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On Compatible Strain with Reference to Biomechanics
Linköping University, Department of Management and Engineering, Mechanics . Linköping University, The Institute of Technology.ORCID iD: 0000-0001-8460-0131
Linköping University, Department of Management and Engineering, Mechanics . Linköping University, The Institute of Technology.
2005 (English)In: Zeitschrift für Angewandte Mathematik und Mechanik, ISSN 0044-2267, Vol. 85, no 6, 440-448 p.Article in journal (Refereed) Published
Abstract [en]

In previous studies, residual stresses and strains in soft tissues have been experimentally investigated by cutting the material into pieces that are assumed to become stress free. The present paper gives a theoretical basis for such a procedure, based on a classical theorem of continuum mechanics. As applications of the theory we study rotationally symmetric cylinders and spheres. A computer algebra system is used to state and solve differential equations that define compatible strain distributions. A mapping previously used in constructing a mathematical theory for the mechanical behavior of arteries is recovered as a corollary of the theory, but is found not to be unique. It is also found, for a certain residual strain distribution, that a sphere can be cut from pole to pole to form a stress and strain free configuration.

Place, publisher, year, edition, pages
2005. Vol. 85, no 6, 440-448 p.
Keyword [en]
biomechanics, compatible strain, curvature tensor, elasticity, residual stress, soft tissue
National Category
Engineering and Technology
Identifiers
URN: urn:nbn:se:liu:diva-14329DOI: 10.1002/zamm.200410192OAI: oai:DiVA.org:liu-14329DiVA: diva2:23256
Available from: 2007-03-16 Created: 2007-03-16 Last updated: 2017-05-15
In thesis
1. Soft Tissue Mechanics with Emphasis on Residual Stress Modeling
Open this publication in new window or tab >>Soft Tissue Mechanics with Emphasis on Residual Stress Modeling
2007 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis concerns residual stress modeling in soft living tissues. The word living means that the tissue interacts with surrounding organs and that it can change its internal properties to optimize its function. From the first day all tissues are under pressure, due, for example, to gravity, other surrounding organs that utilize pressure on the specific tissue, and the pressure from the blood that circulates within the body. This means that all organs grow and change properties under load, and an unloaded configuration is never present within the body. When a tissue is removed from the body, the obtained unloaded state is not naturally stress free. This stress within an unloaded body is called residual stress. It is believed that the residual stress helps the tissue to optimize its function by homogenizing the transmural stress distribution.

The thesis is composed of two parts: in the first part an introduction to soft tissues and basic modeling is given and the second part consist of a collection of five manuscripts. The first four papers show how residual stress can be modeled. We also derive evolution equation for growth and remodeling and show how residual stress develops under constant pressure. The fifth paper deals with damage and viscosity in soft tissues.

Place, publisher, year, edition, pages
Institutionen för konstruktions- och produktionsteknik, 2007. 27 p.
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1081
Keyword
Mechanics, Residual Stress, Growth, Remodeling
National Category
Mechanical Engineering
Identifiers
urn:nbn:se:liu:diva-8490 (URN)978-91-85715-50-3 (ISBN)
Public defence
2007-04-13, Sal C3, Hus C, Linköpings universitet, Linköping, 10:15 (English)
Opponent
Supervisors
Available from: 2007-03-16 Created: 2007-03-16 Last updated: 2017-05-15
2. Modeling Residual Stress Fields in Soft Tissues with Application to Human Arteries
Open this publication in new window or tab >>Modeling Residual Stress Fields in Soft Tissues with Application to Human Arteries
2004 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

Biomechanics or Mechanics of Biology comprises many different fields. This thesis deals with soft tissues or living tissues and the fact that these materials live in a pressurized environment. This means that an unloaded tissue may not be stress free. The stress in an unloaded body is usually called residual stress. This thesis consist of an introduction to continuum mechanics and to soft tissues, and three research papers.

The first paper deals with a zero stress configuration and the question of compatibility. It is shown that the zero stress configuration not necessarily constitutes a compatible body. The condition for compatibility is analyzed and exemplified on a cylinder and a sphere.

In the second paper a model for residually stressed arteries based on local deformations is developed. The material properties for a human aorta is identified by the solution to an optimization problem. The resulting initial strains show a non constant behavior and this behavior cannot be described by the commonly used opening- angle model.

The last paper is about formulating boundary value problems for initially stressed bodies in three different reference configurations. Firstly, the equilibrium equation and constitutive relation are stated on the unloaded residually stressed body. Secondly, all material points are relieved from stress by a tangent map and the new stress free configuration, which may be incompatible, is used to state the boundary value problem. Thirdly; we use the relieving tangent map to induce a new metric on the body. This induced metric is in general non Euclidean. Finally, we formulate the boundary value problem on this manifold.

Place, publisher, year, edition, pages
Linköping: Linköpings universitet, 2004. 40 p.
Series
Linköping Studies in Science and Technology. Thesis, ISSN 0280-7971 ; 1118
Series
LiU-TEK-LIC, 47
Keyword
Continuum Mechanics, Modeling, Residual Stress, Soft Tissues
National Category
Engineering and Technology
Identifiers
urn:nbn:se:liu:diva-22418 (URN)1633 (Local ID)91-85295-48-5 (ISBN)1633 (Archive number)1633 (OAI)
Available from: 2009-10-07 Created: 2009-10-07 Last updated: 2013-11-12

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Klarbring, AndersOlsson, Tobias

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