly elastic with little growth of porosity. Peak pressure is reached when p_ ¼ 0, or according to Eq. (3), _ The slanting orange line in Fig. 2 has a slope of 106 s1 when 4_ ze. which equals the slope of both porosity curves at the times of peak stress. Note further that in both cases, the porosity at peak stress is still very small and nearly the same for both. Fig. 3 shows the macroscopic mean stress versus porosity for the same cases as in Fig. 2, but plotted in this form, the results may be regarded as a synthetic constitutive law for (tensile) pressure vs. porosity with volumetric strain rate as a parameter. The falling curve indicates
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Fig. 2. Mean stress (tensile pressure) and porosity versus time. Mean stress increases elastically, reaches a peak as porosity accelerates, and then rapidly decreases Pattern is similar for both rate independent and rate dependent materials, but the rate dependent peak is delayed and is higher. Porosity at peak is nearly the same for both cases.
strain softening, which in turn indicates severe instability because of the steepness of the slope. Although actual failure may not occur before the curve reaches its minimum, the first peak stress is an estimate of the spall strength of the material because it is a maximum that can never be reached again even in the absence of failure. Because the porosity is still very small at the first peak, void interaction has not yet played a significant role, and a more detailed model is not yet required. Fig. 4 shows a surface plot of individual void growth versus time and nucleation pressure for the case of a rate dependent material. Although it differs in detail from the case of a rate independent material, as in [1], its major features are similar, and it clearly distinguishes between the time scales of nucleation and growth. As the macroscopic mean stress increases, the thresholds for nucleation of successive groups of voids are exceeded along the yellow line and new voids begin to grow. Nucleation here is driven almost entirely by increasing elastic mean stress. The growth history of the first voids to be nucleated can be seen along the left edge and the top of the ridge. Note the slow initial rate of growth, followed by acceleration to a rapid climb up the left edge, as
Fig. 3. Mean stress (tensile pressure) versus porosity. Plotted in this fashion, the curve is a synthetic constitutive relation between macroscopic quantities at a constant rate of expansion.
dictated by the dynamics of void growth. The successive traces for voids nucleating later are similar in shape to the first trace, but are delayed in time due to later nucleation and do not reach the same size. After mean stress passes its peak and begins to decline, the driving stress for the last voids nucleated falls below their critical level so that voids stop growing in reverse order to their sequence of initiation. The contribution of the various void histories to porosity is not readily apparent because of the differing statistical weights given to the various traces. Nevertheless, the figure makes a vivid distinction between rates of nucleation and rates of growth, and it illustrates how the statistical distribution of nucleation stresses is the modulating influence that gives structure to their combined effects. Hidden within this picture is the more subtle influence of actual material rate dependence. Fig. 5 shows plots of the estimated spall strength vs. the volumetric strain rate for four levels of rate dependence and compares them with the rate independent case. The case for extremely low rate sensitivity agrees essentially with the rate independent case, and increasing rate dependence raises the spall strength above the rate independent case. For infinitesimal strain rate with no porosity, h ¼ 1, and the maximum mean stress would be the smallest critical stress in the distribution, chosen for this paper to be 0.4 GPa as shown in Fig. 2. Therefore, even at the lowest rates shown in Fig. 5, inertia has contributed more to the apparent spall strength than the largest inherent material rate dependence shown. As the strain rate rises, the rate dependent curves stay above the rate independent curve, but the contribution of inertia rises rapidly and contributes an ever-increasing proportion to the increment of strength above the minimum value. Experimental results for polycrystalline Cu [2], labeled Kanel PolyCu, are also plotted in the figure. Computed trends agree reasonably well with the data, at least for the first few decades of strain rate. It has also been found that inclusion of a pressure dependent bulk modulus has only a small influence on the results. 5. Discussion and conclusions Calculations have shown that inclusion of the term ð2=3Þse ðbÞ in Eq. (5) makes little difference in the results. For example, at a loading rate of e_ ¼ 106 s1 the peak mean stresses with and without the extra term are pmax ¼ 3:085 and 3:092 GPa respectively, and the corresponding porosities are 4 ¼ 1:64 103 and 1:70 103 . The implication is that the integrated deviatoric resistance is far more important than the actual value of the remote deviatoric stress. Fig. 5 can be rescaled to reveal other important, but more subtle, results. In Ref. [1] nondimensionalization of the governing equations showed that a nucleation-growth parameter, c ¼ N a3=2 =e_ 3 r3=2 , is the dominant factor in determining peak mean stress, as shown in Fig. 8 in that paper. Rescaling in this paper in the same way as in [1] would again reveal the dominant effect of c on the results. Scaling all stresses with sY rather than a, as in [1], is also a fully consistent approach, p and if we plot spall stress in this paper against ffiffiffiffiffiffiffiffiffiffi _ khN1=3 e= sY =r, Fig. 5 would look exactly the same but with a rescaled horizontal axis. Now however, all other things remaining the same, the nondimensionalization tells us that any change that makes k either increase or decrease would have the same effect as increasing or decreasing the volumetric strain rate in the same proportion. For example, a fourfold increase of the density (or fourfold decrease of the yield stress) while the other components of k are held constant, would have the same effect as doubling the strain rate and would only increase the spall stress by perhaps 10–20%. The log–log plot shows that very large changes in k are required to cause substantial changes in spall stress. Only the density of _ have any potential sites, N, and the volumetric strain rate, e,
T.W. Wright, K.T. Ramesh / International Journal of Impact Engineering 36 (2009) 1242–1249
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Fig. 4. Surface plot of void size over time and nucleation stress. The yellow lines bound the region within which void sizes are nonzero. Nucleation occurs continuously along the lefthand line. Traces of growing voids are shown beginning from the line of nucleation and crossing the figure to the right. After a short delay from nucleation, each group of voids grows rapidly until pressure first peaks, then falls and shuts down void growth in reverse order to initiation.
physically reasonable possibility of changing by orders of magnitude, and N must change by three orders of magnitude to have the _ same relative effect as a change of just one order of magnitude in e. Extremely large changes in N seem quite reasonable however, because N 1=3 , which has dimension of length, should reflect grain size, as well as characteristic spacing of precipitates, for example, or of any other internal additions or deletions to the basic crystalline structure. Data in [2] qualitatively reflect this observation in that single crystal Cu has higher spall strength than polycrystalline Cu, and deformed single crystal Mo has lower spall strength than undeformed single crystal Mo. See Figs. 5.13 and 5.16 in Ref. [2]. In the first case the absence of grain boundaries greatly reduces the number of potential nucleation sites and increases the characteristic length of separation between sites, and in the second case
Fig. 5. Estimated spall strength versus volumetric strain rate. Results are plotted on dimensional axes for comparison to data from Antoun et al. [2]. As described Section ffiffiffiffiffiffiffiffiffiffi ffi pin _ sY =r. The 5, more appropriate nondimensional axes would be p=sY versus N 1=3 e= computed curves would look exactly the same, but with rescaled axes.
a large number of dislocations increases the number of potential sites and reduces the characteristic length of separation. Furthermore, because changes such as grain size, precipitates, etc. also affect the basic yield strength of the material, it is not physically possible to consider changes in N without also considering changes in sY . In turn changes in yield strength would cause changes in the vertical placement of the figure because the fundamental bifurcation limit scales with tensile stress. After these considerations it appears that the shape of the figure would not change in any essential way if the vertical axis had been chosen to the horizontal axis had been chosen to be be p=sY and pffiffiffiffiffiffiffiffiffiffi _ k ¼ N 1=3 e= sY =r. Such a rescaling would be stable under changes of the principal physical parameters and would permit extension of these results to other ductile materials. We will call k the index of expansion. This proposed rescaling requires only a modest change in the nondimensionalized equations. Compare Eq. (15) with Eqs. (7.1), (7.2), and (7.3) in Ref. [1]. At the highest rates of expansion in Fig. 5 our calculation overestimates the reported data. On the one hand, the overestimation may be simply a defect in the present form of the model. For example, it may be the result of insufficient coupling assumed between the micro and macro mechanical problems. For present purposes we have chosen to regard the total volumetric strain rate as a constant, e_ ¼ vx_ i =vxi , but actually it is the trace of the macroscopic velocity gradient, vx_ i =vxj , which arises from a tensile pulse, and should be determined from a problem that is fully coupled to macroscopic momentum transfer, vSij =vxj ¼ r=x_ i . On the other hand, the experimental data at the highest rate in Fig. 5 was obtained from the impact of a laser launched foil striking a thin target. Very high rates are indeed generated, but the shape of the tensile pulse at the spall plane is unlikely to have been well determined because of its extremely short duration, which may well be too short to allow void growth to develop fully to the stage of rapid growth shown in Fig. 4. Thus there are reasons to examine both the calculation and the experimental data with greater care than is presently possible before correlation between the two may be properly assessed at the highest strain rates.
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The model developed in [1] and this paper may be regarded as a pointwise model with porosity and all void sizes having the status of internal variables. From this point of view perhaps the statistical distribution of critical stresses should be thought of as arising from the microstructure in the surrounding region, which affects local conditions at the point through wave propagation and/or shielding from events at neighboring sites. In any event, although the statistical representation of microstructure as used in this paper suggests many qualitatively reasonable results, it should be put on a firmer foundation. Finally, our chief observations are that peak stress is reached when the total budget for rate of volumetric expansion is used up by the growth rate of porosity no matter what the rate dependence of the material may be, see Eq. (3), but the actual value of porosity at peak stress seems to be of little consequence, see Fig. 2. Inherent rate dependence in the material has introduced another time scale into our model besides the externally imposed rate of loading (which drives nucleation) and the independent dynamics of void growth (which is intrinsic to the material). Material rate dependence of the matrix inhibits the earliest void growth and thus delays and increases peak stress, but in all other respects, the general patterns of void nucleation and growth and the interactions with macroscopic stress and total porosity are essentially the same for all levels of rate dependence, including the limiting case of rate independence. In all cases considered inertia dominates material rate dependence in increasing the spall stress, and the dominance increases rapidly with imposed volumetric strain rate.
At some distance from the void, the strain is only elastic. The transition occurs when
3T ¼ 3Y ¼
sY E
or at
rp3 2 ¼ 3Y ln 3 3 rp a3
(A2)
The transition radius is located at rp zað33Y =2Þ1=3 . For r > rp, the strain is then only elastic and 3T z2a3 =3r 3 . For a given initial distance, R, away from a void center, there will be a time tR at which the material first experiences plasticity as the void expands so that se ðR; tR Þ ¼ sY and 3T ðR; tR Þ ¼ 3Y , and after that time the full rate equation applies. For purposes of analysis, let us assume that the right hand side of the rate equation is a known function. Now look for an asymptotic solution that gives se ðR; t tR Þ for given R. With u ¼ se =sY , 3Y ¼ sY =E, and b s ¼ 3_ 0 ðt tR Þ the rate equation becomes
3Y ubs þ u1=m 1 ¼
2a2 abs
R3 þ a3
s ¼ C R; b
(A3)
where R is only a parameter and the function C is assumed to be known. This equation should show a classic boundary layer (see Bender and Orszag [11]) at the initial time because 3Y is a small parameter, 3Y ¼ Oð103 102 Þ. With 3Y ¼ 0, the outer solution s =3Y the equation for the becomes uout ¼ ð1 þ CÞm , and with z ¼ b inner solution becomes
uz þ u1=m 1 ¼ CðR; 3Y zÞ
(A4)
It seems reasonable to assume that C varies slowly on the
z-timescale, and that for a given value of R, it may be taken to be constant, C0 ¼ CðR; 0Þ, as a first approximation. Now the solution may be written as the quadrature
Acknowledgments This research was sponsored by the United States Army Research Laboratory (ARL) and accomplished under the ARMACRTP Cooperative Agreement Numbers DAAD19-01-2-003 and W911NF-06-2-0006. The views expressed in this document are those of the authors and do not represent official policies of ARL or the US Government. The US Government is authorized to reproduce and distribute reprints for Government purposes not withstanding any copyright notation hereon.
Appendix. Asymptotic analysis for elastic/plastic transition near expanding voids The total strain rate is the sum of elastic and plastic parts,
3_ T ¼ 3_ e þ 3_ p . Because the material has been assumed to be incompressible, we have r 3 a3 ¼ R3 for each pair of current radii ðr; aÞ corresponding to initial radii ðR; 0Þ. In addition, r 2 r_ ¼ a2 a_ and the total strain rate are given by 3_ T ¼ 2r_ =r. Although incompressible, the material still has shearing elasticity, and the elastic rate for the equivalent tensile stress is 3_ e ¼ s_ e =E. A simple rate law has been assumed for the plastic strain rate, 3_ p ¼ 3_ 0 Cjse =sY j1=m 1D, where 3_ 0 ¼ Oð104 Þ s1 , and the strain rate sensitivity is a small number, m. Plastic strain rate vanishes when se sY , and increases rapidly when se > sY . Rather than trying to deal with the complexities of work hardening and thermal effects, the strength is simply taken to be constant, implicitly assuming that prior processes have saturated material strength. With the above considerations taken into account the rate equation now becomes
1=m se 2r_ a2 a_ 1 ¼ ¼ 2 3 þ 3_ 0 sY r E R þ a3
s_ e
(A1)
du
¼ dz C0 u1=m 1
(A5)
and because u 1 is expected to be a small positive quantity, we write u ¼ 1 þ z where z is a small quantity. Consequently u1=m ¼ ð1 þ zÞ1=m zez=m ½1 þ Oðz2 =2mÞ where lnð1 þ zÞzz 1=2z2 has been used. With this approximate change of variable the quadrature for the inner solution may be evaluated, then reconverted to its original variables as
sin 1 þ C0 e w sY 1 þ C0 eð1þC0 Þz=m
m
!ð1 þ C0 Þ
m
(A6)
z/N
The complete asymptotic solution may be written as uwuout þ uin uint where uint is the intermediate solution given by uint ¼ ð1 þ C0 Þm . This is the outer limit of the inner solution and the inner limit of the outer solution. Thus, the complete asymptotic solution is given by
se 1 þ C0 wð1 þ CÞm þ sY 1 þ C0 eð1þC0 Þ3_ 0 ðttR Þ=3Y m
m
ð1 þ C0 Þm
(A7)
At t ¼ tR we have C ¼ C0 and u ¼ 1, and for t[tR, we have u ¼ ð1 þ CÞm . Furthermore, even though the stress rate initially continues as if fully elastic (at u ¼ 1 we have s_ e ¼ E3_ T , according to the ODE), the transition is expected to be fast, because the exponent rapidly becomes a large negative number once viscoplasticity begins at t ¼ tR for the material point R, even when C0 is close to zero. For example, for the values of material constants used in this paper the exponential term should decay in 5 ns or so, and for larger C0, the more rapid the decay. The essential argument in this Appendix is unchanged if initial void sizes are finite, and it is independent of applied volumetric
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_ However, the larger e_ is, the more rapidly peak stress strain rate, e. is reached, so that it may not then be fully consistent to ignore the decay time in the elastic/plastic transition. This is expected to make little difference in the final result.
References [1] Wright TW, Ramesh KT. Dynamic void nucleation and growth in solids: a selfconsistent statistical theory. J Mech Phys Solids 2008;56:336–59. [2] Antoun T, Seaman L, Curran DR, Kanel GI, Razorenov SV, Utkin AV. Spall fracture. New York: Springer; 2003. [3] Ramesh KT. High-strain-rate and impact techniques. In: Sharpe Jr WN, editor. Handbook of experimental mechanics. Springer; 2008. [4] Meyers MA. Dynamic deformation and failure. In: Meyers MA, Armstrong RW, Kirchner HOK, editors. Mechanics and materials. J. Wiley; 1999. p. 565. Fig.14.52, Copyright Wiley-VCH Verlag GmbH & Co. KGaA. Reproduced with permission.
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[5] Ball JM. Discontinuous equilibrium solutions and cavitation in nonlinear elasticity. Phil Trans Roy Soc London A 1982;306:557–611. [6] Huang Y, Hutchinson JW, Tvergaard V. Cavitation instabilities in elastic–plastic solids. J Mech Phys Solids 1991;39:223–41. [7] Wu XY, Ramesh KT, Wright TW. The dynamic growth of a single void in a viscoplastic material under transient hydrostatic loading. J Mech Phys Solids 2003a;51:1–26. Wu XY, Ramesh KT, Wright TW. The effects of thermal softening and heat conduction on the dynamic growth of voids. Int J Solids Struct 2003b;40:4461–78. [8] Molinari A, Mercier S. Micromechanical modelling of porous materials under dynamic loading. J Mech Phys Solids 2001;49:1497–516. [9] Christensen RM. A critical evaluation for a class of micromechanics models. J Mech Phys Solids 1990;38:379–404. [10] Knowles JK, Jakub MT. Finite dynamic deformations of an incompressible medium containing a spherical cavity. Arch Rat Mech Anal 1965;18: 367–78. [11] Bender CM, Orszag SA. Advanced mathematical methods for scientists and engineers. New York: McGraw-Hill; 1978.
Contents lists available at ScienceDirect
International Journal of Impact Engineering journal homepage: www.elsevier.com/locate/ijimpeng
Statistically informed dynamics of void growth in rate dependent materials T.W. Wright a, b, K.T. Ramesh b, * a b
U.S. Army Research Laboratory, 4600 Deer Creek Loop, Aberdeen Proving Ground, MD 21005, USA Mechanical Engineering, Johns Hopkins University, Baltimore, MD 21218, USA
a r t i c l e i n f o
a b s t r a c t
Article history: Received 27 August 2008 Received in revised form 1 May 2009 Accepted 17 May 2009 Available online 27 May 2009
An inherently rate dependent material model is used to model the nucleation and growth of voids in metals, leading to spall fracture. Intrinsic material rate dependence introduces a third time scale in addition to those introduced by rates of nucleation and growth. Material rate dependence does increase the spall strength of metals, but it is not nearly as important as local inertia in doing so. Ó 2009 Elsevier Ltd. All rights reserved.
Keywords: Spall Voids Nucleation Growth Local inertia Rate dependence
1. Introduction The rapid cooperative growth of voids within materials is a classic failure mechanism, and it is particularly important under impact loading. Impact events typically produce large amplitude compressive stress waves, which reflect from free surfaces. The interaction of a reflected wave with other waves can generate nearly hydrostatic tensile stress at high loading rates, resulting in rapid nucleation, growth and interaction of voids. The result often is internal rupture through the process known as spallation. In previous work the authors have developed a statistically informed model for the cooperative nucleation, growth and early interaction of voids in a plastically deforming metal [1]. That work focused on rate independent materials, but most metals are well known to be rate-sensitive at the high strain rates associated with impact. Spall data from many uniaxial strain experiments performed on many different materials have been represented in the form m _ sspall ¼ AðV=V (see Antoun et al., [2]), where A and m are 0Þ _ material constants, and V=V 0 is the maximum volumetric strain rate at the spall plane as estimated from data collected at a free surface. Ref. [2] also provides an excellent introduction to the vast literature on spall fracture, which we could not begin to summarize
* Corresponding author. Department of Mechanical Engineering, Johns Hopkins University, Baltimore, MD 21218, USA. Tel: þ1 410 516 7735; fax: þ1 410 516 7254. E-mail address: [email protected] (K.T. Ramesh). 0734-743X/$ – see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijimpeng.2009.05.007
here. However, the apparent rate sensitivity, m, as used above for spall and determined experimentally, is generally much higher than that known from other experiments designed to produce high-rate, homogeneous straining in a small volume of material. Examples of such tests are the split Hopkinson pressure or torsion bar (due to Kolsky) and the pressure/shear experiment (due to Clifton). For a brief discussion of these and other techniques, see Ramesh [3]. The object of this paper is to examine the role of true material rate dependence in contributing to spall strength. As in [1] previously, we postulate that spall strength is not in itself a fundamental property of a material, but rather it is the result of a deformation process that is fully controlled by the bulk homogeneous properties of the intact matrix material, including material rate dependence. In most ductile metals the process underlying spall is void nucleation and growth as the material responds to rapid expansion, e.g., see Meyers [4]. Voids formed under dynamic conditions often seem rapidly to become spherical in shape and to stay approximately spherical during expansion, as in Fig. 1. The figure also shows the preference for voids to form on or near grain boundaries, but voids can also form on other disruptions of the fundamental material lattice or within the bulk lattice if the local mean stress is strong enough. Spall strength can be much greater than the threshold for void nucleation, which is itself often four or more times larger than the ordinary tensile yield strength. From the continuum point of view nucleation represents a bifurcation in the response to hydrostatic
T.W. Wright, K.T. Ramesh / International Journal of Impact Engineering 36 (2009) 1242–1249
1243
because the homogenized quantities are work conjugate, within an error term, provided the kinetic energy relative to the c.g. is retained:
Sij Dij ¼
sij dij
þ
K_ rel
1 Vtotal
#
x_ i Dij xj
Sext
ðsik Sik Þnk ds
Fig. 1. Voids at grain boundaries in Cu. Dynamically formed voids tend to be spherical even when close to another void [4]. Reproduced with permission.
tension. Ball [5] demonstrated the existence of hydrostatic bifurcation for nonlinear elastic materials, and Huang et al. [6] obtained an explicit formula for the bifurcation stress for elastic/plastic materials. Wu et al. [7] later showed that the calculated value for many materials corresponded roughly to the lowest level of experimentally observed spall stress in those same materials. We will refer to the bifurcation stress as a critical stress. It is not unusual for spall strength to be 10 times the static tensile yield stress, which means that the tensile pressure at spallation can be nearly an order of magnitude larger than any measure of deviatoric stress. We demonstrate that when local inertia around growing voids, together with intrinsic bulk material rate dependence, is taken into account, spall data can be accurately estimated over several decades of the rate of volumetric expansion, and that in all cases examined local inertia around the voids accounts for a greater proportion of the increase than does intrinsic material rate dependence. Thus, inertia is the main contributor to the apparently higher rate sensitivity in spall. 2. Background For clarity we repeat the chief elements of the theory as presented in [1]. Following Molinari and Mercier [8], but using a position vector x relative to the center of mass of a material volume element rather than relative to a fixed frame, we define the macroscopic or homogenized strain rate and stress in terms of average fields within a representative volume element or RVE as follows: 1 # x_ ði njÞ ds ¼ Vtotal
1 Dij ðx;tÞhVtotal
Sext 1 Sij ðx;tÞhVtotal
# xðj tiÞ ds ¼ Sext
Z
dij dv þ
Vsol 1 Vtotal
X
#
!
x_ ði njÞ ds
voids Svoid
! o € sij þ rxðj xiÞ dv
Z n
(1)
Vsolid
In Eq. (1) x is the vectorial position of the center of gravity (c.g.) of the RVE, x is the position vector relative to the c.g., Vtotal and Sext are the total volume and exterior material surface respectively of the RVE, n is the exterior normal on Sext or on the voids. Tractions on the exterior surface and the stress and strain rate within the RVE are given by ti ¼ sij nj ; sij ; and dij, respectively. Thus the homogenized strain rate and stress are defined completely by particle velocities relative to the c.g. and by tractions, both taken only on the exterior surface of the RVE, and by the geometrical properties of the RVE itself. The right hand sides of Eq. (1) follow exactly from the definitions, given sufficient smoothness of the fields and surfaces. As shown by the following identity, the definitions can be useful
(2)
where CC$DD indicates the volumetric average of the quantity within the double brackets. Again the expression is exact given sufficient smoothness. However, its usefulness does require that the last term is sufficiently small, which we assume in this paper. With the porosity 4 defined as the void volume divided by the total volume in an RVE of total mass m, the macroscopic density of the RVE may be defined as rhm=Vtotal . We also define the average density of just the solid material in the RVE, taken by itself, as b r hm=Vsolid , and therefore 1 4 ¼ Vsolid =Vtotal ¼ r=br . Macroscopic conservation of mass is given by r_ =r þ vx_ i =vxi ¼ 0. With total rate of volumetric expansion given by e_ ¼ vx_ i =vxi, conservar_ =br 4_ =ð1 4Þ þ e_ ¼ 0. From the tion of mass demands that b r it seems reasonable to associate the first term with definition of b elastic expansion in the macroscopic porous material. Elastic dilation of the matrix material will increase the size of voids in approximately the same degree as the matrix itself, but have little if any effect on the porosity (void volume per unit total volume). It follows then that change in porosity should be associated with plastic deformation around voids, and that therefore, it is an essentially incompressible process within the matrix material. With these ideas in mind, and with the macroscopic mean stress given by p ¼ ð1=3ÞSii, a constitutive law for the elastic rate of _ r_ =br ¼ p=kð 4; pÞ, and the macroscopic expansion may be written b balance of mass becomes
p_ kð4; pÞ
þ
4_ ¼ e_ 14
(3)
The instantaneous bulk modulus for the homogenized RVE is k, which may depend on the porosity and the mean stress. Eq. (3) shows that volumetric expansion will force a competition between elastic expansion of the porous macroscopic material and more essential changes in porosity. The macroscopic mean stress driving the RVE is p, and for the homogenized bulk modulus we use
14 1 þ ð3k=4mÞ4
k ¼ k
(4)
where k; m are the standard bulk and shear moduli, respectively, for the matrix material. The terms in the square brackets modify the elastic bulk modulus to account for porosity according to a selfconsistent elasticity argument cited by Christensen [9]. The homogenized bulk modulus is predicted to decrease with increasing porosity, which would be expected, but the quantitative accuracy of these terms is not known. In the essential part of our calculations porosity remains much less than a few percent, so the variation of the bulk modulus is unlikely to be a critical issue. The simplest model of a process for increasing porosity is radial expansion of an incompressible, thick walled, hollow sphere, with inner radius a and outer radius b, which is driven by radial tractions only on its outer surface. The surfaces of the voids will be traction free. Incompressibility is captured by the expression r 3 a3 ¼ R3 A3 where upper case denotes initial values, lower case denotes current values, and a ¼ aðA; tÞ and r ¼ rðR; tÞ, respectively. The _ 2, particle velocity and acceleration may be expressed as r_ ¼ a2 a=r 2 2 2 2 2 2 5 _ _ =r . Radial integration of the þ 2aa_ =r 2ða aÞ and r_ ¼ a a=r equation of motion through the wall of the hollow sphere results in the equation
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raa€ 1
Zb a 3 a 1 a4 2 2 2se _ ¼ p se ðbÞ þr a dr 2 þ r b 2 b 2 b4 3
(5)
a
p 2=3se ðbÞ.
where srr ðbÞ ¼ In our case Eq. (5) will be interpreted to represent void growth in a material that is embedded in a porous material whose macroscopic porosity 4 is exactly the same as the local porosity, a3 =b3 , thus a=b ¼ 41=3 . Previously in [1] we had omitted the term ð2=3Þse ðbÞ on the right hand side of Eq. (5) on the grounds that it is much less than p, but here for consistency in developing the growth equation, we leave it in place. As reported later in Section 5, actual calculations show that it has an insignificant effect. Eq. (5) is essentially the equation of motion given by Knowles and Jakub [10] for an elastic incompressible material, but here it is interpreted as applying to an elastic/plastic, incompressible material where the effective tensile stress in spherical expansion is
pffiffiffiffiffiffiffi
se h 3J2 ¼ jsrr sqq j:
(6)
In expansion srr sqq < 0. The right hand side of Eq. (5) represents the difference between the driving macroscopic mean stress, p, and the total resistive effect of an elastic/plastic deviatoric stress, Rb Ihð2=3Þse ðbÞ þ a ð2se =rÞdr. We will require that a_ 0, thus eliminating any consideration of reverse plastic flow (collapsing rather than expanding voids). To ensure this requirement, we impose the joint conditions that if ða_ ¼ 0ÞWðp I 0Þ, then the r.h.s. of Eq. (5) is set equal to zero so that a_ ¼ 0 as well, and void growth stops completely; otherwise the r.h.s. is equal to p I. Thus, if a_ > 0, then the r.h.s. is as written in Eq. (5), even if p I is negative. But if a_ ¼ 0, then the r.h.s. as written in Eq. (5) is not applied until p I becomes positive. With these conditions a_ can never be negative. Finally, as a surrogate for microstructure, which necessarily introduces many defects and stress concentrations into an otherwise perfect crystalline lattice, we assume that there are many sites where a void might nucleate and that there is a corresponding distribution of critical stresses for nucleation. Once voids have nucleated, the total void volume density per unit of solid material in the RVE we denote as y. The contribution from all voids with size ac , which in the absence of porosity have nucleation stress pc , are weighted by a distribution function, gðpc p0c Þ, where p0c is the lowest critical stress for any void in the material. The increment of volume contributed by all voids of radius ac is 4=3pa3c dn, where R dn ¼ Ngðpc Þdpc and dn ¼ N. Here N is the number of potential nucleation sites per unit volume of matrix material. In this article we use only gðpc Þdpc ¼ 2CxDexpðx2 Þdx, where x ¼ ðpc p0c Þ=a. This is a Weibull distribution with modulus 2, stress scale a, and lower limit p0c . Using the relationships between void volume density and porosity, ð1 þ yÞð1 4Þ ¼ 1, and their rates y_ ¼ 4_ =ð1 4Þ2 we arrive at an equation for the rate of growth of porosity
ZN 4_ ¼ 4pNð1 4Þ a2c a_ c gðpc Þdpc 2
or it may also depend on a measure of effective strain rate, Ac , as explained below. Note that the critical stress, pc , which is the variable of integration in Eq. (7), is also used in determining the total deviatoric resistance in Eq. (5), as shown below. Before continuing our discussion of the right hand side of Eq. (5) and introducing a rate model, it is necessary to examine the assumed kinematics of the thick walled sphere further. The spatial representation of the strain rate tensor in spherical coordinates for this case is diagonal with entries ðvr_ =vr; r_ =r; r_ =rÞ. Because the matrix material is assumed to be incompressible, the sum of the three rates must vanish, so the first term equals 2r_ =r. Consequently the total effective deviatoric strain rate that is work conjugate to the effective stress above is given by
3_ T h
rffiffiffiffiffiffiffiffiffiffiffiffiffiffi r_ 2 a3 a_ dij dij ¼ 2 ¼ 2 3 r 3 r a
(8)
The right hand side of Eq. (8) may also be written as 3 A3 þ a3 Þ so the total strain at a fixed value of R may be _ 2a2 a=ðR found by integration to be
3T ¼
2 R3 A3 þ a3 2 r3 ¼ ln ln 3 3 r 3 a 3 þ A3 R3
(9)
Assuming that initial void sizes are small enough to be ignored, we set A ¼ 0. At a large enough distance from the void the total strain will be less than the elastic yield strain, 3Y ¼ sY =E, and the transition will occur at approximately a3 =rp3 ¼ 33Y =2. If As0, first yield will occur at the void surface when ða AÞ=Az1=23Y , but as the void grows the elastic/plastic transition will occur at approximately rp3 za3 ð1 A3 =a3 Þð2=33Y Þ, which will rapidly approach the first approximation for A ¼ 0. 3. Rate model The main difference between this and the previous work is the construction of the deviatoric resistance, which now is modulated by the rate sensitive nature of the material. For present purposes it is sufficient to model plastic strain rate within each hollow sphere by the simple expression 3_ p ¼ 3_ 0 Cjse =sY j1=m 1D, where the strain rate sensitivity is small, m 1, 3_ 0 ¼ Oð104 Þ s1 is a characteristic constant, and sY is the static yield stress in tension. In general sY will evolve with the history of deformation and temperature at each point in the material, but for purposes of this paper, we assume that work hardening has saturated before rapid expansion begins, and that sY is constant. Recalling that the total strain rate within an incompressible hollow _ 3, and that the elastic plastic sphere is given by 3_ T ¼ 2r_ =r ¼ 2a2 a=r 3 3 boundary is located at rp =a z2=33Y , it is shown in Appendix A that the effective stress in the elastic and plastic regions, respectively, is
selas e =sY z2x=33Y ; (7)
p0c
Eqs. (3), (5), and (7) have the same overall structure as that established previously for rate independent materials [1]. Eq. (5) will be applied to every void with internal radius, ac , effective external radius, bc ¼ ac 41=3 , and critical or nucleation stress, pc . Here the cumulative porosity 4 from all voids in the RVE has been used to generate an effective outer radius for each hollow sphere. As before, the integrated momentum equation is driven by the excess of mean stress over the total deviatoric resistance in a thick-walled R a 41=3 ð2se =rÞdr, but hollow sphere, Ið4; Ac ; pc Þh2=3se ðac 41=3 Þ þ acc now the equivalent stress, se , may be due only to elastic deformation
x ¼ a3c =r 3 33Y =2 m 1 þ A R xR m splas s = wð1 þ A xÞ þ c Y e 1 þ AR xR eð1 þ AR xR Þ3_ 0 ðt tR Þ=3Y m ð1 þ AR xR Þm !ð1 þ Ac xÞm ;
t[tR
x ¼ a3c =r 3 > 33Y =2
(10)
where Ac ¼ 2a_ c ðtÞ=3_ 0 ac ðtÞ, AR ¼ 2a_ c ðtR Þ=3_ 0 ac ðtR Þ, and tR is the reference time at which plasticity first begins at particles with initial plas radius R. The expression for se has been found from an asymptotic boundary layer analysis, described in Appendix A. Note that the full asymptotic expression is continuous at t ¼ tR . However, we will use
T.W. Wright, K.T. Ramesh / International Journal of Impact Engineering 36 (2009) 1242–1249
only the limiting expression, which is not continuous at rp . This procedure will tend to overestimate deviatoric resistance slightly by transitioning too rapidly to full viscoplastic flow. After taking account of both elastic and viscoplastic responses, we find that the total resistance in Eq. (5) for each void may now be expressed as:
2 e s ðbc Þþ 3
¼
Zbc ac
2se dr r
8 ( ) Z1 > > > 2s m 1 > ð1þAc xÞ x dx ; > Y 1þ > > <3
4 33Y =2
33Y =2
( ) > Z1 > > 2 > m m 1 > > sY ð1þAc 4Þ þ ð1þAc xÞ x dx ; 4 >33Y =2 > :3
homogeneous material, the number p0c is a calibrated value for the actual lowest value for void nucleation in a real material. We would expect that j1 p0c =p0 j 1 in most cases. Thus the range of pc represents all the possible values at which a void could nucleate at various sites in a material, and the distribution function gðpc p0c Þ represents the probability of void nucleation at pc . The resulting set of Eqs. (3), (5), and (7) thus provides a means for representing an inhomogeneous material with microstructure. Voids nucleate over a range of stresses, each grows according to its own dynamics, but all voids contribute to the total porosity, and the driving tensile pressure, the total porosity, and the growth of all voids, are fully coupled dynamically. Nondimensionalization of Eqs. (3), (5), and (7) brings out other features. Our approach in this paper is slightly different from that used in [1]. We choose the following:
~ ¼ ea= _ X ¼ p=sY ; a
4
(11) where m is now only the material strain rate sensitivity in the matrix material. The lower limit in the integral corresponds to the plastically deforming points most distant from the cavity and the upper limit to the surface of the cavity. The integrand in Eq. (11) is positive and decreases monotonically as x increases within the range of integration. Therefore, even though the normalized strain rate, Ac x, is highest near the surface of the cavity (the upper limit of the integral), it is not those points that contribute the most to the total deviatoric resistance. Rather, it is the points that are closest to the elastic plastic boundary (the lower limit) that contribute the most because they represent a larger volume of matrix material. Consequently we expect that strong interaction between voids may occur either as their outer plastic zones collide or certainly as the voids coalesce to become larger voids. We do not attempt to model void collisions explicitly, and as we shall see, estimation of spall strength does not appear to depend on it, although subsequent fracture and fragmentation certainly do. There are two obvious limits for the integrals in Eq. (11) that can be evaluated explicitly, namely, the rate independent limit ðm/0 or A/0Þ and the high rate limit ðA4[1Þ. These are
Z1 4
8 < ln41 ; dx ð1 þ AxÞm / : m 1 x A m ð1 4m Þ;
as A/0
(12)
for A4[1
hð4; Ac Þ ¼
( ) Z1 2 e 2 s ðbc Þ þ ðse =xÞdx : 3 3
(13)
4
We now represent the total deviatoric resistance in Eq. (5) for any void in the distribution as
Ið4; Ac ; pc Þ ¼ pc hð4; Ac Þ;
p0c pc < N
pffiffiffiffiffiffiffiffiffiffi sY =r; 3 ¼ sY =k
_ t0 Þ; x ¼ pc p0c s ¼ eðt
.
a
which reduces the three equations to:
3k 4 þ 4s ¼ 1 4 4m
a 3 1 p0 ~2s ¼ X ~a ~ss þ x þ c hð4;AÞ 1 41=3 a 241=3 þ 4 4=3 a sY sY 2 2 ZN 4s ¼ 4pk3 ð1 4Þ2 a~2 a~s gðxÞdx (15)
3Xs 1 þ
0
pffiffiffiffiffiffiffiffiffiffi ~s =3_ 0 a ~ and k ¼ N 1=3 e= _ sY =r. Writing the In Eq. (15) A ¼ 2e_ a equations in this form reveals several physical parameters, the most important of which are 3 ¼ Oð102 Þ and k ¼ Oð103 104 Þ. Initial conditions are: Xð0Þ ¼ p0c =sY ; 4ð0Þ ¼ 0; ac ð0Þ ¼ a_ c ð0Þ ¼ 0. 4. Results and comparison to rate independent case Figs. 2–5 summarize the major differences between our previous rate independent results for Cu and the current results with material rate dependence included. The properties we have used for polycrystalline Cu are:
Cu : r ¼ 8930 kg=m3 ; k ¼ 140 GPa ; m ¼ 47:5 GPa ;
These two cases become identical as m/0, but if Ax ¼ Oð1Þ somewhere within the limits of integration, then for ms0, neither approximation is valid over the whole range. However, the rate independent limit is a lower bound for all cases so in calculations we use the maximum of the two limiting cases. When 4 ¼ 0, the rate independent value of Eq. (11) reduces to 2=3sY ð1 þ ln 2=33Y Þ, which we denote as p0 and use to normalize the total resistance,
p1 0
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(14)
Whereas the number called p0 above is the calculated estimate of the tensile pressure for spherical bifurcation at which a first void can appear spontaneously at some point in a nominally
E ¼ 128 GPa ; sY ¼ 330 MPa ; N ¼ 1015 m3 ; p0c ¼ 0:4 GPa ; a ¼ 1:0 GPa Fig. 2 shows examples of evolving macroscopic mean stress and porosity for rate independent and rate dependent materials with total volumetric rates of expansion, e_ ¼ 106 s1 . The first nucleation of voids begins at 0.4 GPa, but because growth begins relatively slowly (initially each void radius grows as the 3/2 power of the time since nucleation), porosity is extremely small until about 10 ns for the rate independent material, and about 18 ns for the rate dependent one. During this early period, the mean stress increase is nearly elastic with little growth of porosity. Peak pressure is reached when p_ ¼ 0, or according to Eq. (3), _ The slanting orange line in Fig. 2 has a slope of 106 s1 when 4_ ze. which equals the slope of both porosity curves at the times of peak stress. Note further that in both cases, the porosity at peak stress is still very small and nearly the same for both. Fig. 3 shows the macroscopic mean stress versus porosity for the same cases as in Fig. 2, but plotted in this form, the results may be regarded as a synthetic constitutive law for (tensile) pressure vs. porosity with volumetric strain rate as a parameter. The falling curve indicates
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Fig. 2. Mean stress (tensile pressure) and porosity versus time. Mean stress increases elastically, reaches a peak as porosity accelerates, and then rapidly decreases Pattern is similar for both rate independent and rate dependent materials, but the rate dependent peak is delayed and is higher. Porosity at peak is nearly the same for both cases.
strain softening, which in turn indicates severe instability because of the steepness of the slope. Although actual failure may not occur before the curve reaches its minimum, the first peak stress is an estimate of the spall strength of the material because it is a maximum that can never be reached again even in the absence of failure. Because the porosity is still very small at the first peak, void interaction has not yet played a significant role, and a more detailed model is not yet required. Fig. 4 shows a surface plot of individual void growth versus time and nucleation pressure for the case of a rate dependent material. Although it differs in detail from the case of a rate independent material, as in [1], its major features are similar, and it clearly distinguishes between the time scales of nucleation and growth. As the macroscopic mean stress increases, the thresholds for nucleation of successive groups of voids are exceeded along the yellow line and new voids begin to grow. Nucleation here is driven almost entirely by increasing elastic mean stress. The growth history of the first voids to be nucleated can be seen along the left edge and the top of the ridge. Note the slow initial rate of growth, followed by acceleration to a rapid climb up the left edge, as
Fig. 3. Mean stress (tensile pressure) versus porosity. Plotted in this fashion, the curve is a synthetic constitutive relation between macroscopic quantities at a constant rate of expansion.
dictated by the dynamics of void growth. The successive traces for voids nucleating later are similar in shape to the first trace, but are delayed in time due to later nucleation and do not reach the same size. After mean stress passes its peak and begins to decline, the driving stress for the last voids nucleated falls below their critical level so that voids stop growing in reverse order to their sequence of initiation. The contribution of the various void histories to porosity is not readily apparent because of the differing statistical weights given to the various traces. Nevertheless, the figure makes a vivid distinction between rates of nucleation and rates of growth, and it illustrates how the statistical distribution of nucleation stresses is the modulating influence that gives structure to their combined effects. Hidden within this picture is the more subtle influence of actual material rate dependence. Fig. 5 shows plots of the estimated spall strength vs. the volumetric strain rate for four levels of rate dependence and compares them with the rate independent case. The case for extremely low rate sensitivity agrees essentially with the rate independent case, and increasing rate dependence raises the spall strength above the rate independent case. For infinitesimal strain rate with no porosity, h ¼ 1, and the maximum mean stress would be the smallest critical stress in the distribution, chosen for this paper to be 0.4 GPa as shown in Fig. 2. Therefore, even at the lowest rates shown in Fig. 5, inertia has contributed more to the apparent spall strength than the largest inherent material rate dependence shown. As the strain rate rises, the rate dependent curves stay above the rate independent curve, but the contribution of inertia rises rapidly and contributes an ever-increasing proportion to the increment of strength above the minimum value. Experimental results for polycrystalline Cu [2], labeled Kanel PolyCu, are also plotted in the figure. Computed trends agree reasonably well with the data, at least for the first few decades of strain rate. It has also been found that inclusion of a pressure dependent bulk modulus has only a small influence on the results. 5. Discussion and conclusions Calculations have shown that inclusion of the term ð2=3Þse ðbÞ in Eq. (5) makes little difference in the results. For example, at a loading rate of e_ ¼ 106 s1 the peak mean stresses with and without the extra term are pmax ¼ 3:085 and 3:092 GPa respectively, and the corresponding porosities are 4 ¼ 1:64 103 and 1:70 103 . The implication is that the integrated deviatoric resistance is far more important than the actual value of the remote deviatoric stress. Fig. 5 can be rescaled to reveal other important, but more subtle, results. In Ref. [1] nondimensionalization of the governing equations showed that a nucleation-growth parameter, c ¼ N a3=2 =e_ 3 r3=2 , is the dominant factor in determining peak mean stress, as shown in Fig. 8 in that paper. Rescaling in this paper in the same way as in [1] would again reveal the dominant effect of c on the results. Scaling all stresses with sY rather than a, as in [1], is also a fully consistent approach, p and if we plot spall stress in this paper against ffiffiffiffiffiffiffiffiffiffi _ khN1=3 e= sY =r, Fig. 5 would look exactly the same but with a rescaled horizontal axis. Now however, all other things remaining the same, the nondimensionalization tells us that any change that makes k either increase or decrease would have the same effect as increasing or decreasing the volumetric strain rate in the same proportion. For example, a fourfold increase of the density (or fourfold decrease of the yield stress) while the other components of k are held constant, would have the same effect as doubling the strain rate and would only increase the spall stress by perhaps 10–20%. The log–log plot shows that very large changes in k are required to cause substantial changes in spall stress. Only the density of _ have any potential sites, N, and the volumetric strain rate, e,
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Fig. 4. Surface plot of void size over time and nucleation stress. The yellow lines bound the region within which void sizes are nonzero. Nucleation occurs continuously along the lefthand line. Traces of growing voids are shown beginning from the line of nucleation and crossing the figure to the right. After a short delay from nucleation, each group of voids grows rapidly until pressure first peaks, then falls and shuts down void growth in reverse order to initiation.
physically reasonable possibility of changing by orders of magnitude, and N must change by three orders of magnitude to have the _ same relative effect as a change of just one order of magnitude in e. Extremely large changes in N seem quite reasonable however, because N 1=3 , which has dimension of length, should reflect grain size, as well as characteristic spacing of precipitates, for example, or of any other internal additions or deletions to the basic crystalline structure. Data in [2] qualitatively reflect this observation in that single crystal Cu has higher spall strength than polycrystalline Cu, and deformed single crystal Mo has lower spall strength than undeformed single crystal Mo. See Figs. 5.13 and 5.16 in Ref. [2]. In the first case the absence of grain boundaries greatly reduces the number of potential nucleation sites and increases the characteristic length of separation between sites, and in the second case
Fig. 5. Estimated spall strength versus volumetric strain rate. Results are plotted on dimensional axes for comparison to data from Antoun et al. [2]. As described Section ffiffiffiffiffiffiffiffiffiffi ffi pin _ sY =r. The 5, more appropriate nondimensional axes would be p=sY versus N 1=3 e= computed curves would look exactly the same, but with rescaled axes.
a large number of dislocations increases the number of potential sites and reduces the characteristic length of separation. Furthermore, because changes such as grain size, precipitates, etc. also affect the basic yield strength of the material, it is not physically possible to consider changes in N without also considering changes in sY . In turn changes in yield strength would cause changes in the vertical placement of the figure because the fundamental bifurcation limit scales with tensile stress. After these considerations it appears that the shape of the figure would not change in any essential way if the vertical axis had been chosen to the horizontal axis had been chosen to be be p=sY and pffiffiffiffiffiffiffiffiffiffi _ k ¼ N 1=3 e= sY =r. Such a rescaling would be stable under changes of the principal physical parameters and would permit extension of these results to other ductile materials. We will call k the index of expansion. This proposed rescaling requires only a modest change in the nondimensionalized equations. Compare Eq. (15) with Eqs. (7.1), (7.2), and (7.3) in Ref. [1]. At the highest rates of expansion in Fig. 5 our calculation overestimates the reported data. On the one hand, the overestimation may be simply a defect in the present form of the model. For example, it may be the result of insufficient coupling assumed between the micro and macro mechanical problems. For present purposes we have chosen to regard the total volumetric strain rate as a constant, e_ ¼ vx_ i =vxi , but actually it is the trace of the macroscopic velocity gradient, vx_ i =vxj , which arises from a tensile pulse, and should be determined from a problem that is fully coupled to macroscopic momentum transfer, vSij =vxj ¼ r=x_ i . On the other hand, the experimental data at the highest rate in Fig. 5 was obtained from the impact of a laser launched foil striking a thin target. Very high rates are indeed generated, but the shape of the tensile pulse at the spall plane is unlikely to have been well determined because of its extremely short duration, which may well be too short to allow void growth to develop fully to the stage of rapid growth shown in Fig. 4. Thus there are reasons to examine both the calculation and the experimental data with greater care than is presently possible before correlation between the two may be properly assessed at the highest strain rates.
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The model developed in [1] and this paper may be regarded as a pointwise model with porosity and all void sizes having the status of internal variables. From this point of view perhaps the statistical distribution of critical stresses should be thought of as arising from the microstructure in the surrounding region, which affects local conditions at the point through wave propagation and/or shielding from events at neighboring sites. In any event, although the statistical representation of microstructure as used in this paper suggests many qualitatively reasonable results, it should be put on a firmer foundation. Finally, our chief observations are that peak stress is reached when the total budget for rate of volumetric expansion is used up by the growth rate of porosity no matter what the rate dependence of the material may be, see Eq. (3), but the actual value of porosity at peak stress seems to be of little consequence, see Fig. 2. Inherent rate dependence in the material has introduced another time scale into our model besides the externally imposed rate of loading (which drives nucleation) and the independent dynamics of void growth (which is intrinsic to the material). Material rate dependence of the matrix inhibits the earliest void growth and thus delays and increases peak stress, but in all other respects, the general patterns of void nucleation and growth and the interactions with macroscopic stress and total porosity are essentially the same for all levels of rate dependence, including the limiting case of rate independence. In all cases considered inertia dominates material rate dependence in increasing the spall stress, and the dominance increases rapidly with imposed volumetric strain rate.
At some distance from the void, the strain is only elastic. The transition occurs when
3T ¼ 3Y ¼
sY E
or at
rp3 2 ¼ 3Y ln 3 3 rp a3
(A2)
The transition radius is located at rp zað33Y =2Þ1=3 . For r > rp, the strain is then only elastic and 3T z2a3 =3r 3 . For a given initial distance, R, away from a void center, there will be a time tR at which the material first experiences plasticity as the void expands so that se ðR; tR Þ ¼ sY and 3T ðR; tR Þ ¼ 3Y , and after that time the full rate equation applies. For purposes of analysis, let us assume that the right hand side of the rate equation is a known function. Now look for an asymptotic solution that gives se ðR; t tR Þ for given R. With u ¼ se =sY , 3Y ¼ sY =E, and b s ¼ 3_ 0 ðt tR Þ the rate equation becomes
3Y ubs þ u1=m 1 ¼
2a2 abs
R3 þ a3
s ¼ C R; b
(A3)
where R is only a parameter and the function C is assumed to be known. This equation should show a classic boundary layer (see Bender and Orszag [11]) at the initial time because 3Y is a small parameter, 3Y ¼ Oð103 102 Þ. With 3Y ¼ 0, the outer solution s =3Y the equation for the becomes uout ¼ ð1 þ CÞm , and with z ¼ b inner solution becomes
uz þ u1=m 1 ¼ CðR; 3Y zÞ
(A4)
It seems reasonable to assume that C varies slowly on the
z-timescale, and that for a given value of R, it may be taken to be constant, C0 ¼ CðR; 0Þ, as a first approximation. Now the solution may be written as the quadrature
Acknowledgments This research was sponsored by the United States Army Research Laboratory (ARL) and accomplished under the ARMACRTP Cooperative Agreement Numbers DAAD19-01-2-003 and W911NF-06-2-0006. The views expressed in this document are those of the authors and do not represent official policies of ARL or the US Government. The US Government is authorized to reproduce and distribute reprints for Government purposes not withstanding any copyright notation hereon.
Appendix. Asymptotic analysis for elastic/plastic transition near expanding voids The total strain rate is the sum of elastic and plastic parts,
3_ T ¼ 3_ e þ 3_ p . Because the material has been assumed to be incompressible, we have r 3 a3 ¼ R3 for each pair of current radii ðr; aÞ corresponding to initial radii ðR; 0Þ. In addition, r 2 r_ ¼ a2 a_ and the total strain rate are given by 3_ T ¼ 2r_ =r. Although incompressible, the material still has shearing elasticity, and the elastic rate for the equivalent tensile stress is 3_ e ¼ s_ e =E. A simple rate law has been assumed for the plastic strain rate, 3_ p ¼ 3_ 0 Cjse =sY j1=m 1D, where 3_ 0 ¼ Oð104 Þ s1 , and the strain rate sensitivity is a small number, m. Plastic strain rate vanishes when se sY , and increases rapidly when se > sY . Rather than trying to deal with the complexities of work hardening and thermal effects, the strength is simply taken to be constant, implicitly assuming that prior processes have saturated material strength. With the above considerations taken into account the rate equation now becomes
1=m se 2r_ a2 a_ 1 ¼ ¼ 2 3 þ 3_ 0 sY r E R þ a3
s_ e
(A1)
du
¼ dz C0 u1=m 1
(A5)
and because u 1 is expected to be a small positive quantity, we write u ¼ 1 þ z where z is a small quantity. Consequently u1=m ¼ ð1 þ zÞ1=m zez=m ½1 þ Oðz2 =2mÞ where lnð1 þ zÞzz 1=2z2 has been used. With this approximate change of variable the quadrature for the inner solution may be evaluated, then reconverted to its original variables as
sin 1 þ C0 e w sY 1 þ C0 eð1þC0 Þz=m
m
!ð1 þ C0 Þ
m
(A6)
z/N
The complete asymptotic solution may be written as uwuout þ uin uint where uint is the intermediate solution given by uint ¼ ð1 þ C0 Þm . This is the outer limit of the inner solution and the inner limit of the outer solution. Thus, the complete asymptotic solution is given by
se 1 þ C0 wð1 þ CÞm þ sY 1 þ C0 eð1þC0 Þ3_ 0 ðttR Þ=3Y m
m
ð1 þ C0 Þm
(A7)
At t ¼ tR we have C ¼ C0 and u ¼ 1, and for t[tR, we have u ¼ ð1 þ CÞm . Furthermore, even though the stress rate initially continues as if fully elastic (at u ¼ 1 we have s_ e ¼ E3_ T , according to the ODE), the transition is expected to be fast, because the exponent rapidly becomes a large negative number once viscoplasticity begins at t ¼ tR for the material point R, even when C0 is close to zero. For example, for the values of material constants used in this paper the exponential term should decay in 5 ns or so, and for larger C0, the more rapid the decay. The essential argument in this Appendix is unchanged if initial void sizes are finite, and it is independent of applied volumetric
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_ However, the larger e_ is, the more rapidly peak stress strain rate, e. is reached, so that it may not then be fully consistent to ignore the decay time in the elastic/plastic transition. This is expected to make little difference in the final result.
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