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Convexity Conditions and Energy Minimization for Highly Deformable Elastic Surfaces
Convexity Conditions and Energy Minimization for Highly Deformable Elastic Surfaces
상세정보
- 자료유형
- 학위논문 서양
- 최종처리일시
- 20250211152129
- ISBN
- 9798384050117
- DDC
- 519
- 서명/저자
- Convexity Conditions and Energy Minimization for Highly Deformable Elastic Surfaces
- 발행사항
- [Sl] : Cornell University, 2024
- 발행사항
- Ann Arbor : ProQuest Dissertations & Theses, 2024
- 형태사항
- 96 p
- 주기사항
- Source: Dissertations Abstracts International, Volume: 86-03, Section: B.
- 주기사항
- Advisor: Healey, Timothy.
- 학위논문주기
- Thesis (Ph.D.)--Cornell University, 2024.
- 초록/해제
- 요약The theory of thin elastic surfaces is a source of many fascinating problems in the calculus of variations. When an elastic surface is deformed, its energy can be roughly decomposed into two parts---a "stretching/membrane energy'' that is non-convex in the derivative of the deformation map and a "bending energy'' that depends on higher order derivatives of the deformation map. The interplay between these two terms gives rise to a variety of interesting phenomena unique to thin elastic objects. We are in part motivated by wrinkling observed in highly stretched polymer sheets. This thesis has three parts: In Chapter 3, we consider a wide class of models known as "Cosserat shells'' and identify a new, physically meaningful convexity condition that leads to the existence of energy minimizers for these models. We argue that this convexity condition is suitable for predicting wrinkling phenomena. In Chapter 4, we focus on a pure membrane version of the model in Chapter 3, which in general is not even rank-one convex. Nevertheless, we prove that it admits energy minimizers when the image surface is constrained to lie on some prescribed embedded oriented surface in ℝ3. Under additional assumptions, we show that the minimizers are homeomorphisms onto their image and are weak solutions to the spatial equilibrium equations. In both Chapter 3 and Chapter 4, we ensure that the minimizers are locally injective/orientation preserving by requiring the energy density function to grow unboundedly as an appropriate notion of the local (signed) area/volume measure approaches zero. In Chapter 5, we study the membrane energy from the viewpoint of 3D to 2D dimension reduction. We embed our 3D variational problems into an appropriate class of parametrized measures and obtain a compactness result as the thickness of the body goes to zero. The putative membrane limit is defined on a class of locally injective gradient Young measures. Our main reason for choosing a Young measure approach for dimension reduction is that it can be used to uniquely identify a non-relaxed membrane energy density. Thus, membrane energy densities obtained this way do not preclude unbounded growth near non locally-injective configurations and can capture fine oscillations of minimizing sequences, like in the case of wrinkling.
- 일반주제명
- Applied mathematics
- 일반주제명
- Mathematics
- 일반주제명
- Theoretical mathematics
- 키워드
- Homeomorphisms
- 기타저자
- Cornell University Applied Mathematics
- 기본자료저록
- Dissertations Abstracts International. 86-03B.
- 전자적 위치 및 접속
- 로그인 후 원문을 볼 수 있습니다.
MARC
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■00520250211152129
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■020 ▼a9798384050117
■035 ▼a(MiAaPQ)AAI31483038
■040 ▼aMiAaPQ▼cMiAaPQ
■0820 ▼a519
■1001 ▼aNair, Gokul Gopan.▼0(orcid)0000-0002-4453-7489
■24510▼aConvexity Conditions and Energy Minimization for Highly Deformable Elastic Surfaces
■260 ▼a[Sl]▼bCornell University▼c2024
■260 1▼aAnn Arbor▼bProQuest Dissertations & Theses▼c2024
■300 ▼a96 p
■500 ▼aSource: Dissertations Abstracts International, Volume: 86-03, Section: B.
■500 ▼aAdvisor: Healey, Timothy.
■5021 ▼aThesis (Ph.D.)--Cornell University, 2024.
■520 ▼aThe theory of thin elastic surfaces is a source of many fascinating problems in the calculus of variations. When an elastic surface is deformed, its energy can be roughly decomposed into two parts---a "stretching/membrane energy'' that is non-convex in the derivative of the deformation map and a "bending energy'' that depends on higher order derivatives of the deformation map. The interplay between these two terms gives rise to a variety of interesting phenomena unique to thin elastic objects. We are in part motivated by wrinkling observed in highly stretched polymer sheets. This thesis has three parts: In Chapter 3, we consider a wide class of models known as "Cosserat shells'' and identify a new, physically meaningful convexity condition that leads to the existence of energy minimizers for these models. We argue that this convexity condition is suitable for predicting wrinkling phenomena. In Chapter 4, we focus on a pure membrane version of the model in Chapter 3, which in general is not even rank-one convex. Nevertheless, we prove that it admits energy minimizers when the image surface is constrained to lie on some prescribed embedded oriented surface in ℝ3. Under additional assumptions, we show that the minimizers are homeomorphisms onto their image and are weak solutions to the spatial equilibrium equations. In both Chapter 3 and Chapter 4, we ensure that the minimizers are locally injective/orientation preserving by requiring the energy density function to grow unboundedly as an appropriate notion of the local (signed) area/volume measure approaches zero. In Chapter 5, we study the membrane energy from the viewpoint of 3D to 2D dimension reduction. We embed our 3D variational problems into an appropriate class of parametrized measures and obtain a compactness result as the thickness of the body goes to zero. The putative membrane limit is defined on a class of locally injective gradient Young measures. Our main reason for choosing a Young measure approach for dimension reduction is that it can be used to uniquely identify a non-relaxed membrane energy density. Thus, membrane energy densities obtained this way do not preclude unbounded growth near non locally-injective configurations and can capture fine oscillations of minimizing sequences, like in the case of wrinkling.
■590 ▼aSchool code: 0058.
■650 4▼aApplied mathematics
■650 4▼aMathematics
■650 4▼aTheoretical mathematics
■653 ▼aCalculus of variations
■653 ▼aNonlinear elasticity
■653 ▼aPartial differential equations
■653 ▼aHomeomorphisms
■690 ▼a0364
■690 ▼a0405
■690 ▼a0642
■71020▼aCornell University▼bApplied Mathematics.
■7730 ▼tDissertations Abstracts International▼g86-03B.
■790 ▼a0058
■791 ▼aPh.D.
■792 ▼a2024
■793 ▼aEnglish
■85640▼uhttp://www.riss.kr/pdu/ddodLink.do?id=T17163049▼nKERIS▼z이 자료의 원문은 한국교육학술정보원에서 제공합니다.
