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Damage evolution of ductile material under impact loading
MECHANICS RESEARCH COMMUNICATIONS
Mechanics Research Communications 33 (2006) 26–34 www.elsevier.com/locate/mechrescom
Damage evolution of ductile material under impact loading Changsuo Zhang a
a,*
, Steve Zou b, Baoping Zhang
c
Department of Mining Engineering, Taiyuan University of Technology, Taiyuan 030024, China b Department of Mining Engineering, Dalhousie University, Halifax, Canada B3J 2X4 c Beijing Institute of Technology, Beijing 100081, China Available online 16 June 2005
Abstract Nucleation, growth and coalescence of micro-voids result in the fracture of materials. Most mathematical models neglect nucleation and introduce initial damage, assuming it as a material constant. However, the original damage, which is formed during material working, is a material constant. The initial damage is a model parameter and depends on the load. Apparently, the predictability of such a model is poor. This paper made comparison and analysis of the four classical void growth models and showed their similarities. At the beginning of damage evolution, all the models follow a linear relationship in the form c_ ¼ kc, where c is the size of micro voids and k is a parameter which relates the material and loading condition. With the concept of statistical microdamage and the assumption of uniform void radius for new voids, a damage evolution equation was deduced based on the above void growth model. With this equation the effects of nucleation and growth at the beginning of the damage stage on the whole process of damage evolution can be calculated. The transition time from the nucleation dominant phase to the growth dominant phase can be determined. When the transition time is applied to the damage failure model of ductile material proposed by Johnson, the initial damage (f0), a model parameter in the original model, can also be determined. The results of the derived damage evolution equation agree well with the previous research results. Ó 2005 Published by Elsevier Ltd. Keywords: Micro-statistical damage; Initial damage; Growth model
1. Introduction Many experiments and metallographic observations for metals at ductile failure show that fracture in ductile materials generally results from the nucleation, growth and coalescence of micro-voids. In these three sequential stages, nucleation is the predominant process at the first stage of damage evolution. During the middle and the last stages of damage evolution, the effect of nucleation is relatively small and can be *
Corresponding author.
0093-6413/$ - see front matter Ó 2005 Published by Elsevier Ltd. doi:10.1016/j.mechrescom.2005.05.004
C. Zhang et al. / Mechanics Research Communications 33 (2006) 26–34
27
neglected when compared with void growth and coalescence. Void growth lasts the longest period of time and is therefore the most important of the three stages. Coalescence predominates the process in the last stage and is very short in time. In order to simplify development of mathematical models for failure of ductile materials, two parameters are introduced in most models, namely the initial damage (f0) and the critical damage (fc). The initial damage (f0) is used to replace the nucleation process and the critical damage (fc) to replace the coalescence process. In fact, most models are focused on the void growth stage. When the damage degree reaches the value of fc, the onset of the final coalescence process begins. Due to the fact that coalescence is too quick to be observed directly, using fc to replace coalescence seems reasonable. During the past thirty years, many researchers have developed their own theoretical models. By investigating the behavior of dynamic damage and fracture in a solid, Seaman et al. (1976) and Curran et al. (1977, 1987) developed a nucleation and growth (NAG) model. The NAG model has sufficient generality to include the statistical distribution of one or more variables such as porosity and void density. However, the model is only based on statistical observations so it needs numerous parameters that are difficult to obtain. Rice and Tracey (1969) investigated the growth of an isolated cylindrical void and a spherical void embedded in infinite media. They proposed a void growth model based on an unlimited matrix. Gurson (1977) presented a creative concept of cell model, which possesses a finite volume and contains a spherical void. He then theoretically developed a plastic potential function for porous materials. Tvergaard (1982) introduced two adjustable parameters, q1 and q2, and considered the interaction between the voids to improve the original Gurson model. Carroll and Holt (1972) established static and dynamic pore-collapse relation for ductile porous materials. Johnson (1981) applied Carroll and HoltÕs approach and presented a void growth model. However, most of these models adopted the idea of initial damage instead of nucleation. This approximation of the nucleation process greatly simplifies the numerical modeling of ductile fracture. Unfortunately, it has shown that the initial damage f0 is not a material constant. Our numerical results reveal that f0 varies with the stress. In JohnsonÕs void growth model, for example, nucleation effect was replaced by initial damage and was regarded as a dimensionless material constant whose value lies between 103 and 104. In fact, initial damage and original damage are different. Original damage is some micro defects that are formed during material working, and is therefore a material constant. On the other hand, initial damage is used to replace nucleation effect and it is a model parameter that depends on both the stress and material. Fig. 1 shows free-surface velocity of scab of different initial damages in JohnsonÕs model. From Fig. 1, when the initial damage changes from 0.0003 to 0.003, the signal of spalling has changed and the curve after spalling shifted greatly, sometimes without spalling. On the other hand, application of micromechanical models should enable attainment of so-called transferability of model parameters to different geometries and different loading conditions. In other words, if the initial damage is a material parameter, it should be a constant for the same material, but this is not true. Taking OFHC copper as an example,
800
Vf (m/s)
600
1 2 3
400
4
experiment data 1 2 3 4
200
0.0
0.5
1.0
f0=0.003 f0=0.001 f0=0.0006 f0=0.0003
1.5
t [µs] Fig. 1. Free-surface velocity of scab in different initial damages (after Seaman et al., 1976).
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C. Zhang et al. / Mechanics Research Communications 33 (2006) 26–34
a good agreement is obtained with the initial damage of 0.0002 with lower impacting velocity, but big deviation occurs when the model is applied to higher impacting velocity with the same initial damage. This shows that the initial damage in all of void growth models is a model parameter not a material constant. It is therefore difficult to use this model to predict the fracture of ductile material. In the early stage of damage evolution, the distance between voids is relatively large and the interaction of voids can be ignored. Thus the statistical relation between different voids is independent. In such case, damage has little influence on the local stress. As a result, it may be easy to get an analytical solution. In this article, we will apply the theory of statistical micro-damage to calculate the effect of nucleation on the whole process of damage evolution and to establish the value of initial damage f0. 2. Description of statistical micro-voids evolution A framework of statistical micro damage mechanics was established by Bai et al. (1991, 2000, 2001). They developed an evolution equation (Eq. (1)) of density of micro-voids number when they investigated damages under plane impacting loading. 8 oðn_cÞ on > < ot þ oc ¼ nN ðaÞ ð1Þ nN ¼ nN ðc; tÞ ðbÞ > : c_ ¼ c_ ðcÞ ðcÞ Eqs. (1b) and (1c) are nucleation and growth equations, respectively, where n is the number density, c is the radius of micro-voids and nN is nucleation rate density of micro-void. In our analysis, Eq. (1) is used to describe the damage evolution process of micro-voids. From experiment it is known that the volume of a micro-void is very small after its nucleation and the conformation of radius is ideal. In order to simplify computation, we assume that the newly nucleated voids have the same size and the radius equals to b. In fact, in many models such as NAG model and nucleation model proposed by Chu and Needleman (1980), the volumes of the new nucleation voids are supposed to be the same. The nucleation equation therefore becomes on oð_cnÞ þ ¼ qðtÞdðc bÞ ot oc where q(t) is the number of nucleated voids within a unit time, d(c b) is the Dirac function of c. Integrating Eq. (2) and let e ! 0 gives Z bþe b þ e on dc þ c_ n ¼ qðtÞ be be ot : As on ¼qðtÞ is limited and n(c, t) = 0 if c < b, Eq. (3) becomes ot ð_cnÞc¼b ¼ qðtÞ
ð2Þ
ð3Þ
ð4Þ
Based on the theory of differential equation, when the radii of micro-voids are the same the original Eq. (1) becomes 8 on oðn_cÞ > > þ ¼ 0 ðaÞ > > ot oc > > > > < ð_cnÞc¼b ¼ qðtÞ ðbÞ ð5Þ c_ ¼ c_ ðcÞ ðcÞ > > > > > nðc; tÞt¼0 ¼ 0 ðdÞ > > > : nðc; tÞc¼1 ¼ 0 ðeÞ
C. Zhang et al. / Mechanics Research Communications 33 (2006) 26–34
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where b is the radius of micro-void when it nucleates. The general solution of Eq. (5a) (Bai et al., 1999) is Z c 1 dc nðc; tÞ ¼ w t ð6Þ c_ c_ b The format of function W can be determined based on the boundary condition ð_cnÞc¼b ¼ qðtÞ. Therefore, if the void growth equation c_ ¼ c_ ðcÞ and void nucleation equation q(t) are given, the damage evolution equation can be deduced.
3. Void growth model In the following, we will analyze four classical void growth models to deduce the general growth model at the early stage of damage evolution. 3.1. NAG model Curran and co-workers (Seaman et al., 1976; Curran et al., 1977, 1987) developed a nucleation and growth (NAG) model. From the statistical observation they got the increment of damage (DVvg) during Dt from void growth DV vg
3 P S P g0 Dt V v0 ¼ V v0 exp 4 g
ð7Þ
where g is the material viscosity, Pg0 is material threshold for growth. According to the differential theory, if is small, it can be simplified approximatively to DV vg 3 P S P g0 ¼ V v0 4 Dt g then we have 3 P S P g0 V_ ¼ V 4 g If radius R is used in the above equation, the void growth equation becomes 1 P S P g0 R_ ¼ R 4 g
ð8Þ
3.2. Rice and Tracey void growth model For the case of an empty spherical void of radius R in a remote uniform plastic strain rate field Dpij , Rice and Tracey developed a growth model of micro-void
_R ¼ 5 Dpk þ 3 rm Dp R ð9Þ 3 4 rM where Dpk is the principal strain rates, rm is the mean normal stress, and rM is the yield stress.
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C. Zhang et al. / Mechanics Research Communications 33 (2006) 26–34
3.3. Gurson and Tvergaard yield function The classical yield function for porous solids was presented by Gurson (1977) and modified by Tvergaard (1982) and it takes the form 2 3 rm r 2 2 q ¼ ð1 þ q1 C v Þ 2q1 C v cosh ð10Þ M M 2 2r r where Cv = Vv/(Vv + VM), Vv is the volume of voids and VM is volume of the matrix. Using the above formula and considering a relatively small Cv, Ragab (2004) deduced the void growth equation V_ v 3 3 rm q ¼ q q sinh e_ 1 2 2 rY Vv 2 1 2 If radius R is used as variable, the above equation becomes _R ¼ 1 q1 q2 sinh 3 q2 rm e_ 1 R 2 2 rY
ð11Þ
3.4. JohnsonÕs void growth model Based on the work of Carroll and Holt (1972) and Johnson (1981) presented a void growth model. In JohnsonÕs model, the presence of voids is expressed in terms of the distension ratio a, which is related to the porosity / through the expression / = 1 1/a. The growth rate of is given by ( a_ ¼ 0 if Dp P 0 ð12Þ ða0 1Þ2=3 1=3 a_ ¼ g a a 1 DP if Dp < 0 where DP is the driving stress, and a0 is the initial damage. When a is very small, just as in the early stage of damage evolution, the above growth equation becomes 2=3
a_ ¼
ða0 1Þ g
aDP
ð13Þ
In all of the four classical void growth models, whether the void is surrounded by infinite matrix (Rice and TraceyÕs), or by a holly ball cell (GursonÕs and JohnsonÕs), or represented by pure statistical observation result (NAG), the void growth model can be represented by a linear equation when the damage degree is relatively small. The following generalized growth equation can therefore be used at the early stage of damage evolution: c_ ¼ kc ð14Þ 4. Damage evolution equation In the NAG model, the following nucleation equation was presented by Seaman et al. (1976), and Curran et al. (1977, 1987): 8 h i < N_ ¼ N_ exp P S P no if P S > P n0 0 P1 ð15Þ : _ N ¼0 if P S 6 P n0 where N_ 0 and P1 are material constants, PS is tensile pressure and Pn0 is nucleation threshold of material.
C. Zhang et al. / Mechanics Research Communications 33 (2006) 26–34
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In the above equation, except N_ and PS, the other items are constants. Therefore, the nucleation equation can be simplified as qðtÞ ¼ gH ðtÞ
ð16Þ
where g and k in Eq. (14) are constants and depend on the material and stress, and H ðtÞ ¼
0
if t < 0
1
if t P 0
Considering the boundary condition, the function w in Eq. (6) becomes wðtÞ ¼ gH ðtÞ
ð17Þ
Therefore, the number density of the micro-voids is g
kc
if t P 1=kðln c ln bÞ
0
if t < 1=kðln c ln bÞ
nðc; tÞ ¼
ð18Þ
It can be proved that Eq. (18) satisfies the boundary and the initial conditions, and it is the solution to Eq. (5) The final aim of statistical damage analysis is to discover the relationship between the process of damage evolution in the material and its mechanics performance. Thus it is necessary to obtain the damage variable based on statistical average. The general damage function is Z 1 fm ðtÞ ¼ cm nðc; tÞ dc ð19Þ b
If the volume of micro-voids is regarded as damage variable, then m = 3, f(t) is the whole volume of micro-voids in the material, so above general damage function becomes Z 1 4 f ðtÞ ¼ p c3 nðc; tÞ dc ð20Þ 3 b At a given time, the biggest void in the material is the void which nucleates at t = 0, therefore Z B dc t¼ c_ b where B is the radius of the biggest void. The volume of the void at time t is Z Z 4p B g 3 4p b expðktÞ g 3 4pg 3 f ðtÞ ¼ c dc ¼ c dc ¼ b ðexpð3ktÞ 1Þ 3 b kc 3 b kc 9k
ð21Þ
ð22Þ
Assuming k and g are constants and differentiating Eq. (22) with respect to time t, gives df 4 ¼ pgb3 expð3ktÞ dt 3
ð23Þ
and the following damage evolution equation is obtained 4 f_ ¼ 3kf þ pgb3 3
ð24Þ
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C. Zhang et al. / Mechanics Research Communications 33 (2006) 26–34
5. Discussion and conclusion 5.1. Initial damage Considering the right hand side of Eq. (24), the first item is the contribution of nucleation and the second item is the contribution of growth. The contribution of nucleation over the total volume of voids is represented by DfN 1 ¼ Df ðtÞ expð3ktÞ
ð25Þ
where DfN is the nucleation contribution during time Dt and Df(t) is the increment of the total void volume during Dt. From Eq. (25), the contribution of nucleation to the overall damage decreases greatly with time. As a result, many models can get good results through adjusting initial damage without considering nucleation. In order to calculate the effects of growth and nucleation during damage evolution, the damage evolution Eq. (24) is embedded in one-dimensional hydrodynamic simulations. Here we used normalized volume as the label of vertical axes in Fig. 2 which is the ratio between whole volume of micro-voids and the volume of original voids. When the calculation is done, k and g are calculated with Eqs. (8) and (15) respectively and the related material parameters and initial void radius (b = 104) come from NAG model. The result is shown in Fig. 2 for two impacting velocities. It is evident that at the beginning of damage evolution the damage is mostly caused by the effects of nucleation. After a period of time growth becomes dominant in the damage and nucleation can be ignored. The transition time between the two stages is very short and is dependent upon the impacting velocity. The higher the impacting velocity, the shorter the transition time. Assuming that the contribution of nucleation can be ignored when it accounts for only one third of the damage, the transition time is given by t¼
logð3Þ ð3kÞ
ð26Þ
Considering Eq. (24), the initial damage can be estimated as f0 ¼ 4p=3gtb3
ð27Þ
Fig. 2. Damage accumulation of spalling surface under different impacting velocity.
C. Zhang et al. / Mechanics Research Communications 33 (2006) 26–34
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Substituting Eq. (26) into Eq. (27), the initial damage is given by f0 ¼ ð4p=3Þ logð3Þ=ð3kÞgb3
ð28Þ
The above equation indicates that the initial damage is a function of the material and stress. 5.2. Linear nucleation law At time t, the total number of voids in the material is Z 1 Z b expðktÞ q dc ¼ qt N ðtÞ ¼ nðc; tÞ dc ¼ kc b b
ð29Þ
Eq. (29) shows that the number of voids in the material depends only on the nucleation, and it increases linearly with time. This relationship agrees with the experimental observations of Bai et al. (1991) and Curran et al. (1987). In their experiment, they counted microdamage and found that the nucleation rate in planar impact tests is time independent. So if annihilation and connection of microdamage are ignored at the very beginning of damage evolution, the total number of voids in the material should increase linearly with time. 5.3. Comparison with other damage evolution equations In the NAG model the damage accumulation at time t + Dt is f ðt þ DtÞ ¼ f ðtÞ expð3kDtÞ þ 8pqDtb3
ð30Þ
From Eq. (24), the damage accumulation at time t + Dt is 4 f ðt þ DtÞ ¼ f ðtÞð1 þ 3kDtÞ þ pqDtb3 3
ð31Þ
The above two equations are only identical when Dt ! 0. If the damage degree (D) is used to describe the material failure degree D ¼ f =ðf þ V S Þ where VS is the volume of the matrix material, which is assumed uncompressible and will not change its volume with time during damage evolution, i.e., oVS/ot = 0. Differentiating the above equation against time, we have D_ ¼ oD=ot ¼ ð1 DÞf_ =ðf þ V S Þ
ð32Þ
Neglecting the effect of nucleation and substituting Eq. (24) into Eq. (32) we get D_ ¼ 3kð1 DÞD
ð33Þ
The above damage degree evolution equation is similar to the equation developed by Feng et al. (1997). Damage and fracture of material result from damage nucleation, growth and coalescence. Nucleation, growth and coalescence have different effects at different stages during damage evolution. Nucleation is dominant at the beginning of damage evolution. After a period of time the growth becomes dominant, and nucleation can be ignored. The transition time between the two stages is related to the material and loading conditions. Given the fact that the duration of nucleation is much shorter than the duration of growth, it is reasonable to use initial damage to replace nucleation, but the initial damage must take the effects of loading into consideration.
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C. Zhang et al. / Mechanics Research Communications 33 (2006) 26–34
Acknowledgements The authors would like to thank the technician C.L. Liu and the Ph.D. student T. Jiao at the explosive experiment laboratory in Beijing Institute of Technology for their assistance. They also appreciate the two graduate students, Mr. W. Megenda and P. Frempong at Dalhousie University for their assistance in editing and proofreading the manuscript.
References Bai, Y.L., Ke, F.J., Xia, M.F., 1991. Formulation of statistical evolution of microcracks in solids. Acta Mech. Sinica 7, 59–66. Bai, J., Xia, M.F., Ke, F.J., Bai, Y.L., 1999. Properties of the statistical damage evolution equation and its numerical simulation. Acta Mech. Sinica 31, 38–48 (in Chinese). Bai, Y.L., Bai, J., LI, H.L., Xia, M.F., Ke, F.J., 2000. Damage evolution, localization and failure of solids subjected to impact loading. Int. J. Impact Eng. 24, 685–701. Bai, Y.L., Xia, M.F., Ke, F.J., Li, H.L., 2001. Statistical microdamage mechanics and damage field evolution. Theor. Appl. Fract. Mech. 37, 1–10. Carroll, M.M., Holt, A.C., 1972. Static and dynamic pore-collapse relations for ductile porous materials. J. Appl. phys. 43 (4), 1626– 1636. Chu, C.C., Needleman, A., 1980. Void nucleation effects in biaxially stretched sheets. J. Eng. Mater. Technol. 102, 247–256. Curran, D.R., Seaman, L., Shockey, D.A., 1977. Dynamic fracture in solids. Phys. Today 30, 46–55. Curran, D.R., Seaman, L., Shockey, D.A., 1987. Dynamic failure of solids. Phys. Rep. 147, 254–388. Feng, J., Jing, F., Zhang, G., 1997. Dynamic ductile fragmentation and the damage function model. J. Appl. Phys. 81 (6), 2575–2578. Gurson, A.L., 1977. Continuum theory of ductile rupture by void nucleation and growth-I. Yield criteria and flow rules for porous ductile media. Eng. Mater. Technol. 99, 2–15. Johnson, J.N., 1981. Dynamic fracture and spallation in ductile solids. J. Appl. Phys. 52 (4), 2812–2825. Ragab, 2004. Application of an extended void growth model with strain hardening and void shape evolution to ductile fracture under axisymmetric tension. Eng. Fract. Mech. 71, 1515–1534. Rice, J.R., Tracey, D.M., 1969. On the ductile enlargement of voids in triaxial stress fields. J. Mech. Phys. Solids 17 (3), 201–217. Seaman, L., Curran, D.R., Shockey, D.A., 1976. Computational models for ductile and brittle fracture. J. Appl. Phys. 47 (11), 4814– 4824. Tvergaard, V., 1982. Ductile fracture by cavity nucleation between larger voids. J. Mech. Phys. Solids 30, 265–286.
Mechanics Research Communications 33 (2006) 26–34 www.elsevier.com/locate/mechrescom
Damage evolution of ductile material under impact loading Changsuo Zhang a
a,*
, Steve Zou b, Baoping Zhang
c
Department of Mining Engineering, Taiyuan University of Technology, Taiyuan 030024, China b Department of Mining Engineering, Dalhousie University, Halifax, Canada B3J 2X4 c Beijing Institute of Technology, Beijing 100081, China Available online 16 June 2005
Abstract Nucleation, growth and coalescence of micro-voids result in the fracture of materials. Most mathematical models neglect nucleation and introduce initial damage, assuming it as a material constant. However, the original damage, which is formed during material working, is a material constant. The initial damage is a model parameter and depends on the load. Apparently, the predictability of such a model is poor. This paper made comparison and analysis of the four classical void growth models and showed their similarities. At the beginning of damage evolution, all the models follow a linear relationship in the form c_ ¼ kc, where c is the size of micro voids and k is a parameter which relates the material and loading condition. With the concept of statistical microdamage and the assumption of uniform void radius for new voids, a damage evolution equation was deduced based on the above void growth model. With this equation the effects of nucleation and growth at the beginning of the damage stage on the whole process of damage evolution can be calculated. The transition time from the nucleation dominant phase to the growth dominant phase can be determined. When the transition time is applied to the damage failure model of ductile material proposed by Johnson, the initial damage (f0), a model parameter in the original model, can also be determined. The results of the derived damage evolution equation agree well with the previous research results. Ó 2005 Published by Elsevier Ltd. Keywords: Micro-statistical damage; Initial damage; Growth model
1. Introduction Many experiments and metallographic observations for metals at ductile failure show that fracture in ductile materials generally results from the nucleation, growth and coalescence of micro-voids. In these three sequential stages, nucleation is the predominant process at the first stage of damage evolution. During the middle and the last stages of damage evolution, the effect of nucleation is relatively small and can be *
Corresponding author.
0093-6413/$ - see front matter Ó 2005 Published by Elsevier Ltd. doi:10.1016/j.mechrescom.2005.05.004
C. Zhang et al. / Mechanics Research Communications 33 (2006) 26–34
27
neglected when compared with void growth and coalescence. Void growth lasts the longest period of time and is therefore the most important of the three stages. Coalescence predominates the process in the last stage and is very short in time. In order to simplify development of mathematical models for failure of ductile materials, two parameters are introduced in most models, namely the initial damage (f0) and the critical damage (fc). The initial damage (f0) is used to replace the nucleation process and the critical damage (fc) to replace the coalescence process. In fact, most models are focused on the void growth stage. When the damage degree reaches the value of fc, the onset of the final coalescence process begins. Due to the fact that coalescence is too quick to be observed directly, using fc to replace coalescence seems reasonable. During the past thirty years, many researchers have developed their own theoretical models. By investigating the behavior of dynamic damage and fracture in a solid, Seaman et al. (1976) and Curran et al. (1977, 1987) developed a nucleation and growth (NAG) model. The NAG model has sufficient generality to include the statistical distribution of one or more variables such as porosity and void density. However, the model is only based on statistical observations so it needs numerous parameters that are difficult to obtain. Rice and Tracey (1969) investigated the growth of an isolated cylindrical void and a spherical void embedded in infinite media. They proposed a void growth model based on an unlimited matrix. Gurson (1977) presented a creative concept of cell model, which possesses a finite volume and contains a spherical void. He then theoretically developed a plastic potential function for porous materials. Tvergaard (1982) introduced two adjustable parameters, q1 and q2, and considered the interaction between the voids to improve the original Gurson model. Carroll and Holt (1972) established static and dynamic pore-collapse relation for ductile porous materials. Johnson (1981) applied Carroll and HoltÕs approach and presented a void growth model. However, most of these models adopted the idea of initial damage instead of nucleation. This approximation of the nucleation process greatly simplifies the numerical modeling of ductile fracture. Unfortunately, it has shown that the initial damage f0 is not a material constant. Our numerical results reveal that f0 varies with the stress. In JohnsonÕs void growth model, for example, nucleation effect was replaced by initial damage and was regarded as a dimensionless material constant whose value lies between 103 and 104. In fact, initial damage and original damage are different. Original damage is some micro defects that are formed during material working, and is therefore a material constant. On the other hand, initial damage is used to replace nucleation effect and it is a model parameter that depends on both the stress and material. Fig. 1 shows free-surface velocity of scab of different initial damages in JohnsonÕs model. From Fig. 1, when the initial damage changes from 0.0003 to 0.003, the signal of spalling has changed and the curve after spalling shifted greatly, sometimes without spalling. On the other hand, application of micromechanical models should enable attainment of so-called transferability of model parameters to different geometries and different loading conditions. In other words, if the initial damage is a material parameter, it should be a constant for the same material, but this is not true. Taking OFHC copper as an example,
800
Vf (m/s)
600
1 2 3
400
4
experiment data 1 2 3 4
200
0.0
0.5
1.0
f0=0.003 f0=0.001 f0=0.0006 f0=0.0003
1.5
t [µs] Fig. 1. Free-surface velocity of scab in different initial damages (after Seaman et al., 1976).
28
C. Zhang et al. / Mechanics Research Communications 33 (2006) 26–34
a good agreement is obtained with the initial damage of 0.0002 with lower impacting velocity, but big deviation occurs when the model is applied to higher impacting velocity with the same initial damage. This shows that the initial damage in all of void growth models is a model parameter not a material constant. It is therefore difficult to use this model to predict the fracture of ductile material. In the early stage of damage evolution, the distance between voids is relatively large and the interaction of voids can be ignored. Thus the statistical relation between different voids is independent. In such case, damage has little influence on the local stress. As a result, it may be easy to get an analytical solution. In this article, we will apply the theory of statistical micro-damage to calculate the effect of nucleation on the whole process of damage evolution and to establish the value of initial damage f0. 2. Description of statistical micro-voids evolution A framework of statistical micro damage mechanics was established by Bai et al. (1991, 2000, 2001). They developed an evolution equation (Eq. (1)) of density of micro-voids number when they investigated damages under plane impacting loading. 8 oðn_cÞ on > < ot þ oc ¼ nN ðaÞ ð1Þ nN ¼ nN ðc; tÞ ðbÞ > : c_ ¼ c_ ðcÞ ðcÞ Eqs. (1b) and (1c) are nucleation and growth equations, respectively, where n is the number density, c is the radius of micro-voids and nN is nucleation rate density of micro-void. In our analysis, Eq. (1) is used to describe the damage evolution process of micro-voids. From experiment it is known that the volume of a micro-void is very small after its nucleation and the conformation of radius is ideal. In order to simplify computation, we assume that the newly nucleated voids have the same size and the radius equals to b. In fact, in many models such as NAG model and nucleation model proposed by Chu and Needleman (1980), the volumes of the new nucleation voids are supposed to be the same. The nucleation equation therefore becomes on oð_cnÞ þ ¼ qðtÞdðc bÞ ot oc where q(t) is the number of nucleated voids within a unit time, d(c b) is the Dirac function of c. Integrating Eq. (2) and let e ! 0 gives Z bþe b þ e on dc þ c_ n ¼ qðtÞ be be ot : As on ¼qðtÞ is limited and n(c, t) = 0 if c < b, Eq. (3) becomes ot ð_cnÞc¼b ¼ qðtÞ
ð2Þ
ð3Þ
ð4Þ
Based on the theory of differential equation, when the radii of micro-voids are the same the original Eq. (1) becomes 8 on oðn_cÞ > > þ ¼ 0 ðaÞ > > ot oc > > > > < ð_cnÞc¼b ¼ qðtÞ ðbÞ ð5Þ c_ ¼ c_ ðcÞ ðcÞ > > > > > nðc; tÞt¼0 ¼ 0 ðdÞ > > > : nðc; tÞc¼1 ¼ 0 ðeÞ
C. Zhang et al. / Mechanics Research Communications 33 (2006) 26–34
29
where b is the radius of micro-void when it nucleates. The general solution of Eq. (5a) (Bai et al., 1999) is Z c 1 dc nðc; tÞ ¼ w t ð6Þ c_ c_ b The format of function W can be determined based on the boundary condition ð_cnÞc¼b ¼ qðtÞ. Therefore, if the void growth equation c_ ¼ c_ ðcÞ and void nucleation equation q(t) are given, the damage evolution equation can be deduced.
3. Void growth model In the following, we will analyze four classical void growth models to deduce the general growth model at the early stage of damage evolution. 3.1. NAG model Curran and co-workers (Seaman et al., 1976; Curran et al., 1977, 1987) developed a nucleation and growth (NAG) model. From the statistical observation they got the increment of damage (DVvg) during Dt from void growth DV vg
3 P S P g0 Dt V v0 ¼ V v0 exp 4 g
ð7Þ
where g is the material viscosity, Pg0 is material threshold for growth. According to the differential theory, if is small, it can be simplified approximatively to DV vg 3 P S P g0 ¼ V v0 4 Dt g then we have 3 P S P g0 V_ ¼ V 4 g If radius R is used in the above equation, the void growth equation becomes 1 P S P g0 R_ ¼ R 4 g
ð8Þ
3.2. Rice and Tracey void growth model For the case of an empty spherical void of radius R in a remote uniform plastic strain rate field Dpij , Rice and Tracey developed a growth model of micro-void
_R ¼ 5 Dpk þ 3 rm Dp R ð9Þ 3 4 rM where Dpk is the principal strain rates, rm is the mean normal stress, and rM is the yield stress.
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3.3. Gurson and Tvergaard yield function The classical yield function for porous solids was presented by Gurson (1977) and modified by Tvergaard (1982) and it takes the form 2 3 rm r 2 2 q ¼ ð1 þ q1 C v Þ 2q1 C v cosh ð10Þ M M 2 2r r where Cv = Vv/(Vv + VM), Vv is the volume of voids and VM is volume of the matrix. Using the above formula and considering a relatively small Cv, Ragab (2004) deduced the void growth equation V_ v 3 3 rm q ¼ q q sinh e_ 1 2 2 rY Vv 2 1 2 If radius R is used as variable, the above equation becomes _R ¼ 1 q1 q2 sinh 3 q2 rm e_ 1 R 2 2 rY
ð11Þ
3.4. JohnsonÕs void growth model Based on the work of Carroll and Holt (1972) and Johnson (1981) presented a void growth model. In JohnsonÕs model, the presence of voids is expressed in terms of the distension ratio a, which is related to the porosity / through the expression / = 1 1/a. The growth rate of is given by ( a_ ¼ 0 if Dp P 0 ð12Þ ða0 1Þ2=3 1=3 a_ ¼ g a a 1 DP if Dp < 0 where DP is the driving stress, and a0 is the initial damage. When a is very small, just as in the early stage of damage evolution, the above growth equation becomes 2=3
a_ ¼
ða0 1Þ g
aDP
ð13Þ
In all of the four classical void growth models, whether the void is surrounded by infinite matrix (Rice and TraceyÕs), or by a holly ball cell (GursonÕs and JohnsonÕs), or represented by pure statistical observation result (NAG), the void growth model can be represented by a linear equation when the damage degree is relatively small. The following generalized growth equation can therefore be used at the early stage of damage evolution: c_ ¼ kc ð14Þ 4. Damage evolution equation In the NAG model, the following nucleation equation was presented by Seaman et al. (1976), and Curran et al. (1977, 1987): 8 h i < N_ ¼ N_ exp P S P no if P S > P n0 0 P1 ð15Þ : _ N ¼0 if P S 6 P n0 where N_ 0 and P1 are material constants, PS is tensile pressure and Pn0 is nucleation threshold of material.
C. Zhang et al. / Mechanics Research Communications 33 (2006) 26–34
31
In the above equation, except N_ and PS, the other items are constants. Therefore, the nucleation equation can be simplified as qðtÞ ¼ gH ðtÞ
ð16Þ
where g and k in Eq. (14) are constants and depend on the material and stress, and H ðtÞ ¼
0
if t < 0
1
if t P 0
Considering the boundary condition, the function w in Eq. (6) becomes wðtÞ ¼ gH ðtÞ
ð17Þ
Therefore, the number density of the micro-voids is g
kc
if t P 1=kðln c ln bÞ
0
if t < 1=kðln c ln bÞ
nðc; tÞ ¼
ð18Þ
It can be proved that Eq. (18) satisfies the boundary and the initial conditions, and it is the solution to Eq. (5) The final aim of statistical damage analysis is to discover the relationship between the process of damage evolution in the material and its mechanics performance. Thus it is necessary to obtain the damage variable based on statistical average. The general damage function is Z 1 fm ðtÞ ¼ cm nðc; tÞ dc ð19Þ b
If the volume of micro-voids is regarded as damage variable, then m = 3, f(t) is the whole volume of micro-voids in the material, so above general damage function becomes Z 1 4 f ðtÞ ¼ p c3 nðc; tÞ dc ð20Þ 3 b At a given time, the biggest void in the material is the void which nucleates at t = 0, therefore Z B dc t¼ c_ b where B is the radius of the biggest void. The volume of the void at time t is Z Z 4p B g 3 4p b expðktÞ g 3 4pg 3 f ðtÞ ¼ c dc ¼ c dc ¼ b ðexpð3ktÞ 1Þ 3 b kc 3 b kc 9k
ð21Þ
ð22Þ
Assuming k and g are constants and differentiating Eq. (22) with respect to time t, gives df 4 ¼ pgb3 expð3ktÞ dt 3
ð23Þ
and the following damage evolution equation is obtained 4 f_ ¼ 3kf þ pgb3 3
ð24Þ
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5. Discussion and conclusion 5.1. Initial damage Considering the right hand side of Eq. (24), the first item is the contribution of nucleation and the second item is the contribution of growth. The contribution of nucleation over the total volume of voids is represented by DfN 1 ¼ Df ðtÞ expð3ktÞ
ð25Þ
where DfN is the nucleation contribution during time Dt and Df(t) is the increment of the total void volume during Dt. From Eq. (25), the contribution of nucleation to the overall damage decreases greatly with time. As a result, many models can get good results through adjusting initial damage without considering nucleation. In order to calculate the effects of growth and nucleation during damage evolution, the damage evolution Eq. (24) is embedded in one-dimensional hydrodynamic simulations. Here we used normalized volume as the label of vertical axes in Fig. 2 which is the ratio between whole volume of micro-voids and the volume of original voids. When the calculation is done, k and g are calculated with Eqs. (8) and (15) respectively and the related material parameters and initial void radius (b = 104) come from NAG model. The result is shown in Fig. 2 for two impacting velocities. It is evident that at the beginning of damage evolution the damage is mostly caused by the effects of nucleation. After a period of time growth becomes dominant in the damage and nucleation can be ignored. The transition time between the two stages is very short and is dependent upon the impacting velocity. The higher the impacting velocity, the shorter the transition time. Assuming that the contribution of nucleation can be ignored when it accounts for only one third of the damage, the transition time is given by t¼
logð3Þ ð3kÞ
ð26Þ
Considering Eq. (24), the initial damage can be estimated as f0 ¼ 4p=3gtb3
ð27Þ
Fig. 2. Damage accumulation of spalling surface under different impacting velocity.
C. Zhang et al. / Mechanics Research Communications 33 (2006) 26–34
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Substituting Eq. (26) into Eq. (27), the initial damage is given by f0 ¼ ð4p=3Þ logð3Þ=ð3kÞgb3
ð28Þ
The above equation indicates that the initial damage is a function of the material and stress. 5.2. Linear nucleation law At time t, the total number of voids in the material is Z 1 Z b expðktÞ q dc ¼ qt N ðtÞ ¼ nðc; tÞ dc ¼ kc b b
ð29Þ
Eq. (29) shows that the number of voids in the material depends only on the nucleation, and it increases linearly with time. This relationship agrees with the experimental observations of Bai et al. (1991) and Curran et al. (1987). In their experiment, they counted microdamage and found that the nucleation rate in planar impact tests is time independent. So if annihilation and connection of microdamage are ignored at the very beginning of damage evolution, the total number of voids in the material should increase linearly with time. 5.3. Comparison with other damage evolution equations In the NAG model the damage accumulation at time t + Dt is f ðt þ DtÞ ¼ f ðtÞ expð3kDtÞ þ 8pqDtb3
ð30Þ
From Eq. (24), the damage accumulation at time t + Dt is 4 f ðt þ DtÞ ¼ f ðtÞð1 þ 3kDtÞ þ pqDtb3 3
ð31Þ
The above two equations are only identical when Dt ! 0. If the damage degree (D) is used to describe the material failure degree D ¼ f =ðf þ V S Þ where VS is the volume of the matrix material, which is assumed uncompressible and will not change its volume with time during damage evolution, i.e., oVS/ot = 0. Differentiating the above equation against time, we have D_ ¼ oD=ot ¼ ð1 DÞf_ =ðf þ V S Þ
ð32Þ
Neglecting the effect of nucleation and substituting Eq. (24) into Eq. (32) we get D_ ¼ 3kð1 DÞD
ð33Þ
The above damage degree evolution equation is similar to the equation developed by Feng et al. (1997). Damage and fracture of material result from damage nucleation, growth and coalescence. Nucleation, growth and coalescence have different effects at different stages during damage evolution. Nucleation is dominant at the beginning of damage evolution. After a period of time the growth becomes dominant, and nucleation can be ignored. The transition time between the two stages is related to the material and loading conditions. Given the fact that the duration of nucleation is much shorter than the duration of growth, it is reasonable to use initial damage to replace nucleation, but the initial damage must take the effects of loading into consideration.
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Acknowledgements The authors would like to thank the technician C.L. Liu and the Ph.D. student T. Jiao at the explosive experiment laboratory in Beijing Institute of Technology for their assistance. They also appreciate the two graduate students, Mr. W. Megenda and P. Frempong at Dalhousie University for their assistance in editing and proofreading the manuscript.
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