PSI - Issue 61

Orhun Bulut et al. / Procedia Structural Integrity 61 (2024) 3–11

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O. Bulut et al. / Structural Integrity Procedia 00 (2024) 000–000

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2. Theory and Methodology

2.1. Crystal plasticity model

A rate-dependent finite strain local crystal plasticity model is employed in this study. The deformation gradient, F , is multiplicatively decomposed into elastic part ( F e ) and plastic part ( F p ). Plastic velocity gradient, L p , is obtained by the summation of the slip rates in all slip systems,

N α = 1

= ˙ F

˙ γ ( α ) ( m ( α ) ⊗ n ( α ) )

L p

p · ( F p ) − 1

(1)

=

where α defines the slip system, N is the total number of slip systems and ˙ γ is the slip rate. m α and n α denote the slip direction and the normal to the slip plane of the system α , respectively. In this study, for the HCP titanium alloy, 3 basal, 3 prismatic and 12 pyramidal slip systems are considered active. Slip rate is calculated for each system according to the following power law,

= ˙ γ 0 τ ( α )

g ( α ) n

sign( τ ( α ) )

˙ γ ( α )

(2)

where τ ( α ) and g ( α ) denote the resolved shear stress and slip resistance, respectively. Moreover, ˙ γ reference slip rate and the rate sensitivity exponent. The slip resistance evolves according to

0 and n represent the

= β

h αβ ˙ γ β

˙ g ( α )

(3)

where h αβ is the latent hardening matrix which is calculated as h αβ = q αβ h αα , ( α β )

(4)

where q αβ includes latent hardening coe ffi cients and these coe ffi cients are taken as 1 for the analyses. For self hardening, the law proposed in Peirce et al. (1982) is employed given by the following equations:

2

h 0 γ g s − g 0

h αα = h ( γ ) = h

(5)

0 sech

where g 0 and g s are the initial and saturated slip resistances while h 0 is the initial hardening modulus. Since this framework is a rate-dependent crystal plasticity model, all of the slip systems are considered active from the onset. In Tables 1 and 2, elastic constants and crystal plasticity model parameters for slip systems are shown (retrieved from Ozturk et al. (2019) for Ti-7Al). In the context of this study, the rate sensitivity exponent, n, is kept lower than the aforementioned paper to observe rate dependent behavior more clearly. The exponent is planned to be calibrated to experimental data to capture more realistic behavior of the Ti crystals in future studies.

Table 1. Elastic constants C 11 = C 22

C 33

C 12

C 13 = C 23

C 55 = C 66

184GPa

224GPa

107GPa

93.7GPa

55.6GPa

2.2. Coupling with phase field fracture

The phase field fracture model couples the evolution of phase field parameter with both elastic and plastic defor mation through the global minimization of the following energy functional: