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Modeling Residual Stress Fields in Soft Tissues with Application to Human Arteries
Linköping University, Department of Mechanical Engineering. Linköping University, The Institute of Technology.
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 [en]
Continuum Mechanics, Modeling, Residual Stress, Soft Tissues
National Category
Engineering and Technology
Identifiers
URN: urn:nbn:se:liu:diva-22418Local ID: 1633ISBN: 91-85295-48-5 (print)OAI: oai:DiVA.org:liu-22418DiVA: diva2:242731
Available from: 2009-10-07 Created: 2009-10-07 Last updated: 2013-11-12
List of papers
1. On Compatible Strain with Reference to Biomechanics
Open this publication in new window or tab >>On Compatible Strain with Reference to Biomechanics
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.

Keyword
biomechanics, compatible strain, curvature tensor, elasticity, residual stress, soft tissue
National Category
Engineering and Technology
Identifiers
urn:nbn:se:liu:diva-14329 (URN)10.1002/zamm.200410192 (DOI)
Available from: 2007-03-16 Created: 2007-03-16 Last updated: 2017-05-15
2. Modeling initial strain distribution in soft tissues with application to arteries
Open this publication in new window or tab >>Modeling initial strain distribution in soft tissues with application to arteries
2006 (English)In: Biomechanics and Modeling in Mechanobiology, ISSN 1617-7959, E-ISSN 1617-7940, Vol. 5, no 1, 27-38 p.Article in journal (Refereed) Published
Abstract [en]

A general theory for computing and identifying the stress field in a residually stressed tissue is presented in this paper. The theory is based on the assumption that a stress free state is obtained by letting each point deform independently of its adjacent points. This local unloading represents an initial strain, and can be described by a tangent map. When experimental data is at hand in a specific situation, the initial strain field may be identified by stating a nonlinear minimization problem where this data is fitted to its corresponding model response. To illustrate the potential of such a method for identifying initial strain fields, the application to an in vivo pressure–radius measurement for a human aorta is presented. The result shows that the initial strain is inconsistent with the strain obtained with the opening-angle-method. This indicates that the opening-angle-method has a too restrictive residual strain parameterization, in this case.

National Category
Engineering and Technology
Identifiers
urn:nbn:se:liu:diva-14331 (URN)10.1007/s10237-005-0008-8 (DOI)
Available from: 2007-03-16 Created: 2007-03-16 Last updated: 2017-12-13
3. Boundary Value Problems of an Initially Stressed Body
Open this publication in new window or tab >>Boundary Value Problems of an Initially Stressed Body
2004 (English)Report (Other academic)
Abstract [en]

The initial strain in a body can be described by a two-point tensor that maps tangent vectors between a local stress free configuration (manifolds) and the initially strained configuration. The initial strain tensor can be used to form a metric tensor and this metric can be used to induce a third manifold. The boundary value problem of an elastic body is in this paper formulated on these three manifolds: the induced manifold, the initially strained configuration and the local stress free configuration. Moreover, we consider a hyperelastic body with incompressible properties and we show that in that case, the divergence of the first Piola- Kirchhoff stress tensor defined on the initially strained configuration and on the induced manifold coincide. There-fore, we conclude that for an incompressible material, the boundary value problems stated on the initially strained configuration and on the induced manifold are the same.

Place, publisher, year, edition, pages
Linköping: Linköpings universitet, 2004. 17 p.
Series
LITH-IKP-R, ISSN 0281-5001 ; 1348
Keyword
Boundary, Value, Problems Initially Stressed Body
National Category
Engineering and Technology
Identifiers
urn:nbn:se:liu:diva-22424 (URN)1640 (Local ID)1640 (Archive number)1640 (OAI)
Available from: 2009-10-07 Created: 2009-10-07 Last updated: 2017-05-15

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Olsson, Tobias

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