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The numerical simulation of ductile fracture
2 Solution Strategies and Methodology
Compvrers & Srrucrures Vol. 30, No. 4, pp. 817-819. Printed in Great Britain.
THE NUMERICAL
0045.7949/88 53.00 + 0.00 0 1988 Pergamon Press plc
1988
SIMULATION OF DUCTILE FRACTURE
XINWAN BEI, ZHIIIE ZHANG and LIANSHENG ZHOU Department of Applied Physics, National University of Defense Technology,
The People’s Republic of China Abstract-The ductile microstructural fracture mode of Seaman et al. (J. appl. Phys. 47,48144826, 1976) is incorporated into a two-dimensional Lagrangian hydrodynamic-elastic-plastic finite-difference code LZEP discussed elsewhere and ductile fracture phenomena under intense impulsive loading thus can be simulated in great detail. An example that simulates damage situation in an aluminum target after high-velocity impact with an iron projectile is given. The advantages and limitations of this model are briefly discussed: INTRODUCTION
When a solid material is subjected to intense external loading, such as the high-velocity impact of a projectile against a target, compressive stress waves will exist. The compressive waves will be turned into rarefaction waves while reflecting from the free surfaces. When two rarefraction waves meet together, intense tension will usually result in the material. Under the action of the tensile waves of a certain stength, the material will suffer damage. During tension, the damage will develop and finally leads to the failure of the material. The mechanism of damage development under intense dynamic loading is very complicated and worthy of deep study. In recent years the computer simulation of dynamic fracture has greatly promoted scientific research in this field. As a rule, a certain criterion must be added to the computation code in order to simulate the dynamic fracture. There have been two fracture criteria used in the past. One is the criterion of maximum tensile stress and the other is the criterion of cumulative spalling. Generally speaking, the former is applicable for brittle fracture, and the latter for ductile fracture. Although both criteria can reflect the true state to a certain degree, there is a common weakness in them. In these criteria, the stresses can affect the damage, but the damage will not in turn affect the stresses. Seaman’s microstructural fracture model [l], on the other hand, can describe not only the damage developing in the material but also the effect of the damage on the stress conditions. With these improvements the model gives better agreement with experiments than the aforementioned criteria. Referring to [l], the present authors incorporated a ductile fracture subroutine into their two-dimensional Lagrangian hydrodynamicelastic-plastic code LSEP 121; this can simulate ductile fracture under dynamic loading in great detail.
tion of experiment results and computational method. In the experiments, the specimen was impacted with a flyer plate at a certain velocity in each test. The shocked specimens were then recovered, sectioned and polished, and the voids on the crosssectional surface of polished specimens were measured, counted and assembled into an overall surface distribution curve of void number versus void sizes. The surface distribution curves were then transformed into a volumetric distribution curve using a statistical method. The volumetric distributions obtained can be approximated by the equation N,(P) = No exp( - RIP, ),
(1)
where N, is the cumulative number/cm3 of voids with radius larger than R, No is the total number of voids/ cm3 and RI is a parameter of distribution. Combining the experimental and computational results we can obtain the nucleation and growth rates of voids. The nucleation rate function can be calculated as 19 = % expKP, -
PJlP,l P,> P.,,
(2)
where P,, is the threshold value for nucleation, ?& and P, are material constants and P, is the tensile stress in a solid. The growth rate function can be. fitted as d =R(P,-P,,)l(4tl)
P,=-P,,,
(3)
where n is a material constant and Pgo is the threshold value for growth. The total change in void volume is the sum of the changes contributed by the nucleation and growth in the process. The total void volume at the end of each time interval is V, = VMexp[3At(P, - P,,)/(4r7)] + 8x h’AtR2,
(4)
where Vti is the void volume at the beginning of the time interval At, and R, is the nucleation size parameter. The damage material is composed of solid and
THE CONSTITUTION OF MODEL AND FORMULAS
According to [ 11, the ductile microstructural fracture model is developed on the basis of the combina817
XINYIJAN BEI et
818
al.
voids. The pressure and volume in the damaged material are related by V = V, + V”
(5)
P = P,V,/V,
(6)
where subscript s denotes the solid and subscript u the void. P is the average pressure and V is the gross specific volume. The equation of state is
Time= l.OOOlE-5
(set)
Fig. 2. Distortion nets and damage, 10psec after impact. (7) where E is the specific internal energy. The above-mentioned system of eqns (2)-(7) should be solved simultaneously. Hence an iteration procedure is required.
3. Additional criteria and controllable switches. 4. A statement for calling the fracture subroutine. 5. Additional output information on damage. AN EXAMPLE
INSERTION
OF FRACTURE INTO LZEP
OF COMPUTATION
SUBROUTINE
In the previous scalar form of our L2EP code, when the tensile stress in certain cells overrides the nucleation threshold value Pd or these cells have been damaged previously, the deviator stresses and pressure should be treated with the fracture subroutine instead of the general constitutive relation subroutines. Now our code L2EP is a vectorized one, in which these stresses are at first computed with general constitutive relation subroutines in all cells and then the stresses in some cells must be recomputed with the fracture subroutine based on the fracture criteria. It seems that more computer time is required due to the recomputation. However, the major part of the cells usually remains in an undamaged condition, and the computation of stresses with vectorized constitutive relation subroutines in full continual array can speed up the computation by a factor of approximately 20. Such an advantage overrides the disadvantage of the necessary recomputation. Some new features should be added to the code as follows.
The impact process of an iron projectile with an aluminum target at 500 m/set is simulated. The projectile has a diameter of 0.010 m and a length of 0.012 m; the target diameter is 0.028 m and its thickness is 0.007 m. The initial Lagrangian meshes are taken as squares with side lengths of 0.001 m. Figure 1 shows the isobars obtained at 1.Opsec after impact. The interval between two neighboring isobars is 5 kb. Figure 2 shows the damage situation at 10 psec after impact. At this time, the target has been distorted seriously and the damage does not continue afterwards. The damage degrees are described in Fig. 2 by four degrees of shading. Figure 3 shows the relative void volume growing curve in the destroyed cell (16,l). It can be seen that the relative void volume rises linearly and rapidly before arriving at the fracture threshold value. This simulation requires 57 set of CPU time on the YH-1 computer, which is a vector computer manufactured in China. This is about twice as long as the time when the limit tension stress criterion is used, as iteration must be used for each damaged cell in the micro-
1. Additional array specifications for storing the parameters on damage. 2. Additional read-ins for material parameters and control parameters on damage.
000.
Time = l.O062E-6
Fig. 1.
(sec)
Isobars 1.0+ec after impact.
0123rr5 Time (peel
Fig. 3. Relative void volume growing curve in the cell (16,l)
819
The numerical simulation of ductile fracture
structure fracture subroutine cannot be vectorized.
and such a subroutine
parameters for damage are taken from [3]; the results of computation remain to be checked by further experiments. REFERENCES
CONCLUSION
The main advantage of the present model is that it has a solid base in experiments. It involves not only the effect of the stress on the damage of material, but also the reaction of the damage on stress conditions in material. It requires information from experiments with sufficient precision and requires a computer with sufficient storage and computing speed. In the above mentioned example, the material
L. Seaman, D. R. Curran and D. A. Shockey, Computational models for ductile and brittle fracture. J. uppl. Phys. 47, 48144826 (1976). Z. Zhang, X. Bei and L. Zhou, The calculation of elastic-plastic flow on flying plate impact. J. Nat. Vniu. Defensd Technol. 48, 13-34 (1984) (in Chinese). D. A. Shockev. L. Seaman and D. R. Curran. The microstatisticai ‘fracture mechanics approach to dynamic fracture problems. Inr. J. Fracture 27, 145-157 (1985).
Compvrers & Srrucrures Vol. 30, No. 4, pp. 817-819. Printed in Great Britain.
THE NUMERICAL
0045.7949/88 53.00 + 0.00 0 1988 Pergamon Press plc
1988
SIMULATION OF DUCTILE FRACTURE
XINWAN BEI, ZHIIIE ZHANG and LIANSHENG ZHOU Department of Applied Physics, National University of Defense Technology,
The People’s Republic of China Abstract-The ductile microstructural fracture mode of Seaman et al. (J. appl. Phys. 47,48144826, 1976) is incorporated into a two-dimensional Lagrangian hydrodynamic-elastic-plastic finite-difference code LZEP discussed elsewhere and ductile fracture phenomena under intense impulsive loading thus can be simulated in great detail. An example that simulates damage situation in an aluminum target after high-velocity impact with an iron projectile is given. The advantages and limitations of this model are briefly discussed: INTRODUCTION
When a solid material is subjected to intense external loading, such as the high-velocity impact of a projectile against a target, compressive stress waves will exist. The compressive waves will be turned into rarefaction waves while reflecting from the free surfaces. When two rarefraction waves meet together, intense tension will usually result in the material. Under the action of the tensile waves of a certain stength, the material will suffer damage. During tension, the damage will develop and finally leads to the failure of the material. The mechanism of damage development under intense dynamic loading is very complicated and worthy of deep study. In recent years the computer simulation of dynamic fracture has greatly promoted scientific research in this field. As a rule, a certain criterion must be added to the computation code in order to simulate the dynamic fracture. There have been two fracture criteria used in the past. One is the criterion of maximum tensile stress and the other is the criterion of cumulative spalling. Generally speaking, the former is applicable for brittle fracture, and the latter for ductile fracture. Although both criteria can reflect the true state to a certain degree, there is a common weakness in them. In these criteria, the stresses can affect the damage, but the damage will not in turn affect the stresses. Seaman’s microstructural fracture model [l], on the other hand, can describe not only the damage developing in the material but also the effect of the damage on the stress conditions. With these improvements the model gives better agreement with experiments than the aforementioned criteria. Referring to [l], the present authors incorporated a ductile fracture subroutine into their two-dimensional Lagrangian hydrodynamicelastic-plastic code LSEP 121; this can simulate ductile fracture under dynamic loading in great detail.
tion of experiment results and computational method. In the experiments, the specimen was impacted with a flyer plate at a certain velocity in each test. The shocked specimens were then recovered, sectioned and polished, and the voids on the crosssectional surface of polished specimens were measured, counted and assembled into an overall surface distribution curve of void number versus void sizes. The surface distribution curves were then transformed into a volumetric distribution curve using a statistical method. The volumetric distributions obtained can be approximated by the equation N,(P) = No exp( - RIP, ),
(1)
where N, is the cumulative number/cm3 of voids with radius larger than R, No is the total number of voids/ cm3 and RI is a parameter of distribution. Combining the experimental and computational results we can obtain the nucleation and growth rates of voids. The nucleation rate function can be calculated as 19 = % expKP, -
PJlP,l P,> P.,,
(2)
where P,, is the threshold value for nucleation, ?& and P, are material constants and P, is the tensile stress in a solid. The growth rate function can be. fitted as d =R(P,-P,,)l(4tl)
P,=-P,,,
(3)
where n is a material constant and Pgo is the threshold value for growth. The total change in void volume is the sum of the changes contributed by the nucleation and growth in the process. The total void volume at the end of each time interval is V, = VMexp[3At(P, - P,,)/(4r7)] + 8x h’AtR2,
(4)
where Vti is the void volume at the beginning of the time interval At, and R, is the nucleation size parameter. The damage material is composed of solid and
THE CONSTITUTION OF MODEL AND FORMULAS
According to [ 11, the ductile microstructural fracture model is developed on the basis of the combina817
XINYIJAN BEI et
818
al.
voids. The pressure and volume in the damaged material are related by V = V, + V”
(5)
P = P,V,/V,
(6)
where subscript s denotes the solid and subscript u the void. P is the average pressure and V is the gross specific volume. The equation of state is
Time= l.OOOlE-5
(set)
Fig. 2. Distortion nets and damage, 10psec after impact. (7) where E is the specific internal energy. The above-mentioned system of eqns (2)-(7) should be solved simultaneously. Hence an iteration procedure is required.
3. Additional criteria and controllable switches. 4. A statement for calling the fracture subroutine. 5. Additional output information on damage. AN EXAMPLE
INSERTION
OF FRACTURE INTO LZEP
OF COMPUTATION
SUBROUTINE
In the previous scalar form of our L2EP code, when the tensile stress in certain cells overrides the nucleation threshold value Pd or these cells have been damaged previously, the deviator stresses and pressure should be treated with the fracture subroutine instead of the general constitutive relation subroutines. Now our code L2EP is a vectorized one, in which these stresses are at first computed with general constitutive relation subroutines in all cells and then the stresses in some cells must be recomputed with the fracture subroutine based on the fracture criteria. It seems that more computer time is required due to the recomputation. However, the major part of the cells usually remains in an undamaged condition, and the computation of stresses with vectorized constitutive relation subroutines in full continual array can speed up the computation by a factor of approximately 20. Such an advantage overrides the disadvantage of the necessary recomputation. Some new features should be added to the code as follows.
The impact process of an iron projectile with an aluminum target at 500 m/set is simulated. The projectile has a diameter of 0.010 m and a length of 0.012 m; the target diameter is 0.028 m and its thickness is 0.007 m. The initial Lagrangian meshes are taken as squares with side lengths of 0.001 m. Figure 1 shows the isobars obtained at 1.Opsec after impact. The interval between two neighboring isobars is 5 kb. Figure 2 shows the damage situation at 10 psec after impact. At this time, the target has been distorted seriously and the damage does not continue afterwards. The damage degrees are described in Fig. 2 by four degrees of shading. Figure 3 shows the relative void volume growing curve in the destroyed cell (16,l). It can be seen that the relative void volume rises linearly and rapidly before arriving at the fracture threshold value. This simulation requires 57 set of CPU time on the YH-1 computer, which is a vector computer manufactured in China. This is about twice as long as the time when the limit tension stress criterion is used, as iteration must be used for each damaged cell in the micro-
1. Additional array specifications for storing the parameters on damage. 2. Additional read-ins for material parameters and control parameters on damage.
000.
Time = l.O062E-6
Fig. 1.
(sec)
Isobars 1.0+ec after impact.
0123rr5 Time (peel
Fig. 3. Relative void volume growing curve in the cell (16,l)
819
The numerical simulation of ductile fracture
structure fracture subroutine cannot be vectorized.
and such a subroutine
parameters for damage are taken from [3]; the results of computation remain to be checked by further experiments. REFERENCES
CONCLUSION
The main advantage of the present model is that it has a solid base in experiments. It involves not only the effect of the stress on the damage of material, but also the reaction of the damage on stress conditions in material. It requires information from experiments with sufficient precision and requires a computer with sufficient storage and computing speed. In the above mentioned example, the material
L. Seaman, D. R. Curran and D. A. Shockey, Computational models for ductile and brittle fracture. J. uppl. Phys. 47, 48144826 (1976). Z. Zhang, X. Bei and L. Zhou, The calculation of elastic-plastic flow on flying plate impact. J. Nat. Vniu. Defensd Technol. 48, 13-34 (1984) (in Chinese). D. A. Shockev. L. Seaman and D. R. Curran. The microstatisticai ‘fracture mechanics approach to dynamic fracture problems. Inr. J. Fracture 27, 145-157 (1985).