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E.F. Sheka et al.
tions of macroscopic linear theory of elasticity and lay the foundation for the evaluation of micro-macroscopic mechanical parameters such as Young’s modulus (E ∗ ),
the Poisson ratio (ν ∗ ), and so on. Nothing to mention that parameters E and E ∗ as
well as ν and ν ∗ are not the same so that their coincidence is quite accidental. Obviously, the atomistic approach falls in opinion comparing with the continuum one
due to time consuming calculations and, as a result, due to applicability to smaller
objects. However, it possesses doubtless advantages concerning the description of
the mechanical behavior of the object under certain loading (shape changing) as
well as exhibiting the deformation and failure process at the atomic level. A serious
deficiency of both standard approaches is their close links with the theory of elasticity, which drives the graphene into the Procrustean bed of elastic deformation,
depriving it of the right to permanent plastic behavior.
Recently a new atomistic approach has been suggested for the description of
the graphene deformation based on considering the failure and rupture process of
graphene as the occurrence of a mechanochemical reaction [29–32]. A similarity
between the mechanically induced reaction and the first-type chemical ones, first
pointed out by Tobolski and Eyring seventy years ago [33], suggested the use of
a well developed quantum-chemical approach of the reaction coordinate [34] in
the study of the atomic structure transformation under deformation. Firstly applied
to the deformation of poly(dimethylsiloxane) oligomers [35], the approach has revealed a high efficacy in exhibiting elastic, plastic, and superplastic regions of the
uniaxial tension of the oligomer, disclosing the mechanism of its failure and rupture.
It has been successfully applied recently for the description of the uniaxial tension
of both graphene [29, 30] and graphane [31] molecules, thus positioning itself as a
significant part of the molecular theory of graphene [28].
The main point of the approach concerns the reaction coordinate definition.
When dealing with chemical reactions, the coordinate is usually selected among
the internal ones (valence bond, bond angle or torsion angle) or is presented as a
linear combination of the latter. Similarly, mechanochemical internal coordinates
(MICs) are introduced as modified internal coordinates defined in such a way as to
be able to specify the considered deformational modes [35, 36]. Thus, uniaxial tension and contraction are described by linear MICs similar to valence bonds. In the
case of tensile deformation, the benzenoid pattern of graphene sheets and a regular
packing of the units predetermined the choice of either parallel or normal MICs orientation with respect to the chain of C–C bonds. In the rectangular nanographene
sheets and nanoribbons the former orientation corresponds to tensile deformation
applied to the zigzag edges (zigzag mode) while the latter is attributed to the armchair edges (armchair mode). The MIC configurations of the two tensile modes of
the (5, 5) NGr molecule are presented in Fig. 16.1. The molecule lays the foundation of previously performed computational experiments [29–32] and presents a
rectangular fragment of a graphene sheet that is cut along zigzag and armchair edges
and contains 5 benzenoid units along each direction. The deformation proceeds as
a stepwise elongation of the MICs with the increment δL = 0.1 Å at each step so
that the current MIC length constitutes L = L 0 + nδL, where L 0 is the initial length
of the MIC and n counts the number of the deformation steps. The right ends of
E.F. Sheka et al.
tions of macroscopic linear theory of elasticity and lay the foundation for the evaluation of micro-macroscopic mechanical parameters such as Young’s modulus (E ∗ ),
the Poisson ratio (ν ∗ ), and so on. Nothing to mention that parameters E and E ∗ as
well as ν and ν ∗ are not the same so that their coincidence is quite accidental. Obviously, the atomistic approach falls in opinion comparing with the continuum one
due to time consuming calculations and, as a result, due to applicability to smaller
objects. However, it possesses doubtless advantages concerning the description of
the mechanical behavior of the object under certain loading (shape changing) as
well as exhibiting the deformation and failure process at the atomic level. A serious
deficiency of both standard approaches is their close links with the theory of elasticity, which drives the graphene into the Procrustean bed of elastic deformation,
depriving it of the right to permanent plastic behavior.
Recently a new atomistic approach has been suggested for the description of
the graphene deformation based on considering the failure and rupture process of
graphene as the occurrence of a mechanochemical reaction [29–32]. A similarity
between the mechanically induced reaction and the first-type chemical ones, first
pointed out by Tobolski and Eyring seventy years ago [33], suggested the use of
a well developed quantum-chemical approach of the reaction coordinate [34] in
the study of the atomic structure transformation under deformation. Firstly applied
to the deformation of poly(dimethylsiloxane) oligomers [35], the approach has revealed a high efficacy in exhibiting elastic, plastic, and superplastic regions of the
uniaxial tension of the oligomer, disclosing the mechanism of its failure and rupture.
It has been successfully applied recently for the description of the uniaxial tension
of both graphene [29, 30] and graphane [31] molecules, thus positioning itself as a
significant part of the molecular theory of graphene [28].
The main point of the approach concerns the reaction coordinate definition.
When dealing with chemical reactions, the coordinate is usually selected among
the internal ones (valence bond, bond angle or torsion angle) or is presented as a
linear combination of the latter. Similarly, mechanochemical internal coordinates
(MICs) are introduced as modified internal coordinates defined in such a way as to
be able to specify the considered deformational modes [35, 36]. Thus, uniaxial tension and contraction are described by linear MICs similar to valence bonds. In the
case of tensile deformation, the benzenoid pattern of graphene sheets and a regular
packing of the units predetermined the choice of either parallel or normal MICs orientation with respect to the chain of C–C bonds. In the rectangular nanographene
sheets and nanoribbons the former orientation corresponds to tensile deformation
applied to the zigzag edges (zigzag mode) while the latter is attributed to the armchair edges (armchair mode). The MIC configurations of the two tensile modes of
the (5, 5) NGr molecule are presented in Fig. 16.1. The molecule lays the foundation of previously performed computational experiments [29–32] and presents a
rectangular fragment of a graphene sheet that is cut along zigzag and armchair edges
and contains 5 benzenoid units along each direction. The deformation proceeds as
a stepwise elongation of the MICs with the increment δL = 0.1 Å at each step so
that the current MIC length constitutes L = L 0 + nδL, where L 0 is the initial length
of the MIC and n counts the number of the deformation steps. The right ends of