\r\n\tThe output uncertainty in mathematical models can be reduced by using uncertainty and sensitivity analysis.
\r\n\tThis book intends to provide the reader with a comprehensive overview of uncertainty quantification and sensitivity analysis, their implementations and possible applications in different fields of science and technology.
The development of molecular modeling tools to describe and predict the mechanical properties of structural materials reveals an undeniable practical importance. It is now well recognized that such an objective can be achieved through the linking of the structure of materials at the nanoscale or with their performances. At the nanometer scale, anisotropic materials exhibit differences due to the directional arrangements of atomic structure, the force between atoms, such as van der Waals force, typical Coulomb force, and other forces. Nanoscale modeling and mechanical properties by using the density functional theory (DFT), a so-called atomic finite element method (AFEM), and the classical molecular dynamics (MD) method are especially concerned according to the modeling requirement of different crystal structures investigated. Omitting the description of these structures’ importance, elastic moduli are separately calculated by either homogenization or curve fitting of the linear portion of the stress-strain curve by the corresponding numerical simulations. This chapter is committed to introduce the modeling and simulation for calculating mechanical properties (Young’s modulus especially) of typical anisotropic crystal structures using three methods (DFT, AFEM, and MD) mentioned above. It is therefore asked to connect to the nanoscale modeling and continuous pattern of behavior by identifying the relevant output data at small scales and bringing the necessary information to higher scales.
Keywords: nanoscale, crystal structures, anisotropic elasticity, DFT, AFEM, molecular dynamics simulation, nanoindentation, mechanical properties
Nanotechnology is considered “high-tech” since it requires deep knowledge of the system considered at the nanoscale in order to intelligently design. Nanomaterials refer to one dimension in the nanometer range (nanosized particles, atomic clusters etc.) or a basic unit at least among a three-dimensional space (nanotubes, ultrathin films, multilayers, superlattices, etc.). Nanoscience and technology turns up in the late 1980; its basic meaning is to understand and transform the nature in the nanometer size range (10−10–10−7 m), directly through the manipulation and arrangements of atoms. Material properties mainly include chemical properties, physical properties (density, viscosity, particle size, specific heat, thermal conductivity, linear expansion coefficient, etc.), and mechanical properties (elasticity, hardness, strength, plasticity, ductility, toughness, impact resistance, fatigue limit, etc.). At nanoscale, the crystalline material with the smallest unit holds anisotropy, meaning that crystal has different properties in different directions.
With the development of material science and technology, the scale of material has expanded from macro to micro, also from micro to nanoscale. Nanostructure scientific research at nanometer scale includes elaboration, observation, characterization, and analysis. Modeling of miniaturized materials has a lot of new properties, where characteristic space and time scales correspond to typical simulation methods. Modeling and applications of structures at typical space/time scales  are shown in Figure 1.
From Figure 1, for electronic structure, DFT-QHA method is commonly used, while for atomistic structure, Monte Carlo method or molecular dynamics is used. However, real materials are never perfectly isotropic. At nanoscale, the crystal structure reorientations are not a direct result of the applied stress but are a geometrical requirement. Bulk anisotropy due to crystal orientation is therefore induced by plastic strain and is only indirectly related to stress. Recrystallization during annealing usually changes the crystallographic texture but does not cause randomness . When the grain size is reduced to the nanoscale, its Young’s modulus and hardness have great changes. The microscopic deformation mechanism is very complex for nanograins, which depends on the shape of the crystal orientation, surface effect, the substrate effect, the grain boundaries affect, etc. In some cases (e.g., composite materials, single crystal), the differences in properties for different directions are so large that one cannot assume isotropic behavior, which leads to another behavior—anisotropic.
Anisotropic behavior is widespread among the variety of materials, and it has a great meaning in practical engineering application in material science field. Anisotropy is the property of being directionally dependent, as opposed to isotropy, which implies identical properties in all directions. It can be defined as a difference, when measured along different axes, in a material’s physical or mechanical properties. From the plastic performance perspective, the anisotropy of plastic deformation of the material shows the performance of texture, ears, and other aspects. Mechanical working produces preferred orientations or crystallographic textures. The primary cause of anisotropy of plastic properties is preferred orientation of grains. In contrast, anisotropy of fracture behavior is largely governed by mechanical alignment of inclusions, voids, and grain boundaries. However, from the elastic performance perspective of crystal structure, the mechanism of elastic anisotropy behavior is still not being recognized, especially for knowing the connection between atomic bonds and mechanical properties of anisotropic crystals structures at nanoscale. Therefore, a comprehensive understanding of the intrinsic nature and the anisotropy behavior at nanoscale has become an urgent task.
Essentially, material properties due to anisotropy are closely related to its internal structure and deformation conditions in the research field of materials science. The microstructure provides a link between processing (how a material is made) and properties (how a material behaves) . It relates with the thermal, electrical, and mechanical properties of a crystal . Alongside the experiment and theory, numerical simulation is a way additional access to the understanding of physical systems. Indeed, it can calculate experimentally measurable quantities and predict properties that are inaccessible in the laboratory or in the model to be validated. In general, both the chemical structure and the microstructure of a material control its properties, of which the chemical structure is relatively fixed and the microstructure depends strongly on how the material is made. However, from mesoscopic level to atomic scale, the drop of material scale makes the contained atoms of nanometer system greatly reduced; besides, macroscopic quasistationary continuous band disappeared, thus showing the energy level separation. The quantum size effect makes physical properties of nanosystem different from other conventional materials, leading to many novel features. At the nanometer scale, anisotropic materials exhibit differences due to these directional arrangements of atomic structure, the force between atoms, such as van der Waals force (caused by hydrogen bonding, ionic bond), typical Coulomb force (caused by charge), and other forces caused by covalent bond, atomic distortion, defects, hydrophilic, etc. Numerical simulation of anisotropic plastic behavior has been investigated by Yonggang  in Harvard University, while the numerical simulation and elastic properties of crystal structure at nanoscale have not been investigated systematically.
Although material performance is not the same in different directions in crystal, however, it has a strict symmetry. Anisotropic behavior of crystalline/structure can be reflected in the constitutive relation, where the elastic coefficient matrix corresponding to crystal system is certain. For different crystal/system, the number of cubic crystal has the least number of 3, hexagonal crystal 6, monoclinic crystal 13, and triclinic crystal with the most number of 21. Elasticity coefficient has important practical significance on the scientific basis and engineering applications of the material. Different crystal systems can be characterized exclusively by their symmetries. Each crystal has a certain level of crystal symmetry with corresponding different number of independent elastic coefficients. By obtaining elastic coefficient matrix, homogenized elastic properties (bulk, shear, and Young’s moduli) of polycrystals at larger scale can be determined, using the Voigt-Reuss-Hill estimation, for example. Because of complex issues mentioned above, for different anisotropic structure, selecting the appropriate modeling approach is particularly critical.
With the development of the modeling and computer technology, the scale of material has expanded from macro to micro, also from micro to nanoscale. Nanomechanics aims to study fundamental mechanical properties of material structure at the nanoscale. Nanoscale modeling methods mainly include quantum molecular (QM), Monte Carlo (MC), molecular (structural) mechanics (MM) , molecular dynamics (MD) , nonlocal continuum theory , the ab initio calculation , tight-binding molecular dynamics (TBMD), and the density functional theory (DFT) . Typical application, different space scale, the corresponding time scales, and simulation methods in computational materials science fields have been discussed by Raabe [1, 10]. Molecular simulation is also known as molecular modeling, which refers to theoretical methods and computational techniques to model or imitate the behavior of molecular. The model size at different scales has gradually decreased to continuum and mesoscopic level (material model diameter greater than 10−4 m), the mesolevel (about 10−6–10−4 m), the microlevel (about 10−7–10−6 m), and atomic/nanolevel (about 10−10–10−7 m). Figure 2 shows typical scales of material computational mechanics.
From Figure 2, scales in computational mechanics field can be divided into four scales: the macro (macroscale), mesoscopic (mesoscale), microscopic (microscale), and nano (nanoscale). Correspondingly, the continuum mechanics is commonly used to solve the macroscale structures, where finite element method can be used. For molecular dynamics for microscale, mesoscale, and nanoscale structures, for example, the discrete model can be an option at mesoscale. For quantum mechanics for both nanoscale and atomic scale structure, for example, density functional theory can be an effective option.
During the last decade, nanomechanics has emerged on the crossroads of classical mechanics, solid-state physics, statistical mechanics, materials science, and quantum chemistry. Numerical methods represent the most versatile computational method for the various engineering disciplines, and the scale of material modeling is gradually transited from bulk scale to nanoscale in Figure 3.
From Figure 3, the simulation approach for the structure in each domain is not all the same. The internal distance of each