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Analysis of large deformation and fracture of axisymmetric tensile specimens
0013.7944/91 $3.00 + 0.00 0 1991 Pergamon Press plc.
Engineering Frac~re Mechanics Vol. 39, No. 5, pp. 851457, 1991 Printed in Great Britain.
ANALYSIS OF LARGE DEFORMA~ON FRACTURE OF AXISYMMETRIC TENSILE SPECIMENS
AND
K. S. ZHANG and C. Q. ZHENG Box 350, Department of Applied Mechanics, Northwestern Polytechnical University, Xian, Shaanxi 710072, P.R.C. Abstract-For a reasonable rupture criterion, if it is able to @ve expression to the mechanism of a material’s damage and fracture, it should not only be suitable to describe the processes of fracture initiation and crack propagation in a cracked body, but also suitable to describe that of crack initiation and crack propagation in a non-cracked body. From this viewpoint, a damage and fracture model for ductile material has been suggested based on the analysis of microscopic mechanics by the authors of this paper [K. S. Zhang, Eng. D. Thesis (1988); K. S. Zhang and C. Q. Zheng, On the Research with Meso-mechtmics of Ductile Fracture and Its Applications, pp. 84-98 (1988); Zngng Frucrure Mech. ?7,621-629 (1990)]. Gn the basis of this model, a new model called the combinatory work density model has been developed [K, S. Zhang, C. Q. Zheng and N. S. Yang, Proc. IiX4-6j. in the present paper, this new model is adopted to describe the damage and fracture of material, and to carry out the computer emotion of the processes of deformation, crack initiation and crack propagation for a round smooth tensile specimen and a round notched tensile specimen under conditions which are closer to the actual case, by using the large elastic-plastic deformation finite element method. The simulated results represent the experimental processes fairly reasonably, and are in very good agreement with test results. This further con8rms the reasonableness and correctness of the ~mbinato~ work density model.
1. I~RODU~ON IN ORDERto develop a mechanical model for describing crack initiation and its propagation in ductile materials based on the micro-mechanism of damage and fracture, a number of studies have been carried out[l-81. The ojbective is to find, by research on ~cro-m~hanics, the failure criterion that is closest to the physical essence of material failure and which can be used to describe and predict the damage and fracture of engineering elastic-plastic material. So far, most research considers that there are two very different problems of failure in a non-cracked body and in a cracked body, so they are more often than not judged by using different failure criteria. However, from the microscopic viewpoint, for a very small material unit, its failure criterion should be the same if its failure mechanism is the same, no matter where the unit is located. The failure ocurring is only dependent on the material’s resistence to rupture, the deformation degree, the deformation history and the stress-state (for the conditions of constant temperature and monotonic quasi-static loading). On this basis, the “microvoid multistage nucleating model” has been suggested according to the analysis on micro-m~hanism[9-111. By using this model, combined with the relative voidcontained material’s constitutive equation, the ductile damage and fracture can be reasonably described and predicted for elastic-plastic engineering material. Subsequently, a new model called the “combinatory work density model” was obtained[l2] from the improvement of the “microvoid multis~ge nucleating model” by the author. In the new model, the Prandtl-Reuss constitutive equation is adopted instead of the void-contained material’s equation; thus it is very convenient for application in practice[ 121. In ref. [12J, computer simulation analyses on the processes of deformation and rupture for a series of a~s~et~c specimens made of 40 Cr steel have been carried out by using the large elastic-plastic deformation finite element method. From this the parameters of the combinatory work density model for the material have been determined and the model’s reasonableness and correctness has been confirmed. Furthermore, with the help of the new model, the simulation and prediction of the processes of deformation and fracture initiation (including small crack propagation) of TPB specimens of 40 Cr steel have been successfully carried out. It has also been 851
852
K. S. ZHANG and C. Q. ZHENG
verified that the model is not only suitable for the failure analysis of a non-cracked body but also for a cracked body. In the present paper, the combinatory work density model, combined with the large elastic-plastic deformation finite element method, is adopted to describe a material’s damage and fracture. Computer simulation of the processes of deformation (necking), crack initiation and propagation of a round smooth tensile specimen and a round notched tentile specimen made of 40 Cr steel has also been carried out. All the results obtained are in good agreement with experimental data which were determined separately, and they can represent the actual processes in experiments fairly well. So, the reasonableness and correctness of the new model are confirmed. 2. MODEL OF FRACTURE MECHANICS The ~mbinato~ work density modelf 121is adopted in this paper. This model is of microscopic physical meaning, but takes advantage of the Prandtl-Reuss equation instead of the void-contained material equation. Since the determination of the model’s parameters do not rely upon microscopic tests, it is very convenient in application. The material failure criterion of the ~ombinato~ work density model is that the rupture occurs in a material unit if the combinatory work density W of the unit reaches a certain critical value. That is, w= w,
(1)
where W, is the fracture combinatory work density, which is regarded as a material constant; the combinatory work density parameter W can be determined by W=
Ws+FpWvp+FEWv~
(2)
where
, WIT=
=kk dkkdt s 0
(5)
and Wl,, WV, and WV, are plastic shape change work density, reference plastic volume change work density and reference elastic volume change work density respectively. They refer respectively to the degree of the plastic shape change, the development of microvoids and the elastic volume variation of a material unit. The coefficients Fp and FE are the plastic combinatory ~effi~ent and the elastic combinatory coefficient respectively; Fp reflects the void expanding ability in a material and FE reflects the influence of elastic volume dilatation on the material’s damage and fracture. These two coefficients can be calibrated by combining tests with computer simulation. ai is the deviatoric part of the Cauthy stress tensor a#; D$ is the plastic part of the rate of deformation tensor D,,; E is defined by
In the case of ductile fracture, plastic deformation and microvoid development (ductile damage) are considered as the main reasons of the material failure, and the influence of elastic volume dilatation on the material damage and fracture is considered to be negligible. That is, eq. (1) can be rewritten as W= Thus,
W,-kF,W,,.
the method of analysis can be simplified further.
(6)
853
Analysis of large deformation and fracture of axisymmetric tensile specimens Table 1, The chemical composition of 40 Cr steel (wt%) C
Si
Mn
P
s
CU
Ni
Cr
Ti
Al
W
0.41
0.31
0.64
0.024
0.01
0.09
<0.08
0.90
0.055
0.020
Table 2. The mechanical properties of 40Cr steel at room temperature co., (MPa)
a, (MPa)
6, (%)
$ (X)
E (GPa)
930
1024
16.9
57.8
205
3. MATERIAL
AND SPECIMENS
The chemical composition of 40 Cr steel and its mechanical properties at room temperature are shown in Tables 1 and 2 respectively. Figure 1 shows the true stress-logarithmic strain curve which has been modified by Bridgman’s formulas[l3]. Figure 2 shows the geometry and size of the round smooth specimen and the round notched specimen. According to ref. [12], the parameters FP and W, of the combinatory work density model for 40 Cr steel are:
FP = 0.973,
W, = 1448 MPa.
4. NUMERICAL
ANALYSIS
The numerical computations are carried out by using the large elastic-plastic deformation finite element program which is based on the updated Lagrangian description and designed by the author of this paper[l3]. All computations in this paper were executed by an IBM-4381 computer. 4.1. Analysis of the deformation and fracture of a round smooth specimen By taking account of the symmetry, the finite element mesh of the computed model can be divided as in Fig. 3. It consists of 126 eight-node quadrilateral isoparametric elements and 443 nodes. The computer simulation assumes loading is by controlled displacement, and the loading position is the same as that of the actual specimen (cf. Fig. 3).
1 0
I
I
I
0.2
0.4
0.6
I 0.6
6
Fig. I. The stress-strain
curve of 40Cr steel.
Fig. 2. The dimensions of the round smooth specimen and the round notched specimen.
854
K. S. WANG
and C. Q. ZHENG
Fig. 3. The finite element mesh of the round smooth specimen.
4.1.1. Method forsimulation ofnecking. In general, necking will take place in a round smooth tensile specimen of a ductile material during large elastic-plastic deformation, and it will lead to the deformation of the specimen in the neck region and finally to rupture at the neck section. Since necking has a large influence on the specimen’s deformation and fracture, the computer simulation of the necking process is very important. Commonly, the analysis of necking by using the finite element method used to only take the specimen’s smooth part into account[l3,14]. In this way the assumption of an initial defect (on the surface or on the inside of the specimen) or a rigid displacement condition at the end of the smooth part will be taken, so that necking can be represented during computer simulation. Both methods are somewhat ind~te~inable and arbitrary. On the one hand, the result will lead to evident error when the various defect assumption is taken, and no one knows what defect is proper. On the other hand, applying rigid displacement conditions to the end of the specimen’s smooth section will lead to a greater difference from the real case, expecially for a material which has a larger uniform deformation before necking occurs. In the present paper, in order to be closer to the actual case, the whole specimen is taken into consideration, and neither the initial defect nor the rigid displacement condition are employed. 4.12. method for s~uiatio~s of crack ~~t~io~ and propagation. For the processes of crack initiation and propagation in the specimen’s neck section, the computer simulation is carried out by release of the correlative node’s boundary conditions: (1) for crack initiation, the condition to node release is that the mean combinatory work density W of the nearest two Gauss points located at the two sides of the node amounts to the fracture ~ombinato~ work density W,; (2) for crack propagation, the condition to release the node at the crack tip is that the combinatory work density W of the Gauss point, which is the nearest to that in front of the crack tip, reaches W,; (3) when a node is released, a force which is equal to the node’s axial force but in the opposite direction will be applied to the node by the program’s equilib~um iteration. 4.1.3. Numerical results. According to the computer simulation for the round smooth specimen, the following results can be obtained. The relation of the applied load P with the variation AD of neck redius D and the relation of P with the axial extension AL of the section L are shown in Fig. 4(a, b). They are in good agreement with the test results[lSj. From these, it is known that the computer simulation of the tensile process of the round smooth specimen represents the necking process in very good agreement with the real process, although the defect assumption and the rigid displacement conditions are not taken into account at all. So, the method adopted in this paper can be considered to be more reasonable for necking analysis than the above mentioned two methods, because this method does not involve the indeterminable and arbitrary fact into the computation. The P-AD and P-AL curves (cf. Fig. 4a and b) after the crack initiates in the neck section can also be obtained by computer simulation; they represent the actual fracture process after the crack initiates. Figure 4(c) shows the simulated propagations of the centred crack at the neck section in a short time, which are very difficult to observe by experiment. It is necessary to point out that during the last stage of fast crack propagation there is an error in the numerical results, since the influence of the viscoplasticity and the dynamic response are not taken into account. 4.2. Analyst of the ~formation and fract~e of a round notched spe~~en Since there is a strong stress concentration near the notch of the notched specimen, the influence of the part of the specimen held by machine on the stress distribution near the notch region can be neglected. By taking account of the symmetry, the finite element mesh of the
855
Analysis of large deformation and fracture of axisymmetric tensile specimens (b)
(a)
0
m
I
4.0
1.0
0
AD(mm)
I
I
6.0
a0
AL (mm)
ci !
-
I
40
I
2.0
c
Ao 2.4mm-Aa
3.0mm
(cl
Fig. 4. (a) P-AD curve. (b) P-AL curve. (c) Centred crack propagation in the neck section.
computed model can be divided as shown in Fig. 5. It consists of 74 eight-node quadrilateral isoparametric elements and 265 nodes. The computer simulation still assumes that loading is by controlled displacement. The method of simulating crack initation and propagation of the specimen is just the same as in Section 4.1. In Fig. 6(a, b), the relations of P-AD and P-AL of the notched specimen by computer simulation are presented; they are also in very good agreement with experimental curves. From this figure, it can be seen that with increasing specimen deformation, the outer loading is reduced rapidly after the centred crack has initiated. As in the smooth specimen, this is the reason that the crack propagates at high speed, so that the bearing area in the neck section is reduced very quickly. This appearance is very difficult to record by test machine since the process takes place in such a very short time. Compared with the case of the smooth specimen after the l/2
L
Fig. 5. The finite element mesh of the round notched specimen.
856
IL S. ZHANG and C. Q. ZHENG
0
I
0.2
I
I
I
I
04
0.6
ae
1.0
I 1.2
AD (mm}
AL(mm) (Cl
(bl Fig. 6. (a) P-AD curve. (b) P-AL curve. (c} Centred crack pro~~tion
in the neck section.
centred crack initiated, the reduced rates of the notched specimen’s P-AD and P-AL curves are less than those of the smooth specimen. This means that the resistance to crack propagation in a notched specimen is larger than that in a smooth specimen, because in the whole neck section of the smooth specimen the level of co~binato~ work density is very high, and the crack is very easy to propagate once it has initiated. Figure 6(c) shows the simulative extensions of the centred crack at the neck section in a notched specimen during a short time period. 5. DISWSSION The rupture processes of a round smooth tensile specimen and a round notched tensile specimen are neglected when research into the fracture problems of a material is carried out. After the fracture mechanics have been discussed, attention is focussed on the fracture of the crack which exists in the structure, and the interest in crack initiation in the non-cracked body is decreased. In fact, there is some relation between the two kinds of fracture. In order to realize this relation, it is necessary to further investigate the whole process of damage and fracture of materials based on the mechanism which is closer to the actual physical process. The investigation of the whole processes of deformation and fracture for the round smooth specimen and the round notched specimen is important in two ways: (1) there must be some relation in essence between the process in a non-cracked body with that in a cracked body, and a mechanical model, if it reflects the physical essence of a material’s fracture, should be verified in such a simple specimen as smooth or notched; (2) this research is wo~hw~le for en~n~~ng practice, because the failure in the round smooth tensile specimen and that in the round notched tensile specimen also represent the fracture behavioui of some engineering structures. In this paper, by using the combinatoj work density model combined with the large elastic-plastic deformation finite element method, a computer simulation representing the
Analysis of large defo~ation
and fracture of axisymmettic tensile specimens
857
experimental processes of deformation and fracture for two kinds of specimen can be made. These results show that the combinatory work density model is superior in the study of crack initiation and propagation for a non-cracked body. Considering ref. [12). it is known that this model is not only suitable for a cracked body but also for a non-cracked body. This gives it an advantage over the theories of the J-integraf and COD, which are in common use.
6. CONCLUSION According to the research of this paper on the combinatory work density model, the following conclusions can be obtained. (1) The numerical simulation for the whole process of crack initiation and propagation in non-cracked tensile specimens can be carried out using this model. (2) Since this model is not only suitable for the analysis of fracture initiation of a cracked specimen[l2], but also for crack initiation and propagation of a non-cracked specimen, it may be more useful in future applications than the J-integral and COD methods. REFERENCES F. A. McClintock, A criterion for ductile fracture by the growth of holes. J. uppl. Mech. 36, 363-371 (1968). J. R. Rice and D. M. Tracev. On the ductile enlargement of voids in triaxial stress fields. J. Me& Phvs. Soli& 17, 201-217 (1969). *’ R. D. Thomson and J. W. Hancock, Ductile fracture by void nucleation, growth and coalescence. Inr. J. Fruclure 26,99-l 12 (1984). F. M. Reremin, Experimental and numerical study of the different stages of ductile rupture: application to crack initiation and stable crack growth, in Three-dimensional Constitutive Reiations and Ductile Fracture, pp 185-205. North-Holland, Amsterdam (198 I). A. S. Argon, 1. Im and R. Safoglu, Cavity formation from inclusions in ductile fracture. Melall. Trans. 6A, 839-851 f19751. .-‘-I.
C. Q. Zheng and J. C. Radon, The formation of voids in the ductile fracture of low alloy steel. Proc. ht. Symp. on Fracture Mechanics. Reiiina. on. 1052-1056. Science Press (1983). - -.__ A. L. Gurson, ~n~inu~ theory of ductile rupture by void nucleation and growth Part I-Yield criteria and flow rules for porous ductile media. J. Engng Muter. Technol. 99, 2-15 (1977). V. Tvergaard and A. Needleman, Analysis of cup-cone fracture in round tensile bar. Actu Mefall. 32, 157-169 (1984). K. S. Zhang and C. Q. Zheng, Microvoid multistage nucleating model with the correlative constitutive equation, in On the Research with Meso-mechanics of Ductile Fracture and Its Applications (Edited by C. Q. Zheng), pp. 84-98. NPU Press (1988). K. S. Zhang, On the studies of micro-mechanics for ductile damage and fracture. Thesis of NPU for Eng. D. (1988). K. S. Zhang and C. Q. Zheng, Microvoid m&i&age nucleating model and its apphcation in analyses of micro damage and fracture. Engng Fracture Mech. 37, 621-629 (1990). K. S. Zhang, C. Q. Zheng and N. S. Yang, A new model for predicting fracture of ductile material. Proc. fCM-6. K. S. Zhang, R. J. Ma, L. S. Hua and C. Q. Zheng, On the analysis of axisymmetric tensile specimen using large elastic-plastic deformation finite element method. NPV J. 6, 331-341 (1988). X. Ji, J. J. Yin and Q. Tang, Analysis on necking by using finite element method. Acra Mech. Sol. 4, 532-542 (1983). K. Wang, Analysis on the influence of temperature and stress state on ductilebrittle transition. Thesis of NPU for Eng. M. (1990). (Recejved 3 July 1990)
ERd
W-E
Engineering Frac~re Mechanics Vol. 39, No. 5, pp. 851457, 1991 Printed in Great Britain.
ANALYSIS OF LARGE DEFORMA~ON FRACTURE OF AXISYMMETRIC TENSILE SPECIMENS
AND
K. S. ZHANG and C. Q. ZHENG Box 350, Department of Applied Mechanics, Northwestern Polytechnical University, Xian, Shaanxi 710072, P.R.C. Abstract-For a reasonable rupture criterion, if it is able to @ve expression to the mechanism of a material’s damage and fracture, it should not only be suitable to describe the processes of fracture initiation and crack propagation in a cracked body, but also suitable to describe that of crack initiation and crack propagation in a non-cracked body. From this viewpoint, a damage and fracture model for ductile material has been suggested based on the analysis of microscopic mechanics by the authors of this paper [K. S. Zhang, Eng. D. Thesis (1988); K. S. Zhang and C. Q. Zheng, On the Research with Meso-mechtmics of Ductile Fracture and Its Applications, pp. 84-98 (1988); Zngng Frucrure Mech. ?7,621-629 (1990)]. Gn the basis of this model, a new model called the combinatory work density model has been developed [K, S. Zhang, C. Q. Zheng and N. S. Yang, Proc. IiX4-6j. in the present paper, this new model is adopted to describe the damage and fracture of material, and to carry out the computer emotion of the processes of deformation, crack initiation and crack propagation for a round smooth tensile specimen and a round notched tensile specimen under conditions which are closer to the actual case, by using the large elastic-plastic deformation finite element method. The simulated results represent the experimental processes fairly reasonably, and are in very good agreement with test results. This further con8rms the reasonableness and correctness of the ~mbinato~ work density model.
1. I~RODU~ON IN ORDERto develop a mechanical model for describing crack initiation and its propagation in ductile materials based on the micro-mechanism of damage and fracture, a number of studies have been carried out[l-81. The ojbective is to find, by research on ~cro-m~hanics, the failure criterion that is closest to the physical essence of material failure and which can be used to describe and predict the damage and fracture of engineering elastic-plastic material. So far, most research considers that there are two very different problems of failure in a non-cracked body and in a cracked body, so they are more often than not judged by using different failure criteria. However, from the microscopic viewpoint, for a very small material unit, its failure criterion should be the same if its failure mechanism is the same, no matter where the unit is located. The failure ocurring is only dependent on the material’s resistence to rupture, the deformation degree, the deformation history and the stress-state (for the conditions of constant temperature and monotonic quasi-static loading). On this basis, the “microvoid multistage nucleating model” has been suggested according to the analysis on micro-m~hanism[9-111. By using this model, combined with the relative voidcontained material’s constitutive equation, the ductile damage and fracture can be reasonably described and predicted for elastic-plastic engineering material. Subsequently, a new model called the “combinatory work density model” was obtained[l2] from the improvement of the “microvoid multis~ge nucleating model” by the author. In the new model, the Prandtl-Reuss constitutive equation is adopted instead of the void-contained material’s equation; thus it is very convenient for application in practice[ 121. In ref. [12J, computer simulation analyses on the processes of deformation and rupture for a series of a~s~et~c specimens made of 40 Cr steel have been carried out by using the large elastic-plastic deformation finite element method. From this the parameters of the combinatory work density model for the material have been determined and the model’s reasonableness and correctness has been confirmed. Furthermore, with the help of the new model, the simulation and prediction of the processes of deformation and fracture initiation (including small crack propagation) of TPB specimens of 40 Cr steel have been successfully carried out. It has also been 851
852
K. S. ZHANG and C. Q. ZHENG
verified that the model is not only suitable for the failure analysis of a non-cracked body but also for a cracked body. In the present paper, the combinatory work density model, combined with the large elastic-plastic deformation finite element method, is adopted to describe a material’s damage and fracture. Computer simulation of the processes of deformation (necking), crack initiation and propagation of a round smooth tensile specimen and a round notched tentile specimen made of 40 Cr steel has also been carried out. All the results obtained are in good agreement with experimental data which were determined separately, and they can represent the actual processes in experiments fairly well. So, the reasonableness and correctness of the new model are confirmed. 2. MODEL OF FRACTURE MECHANICS The ~mbinato~ work density modelf 121is adopted in this paper. This model is of microscopic physical meaning, but takes advantage of the Prandtl-Reuss equation instead of the void-contained material equation. Since the determination of the model’s parameters do not rely upon microscopic tests, it is very convenient in application. The material failure criterion of the ~ombinato~ work density model is that the rupture occurs in a material unit if the combinatory work density W of the unit reaches a certain critical value. That is, w= w,
(1)
where W, is the fracture combinatory work density, which is regarded as a material constant; the combinatory work density parameter W can be determined by W=
Ws+FpWvp+FEWv~
(2)
where
, WIT=
=kk dkkdt s 0
(5)
and Wl,, WV, and WV, are plastic shape change work density, reference plastic volume change work density and reference elastic volume change work density respectively. They refer respectively to the degree of the plastic shape change, the development of microvoids and the elastic volume variation of a material unit. The coefficients Fp and FE are the plastic combinatory ~effi~ent and the elastic combinatory coefficient respectively; Fp reflects the void expanding ability in a material and FE reflects the influence of elastic volume dilatation on the material’s damage and fracture. These two coefficients can be calibrated by combining tests with computer simulation. ai is the deviatoric part of the Cauthy stress tensor a#; D$ is the plastic part of the rate of deformation tensor D,,; E is defined by
In the case of ductile fracture, plastic deformation and microvoid development (ductile damage) are considered as the main reasons of the material failure, and the influence of elastic volume dilatation on the material damage and fracture is considered to be negligible. That is, eq. (1) can be rewritten as W= Thus,
W,-kF,W,,.
the method of analysis can be simplified further.
(6)
853
Analysis of large deformation and fracture of axisymmetric tensile specimens Table 1, The chemical composition of 40 Cr steel (wt%) C
Si
Mn
P
s
CU
Ni
Cr
Ti
Al
W
0.41
0.31
0.64
0.024
0.01
0.09
<0.08
0.90
0.055
0.020
Table 2. The mechanical properties of 40Cr steel at room temperature co., (MPa)
a, (MPa)
6, (%)
$ (X)
E (GPa)
930
1024
16.9
57.8
205
3. MATERIAL
AND SPECIMENS
The chemical composition of 40 Cr steel and its mechanical properties at room temperature are shown in Tables 1 and 2 respectively. Figure 1 shows the true stress-logarithmic strain curve which has been modified by Bridgman’s formulas[l3]. Figure 2 shows the geometry and size of the round smooth specimen and the round notched specimen. According to ref. [12], the parameters FP and W, of the combinatory work density model for 40 Cr steel are:
FP = 0.973,
W, = 1448 MPa.
4. NUMERICAL
ANALYSIS
The numerical computations are carried out by using the large elastic-plastic deformation finite element program which is based on the updated Lagrangian description and designed by the author of this paper[l3]. All computations in this paper were executed by an IBM-4381 computer. 4.1. Analysis of the deformation and fracture of a round smooth specimen By taking account of the symmetry, the finite element mesh of the computed model can be divided as in Fig. 3. It consists of 126 eight-node quadrilateral isoparametric elements and 443 nodes. The computer simulation assumes loading is by controlled displacement, and the loading position is the same as that of the actual specimen (cf. Fig. 3).
1 0
I
I
I
0.2
0.4
0.6
I 0.6
6
Fig. I. The stress-strain
curve of 40Cr steel.
Fig. 2. The dimensions of the round smooth specimen and the round notched specimen.
854
K. S. WANG
and C. Q. ZHENG
Fig. 3. The finite element mesh of the round smooth specimen.
4.1.1. Method forsimulation ofnecking. In general, necking will take place in a round smooth tensile specimen of a ductile material during large elastic-plastic deformation, and it will lead to the deformation of the specimen in the neck region and finally to rupture at the neck section. Since necking has a large influence on the specimen’s deformation and fracture, the computer simulation of the necking process is very important. Commonly, the analysis of necking by using the finite element method used to only take the specimen’s smooth part into account[l3,14]. In this way the assumption of an initial defect (on the surface or on the inside of the specimen) or a rigid displacement condition at the end of the smooth part will be taken, so that necking can be represented during computer simulation. Both methods are somewhat ind~te~inable and arbitrary. On the one hand, the result will lead to evident error when the various defect assumption is taken, and no one knows what defect is proper. On the other hand, applying rigid displacement conditions to the end of the specimen’s smooth section will lead to a greater difference from the real case, expecially for a material which has a larger uniform deformation before necking occurs. In the present paper, in order to be closer to the actual case, the whole specimen is taken into consideration, and neither the initial defect nor the rigid displacement condition are employed. 4.12. method for s~uiatio~s of crack ~~t~io~ and propagation. For the processes of crack initiation and propagation in the specimen’s neck section, the computer simulation is carried out by release of the correlative node’s boundary conditions: (1) for crack initiation, the condition to node release is that the mean combinatory work density W of the nearest two Gauss points located at the two sides of the node amounts to the fracture ~ombinato~ work density W,; (2) for crack propagation, the condition to release the node at the crack tip is that the combinatory work density W of the Gauss point, which is the nearest to that in front of the crack tip, reaches W,; (3) when a node is released, a force which is equal to the node’s axial force but in the opposite direction will be applied to the node by the program’s equilib~um iteration. 4.1.3. Numerical results. According to the computer simulation for the round smooth specimen, the following results can be obtained. The relation of the applied load P with the variation AD of neck redius D and the relation of P with the axial extension AL of the section L are shown in Fig. 4(a, b). They are in good agreement with the test results[lSj. From these, it is known that the computer simulation of the tensile process of the round smooth specimen represents the necking process in very good agreement with the real process, although the defect assumption and the rigid displacement conditions are not taken into account at all. So, the method adopted in this paper can be considered to be more reasonable for necking analysis than the above mentioned two methods, because this method does not involve the indeterminable and arbitrary fact into the computation. The P-AD and P-AL curves (cf. Fig. 4a and b) after the crack initiates in the neck section can also be obtained by computer simulation; they represent the actual fracture process after the crack initiates. Figure 4(c) shows the simulated propagations of the centred crack at the neck section in a short time, which are very difficult to observe by experiment. It is necessary to point out that during the last stage of fast crack propagation there is an error in the numerical results, since the influence of the viscoplasticity and the dynamic response are not taken into account. 4.2. Analyst of the ~formation and fract~e of a round notched spe~~en Since there is a strong stress concentration near the notch of the notched specimen, the influence of the part of the specimen held by machine on the stress distribution near the notch region can be neglected. By taking account of the symmetry, the finite element mesh of the
855
Analysis of large deformation and fracture of axisymmetric tensile specimens (b)
(a)
0
m
I
4.0
1.0
0
AD(mm)
I
I
6.0
a0
AL (mm)
ci !
-
I
40
I
2.0
c
Ao 2.4mm-Aa
3.0mm
(cl
Fig. 4. (a) P-AD curve. (b) P-AL curve. (c) Centred crack propagation in the neck section.
computed model can be divided as shown in Fig. 5. It consists of 74 eight-node quadrilateral isoparametric elements and 265 nodes. The computer simulation still assumes that loading is by controlled displacement. The method of simulating crack initation and propagation of the specimen is just the same as in Section 4.1. In Fig. 6(a, b), the relations of P-AD and P-AL of the notched specimen by computer simulation are presented; they are also in very good agreement with experimental curves. From this figure, it can be seen that with increasing specimen deformation, the outer loading is reduced rapidly after the centred crack has initiated. As in the smooth specimen, this is the reason that the crack propagates at high speed, so that the bearing area in the neck section is reduced very quickly. This appearance is very difficult to record by test machine since the process takes place in such a very short time. Compared with the case of the smooth specimen after the l/2
L
Fig. 5. The finite element mesh of the round notched specimen.
856
IL S. ZHANG and C. Q. ZHENG
0
I
0.2
I
I
I
I
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0.6
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1.0
I 1.2
AD (mm}
AL(mm) (Cl
(bl Fig. 6. (a) P-AD curve. (b) P-AL curve. (c} Centred crack pro~~tion
in the neck section.
centred crack initiated, the reduced rates of the notched specimen’s P-AD and P-AL curves are less than those of the smooth specimen. This means that the resistance to crack propagation in a notched specimen is larger than that in a smooth specimen, because in the whole neck section of the smooth specimen the level of co~binato~ work density is very high, and the crack is very easy to propagate once it has initiated. Figure 6(c) shows the simulative extensions of the centred crack at the neck section in a notched specimen during a short time period. 5. DISWSSION The rupture processes of a round smooth tensile specimen and a round notched tensile specimen are neglected when research into the fracture problems of a material is carried out. After the fracture mechanics have been discussed, attention is focussed on the fracture of the crack which exists in the structure, and the interest in crack initiation in the non-cracked body is decreased. In fact, there is some relation between the two kinds of fracture. In order to realize this relation, it is necessary to further investigate the whole process of damage and fracture of materials based on the mechanism which is closer to the actual physical process. The investigation of the whole processes of deformation and fracture for the round smooth specimen and the round notched specimen is important in two ways: (1) there must be some relation in essence between the process in a non-cracked body with that in a cracked body, and a mechanical model, if it reflects the physical essence of a material’s fracture, should be verified in such a simple specimen as smooth or notched; (2) this research is wo~hw~le for en~n~~ng practice, because the failure in the round smooth tensile specimen and that in the round notched tensile specimen also represent the fracture behavioui of some engineering structures. In this paper, by using the combinatoj work density model combined with the large elastic-plastic deformation finite element method, a computer simulation representing the
Analysis of large defo~ation
and fracture of axisymmettic tensile specimens
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experimental processes of deformation and fracture for two kinds of specimen can be made. These results show that the combinatory work density model is superior in the study of crack initiation and propagation for a non-cracked body. Considering ref. [12). it is known that this model is not only suitable for a cracked body but also for a non-cracked body. This gives it an advantage over the theories of the J-integraf and COD, which are in common use.
6. CONCLUSION According to the research of this paper on the combinatory work density model, the following conclusions can be obtained. (1) The numerical simulation for the whole process of crack initiation and propagation in non-cracked tensile specimens can be carried out using this model. (2) Since this model is not only suitable for the analysis of fracture initiation of a cracked specimen[l2], but also for crack initiation and propagation of a non-cracked specimen, it may be more useful in future applications than the J-integral and COD methods. REFERENCES F. A. McClintock, A criterion for ductile fracture by the growth of holes. J. uppl. Mech. 36, 363-371 (1968). J. R. Rice and D. M. Tracev. On the ductile enlargement of voids in triaxial stress fields. J. Me& Phvs. Soli& 17, 201-217 (1969). *’ R. D. Thomson and J. W. Hancock, Ductile fracture by void nucleation, growth and coalescence. Inr. J. Fruclure 26,99-l 12 (1984). F. M. Reremin, Experimental and numerical study of the different stages of ductile rupture: application to crack initiation and stable crack growth, in Three-dimensional Constitutive Reiations and Ductile Fracture, pp 185-205. North-Holland, Amsterdam (198 I). A. S. Argon, 1. Im and R. Safoglu, Cavity formation from inclusions in ductile fracture. Melall. Trans. 6A, 839-851 f19751. .-‘-I.
C. Q. Zheng and J. C. Radon, The formation of voids in the ductile fracture of low alloy steel. Proc. ht. Symp. on Fracture Mechanics. Reiiina. on. 1052-1056. Science Press (1983). - -.__ A. L. Gurson, ~n~inu~ theory of ductile rupture by void nucleation and growth Part I-Yield criteria and flow rules for porous ductile media. J. Engng Muter. Technol. 99, 2-15 (1977). V. Tvergaard and A. Needleman, Analysis of cup-cone fracture in round tensile bar. Actu Mefall. 32, 157-169 (1984). K. S. Zhang and C. Q. Zheng, Microvoid multistage nucleating model with the correlative constitutive equation, in On the Research with Meso-mechanics of Ductile Fracture and Its Applications (Edited by C. Q. Zheng), pp. 84-98. NPU Press (1988). K. S. Zhang, On the studies of micro-mechanics for ductile damage and fracture. Thesis of NPU for Eng. D. (1988). K. S. Zhang and C. Q. Zheng, Microvoid m&i&age nucleating model and its apphcation in analyses of micro damage and fracture. Engng Fracture Mech. 37, 621-629 (1990). K. S. Zhang, C. Q. Zheng and N. S. Yang, A new model for predicting fracture of ductile material. Proc. fCM-6. K. S. Zhang, R. J. Ma, L. S. Hua and C. Q. Zheng, On the analysis of axisymmetric tensile specimen using large elastic-plastic deformation finite element method. NPV J. 6, 331-341 (1988). X. Ji, J. J. Yin and Q. Tang, Analysis on necking by using finite element method. Acra Mech. Sol. 4, 532-542 (1983). K. Wang, Analysis on the influence of temperature and stress state on ductilebrittle transition. Thesis of NPU for Eng. M. (1990). (Recejved 3 July 1990)
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