PSI - Issue 68

Hande Vural et al. / Procedia Structural Integrity 68 (2025) 573–580 Vural et al. / Procedia Structural Integrity 00 (2024) 000–000

575

3

dependent on the strain rate. The extended Voce rule is expressed as follows: σ y = σ 0 + q 1 1 − e ( − b 1 ¯ ε p ) + q 2 1 − e ( − b 2 ¯ ε p ) 1 + C ln ˙¯ ε p ˙ ε 0

(1)

where σ 0 is the initial yield stress (MPa) and ¯ ε p is the equivalent plastic strain. q i and b i are the specific material parameters associated with the Voce rule. The strain rate sensitivity of the material is defined by C , while ˙ ε 0 is the reference strain rate ( s − 1 ). In this study, eight di ff erent damage models, along with the JC damage model, are utilized to assess the ballistic performance of Armox 500T in a multi-layer target. The JC model is frequently preferred in ballistic impact studies and is based on the e ff ects of stress triaxiality, temperature and strain rate. The other models can predict failure based on equivalent stress or principal stresses. While the JC model has three di ff erent damage parameters that need to be calibrated, the other models have only one damage parameter and these models can be characterized using a single tensile or compression test. All damage criteria and related equations are given in Table 1.

Table 1: Summary of selected typical uncoupled damage criteria.

Criterion

Damage relation

D = D =

¯ ε p

1 D 1 + D 2 exp ( D 3 η )

Johnson-Cook (JC)

d ¯ ε p

0 1 C 1 1 C 2 1 C 3 1 C 4 1 C 5 1 C 6 1 C 7 1 C 8 1 C 9 1 C 10

¯ ε p 0 ⟨ ¯ ε p ¯ ε p 0 ( ¯ ε p 0 ¯ ε p 0

1 3 ⟩ d ¯ ε p d ¯ ε p sinh 3 σ m 2¯ σ d ¯

Ayada-m

σ 1 ¯ σ

+

2 σ 1 3( σ 1 − σ m )

Brozzo

D = D = D = D = D = D = D = D = D =

Le-Roy (LR)

σ 1 − σ m ) d ¯ ε p

¯ σ

¯ σ d ¯

√ 3 2(1 − n ) ¯ σ d ¯

√ 3 2(1 − n )

McClintock (MC)

σ 1 − σ 2

3 4

σ 1 + σ 2

ε p

+

⟨ σ 1 ⟩

OH

ε p

0 ¯ ε p 0 exp ¯ ε p 0 ⟨ ¯ ε p 0 ¯ σ d ¯ ε p

Rice-Trace (RT)

ε p

CL

σ 1 ⟩ d ¯ ε p

Freudenthal

¯ ε p 0 ⟨ ¯ ε p

Ayada

σ m ¯ σ ⟩ d ¯ σ 1 ¯ σ ⟨ 1

ε p

Ko-Huh (KH)

σ m ¯ σ ⟩ d ¯

+ 3

ε p

0

Within these models, σ m represents the mean stress, calculated as σ m = ( σ 11 + σ 22 + σ 33 ) / 3, while σ 1 and σ 2 are the principal stresses. The von Mises equivalent stress is denoted by ¯ σ . The critical damage values, C i , are determined through experimental calibration, and ⟨ . ⟩ refers to the Macauley bracket. The obtained critical damage values are used to normalize the damage accumulation rule of the criteria. In this way, it is assumed that the damage occurs when the D value reaches 1.

2.3. Plasticity and Damage Parameters Calibration

The plasticity model for the specimen is implemented in Abaqus / Implicit through a user-defined hardening sub routine (UHARD). The plasticity parameters from Go¨c¸men et al. (2023) are applied, while the force-displacement data for the tensile tests is obtained from Poplawski et al. (2020). The material properties are summarized in Table 2. The density, Young’s modulus, and Poisson’s ratio of Armox 500T are assumed to be 7.85 g / cm 3 , 201 GPa, and 0.33, respectively. The accuracy of these plasticity parameters and the FE model of the tensile test is validated by comparing the force displacement curve with experimental data, as shown in Figure 1b. The black dashed line represents the experimental data from the literature, while the orange line corresponds to the FE results presented in this study. The results indicate that the plasticity model aligns well with the experimental data. Additionally, the Johnson-Cook (JC) damage model, calibrated by Go¨c¸men et al. (2023) and with parameters provided in Table 2, is employed for comparison with other damage criteria.