An introduction and tutorial on multiple-scale analysis in solids

The approximate solution is numerically calculated to yield comparisons with the uncoiled case. It is found that the coiling effects are long-wave in character and the results do not differ significantly from the corresponding straight bow model. In mathematics and physics, multiple-scale analysis comprises techniques used to construct uniformly valid approximations to the solutions of perturbation problems, both for small as well as large values of the independent https://wizardsdev.com/ variables. This is done by introducing fast-scale and slow-scale variables for an independent variable, and subsequently treating these variables, fast and slow, as if they are independent. In the solution process of the perturbation problem thereafter, the resulting additional freedom – introduced by the new independent variables – is used to remove secular terms. The latter puts constraints on the approximate solution, which are called solvability conditions.

In the latter case, multiple-scale analysis elucidates the connection between weak-coupling perturbative and semiclassical nonperturbative aspects of the wave function. In multiple scale method, the independent variable will be replaced by several variables, each with a scaled down speed of variation. Replacing the independent variable, makes a nonlinear ordinary differential equation to be transformed to a series of linear partial differential equations. Combination of the solutions of the linear partial differential equations make the approximate solution of the original nonlinear equation.

A Laplace-Transform multiple-scale procedure for the asymptotic solution of weakly non-linear partial differential equations

Also, it is shown that by defining the thermostat in the atomistic part, wave reflections are eliminated at the interface of atomic and continuum domains. It is shown that by selecting appropriate dimensions of the atomic domain, there is no need to use nonlinear elasticity in the continuum region. Also, hardness is more affected by sample size than the elastic modulus. The method of multiple scales is applied to the analysis of a curved box model for the spirally coiled cochlea. The fluid motion is fully three-dimensional and the basilar membrane movement is represented by a single mode of deflection. Coiling parameters for the human cochlea are determined from the experimental data of von Békézy [Experiments in Hearing (McGraw-Hill, New York, 1960)].

• The Young’s modulus, shear modulus and density of porous materials are assumed to vary through the thickness direction based on the assumption of a common mechanical feature of the open-cell foam.
• Relevant to disciplines across engineering and physics, the asymptotic method combined with the multiple scale method is shown to be an efficient and intuitive way to approach mechanics.
• Exploiting a multi bias scheme for a single graphene layer provide opportunity to affect device reaction via bias itself and patterns period simultaneously which increase adjustability of device response.
• The traction over the crack is included as a unknown field in the equations system of the problem, and the jump displacement across the discontinuity is obtained with a cohesive constitutive relation (traction-separation law).
• In this paper, a hierarchical RVE-based continuum-atomistic multi-scale procedure is developed to model the nonlinear behavior of nano-materials.

Therefore, the present method is not only valid for weakly nonlinear damped forced systems, but also gives better result for strongly nonlinear systems with both small and strong damping effect. However, this introduces possible ambiguities in the perturbation series solution, which require a careful treatment (see Kevorkian & Cole 1996; Bender & Orszag 1999). Nevertheless, if $\epsilon$ is small, you can consider $\fracz$ as a perturbation term. W.K. Liu, E.G. Karpov, S. Zhang, H.S. Park, An introduction to computational nano mechanics and materials, Comput…. Alternatively, modern approaches derive these sorts of models using coordinate transforms, like in the method of normal forms, as described next.

RELATED PAPERS

In this paper, two simplified concurrent multiscale methods, one with handshake region and another without handshake region are used to investigate the nanoindentation process on a single crystal of Al at room temperature. The multiscale models are validated by observing reasonably well similarities in the load–depth curves obtained from multiscale and full MD simulations. Refining the element size down to atomic spacing resulted in high computational efforts while the analysis results do not improve significantly.

This agrees with the nonlinear frequency changes found by employing the Lindstedt–Poincaré method. Two-scale convergence is adapted to the homogenization of photonic quasi-periodic structures such as Penrose tilings and the vector Maxwell system to generate quasi crystals by considering a periodic structure in an upper-dimensional space. Sensitivity analysis of the frequency response of a piecewise linear system in a frequency island.

Two‐scale cut‐and‐projection convergence; homogenization of quasiperiodic structures

In addition, we found that the dual term of nonlinearity is essential to obtain the class of analytic solution. This extension greatly enriches the predictive capacity of the multiscale approach for simulating saturated granular soils relevant to a wide variety of important civil/mining applications. The new approach is first benchmarked by closed-form solutions to classical 1D and 2D consolidation problems.

The new model still satisfies the fundamental energy conservative property as the original models. We then apply the energy method to prove the well-posedness of the model under the solitary wave hypothesis. Some categories of exact solitary wave solutions of the model are described by using the Ansatz method.

Concurrent coupling of length scales: methodology and applications

However, less research is reported on the applicability of the different SIF-based criteria when they are used to analyze the crack propagation process of concrete with different strength grades. With this objective in mind, three-point bending and four-point shear tests were conducted in this study on C20, C50 and C80 grade concrete to measure the initial fracture toughness, fracture energy, load-crack mouth opening/sliding displacement (CMOD/CMSD). The results indicated that the difference between the peak loads from experiment and from the analysis based on the nil SIF criterion with KII approximately increases with the increase of the concrete strength. By contrast, the predicted peak load and P-CMOD/CMSD curves adopting the initial fracture toughness-based criterion with KII showed better agreement with experimental results for the different concrete strength.

Mathematics research from about the 1980s proposes that coordinate transforms and invariant manifolds provide a sounder support for multiscale modelling . The terahertz frequency band becomes a growth platform of various applications from medical imaging to indoor communications. Emerging new materials such as graphene and developing reliable models paved the design way for graphene-based microstructures. This paper proposes a relatively comprehensive design methodology for graphene-based multi-layers structures. The procedure includes forming device geometry, finding graphene patterns, material types, and optimizing control parameters.

Exploiting a multi bias scheme for a single graphene layer provide opportunity to affect device reaction via bias itself and patterns period simultaneously which increase adjustability of device response. Also using two different graphene patterns turns the device complex regarding design multi-scale analysis optimizations and simulations. So a well-known and simple circuit representation is used to design the proposed methodology and the proposed device. Knowing equivalent circuit models for the device elements triggers developing an evolutionary algorithm to search for a desirable response.