Linear Elasticity of Elastic Circular Inclusions Part 2/Lineare Elastizitatstheorie Bei Kreisrunden Elastischen Einschlussen Teil 2

The book provides real analytic solutions for the "Disc with Circular Inclusion" under normal and shear force at plane-strain state, validated by numeric FEM solution results. The solutions are in terms of mechanical quantities and can be used to study the behavior of the disc under different loading conditions.

Format: Paperback / softback
Length: 100 pages
Publication date: 05 June 2021
Publisher: Springer

This new and revised edition presents the authentic analytic solutions for the "Disc with Circular Inclusion" under normal- and shear force at plane-strain state. The accompanying solution process, which was developed in accordance with the principle of statically indeterminate systems, is thoroughly documented. The solutions are presented in terms of mechanical quantities (deformations, strains, and stresses). Due to the superposition of the solutions for normal force in the x- and y-direction and shear force, the plane strain-stress relation can be formulated. The validation of the real analytic solutions is carried out by numerical FEM solution results. Comparing the results of the finite and infinite disc, there is, however, a very high correspondence of all mechanical quantities. Therefore, it can be assumed that the real analytic solutions are the exact solutions.

Introduction:
The study of the behavior of solid bodies under various loading conditions is an essential aspect of mechanical engineering. One such loading condition is plane strain, where the body is subjected to a combination of normal and shear forces. In this context, the "Disc with Circular Inclusion" problem is of particular interest as it involves the analysis of a disc with a circular inclusion subjected to plane strain conditions.

Previous Work:
Previous studies on the "Disc with Circular Inclusion" problem have primarily focused on the use of finite element methods (FEM) to obtain numerical solutions. While FEM is a powerful tool for modeling and analyzing complex structures, it has certain limitations, such as the need for a large number of elements and the potential for numerical instability.

Objective:
The objective of this new and revised edition is to present the authentic analytic solutions for the "Disc with Circular Inclusion" problem under normal- and shear force at plane-strain state. The associated solution process, which was developed according to the principle of statically indeterminate systems, is documented extensively. The solutions are given in terms of mechanical quantities (deformations, strains, and stresses).

Solution Process:
The solution process for the "Disc with Circular Inclusion" problem involves the following steps:

• Define the geometry of the problem: The disc with a circular inclusion is modeled as a two-dimensional solid body. The dimensions of the disc and the radius of the circular inclusion are specified.
  
• Apply the boundary conditions: The disc is subjected to plane strain conditions, which means that the normal stress and shear stress are zero at the boundary of the disc. The circular inclusion is also subjected to plane strain conditions, which means that its displacement and rotation are constrained to be zero.
  
• Develop the governing equations: The governing equations for the "Disc with Circular Inclusion" problem are derived using the principle of statically indeterminate systems. The equations involve the deformation, strain, and stress fields in the disc and the circular inclusion.
  
• Solve the governing equations: The governing equations are solved using analytical methods, such as the method of characteristics or the method of finite differences. The solutions are obtained in terms of mechanical quantities (deformations, strains, and stresses).
  
• Validate the solutions: The validity of the analytic solutions is verified by comparing them with numerical FEM solution results. The comparison is carried out by computing the stresses and strains at different points in the disc and the circular inclusion.
  

Results:
The real analytic solutions for the "Disc with Circular Inclusion" problem under normal- and shear force at plane-strain state are presented in this new and revised edition. The solutions are given in terms of mechanical quantities (deformations, strains, and stresses).

• Deformations: The deformations of the disc and the circular inclusion are computed using the real analytic solutions. The results show that the deformations are non-linear and depend on the radius of the circular inclusion and the normal stress.
  
• Strains: The strains of the disc and the circular inclusion are computed using the real analytic solutions. The results show that the strains are non-linear and depend on the radius of the circular inclusion and the normal stress.
  
• Stresses: The stresses of the disc and the circular inclusion are computed using the real analytic solutions. The results show that the stresses are non-linear and depend on the radius of the circular inclusion and the normal stress.
  

Conclusion:
This new and revised edition presents the authentic analytic solutions for the "Disc with Circular Inclusion" problem under normal- and shear force at plane-strain state. The associated solution process, which was developed according to the principle.

Weight: 177g
Dimension: 234 x 156 x 6 (mm)
ISBN-13: 9783030723965
Edition number: 2nd 2021 ed.