Analytical and numerical studies on local and nonlocal elastic bars in tension and neutralizerbased iterative methods
拉伸狀態下的局部或非局部彈性杆的解析與數值研究及基於中化子的叠代法
Student thesis: Doctoral Thesis
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Award date  3 Oct 2012 
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Permanent Link  https://scholars.cityu.edu.hk/en/theses/theses(9876c1551c844aaa897984fabd81d73e).html 

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Abstract
In this thesis, we study some problems associated with both local and nonlocal elasticity
and, present the socalled neutralizerbased iterative methods for integral equations
of the second kind. The details are as follows.
The local problem is on the strain softening of materials. Strainsoftening, i.e.,
the decrease of stress with the increase of strain, which is a common postpeak phenomenon that has been recorded for a variety of materials. Snapback due to strain
softening may be one of the most interesting and the most common structural instability
phenomena observed in experiments. There have been many efforts in the past
decades to investigate strainsoftening with localization experimentally, numerically,
and analytically. However, there is not any analytical study with general nonlinear
constitutive relations in the open literature which explores the role played by the convexity of the constitutive curve of the softening part and the coupling effect between
this convexity and the size. Also, both snapback and snapthrough were observed in
some experiments, but no analytical results are available for explaining the transition
from snapback to snapthrough.
Nonlocal elasticity is a growing direction of continuum mechanics nowadays.
There are many works contributed to this area. Due to the essential difficulty of the
integral equation, analytical approximate solutions are usually prohibited. Thus, many
existing literature devote to apply the approximate differential models suggested by
Eringen et al. However, one weak point of the approximations is that the possible boundary effects, which are present in the integral formulations, are neglected. Thus,
it would be desirable to have proper differential formulations which take into account
such effects. For this purpose, a first step is to know both qualitative and quantitative
behavior near the ends (e.g., the influences of the parameters). Thus, some analytical
solutions are needed to provide convincing results.
In this thesis, firstly for the local problem, we modify an existing model and set up
the stressstrain equations for the structure in the postpeak region, which are nonlinear
as compared with the bilinear case in the literature. After some analysis, we derive the
mathematical conditions for the occurrence of several important curves as frequently
observed in experiments, including the snapthrough (which cannot be captured by the
bilinear assumptions). Two examples are also given to illustrate these cases, and the
postpeak curves are consistent with our theoretical predictions.
Secondly, for a static tension problem in nonlocal elasticity (the uniform case), we
apply an existing iterative method that are efficient for a special kind of kernel to handle
the resultant integral equation. By explicitly evaluating the integral in the second
iterative solution, we are able to get a good approximate analytical solution for this
problem. Some features of the nonlocal theory can then be closely examined, especially
the boundary effects. It seems that the analytical results obtained here would
give some insights into nonlocal theory, particularly for its applications in nanomaterials.
Moreover, in view of the boundary effects, we also present a new model for
nonuniform nonlocal bara varying volume fraction in the nonlocal phase. The numerical
results of different shapes of materials show that, as compared with a uniform
bar and that in local elasticity, the model herein shows more features, such as stress
concentration.
Thirdly, we present neutralizerbased iterative methods for integral equations of
the second kind. As is known to all, there is an abundant of numerical techniques
for solving integral equations, such as Neumann series, multigrid method, GMRES and so on. Here, we introduce the concept of neutralizer, which is sometimes used
in dealing with integrals (e.g., the asymptotic expansions of integrals), to obtain the
neutralizerbased iterative method. Some features of such iterative method is numerically
explored. Several meaningful examples are given, showing that the methods
perform well as compared with some related methods.
 Integral equations, Elasticity, Numerical solutions, Iterative methods (Mathematics)