Special Issue on Materials Science on Mille-Feuille Structure
Duality of the Incompatibility Tensor
2020 Volume 61 Issue 5 Pages 875-877
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2020 Volume 61 Issue 5 Pages 875-877
Disclination resulting from kink deformation of the long-period stacking-ordered phase of Mg–Zn–Y alloy affects the strength of the alloy. In this study, we examined theoretical aspects of the kinematic equation of defects, including dislocations and disclinations, based on the dual structure of strain and stress space. Two types of disclination, related to dislocation and bend-twist, were identified. Disclination types were distinguished based on incompatibility. In “dual strain space”, we show that incompatibility is due to a generalized stress function, i.e., the Beltrami stress function, which is a three-dimensional version of the Airy stress function that includes non-diagonal components.
Recently, there has been much interest in the Mg–Zn–Y alloy developed by Kawamura,1) due to its excellent properties including high specific strength.2) This strength property originates from the long-period stacking-ordered (LPSO) phase of the Mg–Zn–Y lattice,3–5) characterized by a particular kink structure.6–8) Kink deformation of the LPSO phase is essential for the excellent mechanical properties demonstrated by the Mg–Zn–Y alloy.9)
Normal kink deformation is part of the dislocation line of the slip plane. However, it has been pointed out that not only dislocations, but also disclinations, affect the strength of the LPSO phase. For instance, Inamura9) showed theoretically that disclination plays an important role in the connection of kink bands, according to the geometric compatibility (rank-1 tensor). The incompatibility tensor has attracted interest for crystal plasticity-based finite element simulations, where kink deformation has been seen in Mg alloys with a mille-feuille-structured LPSO phase.10,11) The relationship between incompatibility and disclination is expected to be key to understanding kink deformation in the LPSO phase. However, there have been few theoretical studies of this phenomenon.
The purpose of this paper is to review the theoretical aspects of the incompatibility tensor using the kinematic equation of defects, accounting for both dislocations and disclinations. To achieve this, we examined the dual structure of strain and stress space based on differential geometry.12,13) This differential geometrical approach is useful for deriving kinematic and continuity equations of defects in a systematic way, and has been applied to various complicated deformations with defect fields, as reported in Earth science.14,15)
The structure of this paper is as follows. In Section 2, we consider incompatibility in strain space and derive the relationship between incompatibility and disclinations to reveal its dual structure, i.e., the stress space. In Section 3, we reconsider the incompatibility in “dual stress space”; our analysis shows that the concept of incompatibility can be extended to generalized stress functions. Conclusions are presented in Section 4.
First, we review the incompatibility in strain space, based on a differential geometrical approach,13) and consider how it is related to disclination. From a geometrical perspective, the strain space structure is characterized by metric, because this is the basic quantity from which other important quantities can be derived, such as curvature. In strain space, the metric and the curvature correspond to distortion and disclination, respectively,12,13) and the geometrical order of the incompatibility corresponds to the curvature.10) From this we can derive the incompatibility using the basic quantity of strain space, i.e., distortion.
In the differential form, distortion β and incompatibility η correspond to 1-form and 3-form, respectively. Given that the exterior derivative operator d maps p-form to (p + 1)-form, the operator d is applied to β twice to derive η. Because dd(⋯) = 0, we cannot obtain the incompatibility simply using the expression ddβ. As such, we assume the following:
| \begin{equation} \eta = d\beta \overleftarrow{d}, \end{equation} | (1) |
| \begin{equation} \eta_{ij} = \epsilon_{jmn}\epsilon_{ikl}\partial_{m}\partial_{k}\beta_{nl}, \end{equation} | (2) |
The next expression also uses the operator d twice: d*dβ, where * is the Hodge star operator in three-dimensional space. In three-dimensional space, the operator * maps p-form to (3-p)-form; thus, the form number of *dβ is [3 − (1 + 1) = 1]. The above expression corresponds to ∇ × (∇ × β) in nabla notation. The index notation of this nabla expression is given by $\epsilon _{imn}\epsilon _{nkl}\partial _{m}\partial _{k}\beta _{lj}$. Note that the index n appears twice in the permutation tensor and it is equal to ∂i∂lβlj − ∂k∂kβij based on the well-known identity $\epsilon _{imn}\epsilon _{nkl} = \delta _{ik}\delta _{ml} - \delta _{il}\delta _{mk}$, where δij is Kronecker delta. Therefore, the index notation of ∇ × (∇ × β) is not equal to the incompatibility (2). However, some papers use this nabla notation for the index notation (2). To avoid any such confusion, here we use only the index notation of the incompatibility (2).
Next, we consider eq. (1) based on the kinematic equation of defects. According to the differential geometrical approach to defects, the dislocation density 2-form α is given by α = dβ + κ, where β is the distortion 1-form defined above, and κ is the bend-twist 2-form.12,13) Substitution of this kinematic equation into eq. (1) gives
| \begin{equation} \eta = \theta^{\alpha} - \theta^{\kappa}, \end{equation} | (3) |
Kadic and Edelen16) divides the dislocations into two types, i.e., translational and disclination-driven dislocations, by noting the homogeneity breaking of translation group T(3) and rotational group SO(3) movements in material space. In this paper, the former and the latter dislocation types are related to Hθα and Hθκ, respectively, where H is the homotopy operator, specifically, the inverse of the exterior derivative operator.17) Since H is a linear operator,17) eq. (3) means that Hη quantifies the difference in contribution between T(3) and SO(3). Hasebe et al.,10) has pointed out that the incompatibility is due to pure deformation and pure rotation. The results imply that the incompatibility provides information about the group related to the defect field, such as T(3) $ \triangleright $ SO(3), a six-parameter gauge group that leaves the Lagrangian invariant of deformation field with defects.18)
In Section 2, we consider the relationship between distortion and incompatibility in strain space. According to previous studies,12,13) strain space has a dual structure, i.e., the stress space. The metric of the stress space is a stress function, similar to the distortion found in strain space. Analogous to eq. (1), we assumed dual incompatibility 3-form in dual stress space, defined by
| \begin{equation} \eta^{\sigma} = d\psi \overleftarrow{d}, \end{equation} | (4) |
| \begin{equation} \eta_{ij}^{\sigma} = \epsilon_{jmn}\epsilon_{ikl}\partial_{m}\partial_{k}\psi_{nl}. \end{equation} | (5) |
To clarify the physical meaning of (5), we examine a particular example of this expression. When we consider the diagonal component of ψ, i.e., n = l, eq. (5) becomes $\eta _{ij}^{\sigma } = \partial _{k}\partial _{k}\delta _{ij}\psi - \partial _{i}\partial _{j}\psi $ where we set ψ11 = ψ22 = ψ33 = ψ for simplicity. Then, we obtain
| \begin{equation} \eta_{12}^{\sigma} = - \partial_{1}\partial_{2}\psi,\quad \eta_{11}^{\sigma} = \partial_{2}\partial_{2}\psi + \partial_{3}\partial_{3}\psi. \end{equation} | (6) |
In two-dimensional space, (6) becomes
| \begin{equation} \eta_{12}^{\sigma} = - \partial_{1}\partial_{2}\psi,\quad \eta_{11}^{\sigma} = \partial_{2}\partial_{2}\psi. \end{equation} | (7) |
This is in agreement with the well-known Airy stress function, in which (6) is a particular component of the three-dimensional version of the Airy stress function, i.e., the Maxwell stress function. Thus, the dual incompatibility in stress space can be recognized as stress. In this case, (5) can be interpreted as the generalized stress function, i.e., the Beltrami stress function. For example, the component of (5) is given by
| \begin{align} &\eta_{12}^{\sigma} = - \partial_{3}\partial_{3}\psi_{12} - \partial_{1}\partial_{2}\psi + \partial_{1}\partial_{3}\psi_{23} + \partial_{2}\partial_{3}\psi_{31},\\ &\eta_{11}^{\sigma} = \partial_{2}\partial_{2}\psi + \partial_{3}\partial_{3}\psi - 2\partial_{2}\partial_{3}\psi_{23}. \end{align} | (8) |
Comparing (8) with (6) and (7), we can confirm that the Beltrami stress function is the three-dimensional version of the Airy stress function with non-diagonal components, ψ12, ψ23 and ψ31.
In Section 2, we showed that the incompatibility in strain space is closely related to the disclination field. In this section, we consider the duality of the incompatibility and derive the Beltrami stress function. The results imply that we should consider the Beltrami stress function in the defect field, including disclination.
Studies on two-dimensional elasticity have shown that the Airy stress function is useful for analyzing deformation, including the dislocation field.19,20) Hasebe et al.,10) uses the Maxwell stress function (a three-dimensional version of the Airy stress function) to model and simulate the dislocation substructure. In future studies, we will consider the Beltrami stress function that generalizes the Maxwell stress function, to include non-diagonal components for analyzing the high-strength kink deformation in the LPSO phase.
Our main conclusions are as follows.
This work was supported by JSPS KAKENHI Grant Number 19H05129.
