2023 Volume 64 Issue 5 Pages 1002-1010
A numerical investigation of the kink strengthening mechanism in long-period stacking ordered magnesium alloy is presented. A higher-order gradient crystal plasticity model is introduced, and the reproducing kernel particle method is adopted for the numerical procedure. The specimen, including a kink band with several kink angles, which is defined as the angle between the inside and outside of the kink band, is considered. A simple shear analysis is conducted to evaluate the kink strengthening due to the kink band. The numerical results suggest two main origins of kink strengthening, namely, orientation and defect strengthening. The former is caused by the spatial distribution of the slip direction due to kink, and the latter is the strengthening induced by crystal defects around the kink boundary. The amplitude of kink strengthening depends on the kink angle, and an optimal kink angle, which maximizes the kink strengthening, might exist. The kink strengthening exhibits a Hall–Petch-like behavior, i.e., the correlation between the inverse of the square root of the kink band width and flow stress is almost linear.
Magnesium alloys with the long-period stacking ordered (LPSO) structure1,2) demonstrate superior strength and are expected as next-generation structural materials. In the LPSO-type magnesium alloy subjected to plastic forming such as extrusion or rolling, a peculiar deformation called kink is observed.3–5) The LPSO-type magnesium alloy that contains kink generally shows much higher strength than a virgin material. Therefore, kink can be considered to induce material strengthening, which is called kink strengthening. Although many experimental studies have been conducted to investigate kink strengthening,6–15) its origin remains unclear. To generalize the governing principle of kink strengthening and to develop a novel material that is strengthened by kink, understanding the kink strengthening mechanism is essential.
Band-shaped kink is sometimes observed in a material strengthened by kink. In LPSO-type magnesium alloys, the active slip system is limited to basal slip system, and kink can be considered as a spatial change in the slip direction of basal slip at the crystalline scale. In contrast to twinning in which the crystal orientation is geometrically determined, kink exhibits a variety of morphologies. At the kink boundary, the basal slip direction discontinuously changes, and the angle between the inside and outside the band can have different values. This angle might affect the amplitude of kink strengthening. Because of nonuniform strain distribution due to kink, a strain gradient exists around the kink band. The existence of a strain gradient suggests that crystal defects exist, such as dislocation or disclination, and the crystal defects induce a size effect of the materials. Therefore, size effect is also an important issue in kink strengthening.
The crystal plasticity theory16,17) is an efficient method of representing the mechanics in the crystalline scale. Although the mechanical behavior of metallic materials is strongly influenced by the size effect at the micrometer scale, the conventional crystal plasticity theories do not consider the size effect because these theories do not take into account the accumulation of dislocation. To describe the size effect at the crystalline scale, a higher-order gradient crystal plasticity model18–20) was proposed. In this model, geometrically necessary dislocations (GNDs) were introduced into a hardening function of the slip system. An additional governing equation that expressed the dislocation density field was derived, and both the displacement and dislocation density fields were simultaneously solved.
The finite element method (FEM), which is the most popular method for solving problems in solid mechanics. Several studies using FEM for evaluating mechanical behavior of LPSO-type magnesium and a kink formation mechanism were reported.21–23) On the other hand, FEM sometimes yields an improper solution in the higher-order gradient crystal plasticity analysis;24) therefore, development of a numerical method to solve higher-order gradient plasticity is necessary. In the present study, the reproducing kernel particle method (RKPM),25,26) which is a kind of meshfree method, is introduced into the higher-order gradient crystal plasticity analysis. To improve the accuracy and stability of the analysis, the stabilized conforming nodal integration (SCNI)27,28) is adopted as a numerical integration scheme. A simple shear problem is considered, and comparison between the conventional Gauss quadrature and SCNI is performed. RKPM with SCNI is found to be more stable and requires lesser computational cost than the conventional method.29)
The main objective of the present study is investigating the kink strengthening mechanism from the mesoscale viewpoint. A numerical study on kink strengthening in LPSO-type magnesium alloy is conducted using the higher-order gradient crystal plasticity, so that the effect of crystal defects on kink strengthening can be considered in the mesoscale mechanics. RKPM is introduced for numerical method to perfume stable and accurate numerical analysis. The effect of kink angle on kink strengthening is quantitatively evaluated by focusing on the kink angle, which is defined as the angle between the inside and outside of kink band, the effect of kink angle on kink strengthening is quantitatively evaluated and the optimal kink angle is presented. The size effect of the kink band is also investigated. Finally, the origin of kink strengthening at the crystalline scale is discussed.
In this study, the higher-order gradient crystal plasticity model proposed by Kuroda and Tvergaard18–20) is introduced as a constitutive model. First, the framework of the model is presented. The velocity gradient L is additively decomposed into the non-plastic and plastic parts.
| \begin{equation} \mathbf{L} = \mathbf{L}^{*} + \mathbf{L}^{\text{p}} \end{equation} | (1) |
| \begin{equation} \mathbf{L}^{\text{p}} = \sum_{\alpha = 1}^{N}\dot{\gamma}^{(\alpha)}(\mathbf{s}^{(\alpha)} \otimes \mathbf{m}^{(\alpha)}) \end{equation} | (2) |
| \begin{equation} \mathbf{D}^{\text{p}} = \frac{1}{2}(\mathbf{L}^{\text{p}} + \mathbf{L}^{\text{pT}}) \end{equation} | (3) |
| \begin{equation} \mathbf{D}^{*} = (\mathbf{L}^{*} + \mathbf{L}^{*\text{T}})/2 \end{equation} | (4) |
| \begin{equation} \mathbf{W}^{*} = (\mathbf{L}^{*} - \mathbf{L}^{*\text{T}})/2 \end{equation} | (5) |
The elastic constitutive law is expressed as
| \begin{equation} \mathring{{\boldsymbol{\sigma}}}^{*} = \dot{{\boldsymbol{\sigma}}} - \mathbf{W}^{*}{\boldsymbol{\sigma}} + {\boldsymbol{\sigma}}\mathbf{W}^{*} = \mathbf{C}:\mathbf{D}^{*} \end{equation} | (6) |
| \begin{equation} \dot{\mathbf{m}}^{(\alpha)} = \mathbf{W} \cdot \mathbf{m}^{(\alpha)} \end{equation} | (7) |
| \begin{equation} \dot{\mathbf{s}}^{(\alpha)} = \mathbf{W} \cdot \mathbf{s}^{(\alpha)} \end{equation} | (8) |
In the present model, the effect of GND is introduced in the following manner. GND represents the dislocation that is necessary to satisfy the consistency condition after the plastic deformation. The GND densities of an α slip system is expressed as follows:
| \begin{equation} \rho_{G(e)}^{(\alpha)} = -\frac{1}{b}\nabla \gamma^{(\alpha)} \cdot \mathbf{s}^{(\alpha)} \end{equation} | (9) |
| \begin{equation} \rho_{G(s)}^{(\alpha)} = \frac{1}{b}\nabla\gamma^{(\alpha)} \cdot \mathbf{l}^{(\alpha)} = -\frac{1}{b}\nabla \gamma^{(\alpha)} \cdot \mathbf{p}^{(\alpha)} \end{equation} | (10) |
To introduce the effect of GND into the crystal plasticity formulation, the slip rate $\dot{\gamma }^{(\alpha )}$ in eq. (2) is assumed in the following form:
| \begin{equation} \dot{\gamma}^{(\alpha)} = \mathop{\text{sgn}}\nolimits (\tau^{(\alpha)} - \tau_{b}^{(\alpha)})\left(\frac{|\tau^{(\alpha)} - \tau_{b}^{(\alpha)}|}{g^{(\alpha)}}\right)^{\frac{1}{m}} \end{equation} | (11) |
| \begin{equation} \dot{g}^{(\alpha)} = \sum_{\beta}h^{(\alpha\beta)}|\dot{\gamma}^{(a)}| \end{equation} | (12) |
| \begin{align} \tau_{b}^{(\alpha)} &= b\lambda\sum_{\beta}S^{(\alpha\beta)}\nabla \rho_{G(e)}^{(\alpha)} \cdot \mathbf{s}^{(\alpha)} \\ &= -\lambda\sum_{\beta}S^{(\alpha\beta)}\mathbf{s}^{(\alpha)} \cdot (\nabla \otimes \nabla \gamma^{(\beta)}) \cdot \mathbf{s}^{(\beta)} \end{align} | (13) |
| \begin{equation} S^{(\alpha\beta)} = \mathbf{s}^{(\alpha)} \cdot \mathbf{s}^{(\beta)} \end{equation} | (14) |
| \begin{equation} \lambda = \tau_{0}L^{2} \end{equation} | (15) |
From eqs. (9) and (10), the equilibrium equations for the GND densities are derived as follows:
| \begin{equation} \frac{1}{b}\nabla\gamma^{(\alpha)} \cdot \mathbf{s}^{(\alpha)} + \rho_{G(e)}^{(\alpha)} = 0 \end{equation} | (16) |
| \begin{equation} \frac{1}{b}\nabla\gamma^{(\alpha)} \cdot \mathbf{p}^{(\alpha)} + \rho_{G(s)}^{(\alpha)} = 0 \end{equation} | (17) |
| \begin{align} \int_{v}\delta \rho \cdot \dot{\rho}_{\text{G(e)}}^{(\alpha)}\text{d}v &= \frac{1}{b}\int_{v}\mathop{\text{grad}}\nolimits \delta \rho \cdot \bar{\mathbf{s}}^{(\alpha)}\dot{\gamma}^{(\alpha)}\text{d}v \\ &\quad + \frac{1}{b}\int_{v}\delta \rho \cdot \text{div}\,\bar{\mathbf{s}}^{(\alpha)}\dot{\gamma}^{(\alpha)}\text{d}v \\ &\quad - \frac{1}{b}\int_{s}\delta \bar{\mathbf{n}} \cdot \bar{\mathbf{s}}^{(\alpha)}\dot{\gamma}^{(\alpha)}\text{d}s \end{align} | (18) |
| \begin{align} \int_{v}\delta \rho \cdot \dot{\rho}_{\text{G(s)}}^{(\alpha)}\text{d}v &= \frac{1}{b}\int_{v}\mathop{\text{grad}}\nolimits \delta \rho \cdot \bar{\mathbf{p}}^{(\alpha)}\dot{\gamma}^{(\alpha)}\text{d}v \\ &\quad + \frac{1}{b}\int_{v}\delta \rho \cdot \text{div}\,\bar{\mathbf{p}}^{(\alpha)}\dot{\gamma}^{(\alpha)}\text{d}v \\ &\quad - \frac{1}{b}\int_{s}\delta \bar{\mathbf{n}} \cdot \bar{\mathbf{p}}^{(\alpha)}\dot{\gamma}^{(\alpha)}\text{d}s \end{align} | (19) |
In eqs. (18) and (19), two types of boundary conditions for the GND density fields can be considered. The first one is prescribed $\dot{\rho }$, which corresponds to the Dirichlet boundary condition. For example, $\dot{\rho } = 0$ indicates that no slip gradient exists on the boundary. The second one is prescribed $\dot{\gamma }^{(\alpha )}$, which indicates a Neumann boundary condition. $\dot{\gamma }^{(\alpha )} = 0$ indicates that dislocation cannot penetrate across the boundary. Equations (18) and (19) are additional governing equations in the present framework. By simultaneously solving them using the ordinary rate form of the virtual work principle, both the displacement and GND densities fields can be simultaneously computed.
2.2 Reproducing kernel particle methodFEM is the most popular method for solving problems in solid mechanics, such as the crystal plasticity analysis. However, it is pointed out that FEM sometimes yields an improper solution in the higher-order gradient crystal plasticity analysis.24) In the present study, RKPM,25,26) which is a kind of meshfree method, is introduced into the higher-order gradient crystal plasticity analysis. Previous study showed that the RKPM provided a stable numerical solution with high accuracy.29)
RKPM is a meshfree method based on the reproducing kernel (RK) approximation. The following equation is obtained using a set of nodes $\{ x_{1},x_{2,}, \cdots ,x_{\textit{NP}}\} $, where NP denotes the total number of nodes.
| \begin{equation} u^{h}(\mathbf{x}) = \sum_{I = 1}^{\mathit{NP}}\Psi_{I}(\mathbf{x})u_{I} \end{equation} | (20) |
| \begin{equation} \Psi_{I}(\mathbf{x}) = \mathbf{H}^{\text{T}}(\mathbf{0})\mathbf{M}^{-1}(\mathbf{x})\mathbf{H}(\mathbf{x} - \mathbf{x}_{I})\phi_{a}(\mathbf{x} - \mathbf{x}_{I}) \end{equation} | (21) |
| \begin{equation} \mathbf{H}^{\text{T}}(\mathbf{0}) = [1\quad 0\quad 0\quad 0\quad 0\quad 0] \end{equation} | (22) |
| \begin{align} \mathbf{H}^{\text{T}}(\mathbf{x} - \mathbf{x}_{I}) &= [1\quad x_{1} - x_{1I}\quad x_{2} - x_{2I}\quad (x_{1} - x_{1I})^{2}\\ &\quad (x_{2} - x_{2I})^{2}] \end{align} | (23) |
| \begin{equation} \mathbf{M}(\mathbf{x}) = \sum_{I = 1}^{\mathit{NP}}\mathbf{H}(\mathbf{x} - \mathbf{x}_{I})\mathbf{H}^{\text{T}}(\mathbf{x} - \mathbf{x}_{I})\phi_{a}(\mathbf{x} - \mathbf{x}_{I}) \end{equation} | (24) |
| \begin{equation} \phi_{a}(\mathbf{x} - \mathbf{x}_{I}) = \varphi_{a}\left(\frac{x_{1} - x_{1I}}{a_{1}}\right)\varphi_{a}\left(\frac{x_{2} - x_{2I}}{a_{2}}\right) \end{equation} | (25) |
| \begin{equation} \varphi_{a}(z) = \begin{cases} \dfrac{2}{3} - 4z^{2} + 4z^{3} & \text{$0 \leq z \leq \dfrac{1}{2}$}\\ \dfrac{4}{3}(1 - z)^{3} & \text{$\dfrac{1}{2} \leq z \leq 1$}\\ 0 & \text{otherwise.} \end{cases} \end{equation} | (26) |
An essential problem in the meshfree methods is the numerical scheme for integration over an analysis domain. In FEM, the finite element can be used for an integration domain. However, no element exists in the meshfree method, and a number of numerical schemes of numerical integration for meshfree methods have been proposed. In the present study, SCNI,27,28) which is a node-based integration scheme, is adopted. It is pointed out that the following condition called integration constraint should be satisfied in a meshfree analysis:
| \begin{equation} \int\limits_{\Omega}^{\wedge}\nabla \Psi_{I}\text{d}\Omega = \int\limits_{\Gamma}^{\wedge}\Psi_{I}\mathbf{n}\text{d}\Gamma \end{equation} | (27) |
In SCNI, the strain-displacement matrix $\tilde{\mathbf{B}}_{I}(\mathbf{x}_{L})$ is computed by the following equation to satisfy the integration constraint:
| \begin{equation} \tilde{\mathbf{B}}_{I}(\mathbf{x}_{L}) = \begin{bmatrix} \tilde{b}_{1I}(\mathbf{x}_{L}) & 0\\ 0 & \tilde{b}_{2I}(\mathbf{x}_{L})\\ \tilde{b}_{2I}(\mathbf{x}_{L}) & \tilde{b}_{1I}(\mathbf{x}_{L}) \end{bmatrix} \end{equation} | (28) |
| \begin{equation} \tilde{b}_{iI}(\mathbf{x}_{L}) = \frac{1}{A_{L}}\int_{\Gamma_{L}}\Psi_{I}\mathbf{n}d\Gamma \end{equation} | (29) |
| \begin{equation} \begin{bmatrix} D_{11}\\ D_{22}\\ 2D_{12} \end{bmatrix} = \sum_{I \in G_{L}}\tilde{\mathbf{B}}_{I}(\mathbf{x}_{L})\mathbf{d}_{I} \end{equation} | (30) |
| \begin{equation} \mathbf{d}_{I} = \begin{bmatrix} \dot{u}_{1I}\\ \dot{u}_{2I} \end{bmatrix} \end{equation} | (31) |
Figures 1(a) and (b) show a schematic illustration of the analysis model and boundary condition. To investigate the essential mechanism of strengthening induced by kink, a rectangular-shaped specimen with a single kink band is assumed. The introduced kink band corresponds to a middle section of ridge-type kink and can be interpreted as the combination of two Hess and Barret type kinks.30) In the LPSO-type magnesium alloy, only basal slip system can be activated. Therefore, in the present analysis, only basal system is introduced, meaning that a single slip system is considered in a two-dimensional analysis. The dashed lines shown in Fig. 1 indicate basal slip plane. Kink is described as a change of the slip direction, and the angle between the slip directions inside and outside the kink band is denoted as θ and is called kink angle. The width of the kink band is $d = \frac{1}{4}H$.
Analysis model. (a) Slip direction and definition of kink angle, (b) boundary conditions.
To represent a simple shear condition, the lower edge of the specimen is fixed, and the prescribed displacement along the horizontal direction is applied to the upper edge up to a 10% shear strain. At the upper and lower boundaries, a dislocation-impenetrable condition is applied, whereas dislocation can penetrate on the left and right boundaries. The plane strain condition is assumed because of the incompressibility of the plastic deformation. The analysis domain is discretized into 32 × 32 nodes both for the displacement and GND density fields. Because the material is assumed to be an LPSO-type magnesium alloy, the Young’s modulus and Poisson’s ratio are 45 GPa and 0.3, respectively. Because a single slip system is considered, matrix h(αβ) in eq. (12) degenerates to scalar h, and the following hardening law is introduced:
| \begin{equation} h = h_{0} \end{equation} | (32) |
Specimens with various kink angles in the range between θ = 5° and 85° are analyzed to investigate the effect of kink angle θ. For comparison, computation of a specimen with “no kink”, which is equivalent to a model with θ = 0°, is also conducted. Here, the length parameter is fixed as L/H = 0.5. The obtained nominal shear stress–strain curves are shown in Fig. 2. The increase in the amount of flow stress strongly depends on kink angle θ. Increasing kink angle θ increases the flow stress up to θ = 45°. Then, the flow stress decreases with the increase in kink angle θ. For all kink angles, the flow stresses are obviously higher than those in the case of no kink. The increase in the flow stress due to the kink band is called kink strengthening in this study.
Nominal stress–nominal strain curves with several kink angles.
The flow stresses at a nominal strain of 10% with a kink angle in the range from θ = 5° to 85° are shown in Fig. 3. The result for θ = 0° corresponds to the no-kink model. To investigate the effect of GND density, an analysis using the conventional crystal plasticity is also conducted. The flow stresses of the specimen with a kink are always higher than those of the no-kink case, meaning that the existence of kink increases the flow stress and results in kink strengthening. The flow stress varies with the kink angle and exhibits the highest value at approximately 50° of kink angle. This result suggests that an optimal kink angle that provides the highest flow stress may exist, and the optimal kink angle in the present study is approximately 50°. This tendency is observed even in the analysis using the conventional crystal plasticity in which no effect of a slip gradient is introduced. However, the present higher-order gradient crystal plasticity gives a much higher flow stress in all ranges of kink angle. The detailed discussion of the origin of the kink strengthening is presented in Section 3.4.
Flow stress at 10% nominal strain with respect to kink angle: comparison with result obtained by conventional crystal plasticity.
The distributions of the GND density and shear stress with several kink angles are shown in Figs. 4 and 5. In Fig. 4, the illustrated GND density is normalized as ρGbH. Figure 4 shows that a concentration of GND density is observed around the kink boundaries in all cases except for the no-kink model, and the amplitude of the GND density depends on the kink angle. Because of the slip direction difference between the inside and outside of the kink band, a gradient of slip exists around the kink boundary, which causes increase in the GND density. In the present higher-order gradient crystal plasticity, the GND density affects the strain hardening, and a higher GND density results in a higher flow stress. As a result, the shear stress inside the kink band increases, as shown in Fig. 5, resulting in kink strengthening. The amplitude of the gradient of slip depends on the difference in the slip directions inside and outside the kink band, and it explains why the kink strengthening depends on the kink angle and that an optimal kink angle may exist.
Distributions of GND density at 10% nominal strain, where GND density is normalized as ρGbH. (a) No kink, (b) θ = 15°, (c) θ = 30°, (d) θ = 45°, (e) θ = 60°, (f) θ = 75°.
Distributions of shear stress at 10% nominal strain. (a) No kink, (b) θ = 15°, (c) θ = 30°, (d) θ = 45°, (e) θ = 60°, (f) θ = 75°.
Next, the size effect of the kink is quantitatively investigated. In the present analysis, length parameter L is varied because a change in L is equivalent to a change in the specimen size, i.e., larger L corresponds to a smaller specimen. The ratio of d/H is fixed at one-fourth, which is the same as that in the analyses presented in Section 3.2. The nominal shear stress–strain curves obtained for several values of L are shown in Fig. 6, where the kink angle is fixed at θ = 45°. For comparison, the result using the conventional crystal plasticity in which no size effect is represented is also shown. The result with the conventional model shows the lowest flow stress. When the higher-order gradient crystal plasticity is adopted, larger L, which corresponds to a narrower kink band, yields a higher flow stress. This result corresponds to the size effect of polycrystalline metals in which smaller crystal grains provide a higher yield stress, which is the so called the Hall–Petch effect.
Nominal stress–nominal strain curves with several length parameters.
Figures 7 and 8 show the distributions of the GND density and shear stress for several length parameter L values. In Fig. 7, the illustrated GND density is normalized as ρGbH. Figure 7 shows that a concentration of GND densities is observed around the kink boundaries in all cases, the same as that shown in Fig. 4, and the amplitude of the GND density depends on the specimen size in which the analysis with smaller L indicates a higher GND density. Note that a higher GND density does not necessarily yield a higher stress. In fact, the shear stress shown in Fig. 8 has the lowest value at L/H = 0.1 and the highest value at L/H = 0.5. Meanwhile, the GND density shown in Fig. 7 has the highest value at L/H = 0.1 and the lowest value at L/H = 0.5. The result shows that stronger kink strengthening is obtained with a narrower kink band.
Distributions of GND density at 10% nominal strain, where GND density is normalized as ρGbH. (a) L/H = 0.1, (b) L/H = 0.3, (c) L/H = 0.5.
Distributions of shear stress at 10% nominal strain. (a) No kink, (b) L/H = 0.1, (c) L/H = 0.3, (d) L/H = 0.5.
For quantitative evaluation of the size effect of the kink band, $(\bar{d})^{ - 1/2}$ with $\bar{d} = d/L$ is introduced as an analogy of the Hall–Petch effect of polycrystalline metals, where d is normalized by length parameter L. The correlation between $(\bar{d})^{ - 1/2}$ and the flow stress at 10% strain is shown in Fig. 9. An almost linear relationship is observed in Fig. 9, which indicates that the size effect of the kink band exhibits Hall–Petch-like behavior, and we can express the relationship between flow stress σ and normalized kink width $\bar{d}$ as
| \begin{equation} \sigma = k(\bar{d})^{-1/2} \end{equation} | (33) |
Scale effect of kink band: correlation between $(\bar{d})^{ - 1/2}$ and flow stress at 10% nominal strain.
Correlation between coefficient k in $\sigma = k(\bar{d})^{ - 1/2}$ and kink angle.
As presented in Sections 3.2 and 3.3, the existence of kink increases the flow stress, which is called kink strengthening. Kink strengthening depends on the kink angle and size of kink, as presented in Sections 3.2 and 3.3, respectively. Kink strengthening can be represented by both the conventional crystal plasticity and present higher-order gradient crystal plasticity. However, the latter provides stronger kink strengthening. This result suggests that kink strengthening could have been caused by at least two reasons. The first reason is represented by both the conventional and present model, and the second one is represented by the present model only.
The kink band is represented by the spatial distribution of crystal orientation in the present framework, and a change in the kink angle corresponds to a difference in the initial orientation. The first origin of kink strengthening, which is represented even by the conventional crystal plasticity, is caused by the difference of initial crystal orientation between inside and outside of kink band; therefore, we call it orientation strengthening.
The essential difference in the higher-order crystal plasticity from the conventional one is that the GND density caused by the gradient of slip is introduced, and the strain hardening is affected by the GND density, which causes a higher GND density to yield a higher flow stress. Therefore, the second origin of kink strengthening, which is represented by the higher-order gradient crystal plasticity only, is caused by the GND density, which is indicated by ρG(e) in eqs. (9) and (16). Note that this term may include not only dislocation but also other crystal defects such as disclination because any type of crystal defect can affect ρG(e) although we call it GND density according to the literature.18–20) Therefore, we refer to this strengthening mechanism as defect strengthening.
According to the result presented in Section 3.2, the contributions of orientation and defect strengthening to the kink strengthening are investigated. The contributions of each strengthening mechanism with respect to the kink angle when the length parameter is L/H = 0.2, L/H = 0.3, L/H = 0.4 or L/H = 0.5 are shown in Fig. 11. Each contribution is evaluated based on the case of no-kink. For instance, when the length parameter is L/H = 0.5, the contributions of the orientation and defect strengthening at θ = 45° are 133% and 132%, respectively, and the flow stress increases to 365% of that in the case with no kink, which represents the summation of the original value (100%), orientation strengthening (133%), and defect strengthening (132%). The magnitude of each strengthening mechanism depends on the kink angle, and the defect strengthening is more dominant when the kink angle is 50° or more. As mentioned in Section 3.2, an optimal kink angle that provides the highest kink strengthening may exist, and the optimal kink angle in the present case is approximately 50°. At approximately θ = 50°, the contributions of both orientation and defect strengthening are maximized. The amplitude of defect strengthening depends on the length parameter L, and a larger L, which means a smaller specimen, gives higher defect strengthening, while the amplitude of orientation strengthening is not affected by the length parameter L. The size effect of kink strengthening shown in Section 3.3 corresponds to this tendency.
Contributions of orientation and defect strengthening to kink strengthening with respect to kink angle, where length parameter is (a) L/H = 0.2, (b) L/H = 0.3, (c) L/H = 0.4 and (d) L/H = 0.5.
The orientation strengthening may be simply evaluated using the Schmid factor of kink band, because this mechanism is essentially resulted by the increasing of shear stress due to crystal orientation. Note that, in contrast, the defect strengthening should be investigated by the present continuum mechanics-based approach using the higher-order gradient crystal plasticity. The defect strengthening depends on analysis conditions, i.e., the boundary conditions, hardening law, specimen size and morphology, and so on, and the present model can evaluate the defect strengthening under arbitrary analysis conditions with the unique length parameter L. The stress field around kink band is not uniform as shown in Figs. 5 and 8, and the breadth of uniform stress field is affected by the length parameter L. It affects the amplitude of defect strengthening. The non-uniform stress field must be computed by a continuum mechanics-based approach. Additionally, the effect of kink angle on the coefficient k, which is one of the important results in the present study, can be represented. This kink angle dependency of k cannot be evaluated by the simple Schmid factor-based calculation.
The present study suggests two main origins of kink strengthening, namely, orientation strengthening and defect strengthening. Both mechanisms cannot be neglected in the kink strengthening of LPSO-type magnesium alloy, and evaluation of the defect strengthening is essential for understanding the kink strengthening mechanism.
In this study, a numerical investigation of the strengthening mechanism of LPSO-type magnesium alloy with kink band is conducted using higher-order gradient crystal plasticity. The conclusions are summarized as follows.
The author appreciates Mr. Daijiro Kamura of Saga University for preliminary computations using the present model. This study was supported by JSPS KAKENHI for Scientific Research on Innovative Areas “MFS Materials Science (grant number 18H05480)”.
