EaglneerLtg Fracture Mechmics Vol. 49, No. 2, pp. 23S241, 1994

0013-7944(94)E0107-R copyright Q 1994 Elscvier science Ltd

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NUMERICAL ANALYSIS OF THE STRESS-STRAIN CURVE AND FRACTURE INITIATION FOR DUCTILE MATERIAL

K. S. ZHANG and Z. H. Ll

P.O. Box 169, Department of Applied Mechanics, Northwestern Polytechnical University, Kin, Shaanxi 710072, P.R.C.

Ahatraet-In the present paper, a way to obtain an accurate stress-strain curve of a ductile material under large plastic deformation (include necking) by tracing the experimental tensile loading-axial displacement curve using computer simulation is suggested. According to this method, the tested load (P)-deformation (AL) curve of the tensile specimen is considered as a target, and the true stress-strain curve of the material will be obtained when the simulated curve P* - AL* is completed using a large elastic-plastic deformation finite element program, by adjusting the trace of the stress-strain curve step by step according to the relative error of the simulated curve P* -AL* with the test one P-AL, under the condition that the relative error is controlled less than a given limit value.

As a verification, in this paper, the method is applied to analyse the large deformation necking process of round smooth tensile specimen made of No. 45 steel. Furthermore, the Combinatorial Work Density Model [K. S. Zhang, C. Q. Zheng and N. S. Yang, Proc. KM-6; K. S. Zhang and C. Q. Zheng. Engng Fracfure Mech. 39,851L857 (1991); Engng Fracture Mech. 39, 859-865 (1991)] is introduced to judge the failure of a round notched tensile specimen made of the same material.

1. INTRODUCTION

UNDER QUASI-STATIC DEFORMATION, the mode I fracture initiation and propagation in an elastic-plastic material can be considered as the results of void nucleation, growth and coalescence. For these micro-processes in materials usually taking place during large elastic-plastic deformation, many researchers managed to find the law of damage and the criteria of fracture of ductile materials from tensile tests, especially from that applied to a round smooth bar as it is characteristic of mode I fracture and can be carried out conveniently. However, it is not so simple as it could be, because only for a very short stage in the total tensile process is the specimen deformed in a uniform way. Necking usually cant be avoided for ductile materials under tension; this will form a neck region with a complex stress-strain state. Thus, there is a difficulty in determining the Q---E curve of material, when people try to solve the problem of damage of material.

Up to now, a lot of research about necking phenomenon has been carried out [l-7), but many problems are still open. Researchers usually obtain the mean stress-strain curve by measuring the change of radius of a tensile specimen; based on that they then get the modified true stress-logarithmic strain curve by applying Bridgmans formulas [l]. But this method isnt convenient because (1) the position of the neck is random and it is very difficult to determine it beforehand; (2) the neck region is usually smooth, so the radial displacement gauge isnt able to be positioned exactly and (3) it needs a series of tests with different loadings to measure the longitudinal radius of curvature of the smallest cross-section in the neck region. In short, it is very difficult to measure it exactly.

To solve these problems, in this paper, a method for searching out a accurate stress-strain curve by using large elastic-plastic deformation finite element computer simulation to trace the experimental tensile loading-axial displacement curve P-AL is suggested. As an example, the stress-strain curve of No. 45 steel has been obtained by this method (this method has been added in the program ALEPDF which was programmed by the first author of this paper). On the basis of this, the processes of deformation and failure of a round notched specimen made of the same material has been predicted exactly by using the program ALEPDF combined with the Combina- torial Work Density Model [g-l 11.

235

236 K. S. ZHANG and Z. H. LI

2. PLASTIC CONSTITUTIVE EQUATION

The Prandtl-Ruesss constitutive equation can be regarded as reasonably describing the stress-strain relation of isotropic metallic material, if the material doesnt yield reversely caused by reverse loading and doesnt deviate from simple loading very seriously.

In the case of large deformation the Prandtl-Ruesss constitutive equation can be written as:

where D, is the rate of deformation tensor, and D, = i (doi/dxj + ~u,/~x,), Vi is the rate of material point, xi is the instantaneous coordinates of material point; a; is the Jaumann rate of Cauchy stress ail; E and v are the Youngs modulus and Poisson rate, respectively; 6, is the Kronecker delta; h is the strain hardening parameter; ah is the deviatoric stress tensor defined by a; = aii - f&a,,; ae is Mises effective stress defined by a, = (i abai) I*. 1 is loading coffection; it is determined by the , loading criteria:

) _ 0 .-

a;DiiO

The inverse form of eqn (1) can be written as:

E a?=-

a,ia;, rl l+v

6*6,,+&s,dk,-'~

2 1+v &I. -af l+gh----- 3 ( )I E

(2)

(3)

3. THE METHOD OF NUMERICAL CALCULATION

To judge the accuracy of a stress-strain curve, one can apply the large elastic-plastic deformation finite element method: this curve can be considered as accurate, on the condition that the simulated deformation and necking vs the varying of load by computation according to this stress-strain curve can represent the experimental process of the smooth tensile specimen under loading reasonably and accurately [7]. Thus it can be regarded that the experimental results can be taken as a target which can be approached by simulation using large deformation finite element calculation according to a stress-strain curve searched forward step by step; this stress-strain curve will be regarded as a true curve if the results approached by computer are in strict agreement with the actual processes.

On the basis of that, the authors of this paper suggest a method which traces the experimental curve of applied load P with the axial extension AL (AL = L - Lo, where L and & are the instantaneous gauge length and the original gauge length of the specimen, respectively) to find out the stress-strain curve of ductile materials. The adopted procedure is written as follows.

(1) Take the stress-strain curve before necking (it can be obtained from the experimental P-AL curve as the deformation before necking is uniform) as the initial effective curve, and divide it into a certain number of smaller stages to approximate the real curve according to the requirement of precision.

(2) Let E, represent the end of the uniform stress-strain curve which can be used to describe the deformation of the specimen before necking, and make an arithmetic series s,,, E,, c2. . . according to the given interval BE (As can be chosen between 0.01 and 0.02).

(3) Calculate a certain number of increment steps according to the uniform stress-strain curve, and obtain the P*-AL* curve before the largest load comes. (P is obtained by integral methods, and the error between the P*-AL* and the experimental curve P-AL is controlled less than 3%).

(4) When the effective strain E, in the specimen by computation equals to or is just greater than (E, _ , + s,)/2, calculate the relative error 6 = (P+ - P)/P by comparing the simulated P+ (AL*) with the tested P (AL*).

(5) Modify the forward trend of the stress-strain curve according to the error 6. (6) Repeat the steps 4 and 5 until AL* reaches the average value of experimental fracture

displacement ALr of specimens.

Numerical analysis of the stress-strain curve

Table I. The chemical compositions of 45 steel (wt%,)

237

C Si Mn P s Cr

0.50 0.25 0.58 0.013 0.027 0.13

Table 2. The mechanical properties of 45 steel at room temperature

6, (MPa) ob (MW 4 WI + WI E (GPa) Y

680 780 24 65 206 0.31

This method mentioned above has the advantage that the progress of tracing the true stress-strain curve during post necking of the round specimen under tension is program-controlled, so it is able to avoid the artificial random factor by manual operation as much as possible.

The numerical computation in this paper is carried out by using the large elastic-plastic deformation finite element program ALEPDF which is programmed on the basis of the updated Lagrangian description. Its reliability has been verified by experiments [7-l 11. All the computations in this paper were executed on an IBM-4381 computer.

4. MATERIAL AND SPECIMENS

The specimens are made of No. 45 steel, and its main chemical constituents and mechanical properties at room temperature are shown in Tables 1 and 2, respectively.

Figure 1 shows the experimental curve of the load P vs displacement AL of the round smooth tensile specimen [ 121.

Figure 2 shows the dimensions of round smooth specimen and notched specimen. All experiments were carried out on an Instron 1196; the loading speed was 0.5 mm/min.

5. NUMERICAL ANALYSIS

5.1. Analysis on the round smooth specimen

By taking account of the symmetry, only one-quarter of the axial section is taken, and the mesh for the round smooth specimen consists of 110 eight-node quadrilateral isoparametric elements and 385 nodes.

To simulate the experiment, the load is applied by controlling displacement, and the loading position is just the same as that of the actual specimen.

From the computer simulation the following results are obtained: (1) The stress-strain curve. By tracing the experimental P-AL curve applying the program

ALEPDF, the stress-strain curve of No. 45 steel is obtained (see Fig. 3), where 0 is the Misess effective stress, E is the effective strain and E = Jr, i: dt.

0 2.5 5.0 7.J 10.0 12.5

AL (mm)

Fig. I. The experimental curve of the tension load vs axial displacement.

238 K. S. ZHANG and Z. H. LI

Fig. 2. The dimensions of the round smooth specimen and notched specimen.

As a comparison, Fig. 3 also shows the nominal stress-strain curve obtained by experiment and the curve modified by Bridgmans method on the basis of the nominal curve. It can be seen that the stress-strain curve obtained by the method suggested in this paper is very close to the result according to Bridgmans formulas.

(2) P*-AL* tune and P*-AD* cume. Figure 4 shows the simulated P*-AL* curve. Comparing it with the experimental P-AL curve, the error of the simulated one is very small (less than 5%), so it can be regarded that the simulated result is fairly accurate. Furthermore, the experimental P-AL curve has a obvious yield platform (see Fig. 4); it is very difficult for computer simulation. However, in this paper, the simulation has been performed very well by using ALEPDF.

The relationship of P* with AD* and the experimental P-AD curve are all shown in Fig. 5, and the difference between them is less than 10%. Considering the test AD isnt very accurate, since the testing point cant be positioned exactly, the P*-AD* curve can be considered as accurate and reliable.

(3) The distribution of the parameters of mechanics on the neck region. The deformation and stress state in the gauge length of the specimen is uniform and uniaxial before the largest load comes. With necking initiating and developing, a neck will be formed on the specimen and the stress state on the neck region will redistribute, thus the stress state will be no longer uniaxial and uniform, because the deformation concentrates rapidly. Figure 6 shows the shape of the neck region simulated by computer when the specimen is going to fracture, and Fig. 7 shows the distribution of the strain, stress and combinatorial work density [8] across the neck section obtained by calculation.

400 - This paper

200 - I I I I I I

0 0.2 0.4 0.6 0.8 1.0 1.2

E

$ 20000 a

15.000 - Experiment

- Simulation 10,000

I I I I I I 0 2.5 5.0 7.5 10.0 12.5

AL (mm)

Fig. 3. The stress-strain curve. Fig. 4. The curves of tension load vs axial displacement.

Numerical analysis of the stress-strain curve 239

a =:I I, , 0 1.0 2.0 3.0 4.0

AD (mm) Fig. 6. The simulated shape of the smooth specimen before

Fig. 5. The curves of tension load vs radial displacement. fracture.

In accordance with the results analysed for the material 400 [9, IO] and No. 20 steel [I I], the plastic combinatorial coefficients Fp for the two materials are 0.973 and 1.026, respectively. The difference of the Fp between the two different materials is so small that it seems that the coefficient Fp may be regarded as a constant for some ductile materials (for example for the materials which would neck deeply before fracture under tension). Taking this assumption, the value of Fp can be considered as 1. Then, the value of the fracture combinatorial work density W, can be determined, and W, = 1420 MPa.

To verify whether this assumption mentioned above is reasonable or not, the following analysis has been carried out.

1.2

1.0 w

. 0.8 6 ; 0.6

$ 0.4

lJa 0.2

0.2 0.4 0.6 0.8 1.0 0 w

0.2 0.4 0.6 0.8 1.0

r/R r/R

Fig. 7. The distribution of the mechanical parameters across the neck

1600 ,-

0 0.2 0.4 0.6 0.8 1.0

r/R

section obtained by simulation.

240 K. S. ZHANG and Z. H. LI

1600 -

18,000 1400

16.000

1200 14,000 2

z 12.000 - 1000 -

h .Z

f 10.000 a 800 -

z 8000 8 - - Experiment Y

600 k

0.4 0.6 0.8 1.0

0 0.2 0.4 0.6 0.8 r/R

Fig. 8. The failure prediction for the notched specimen.

5.2. Analysis on the round notched specimen

AD (mm) Fig. 9. The distribution of the combinatorial work density

on the neck cross-section.

For the round notched specimen, by taking account of the symmetry, only one-quarter of the axial section in the gauge length is taken, and the finite element mesh consists of 79 eight-node quadrilateral isoparametric elements and 282 nodes.

The load is also applied by controlling displacement to simulate the real process. According to the computation applied, the criterion of fracture combinatorial work density

(the value of AD when fracture takes place) is 0.64 mm; the error by comparison with the test one (AD = 0.62 mm) is less than 5% (see Fig. 8) . Meanwhile the error of the AL between the simulated value (0.52 mm) with test one (0.50 mm) is also less 5%. So it seems the assumption is reasonable that and the coefficient Fp may be a constant for the ductile materials which will neck deeply before fracture for the smooth bar under axial tension.

Figure 9 shows the radial distribution of the combinatorial work density Won the neck cross- section. It shows that with the tension increasing, the crack will initate in the centre of the neck section and then propagate in radial direction.

6. CONCLUSION AND DISCUSSION

In this paper, a method is suggested to search out the true stress-strain curve by tracking the experimental tensile load P-axial displacement AL curve of a round smooth bar, using large elastic-plastic deformation finite element computer simulation.

This method has some advantages:

(1) It is only dependent on the experimental curve of tensile load P with axial displacement AL of a round smooth bar, thus the operation of measuring the radial change of the neck of a specimen during the whole process of tension is no longer necessary.

(2) Because the method is program-controlled, it is able to avoid the artificial random factor by manual operation as much as possible.

(3) It is more convenient in practice than Bridgmans method, and is of similar precision.

Furthermore, in the present paper, it is suggested that the coefficient Fp in the combinatorial work density model may be a constant for the ductile materials which will neck deeply before fracture under uniaxial tension. This assumption seems to be reasonable according to the verification of this paper that the reasonable fracture prediction for the notched specimen under tensile loading.

Acknowledgemenr-The authors wish to express their gratitude to Prof. Yang Nan-Sheng and Prof. Zheng Chang-Qing of the Northwestern Polytechnical University for their continual encouragement and enlightening advice throughout this work.

Numerical analysis of the stress-strain curve 241

REFERENCES

[I] P. W. Bridgman. Studies in Large Plastic How and Fracture. Chapters I and 2 (1952). [2] C. Chen. The stress analysis in the neck of a tensile specimen. Collected Papers on Researching Fracture of Me&. _

Metallurgical Industrial Press, Beijing, 169-189 (1978) [in Chinese]. 131 W. H. Chen. Neckina of a bar. fnr. 1. Solids Srrucrures 7. 685-717 (1971). i4j M. Sajc, Necking ofa cylindrical bar in tension. Inr. J. $oli&. Srru~rureslS, 731-742 (1979). [5] V. Tvergaard and A. Needleman, Analysis of the cup-cone fracturs in a round tensile bar. Acra Merall. 32, 157-169

(1984). [6] X. Ji, J. J. Yin and Q. Tang, Analysis of necking by using finite element method. Acre Mech. Sol. Sinicu 532-542 (1983)

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Conf. on Mechanical Behaviour of Materials 4, 263-268 (1991). [9] K. S. Zhang and C. Q. Zheng, Analysis of large deformation and fracture of axisymmetric tensile specimens. Engng

Fracture Mech. 39, 851-857 (1991). [IO] K. S. Zhang and C. Q. Zheng, Computer simulation of fracture initiation and crack propagation of TPB specimens.

Engng Fracture Mech. 39, 859-865 (1991). [I I] K. S. Zhang and C. Q. Zheng, Prediction of the fracture toughness of No. 20 steel-an application of the combinatorial

work density model. Proc. 2nd Symp. on Meso-Mechanics of China, 323-330 (1991) [in Chinese]. [I21 X. Y. Wang, Research on experimental measurement of void growth, Thesis of NPU for Eng. M.Sc. (1992).

(Receiued 20 July 1993)