Convolution method in elastodynamics

Theoretical solid mechanics is mainly based on the equations of motion in terms of displacement fields. Using the constitutive and kinematic equations, the equations of motion that initially involve the stress fields are written in terms of spatial and time derivatives of the displacement field. These equations are at the center of analytical and computational attempts in solid mechanics. However, this approach contains seemingly unresolvable challenges. As an example, in nonlinear elastodynamics, it is well-understood that the Newmark average acceleration method (the trapezoidal rule) fails to conserve the balance of energy and the balance of angular momentum. While several algorithms in the literature attempted to address this challenge, all of them are based on a rather intricate formulation that still falls short in some practical problems. In this dissertation, by contrast, wave phenomena in elastic materials are investigated through the so-called alternative field equations where the convolution product appears in the place of time derivative. In the first step of the study, the stress equations of motion for electromagneto-elastic materials are derived. As the name suggests, in the new framework, the equations of motion are written in terms of the stress field. These equations provide a new foundation to establish numerical methods in terms of stress rather than displacement, which is an appropriate strategy for addressing problems with Neumann boundary conditions. In the next step, based on the alternative field equations, convolutional variational principles for electromagneto-elastic materials are established. Upon convolutional variational principles, a new time-domain finite element method, namely, Convolution Finite Element Method (CFEM), is then introduced for linear elastodynamics problems. It is shown that this new dynamic finite element approach inherits surprising characteristics that cannot be found in the classical methods. Finally, by using the CFEM, a new solution procedure for nonlinear elastodynamics is established. The new algorithm, which is based on the CFEM and Newton-Raphson method, leads to a simple solution procedure while conserving energy and angular momentum.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Theoretical & Applied Mechans

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2022

Author dc:creator

• Amiri Hezaveh, Amirhossein

Contributors dc:contributor

• Ostoja-Starzewski, Martin
• Masud, Arif
• Sehitoglu, Huseyin
• West, Mathew

• Elastodynamics
• Finite Element Method
• Nonlinear Mechanics

Statement dc:rights

• Copyright 2022 Amirhossein Amiri Hezaveh

Language dc:language

en, eng

Handle dc:identifier

https://hdl.handle.net/2142/115941

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F115941