Constitutive and Fracture Models of Q345E Ship Steel Considering Stress State

doi:10.12382/bgxb.2025.0091

Abstract

Abstract:

In order to characterize the mechanical response of Q345E steel under high-speed impact conditions,the quasi-static tensile and Hopkinson compression experiments are conducted to determine the static and dynamic mechanical properties of Q345E steel as well as its fracture characteristics in wide stress triaxiality.Based on the Johnson-Cook (JC) and Cowper-Symonds (CS) models,a JC-CS constitutive model is proposed to characterize the static and dynamic mechanical behaviors of Q345E steel.A three-stage Wierzbicki fracture model is peoposed and the failure parameters in the range of each stress triaxiality are obtained through experiment and numerical simulation.The Abaqus VUMAT subroutine is used to verify the validity of the constitutive and fracture models.The results show that Q345E steel has obvious strain hardening and strain rate effects,and the two kinds of effects are coupled under dynamic loading.The proposed JC-CS constitutive model can accurately characterize the rate-dependent strain hardening and strain rate effects in the range from quasi-static to high strain rate.The fracture strain of Q345E steel is related to the stress triaxiality.When the stress triaxiality is greater than -1/3 and less than 2,the fracture strain decreases first and then increases,and finally tends to zero.The fracture strain becomes infinitely large when the stress triaxiality tends to -1/3.The numerically simulated results demonstrate that the fracture morphology,the failure mode of target plates and the process of penetrating a multi-layer target predicted by the simulation are highly consistent with the experimental results.This indicates that both the proposed JC-CS constitutive model and Wierzbicki fracture model effectively capture the mechanical properties and fracture behavior of Q345E steel as well as its response characteristics under high-speed impact.

Key words: Q345E ship steel, static-dynamic mechanical properties, constitutive model, fracture model, high-speed perforation

CLC Number:

O385

QUAN Xin, DENG Ximin, WU Haijun, YAN Lei, DONG Heng, JIANG Teng, HUANG Fenglei. Constitutive and Fracture Models of Q345E Ship Steel Considering Stress State[J]. Acta Armamentarii, 2025, 46(11): 250091-.

Figures/Tables 50

Fig.1 Specimen structure diagram

Fig.2 Instron 8801 electronic universal material experimental machine

Fig.3 Split Hopkinson pressure bar

Fig.4 DIC schematic diagram

Table 1 Quasi-static experimental loading speed

试样类型	加载速度/(mm·min^-1)	试样类型	加载速度/(mm·min^-1)
C	0.24/2.4/24.0	TS30	0.3
C	0.24/2.4/24.0	TS60	0.4
CS15	0.24	T	1.2
CS30	0.24	R2	0.3
TS0	0.24	R6	0.4

Fig.5 Quasi-static load-displacement curve

Fig.6 Deformation and failure of specimen

Fig.7 Deformation of SHPB specimen with strain rates of 2000s-1,4000s-1 and 6000s-1(top:front view,bottom:top view)

Fig.8 True stress-strain curve of Q345E steel

Table 2 Yield stress of Q345E steel under different strain rates

编号	应变率/ s^-1	屈服应力/ MPa	编号	应变率/ s^-1	屈服应力/ MPa
1	8.58×10^-4	405.1	6	2000	698.1
2	10^-3	419.3	7	4000	771.6
3	10^-2	424.6	8	4000	778.4
4	0.1	430.9	9	6000	815.0
5	2 000	695.0	10	6000	819.2

Fig.9 Relationship between DIF and strain rate

Fig.10 Plastic stress-strain curve of Q345E steel

Fig.11 Fitting results of JC constitutive model

Table 3 Fitting parameters of JC constitutive model

本构模型参数	A_JC	B_JC	n_JC
拟合结果	419.3	617.2	0.45

Fig.12 Strain rate-plastic stress curves under different strains

Table 4 Strain rate parameter fitting of JC-CS model

塑性应变	C_JC-CS	R²
0.07	-0.050	0.899
0.10	-0.049	0.870
0.13	-0.048	0.894

Table 5 JC-CS model parameters of Q345E steel

A_JC-CS/ MPa	B_JC-CS/ MPa	C_JC-CS	n_JC-CS	C_JC-CS/ s^-1	q_JC-CS	$\stackrel{　·}{\epsilon }$₀/s^-1
419.3	617.2	-0.05	0.45	6689.29	2.99	0.001

Fig.13 Comparison of fitting results and experimental data

Fig.14 Finite element model

Fig.15 Grid sensitivity verification(Smooth round bar specimen)

Table 6 Reference stress-strain curve parameters

a/MPa	b	c/MPa	d	k₁	k₂	k₃
576.3	0.784	-186.37	-20.69	0.01	0.23	0.2

Fig.16 Reference stress-strain curve

Fig.17 Load-displacement curve during uniaxial tensile

Fig.18 Comparison of simulated results with quasi-static loading experiments

Fig.19 Plastic strain distribution nephogram of the DIC specimen (left:DIC calculated nephogram, right:plastic strain nephogram)

Fig.20 Maximum principal strain (simulation)

Fig.21 Equivalent plastic strain-stress triaxial curve

Table 7 Mean stress triaxiality and fracture strain of Q345E

试样类型	平均应力三轴度	断裂应变
R2	0.98	0.42
R6	0.71	0.75
TS0	0.19	0.68
TS30	0.39	0.98
TS60	0.34	0.79
T	0.49	1.02
CS15	-0.09	0.99

Table 8 Johnson Cook failure model parameters

D₁	D₂	D₃	D₄	D₅
-0.15	2.36	-1.41	0	0

Fig.22 Mean stress triaxiality-fracture strain curve

Table 9 Parameters of Wierzbicki failure model

D₁	D₂	D₃	D₄	D₅	η₀
0.622	15.400	1.716	-2.164	0.729	0.541

Fig.23 Comparison of simulated (left) and experimental (right) fracture morphologies

Fig.24 Comparison of simulated load-displacement curve with experimental results

Table 10 Stress state parameter in the test field

试样	平均应力三轴度	平均Lode角参数	断裂应变
T	0.49	1	1.02
R2	0.98	1	0.42
R6	0.71	1	0.75
TS0	0.19	0.0517	0.68
TS30	0.39	0.1030	0.98
TS60	0.34	0.7010	0.79
CS15	-0.09	-0.3350	0.99

Fig.25 Relationship anong fracture strain and stress triaxiality and Lode angle parameter

Fig.26 Experimental system

Fig.27 Structural dimensions of projectiles

Fig.28 Target dimensions

Fig.29 Typical perforation processes

Table 11 Change in projectile velocity during perforation

弹体编号	靶网速度/ (m·s^-1)	高速摄影测得速度/(m·s^-1)
弹体编号	靶网速度/ (m·s^-1)	靶1前	相对误差/%	靶1后	靶2后	靶3后
C-1	951	962.54	-1.21	906.89	882.92	854.99
C-2	853	841.54	1.34	793.71	767.51	729.18
C-3	974	946.12	2.86	893.31	865.83	820.03
E-1	938	919.75	1.95	831.78	803.89	758.07
E-2	945	969.90	-2.63	919.53	882.90	845.99
E-3	835	836.29	-0.15	802.49	775.80	733.18

Fig.30 Recovered projectiles

Fig.31 Perforation morphology of C-1

Fig.32 Perforation morphology of E-3

Fig.33 Simulation model and mesh sensitivity verification

Fig.34 Perforation morphology on the back of target(left: experiment results, right: simulation results)

Fig.35 Comparison of experiment and simulation in the process of perforating

Fig.36 Comparison of simulated and experimental attitude angles of C-1

Fig.37 Comparison of simulated and experimental attitude angles of E-3

Table 12 Comparison of experimental and simulated projectile velocities

工况编号	初速/ (m·s^-1)	位置	实验结果/ (m·s^-1)	仿真结果/ (m·s^-1)	相对误差/%
C-1	962.54	靶1	906.89	930.36	2.59
		靶2	882.92	911.27	3.21
		靶3	854.99	892.29	4.36
E-3	836.29	靶1	802.49	799.63	-0.36
		靶2	775.80	781.22	0.70
		靶3	733.18	761.04	3.80

Fig.38 Stress triaxiality of the first layer target when E-3 projectile head perforating

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