Multi‐scale modeling of plastic deformations in nano‐scale materials; Transition to plastic limit

Abstract

1181has attracted much attention toward computational techniques in scientific research communities. The multi-scale modeling of materials in nanoscale, such as carbon nanotubes [1], nano-wires [2,3], thin films [4], bone structures [5,6], and biological materials [7] is an obvious sign of this attraction. Among the various numerical methods used in computational nano-mechanics, the finite element method (FEM) [8], the molecular dynamics [9], and the multi-scale analysis [10] can be mentioned as the most popular techniques. The constitutive model used in FEM and multi-scale methods is usually based on the Cauchy-Born (CB) hypothesis, which is well documented by Park and Liu [11].Because of the size of nanostructures, the ratio of surface area to volume is large and therefore makes the stresses in surface atoms be a couple of magnitudes larger than the atoms in the bulk of the material. This effect, known as the surface effect, is a property of nanostructures. Basically, the surface effect can be attributed to the difference in chemical bonds at the surface and the bulk of the structure. Sander [12] illustrated that the difference between atomic bonds is due to the redistribution of electric charge produced by the reduced atomic coordination in the surface of the structure. A large number of analytical and computational models have been proposed in the literature to study the surface effects in nanostructures. Analytical models are generally based on the definition of a surface-dependent constitutive formulation for continua. The augmented continuum theory by Gurtin and Murdoch [13,14] introduces the concept of surface elasticity tensor into the conventional theory of elasticity. The surface CB model by Park et al. [15] is based on the decomposition of total potential energy of the system into bulk and surface parts that was used to simulate metallic nanowires [16] and silicon nanostructures [17]. Because the model cannot effectively capture the stresses at corners and surfaces of the structure, the boundary CB model was developed by Qomi et al. [18], employing the surface, edge, and corner elements in addition to the conventional bulk element. The linear interpolation employed in their work requires a large number of quadrature points to be used within the boundary elements. Moreover, the method fails to model arbitrary boundary conditions. In order to circumvent the aforementioned deficiencies, Khoei et al. [19] developed the modified boundary CB method that benefits from a local regression in the vicinity of quadrature points, called the radial quadrature method. The method eliminates the necessity to use large number of Gauss points, and it is applicable to arbitrary boundary conditions.The behavior of nanostructures in transition to plastic limit has been studied by Sunyk et al. [20,21]. They employed the relaxation method proposed by Tadmor et al. [22] to capture the plastic phenomena. In their study, additional degrees of freedom were introduced when the material becomes unstable because of the loss of r...

A compatible mixed finite element method for large deformation analysis of two‐dimensional compressible solids in spatial configuration

“…Jahanshahi 17 developed a new integration algorithm for $$$ {J}_2 $$$ plasticity based on a three‐field Hu–Washizu functional and the multiplicative decomposition of deformation gradient into elastic and plastic parts. Khoei and Jahanshahi 18 used the reversed multiplicative decomposition of deformation gradient to extend the classical theory of plasticity to the realm of nano‐structures. Khoei et al 19 studied the homogeneity assumption of deformation gradient in plastic analysis of nano‐structures and proposed a validity surface in stress‐space.…”