Table 1 The extensions of CBR for different types of materials Monolayer crystal film Space-filling material Non-centrosymmetric Centrosymmetric Non-centrosymmetric Centrosymmetric a = F·(A + ) a = F·A All the deformation gradients in monolayer crystal film are based on inscribed surface.
where A and a denote one lattice vector in the undeformed and deformed crystals, respectively; the deformation gradient based on the inscribed surface of F(A/2, ) is function of the position of middle point of undeformed lattice vector A as well as the inner shift vector.
Firstly, Drucker Prager proposed the smooth circular conical surfaces, including the circumscribed, inscribed, and middle ones [2, 3], such that the ideal elasto-plastic constitutive model based on the Drucker Prager criterion was one of the earliest models used for soil and rock and was embedded in some of the earliest commercial finite element programs, such as SAP, ADINA, ANSYS.
However, based on the physical concept of inscribed surface, the present study discovered that the standard CBR can be extended straightly to describe monolayer crystal films accurately.
Based on this congenital advantage of inscribed surface, a new extension of CBR, called inscribed CBR, is proposed to build a rigorous and accurate atomistic continuum theory.
The first one is the diameter of an inscribed cylinder whereas the second one is the result of a numerical simulation based on the 3D image of the strut characterized by X-ray tomography.
The TP permeability and IP permeability of unidirectional fibers are determined first, based on that the minimum distance and the inscribed radius between fibers are adopted as the characteristic lengths for normal and parallel flows, respectively.
We assess CFL conditions based on different element size measures, among which only the radius of the inscribed circle and the shortest height prove appropriate, although they are not totally independent of the triangle shape.
A joint method based on the relative diameter is introduced for evaluating the minimum circumscribed circle and maximum inscribed circle in this article.
The proposed method is an approximate method based on the least square circle solution, and just for the minimum circumscribed circle and maximum inscribed circle criteria.
