Crystal Plasticity Modeling of Hydrogen and Hydrogen-Related Defects in Initial Yield and Plastic Flow of Single-Crystal Stainless Steel 316L

Abstract

Understanding of and accounting for various mechanisms that affect inelastic deformation of crystalline metals in the presence of hydrogen remains an unsettled issue. Macroscopic experimental observations contradict limited atomistic simulations, complicating the situation. In this work, we extend a recent physically based crystal viscoplasticity framework to include constitutive equations with a direct dependence on relevant hydrogen and hydrogen-related defect concentrations. Focusing on initial yield and post-yield strain hardening, we consider hydrogen solute drag on mobile dislocations as well as the role of dilute concentrations of hydrogen-vacancy complexes as obstacles to dislocation motion. Furthermore, the evolution of hydrogen and hydrogen-affected defect concentrations is explicitly considered via evolving hydrogen trap concentrations. The resulting framework is used to investigate hydrogen effects on the quasistatic, monotonic, strain-controlled uniaxial loading of single-crystal stainless steel 316L smooth specimens at room temperature in an attempt to connect atomistic insight and the resulting mesoscale model framework with experimental interpretations. Attributing the primary role of hydrogen in this manner is shown to produce good agreement with experiments in the initial yield and post-yield regime. The dominance of various hydrogen effects mechanisms is discussed.

Appendices

Appendix A

The activation energy, $\Delta G$, associated with dominant short-range barriers to thermally assisted dislocation motion, is expressed as

$$ \Delta G = F_{0} \left( {1 - \left[ {\frac{{\tau_{eff}^{\alpha } }}{{s_{t}^{0} {\raise0.7ex\hbox{$\mu $} \!\mathord{\left/ {\vphantom {\mu {\mu_{0} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\mu_{0} }$}}}}} \right]^{p} } \right)^{q} $$

(A1)

where $F_{0}$ is the activation energy, and $p$ and $q$ are profiling parameters. Parameter $s_{t}^{0}$ is the athermal limit of thermal slip resistance at 0 K, and $\mu$ and $\mu_{0}$ are the shear modulus at temperature $T$ and 0 K, respectively. This relation must reflect the statistical distribution of barriers at the mesoscale via $F_{0}$ and profile parameters $p$ and $q$.

The threshold stress that considers strengthening mechanisms active in the absence of hydrogen, $S^{\alpha }$, accounts for the intrinsic lattice friction, the stress required to bow-out a dislocation, and the dislocation-dislocation self-interactions that result from collinear dislocations in pile-ups. The overall contribution of these three mechanisms can be expressed additively as

$$ S^{\alpha } = S_{0}^{\alpha } + \alpha_{LE} \frac{\mu b}{{2d_{struct} }} + \mu b\left( {A_{ii} \rho_{m}^{\alpha } } \right)^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}} $$

(A2)

where $\alpha_{LE}$ is the dislocation line energy, $f_{w}$ is the volume fraction of dislocation wall substructures, and $d_{struct}$ is the characteristic dimension of dislocation wall substructures. Intrinsic lattice friction $S_{0}^{\alpha }$ is quite small for FCC systems, about 4 MPa in this case, and is dominated by the short range barriers associated with thermal slip resistance $s_{t}^{0}$.

The back stress accounts for the directional internal stress developed due to heterogeneity of plastic deformation within dislocation substructures, ensuring the largely plastic strain in dislocation channels maintains compatibility with the largely elastic strain in dislocation substructure walls. The back stress superimposes on the applied resolved shear stress to form the effective directional over stress. Assuming an Eshelby-type inclusion formulation,[110] the back stress rate can be expressed as

$$ \dot{B}^{\alpha } = \frac{{f_{w} }}{{1 - f_{w} }}\frac{{2\mu (1 - 2S_{1212} )}}{{1 + 4S_{1212} \mu f_{Hill}^{s} }}\dot{\gamma }^{\alpha } $$

(A3)

where $f_{Hill}^{s}$ is the instantaneous macroscopic plastic deformation tangent and $S_{1212}$ is a component of the Eshelby tensor component.[111]

Determination of the dislocation substructure parameters relies on experimentally observed similitude characteristics. Following previous works,[112,113] the characteristic dislocation substructure length scale is determined via

$$ d_{struct} = \frac{{K_{struct} \mu b}}{{\hbox{max} (|\tau^{\alpha } |)}} $$

(A4)

where $K_{struct}$ is the constant of similitude and the max function in the denominator returns the maximum slip system level resolved shear stress in the volume of interest.

The mean free path for dislocation glide is evaluated as

$$ \bar{l} \approx l_{struct} = \eta d_{struct} $$

(A5)

where $\eta$ is a parameter determining the aspect ratio of dislocation substructures. For example, $\eta = 1$ corresponds to dislocation cells while $\eta = 50$ corresponds to more elongated substructures such as dislocation veins. For cyclic loading, $\eta$ is determined by evaluating cyclic plastic strain ranges. In the current study, where specific focus is placed on monotonic loading, this value is assigned a constant value.

The last dislocation substructure parameter to be described is the volume fraction of dislocation dense substructure walls. Here, a phenomenological relationship is leveraged from prior work[114] to define the dislocation substructure wall volume fraction, i.e.,

$$ f_{w} = f_{inf} + (f_{0} - f_{inf} )\exp \left( {\frac{{ - 0.5\gamma^{max} }}{{g_{p} }}} \right) $$

(A6)

where $f_{inf}$, $f_{0}$, and $g_{p}$ are constants that can be estimated by careful substructure characterization using tools such as transmission electron microscopy. Lastly, $\gamma^{max}$ is the maximum shear strain over all slip systems.

The mobile screw dislocation density evolution on slip system $\alpha$ is a result of the competition between dislocation multiplication, annihilation, and cross-slip. Following prior work,[115,116] the overall expression for the dislocation density evolution can be written as

$$ \dot{\rho }_{m}^{\alpha } = \dot{\rho }_{mult}^{\alpha } - \dot{\rho }_{annih}^{\alpha } + \dot{\rho }_{cs} |^{\zeta \to \alpha } - \dot{\rho }_{cs} |^{\alpha \to \zeta } . $$

(A7)

The multiplication rate is defined as

$$ \dot{\rho }_{mult}^{\alpha } = \frac{{k_{multi} }}{{bl_{struct} }}\left| {\dot{\gamma }^{\alpha } } \right| $$

(A8)

where $k_{multi}$ is a constant that depends on the type of expected substructure. The annihilation rate for monotonic loading is defined as

$$ \dot{\rho }_{annih}^{\alpha } = \frac{{2y_{s}^{edge} }}{b}\rho_{m}^{\alpha } \left| {\dot{\gamma }^{\alpha } } \right| $$

(A9)

where $y_{s}^{edge}$ is the annihilation distance for edge dislocations. A second annihilation term formulated by Castelluccio and McDowell[18] to account for anelastic dislocation annihilation upon unloading is dropped in Eq. [A9] as focused is placed on monotonic loading.

The cross-slip term follows a probabilistic formulation from previous work,[117] but considers a local shear stress that accounts for dislocation substructure shielding, i.e.,

$$ \dot{\rho }_{cs}^{\alpha } |^{\alpha \to \zeta } = - v_{G}^{cs} \varphi_{cs} \rho_{m}^{\alpha } \exp \left( { - V_{cs} \frac{{\tau_{III} - \left| {\tau^{\zeta } - B^{\zeta } } \right|}}{{k_{B} T}}} \right) $$

(A10)

$$ \dot{\rho }_{cs}^{\alpha } |^{\zeta \to \alpha } = - v_{G}^{cs} \varphi_{cs} \rho_{m}^{\zeta } \exp \left( { - V_{cs} \frac{{\tau_{III} - \left| {\tau^{\alpha } - B^{\alpha } } \right|}}{{k_{B} T}}} \right) $$

(A11)

where $v_{G}^{cs}$, $\tau_{III}$, $\varphi_{cs}$, and $V_{cs}$ are the characteristic cross-slip frequency, critical cross-slip stress at the onset of stage III hardening, cross-slip efficiency, and cross-slip activation volume, respectively. Indices $\zeta$ and $\alpha$ denote collinear slip systems that permit cross-slip. To account for wall dislocations for hydrogen trapping, we make a simple extension to the MS-CP model and assume $\rho_{i}^{\alpha } = 150\rho_{m}^{\alpha }$. This simple relation is justified for simple loading scenarios as the dislocation density in walls tracks with the mobile dislocation density[118] and the chosen pre-factor generates reasonable wall dislocation densities observed experimentally.[119]

A brief outline of the constitutive equations has been provided. The values used for SS316L are provided in Table A1 with complete details on the MS-CP model found in the original paper.[18]

Table A1 Parameters Related to the MS-CP Model for SS316L at Room Temperature

Appendix B

The functional forms for the values used in Eq. [13] are expressed as[66]

$$ \bar{F}^{max} (\chi_{0} ,\psi ) = \left[ {\left( {0.0033\psi + 0.24} \right)\sqrt {\chi_{0} } + 1/\sqrt {0.518\sqrt \psi + 6.52 \times 10^{ - 4} \psi^{2} } } \right]^{ - 2} $$

(B1)

$$ q_{c} = 1.08\left[ {\frac{2}{\pi }\arctan (0.026\psi^{1.15} )} \right]^{0.279} + 0.075\psi \sqrt {\chi_{0} } $$

(B2)

$$ c' = 0.752\psi $$

(B3)

$$ \psi = \left( {\frac{\mu }{3\pi }\frac{1 + v}{1 - v}\frac{\Delta V}{{k_{B} T}}} \right)^{2} $$

(B4)

where the variables in these equations are defined in Section III.