Linear theory of 2 and 3-monoatomic multilattices: solutions of the shift vector equation

Linear elastic diatomic multilattices: Three-dimensional constitutive modeling and solutions of the shift vector equation

“…Prompt by this important observation in a recent work [11] we lay down a generic constitutive framework for monoatomic multilattices valid for the linear elastic regime in three dimensions that span all 32 crystal classes. Using this generic framework [12], we focus on monoatomic 2-lattices and examine what kind of solutions the theory [11] predicts. In the same work [12], for monoatomic 3-lattices, we find conditions for ruling the shift vectors out of the analysis and when this is not possible, we give conditions for the existence of solutions based on the standard Fredholm alternative theorem [13].…”

“…Using this generic framework [12], we focus on monoatomic 2-lattices and examine what kind of solutions the theory [11] predicts. In the same work [12], for monoatomic 3-lattices, we find conditions for ruling the shift vectors out of the analysis and when this is not possible, we give conditions for the existence of solutions based on the standard Fredholm alternative theorem [13].…”

“…We here expand this analysis to diatomic multilattices: these are multilattices each lattice of which is made of atoms of two possible chemical species [11,12]. We give the most generic expression for the energy for ( n + 1 ) diatomic multilattices and characterize explicitly the tensors present in such an expression for all 122 two-color point groups.…”

“…To be more specific, we focus on the case of diatomic 2- and 3-lattices. We lay down the energy expression for these types of lattices and utilize information from [12] for solving the shift vector equation: we use information from the monoatomic case for cases where calculations are identical. Compared to the monoatomic case [11], here not only holohedral point group is realized by diatomic 2-lattices; this results in cases where the shift vector equation cannot be solved for the shift vector in terms of the strain tensor uniquely so we give conditions for the existence of solutions using the Fredholm alternative [13].…”

“…Section 6 specifies the analysis to diatomic 2- and 3-lattices following closely the monoatomic counterpart approach [11]. For diatomic 2-lattices, we use information [12] in cases when this is possible: these are the cases where the third-order tensor is equal to zero. For those groups that the third-order tensor is not equal to zero, we give conditions for the existence of solutions of the shift vector equation based on the standard Fredholm alternative theorem [13].…”