Micromechanical damage modeling and simulation of punch test

ARTICLE IN PRESS Ocean Engineering 36 (2009) 1158–1163

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Micromechanical damage modeling and simulation of punch test Joonmo Choung  Department of Naval Architecture and Ocean Engineering, Inha University, Incheon, Korea

a r t i c l e in f o

a b s t r a c t

Article history: Received 16 January 2009 Accepted 6 August 2009 Available online 13 August 2009

In order to verify the validity and applicability of the porous plasticity model for the simulation of nonlocal plastic fractures, punch tests are carried out for round steel plates of JIS G3131 SPHC. Incremental tensile coupon tests for the material are also conducted to obtain the plastic mechanical properties. Material parameters required for the porous plasticity model are identified through parametric numerical simulations of the coupon tests. It is proved that that force vs. indentation curves from punch test simulations using the porous plasticity model with the identified material parameters show good agreement with punch test results. It is also verified that simulated fracture shapes in the punch specimens are almost similar to the experimental ones in which fracture occurs along the circumferential edge of indenter where the hydrostatic stress develops highly. & 2009 Elsevier Ltd. All rights reserved.

Keywords: Punch test Shear fracture model Porous plasticity model Hydrostatic stress Stress triaxiality Void volume fraction

1. Introduction It is essential to understand plastic deformation and fracture behaviors of structural steels for rational design based on quantitative damage estimation against accidental limit state (ALS) such as ship-to-ship collisions, ship-to-rock groundings, or explosions in FPSOs. Recalling that the fracture is the final stage of an irreversible plastic deformation process, exact flow stress including initial yield stress are required for large strain and fracture simulations. However, plastic hardening with zero or single slope provided from mill sheets is still frequently employed in many studies, which have been focused on structural rearrangements of bow, side or bottom structures to reduce damage extent under the collision or grounding forces. Most marine structural steels, such as classification steels for ship structures or API steels for offshore structures, are categorized as ductile materials for which plastic deformation process up to fracture typically shows three micromechanical damage stages: nucleation, growth and coalescence of voids. Nucleation of voids usually implies the debonding of inclusions or 2nd phase particles from the continuum, called the steel matrix. On the other hand, growth and coalescence of voids mean enlargement of nucleated voids and weakening of ligaments between enlarged voids, respectively. It is known that the hydrostatic stress in the Cauchy’s stress tensors governs the first two stages and the deviatoric stress the coalescence stage. At the structural locations with high geometric discontinuities such as crack tips or notches, hydro-

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static stress is highly developed during continuous straining. If, therefore, the plastic hardening of a material is assumed to be elastic–perfect–plastic (zero slope hardening) or elastic–linear– plastic (single slope hardening), the stress estimation in real structures can be more inaccurate as plastic strain increases. For this reason, exact plastic hardening data should be taken into account in the simulation of a large strain problem. In the classical shear fracture model based on the von Mises yield function, fracture occurs when the accumulated equivalent plastic strain reaches the designated fracture strain. The shear fracture criterion does not include the hydrostatic stress effect, but most studies have employed the shear fracture model as a fracture criterion due to the simplicity of implementation for numerical simulations. Various fracture strains are found in the published studies. The industrial standard (NORSOK, 2004) suggests 20 percent fracture strain for mild steels, which is extremely conservative considering that logarithmic true fracture strain exceeds 100 percent for most marine structural steels (Choung et al., 2007). Narr et al. (2001) used 70 percent fracture strain, which was obtained from tensile tests. Paik et al. (1999) employed more than 35 percent fracture strains, which was determined from elongation rate of the used material. It is noted that elongation rate has no physical meaning after plastic instability begins and logarithmic true strain should be used. Lehman and Peschmann (2001) tried to express fracture strain as a function of normalized plate thickness. One of the disadvantages of the shear fracture model is that the equivalent plastic strain can be accumulated under compressive stress, and fracture can occur. Urban (2003) and Tornqvist (2003) developed a new fracture criterion, called the RTCL model, by combining Rice–Tracey’s void growth model in tension and Cockcroft–Latham’s shear fracture model in compression. They

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Nomenclature

F dij

eijp en efp s0 seq sY sH su l f

yield potential Kronecker delta plastic strain tensor mean of strain distribution for void nucleation logarithmic true fracture strain initial yield stress von Mises equivalent stress yield stress hydrostatic stress tensile strength thickness–sharpness ratio (Non-dimensional relative curvature) void volume fraction

implemented the RTCL model in commercial explicit code LSDyna and verified the effectiveness of the RTCL model. Choung et al. (2007), who carried out extensive tensile and damage tests for the various marine structural steels, pointed out that the porous plasticity model was suitable for estimation of a plastic deformation process up to fracture of marine structural steels. They also suggested how the material parameters for the porous plasticity model can be identified from tensile tests and simulations. It is thought that punch tests are relatively simple to carry out, but are good to observe the plastic damage and fracture. However, most high-strength marine structural steels are produced in the form of thick plates, and as such are not appropriate for punch tests. In this research, mild steel of JIS G3131 SPHC is chosen for the punch test. In order to obtain the exact hardening data during the whole plastic deformation process, incremental tensile tests have been carried out for a pair of coupons machined from the steel. Material parameters for the porous plasticity model are identified from the comparison of coupon tests and parametric numerical analyses. Indenters with three different radii are used for the punch test. Punch test simulations using the porous plasticity model reveal that the porous plasticity model follows the punch test results well, if proper material constants are identified.

initial void volume fraction ultimate void volume fraction critical void volume fraction void volume fraction at failure volume fraction of nucleated voids total rate of void volume fraction growth rate of void volume fraction nucleation rate of void volume fraction effective void volume fraction f* standard deviation of strain distribution for void sn nucleation q1, q2, q3 material constants initial radius of curvature of striking structure Rstrike initial radius of curvature of struck structure Rstruck t thickness of plate f0 fu fc ff fn f˙ f˙g f˙n

influence of hydrostatic stress at all strain levels, by Tvergaard (1981), to the following form:

F¼



seq sY

2

þ2q1 f cosh



3 q2 sH 2 sY



 ð1 þ q3 f 2 Þ

ð1Þ

When void volume fraction is zero, Eq. (1) reduces to the von Mises yield function with isotropic hardening. As void volume fraction increases, a material is softened by the reduction of the yield surface. Tvergaard and Needleman (1984) modified Eq. (1) by replacing void volume fraction with an effective void volume fraction.

F¼



seq sY

2

þ2q1 f  cosh



3 q2 sH 2 sY



 ð1 þ q3 f 2 Þ

ð2Þ

Eq. (2) only explains growth of existing voids whereas coalescence and nucleation of new voids are neglected. Recalling that coalescence of voids accelerates softening of a material, effective void volume fraction is given by two steps (before and after reaching critical void volume fraction):

2. Theoretical background The theoretical background for micromechanical damage evolution was first established by McClintock (1968) who assumed pre-existing two dimensional array of hollow holes subjected to a remote stress field. In McClintock’s model, the voids (hollow holes) grow and change their shapes depending on the stress state. When the voids contact each other, fracture occurs between the voids. Rice and Tracey (1969) similarly assumed a pre-existing spherical void in a three dimensional matrix subjected to a remote stress field, and derived a model for void growth rate. In the Rice–Tracey model, when the void grows up to a critical size, fracture occurs between voids. Gurson (1977) originally represented plastic flow in a porous medium by assuming that a material behaves as a continuum. The voids appear indirectly through their influence on global flow behavior. The effect of voids is averaged through the cross section, which is assumed to be continuous and homogeneous. The principal difference between the Gurson model and the classical von Mises yield model is that the yield surface in the former contains a small hydrostatic stress dependence, while yielding is definitely independent of hydrostatic stress in the latter. The Gurson’s yield function was later modified to amplify the

1159

f ¼

8 f > > > > fu  fc > > f þ ðf  fc Þ > > < c ff  fc > > > > > > > > :

for f rfc for f 4fc q3

with fu ¼ q1 þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q21  q3 ð3Þ

When the void volume fraction reaches a critical void volume fraction, rapid loss of load carrying capacity of a material is expected. When the void volume fraction reaches a failure void volume fraction, fracture will occur. The total change in void volume fraction is given by Eq. (4) where f˙g and f˙n are porosity rate due to growth of existing voids and newly nucleated voids, respectively. f_ ¼ f_ g þ f_ n

ð4Þ

f_ g ¼ ð1  f Þe_ pij dij

ð5Þ

During progressive straining, the population of voids increases as a result of nucleation of new voids. This was addressed by Needleman and Tvergaard (1984) who suggested that the

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rate in newly nucleated voids was given in Eq. (6). Chu and Needleman (1980) proposed a parameter A for the straincontrolled nucleation process.

1200 3T-S1 Average True 3T-S1 Equivalent True 3T-S2 Average True 3T-S2 Equivalent True

ð6Þ 2

p fn 1 eeq  en A ¼ pffiffiffiffiffiffi EXP 4 sn 2 sn 2p

!2 3 5

ð7Þ

True Stress [MPa]

1000 f_ n ¼ Ae_ p

800

600

400

3. Punch tests 3.1. Material for specimen

200

Specimens are machined from a mild steel of JIS G3131 SPHC with 3 mm thickness. Table 1 and Table 2 show chemical compositions and mechanical properties from mill sheets, respectively.

0

0.4 0.8 Logarithmic True Strain

1.2

Fig. 2. True stress–logarithmic true strain curves determined from incremental tensile tests. (a) Punch specimen. (b) Jig. (c) Indenters.

3.2. Tensile tests Tensile coupons are designed based on ASTM (2004), as shown in Fig. 1. A pair of coupons is prepared regardless of rolling direction. Incremental tensile tests are carried out using 300 kN UTM with a controlled rate of 3.33  104/s. Measured average true stress data (hollow and solid circles in Fig. 2 where two repeated data for specimens 3T-S1 and 3T-S2 are shown) are transformed into equivalent true stresses data (solid and dashed lines in Fig. 2) used for material data for numerical simulation, using the stress correction formula suggested by Choung and Cho (2008). Plastic mechanical properties are summarized in Table 3.

Table 3 Measure plastic properties.

s0 (MPa)

su (MPa)

efp

266.135

537.221

1.066

indenters on fracture strength. Punch tests are carried out six times considering three indenters (Fig. 3(c)) and two punch specimens.

3.3. Specimens for punch tests

4. Numerical simulations

The scantlings of punch specimens are shown in Fig. 3(a), where the specimens are fixed by a strong jig and M12 bolts (Fig. 3(b)). Indenters with radii of 7.5, 15, and 30 mm are also machined in order to identify the influence of sharpness of

4.1. Determination of material constants for the porous plasticity model

Table 1 Chemical compositions of JIS G3131 SPHC (%). C 0.043

Si 0.005

Mn 0.22

P 0.012

S 0.009

Table 2 Typical mechanical properties of JIS G3131 SPHC.

s0 (MPa)

su (MPa)

Minimum Elongation (%)

275

366

38

Fig. 1. Scantlings of tensile coupon (mm).

The porous plasticity model requires many material constants, as shown in Eqs. (2)–(7). The following material constants should be identified: q1, q2, q3, f0, fc, ff, fn, en, and sn, which are alternatively determined through SEM analyses or parametric FE analyses. Recently, with the rapid development of computers, FE analyses are recommended. The values of q1, q2, and q3 for the general structural steels are found to be 1.5, 1.0 and q21 in many studies. It is also usually assumed that f0E0.0, en ¼ 0.3, and sn ¼ 0.1. When void volume fraction reaches a critical level, then it is assumed that fracture follows immediately (ff ¼ fc+0.01). Full factorial design, one of the design of experiments (DOE), is used to identify the remaining two constants of fc and fn. As Tvergaard (1993) pointed out that element size and element type are minor for the tensile test simulations, element size is determined to represent deformed shape of necked geometry. Zhang et al. (1999) also delineated that the stress distribution in the tensile specimen is relatively uniform even after the diffuse necking has appeared, compared with other notched or cracked specimens. In the authors’ experience, there are few differences in results from hexahedron elements between with full and reduced integration schemes. Using Abaqus/Explicit, parametric FE analyses for coupon tests are carried out by changing material constants of critical void volume fraction and newly created void volume fraction, and finally identified constants are listed in

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5.0x102

Nominal Stress [MPa]

4.0x102

φ

3.0x102

2.0x102 Experiment Simulation

1.0x102 Punch specimen 0.0x100 0.0

0.2

0.4 0.6 Area Reduction Rate

0.8

Fig. 5. Comparison of nominal stress–reduction of cross sectional area.

Fig. 5 shows good agreement of nominal stress vs. reduction of cross sectional area curves from real test and simulation. The simulated curve using the identified material parameters traces the experimental one well until close to the fracture, and material constants are consequently proved to be effective. Even though the porous plasticity model well estimates flat fractures due to hydrostatic stress in the core part of tensile specimen, it is not much effective for slant fractures due to shearing stress in the perimeter of the cross section. For this reason, the numerical curve shows a little earlier fracture than does the experimental one.

Jig

4.2. FE modeling of punch specimen

Indenters Fig. 3. Punch specimens, a jig and indenters (mm). Table 4 Identified material constants. q1 1.50

q2 1.00

q3 2.25

f0 0.00

fc 0.05

ff 0.06

fn 0.01

en 0.3

sn 0.1

Instead of axisymmetric analysis, which is more cost-effective, the shell element is used because it is the most popular for simulation of marine structures. Punch specimens are quarterly modeled where indenters are assumed as rigid bodies. For punch simulations, the element sizes are determined according to the indenter radii: 0.75 mm for small indenter (R ¼ 7.5 mm), 1.5 mm for medium indenter (R ¼ 15.0 mm) and 3.0 mm for large indenter (R ¼ 30.0 mm). It is thought that the used element sizes, which are less than or equal to the plate thickness of 3 mm, are small enough to simulate fracture of punch specimens. The friction coefficient between the indenter and the plate specimen is set to be 0.15. Four sets of boundary conditions are applied as shown in Fig. 6, where two symmetry lines, fully fixed line by bolt, and simply supported line by jig edge, are taken into account. 4.3. Simulation results

Fig. 4. Deformed shape of a tensile specimen with fringe of void volume fraction.

Table 4. Fig. 4 shows one eighth FE model with a fringe of void volume fraction where the maximum void volume fraction (VVF) is developed at the center of the specimen.

Even if the porous plasticity model includes a hydrostatic stress term, it can provide differences with experimental results for very sharp notches or stress concentration area, since the identified material constants are determined from smooth flat specimens. In the case of the punch test simulation, stresses in the local contact zone are differently developed according to the relative sharpness between a specimen and an indenter. In an effort to find out values of the newly nucleated void volume fraction, Table 5 represents best fit values of fn with test results according to a new non-dimensional parameter l which implies the ratio of thickness to sharpness between contacting objects. Even though non-dimensional parameter l changes continuously during the plastic deformation, it is noted that this

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paper uses initial values for Rstrike and Rstruck.    1 1   Rstrike Rstruck 

Fully fixed by bolts

l ¼ t

Supported by jig Tz = 0

Symmetry Tx = Ry = 0

All the simulation results using porous plasticity model follow experimental curves well, as delineated in Fig. 7 where results using shear fracture model are compared with results using porous plasticity model. In the case of shear fracture model, while standard value of failure strain by industrial standard NORSOK N-004 is 0.2 for mild steel, values of failure plastic strain best fitting with experiments are 0.85 for indenter R ¼ 7.5 mm, 0.75 for indenter R ¼ 15.0 mm, and 0.60 for indenter R ¼ 30.0 mm. Since hydrostatic stress is more highly developed along the circumferential edge in the contact area, the fractures are observed not in a radial direction but in a tangential direction for both experiment and simulation (see Fig. 8).

y

x

Symmetry Ty = Rx = 0 5. Conclusions

Fig. 6. A quarter symmetry FE model and boundary conditions for punch test simulation. (a) Indenter radius R ¼ 7.5 mm. (b) Indenter radius R ¼ 150 mm. (c) Indenter radius R ¼ 30.0 mm.

Classification steels for ship structures or API steels for offshore structures are categorized as ductile materials for which the macroscopic plastic deformation process up to fracture appears through three equivalent microscopic stages: nucleation, growth and coalescence of the voids. While the hydrostatic stress governs these micromechanical stages, the yield surface in the classical von Mises yield function is not influenced by the hydrostatic stress because yield surface in the porous plasticity yield function can be reduced as void volume fraction in the porous media increases due to the hydrostatic stress.

Table 5 Identified volume fractions of nucleated voids according to thickness–sharpness ratio. Indenter radius

7.5 mm

15.0 mm

30.0 mm

l

0.400 0.010

0.200 0.015

0.100 0.020

6.0x104

1.2x105

Test No.1(3T-S1) Test No.2(3T-S2) Gurson model Shear fracture model (εf=0.20) Shear fracture model (εf=0.85)

Force [N]

4.0x104

2.0x104

Test No.1(3T-S1) Test No.2(3T-S2) Gurson model Shear fracture model (εf=0.20) Shear fracture model (εf=0.75)

8.0x104

4.0x104

0.0x100

0.0x100 0.0

10.0 20.0 30.0 Indentation [mm]

40.0

0.0

Indenter radius R=7.5 mm 2.0x105

20.0 40.0 Indentation [mm] Indenter radius R=150 mm

Test No.1(3T-S1) Test No.2(3T-S2) Gurson model Shear fracture model (εf=0.60) Shear fracture model (εf=0.20)

1.6x105 Force [N]

Force [N]

fn

ð8Þ

1.2x105 8.0x104 4.0x104 0.0x100 0.0

20.0 40.0 60.0 Indentation[mm]

80.0

Indenter radius R=30.0 mm Fig. 7. Comparison of punch force vs. indentation curves.

60.0

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Fracture along Circumference

1163

Fracture along Circumference

Experiment

Simulation

Fig. 8. Comparison of the fractured shapes.

In order to evaluate the effectiveness of the porous plasticity model for fracture problems, punch tests and simulations have been carried out. The test material is the mild steel of JIS G3131 SPHC, and a pair of tensile coupons and six punch specimens are prepared. From incremental tensile tests, precise plastic mechanical properties are obtained. Some material parameters of the porous plasticity model are determined from the literature, and the others are identified through parametric tensile test simulations. Punch tests are conducted using indenters with three kinds of radii. The values of the volume fraction of the nucleated voids according to a parameter (non-dimensional curvature) between contacting objects are proposed from the comparison of tested and simulated fracture strengths.

Acknowledgement This work was also supported by an INHA UNIVERSITY Research Grant. References ASTM E8, 2004. Standard Test Methods of Tension Testing of Metallic Materials. American Society for Testing and Materials. Chu, C., Needleman, A., 1980. Void nucleation effects in biaxially stretched sheets. J. Eng. Mater. Technol. 102, 249–256. Choung, J.M., Cho, S.R., 2008. Study on true stress correction from tensile tests. Journal of Mechanical Science and Technology 22, 1039–1051.

Choung, J.M., Cho, S.R., Yoon, K.Y., 2007. On comparative studies of fracture models for shipbuilding and offshore structural steels. In: Proceedings of the Fourth International Conference on Collision and Grounding of Ships, pp. 177–185. Gurson, A., 1977. Continuum theory of ductile rupture by void nucleation and growth: part 1—yield criteria and flow rules for porous ductile media. ASME Journal of Engineering Materials and Technology 99, 2–15. Lehman, E., Peschmann, J., 2001. Energy absorption by the steel structure of ships in the event of collisions. In: Proceedings of the Second International Conference on Collision and Grounding of Ships, pp. 189–194. McClintock, F.A., 1968. A criterion for ductile fracture by growth of holes. Trans. ASME, J. Appl. Mech. 35, 363–371. Narr, H., Kujala, P., Simonsen, B.C., Ludolphy, H., 2001. Comparison of the crashworthiness of various bottom and side structures. In: Proceedings of the Second International Conference on Collision and Grounding of Ships, pp. 179–188. Needleman, A., Tvergaard, V., 1984. An analysis of ductile rupture in notched bars. Journal of the Mechanics and Physics of Solids 32, 461–490. NORSOK STANDARD N-004, 2004. Design of Steel Structures, Standards Norway. Paik, J.K., Chung, J.Y. Choe, I.H., Thayamballi, A.K., Pedersen, P.T., Wang, G., 1999. On rational design of double Hull Tanker structures against collision. Annual Meeting of SNAME, pp. 323–363. Rice, J.R., Tracey, D.M., 1969. On the ductile enlargement of voids in triaxial stress fields. J. Mech. Phys. Solids. 17, 201–217. Tvergaard, V., 1981. Influence of voids on shear band instabilities under plane strain condition. Int. J. Fract. Mech. 17, 389–407. Tvergaard, V., 1993. Necking in tensile bars with rectangular cross-section. Computer Methods in Applied Mechanics and Engineering 103, 273–290. Tvergaard, V., Needleman, A., 1984. Analysis of the cup–cone fracture in a round tensile bar. Acta Metallurgica 32, 157–169. Tornqvist, R., 2003. Design of crashworthy ship structures, Technical University of Denmark, Ph.D. Thesis. Urban, J., 2003. Crushing and fracture of lightweight structures, Technical University of Denmark, Ph.D. Thesis. Zhang, K.S., Hauge, M., Odegard, J., Thaulow, C., 1999. Determining material true stress–strain curve from tensile specimens with rectangular cross section. Int. J. Solids Struct. 36, 3497–3516.