Linkoping Studies in Science and Technology, Dissertation No. 1081

Soft Tissue Mechanics with

Emphasis on Residual Stress Modeling

Tobias Olsson

Division of MechanicsInstitute of Technology, Linkoping University

SE–581 83, Linkoping, Sweden

Linkoping, February 2007

Cover:Illustration of how growth and remodeling can reduce the stress gradients.The left picture is the stress due to a constant internal pressure, and therightmost figure shows the stress after growth and remodeling.

Printed by:LiU–TryckLinkoping UniversitySE–581 83 Linkoping, SwedenISBN 978–91–85715–50–3ISSN 0345–7524

Distributed by:Institute of Technology, Linkoping UniversityDepartment of Management and EngineeringSE–581 83, Linkoping, Sweden

c© 2007 Tobias OlssonThis document was prepared with LATEX, February 27, 2007

No part of this publication may be reproduced, stored in a retrieval system,or be transmitted, in any form or by any means, electronic, mechanical, pho-tocopying, recording, or otherwise, without prior permission of the author.

Preface

This work has been carried out at the Division of Mechanics at LinkopingUniversity. There are some people that I would like to acknowledge for sup-porting me through these years. First, my supervisor Prof. Anders Klarbring(Linkoping University) for giving me the opportunity to work at the divisionand introducing me to the field of biomechanics. Second, I want to sendmy best wishes to my colleagues, specially Dr. Jonas Stalhand, for fruitfuldiscussions. Third, for arranging my stay in Lisbon, I would like to thankProf. Joao A. C. Martins (I.S.T.). This gave me the chance to work withchallenging problems in soft tissues and taught me how important it is tocollaborate with colleagues in other adherent topics. Also, a special mentiongoes to my colleague Antonio Pinta da Costa (I.S.T.) for making my stay iLisbon the best possible. Finally, I would like to thank my family and friendsfor supporting and believing in me during these years.

This work was partly supported by the Swedish Research Council.

Linkoping, February 2007.

Tobias Olsson

iii

“Mistakes are the portals of discovery.”

—Joyce, James (1882–1941)—

iv

Abstract

This thesis concerns residual stress modeling in soft living tissues. The wordliving means that the tissue interacts with surrounding organs and that itcan change its internal properties to optimize its function. From the first dayall tissues are under pressure, due, for example, to gravity, other surroundingorgans that utilize pressure on the specific tissue, and the pressure from theblood that circulates within the body. This means that all organs grow andchange properties under load, and an unloaded configuration is never presentwithin the body. When a tissue is removed from the body, the obtainedunloaded state is not naturally stress free. This stress within an unloadedbody is called residual stress. It is believed that the residual stress helpsthe tissue to optimize its function by homogenizing the transmural stressdistribution.

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

v

To my mother and father, the best parents one could wish for.

vi

List of Papers

This dissertation consists of a short summary and a collection of five researchpapers:

I Anders Klarbring and Tobias Olsson, On Compatible Strain with Ref-erence to Biomechanics, Zeitschrift fur Angewandte Mathematik undMechanik, 85, 440–448, 2005.

II Anders Klarbring, Tobias Olsson and Jonas Stalhand, Theory of Resid-ual Stresses with Application to an Arterial Geometry, Submitted forpublication 2007.

III Tobias Olsson, Jonas Stalhand and Anders Klarbring, Modeling Ini-tial Strain Distribution in Soft Tissues with Application to Arteries,Biomechanics and Modeling in Mechanobiology, 5, 27–38, 2006.

IV Tobias Olsson and Anders Klarbring, Residual Stresses in Soft Tissuesas a Consequence of Growth and Remodeling, Submitted for publica-tion 2007.

V Tobias Olsson and Joao A. C. Martins, Modeling of Passive Behavior ofSoft Tissues Including Viscosity and Damage, III European Conferenceon Computational Mechanics, C.A. Mota Soares et al. (Eds.), Lisbon,5–9 June, 2006.

The author of this thesis has contributed to the development of the theory,written substantial parts of the text and implemented all the numerical al-gorithms used in the papers.

vii

Contents

Preface iii

Abstract v

List of Papers vii

PART I: INTRODUCTION

1 Background 1

2 Soft Tissues 32.1 Elastic Arteries . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Residual Stress . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Growth and Remodeling . . . . . . . . . . . . . . . . . . . . . 5

3 Mechanics 93.1 Growth and Remodeling . . . . . . . . . . . . . . . . . . . . . 113.2 Damage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2.1 Validation of the Model . . . . . . . . . . . . . . . . . 153.3 Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4 Future Work 19

5 Abstract of Appended Papers 21

PART II: APPENDED PAPERS

Paper I 331 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352 The basic theorem and its use . . . . . . . . . . . . . . . . . . 36

ix

CONTENTS

3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.1 The rotationally symmetric cylinder . . . . . . . . . . . 403.2 The rotationally symmetric sphere . . . . . . . . . . . 44

4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Paper II 531 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552 General theory of a residually stressed body . . . . . . . . . . 58

2.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . 582.2 Balance and constitutive laws . . . . . . . . . . . . . . 612.3 Existence of a stress free compatible reference configu-

ration . . . . . . . . . . . . . . . . . . . . . . . . . . . 623 Arterial geometry . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.1 An identification problem . . . . . . . . . . . . . . . . 653.2 Compatible stress free reference configuration . . . . . 65

4 Riemannian manifold . . . . . . . . . . . . . . . . . . . . . . . 674.1 The tensor m as a metric on B0 . . . . . . . . . . . . 674.2 Determinants, volume elements and densities . . . . . . 69

5 Boundary value problems . . . . . . . . . . . . . . . . . . . . . 705.1 Boundary value problem on B0 with metric m . . . . 705.2 Boundary value problem on B0 with metric G . . . . . 725.3 Boundary value problem on B with metric γ . . . . . 735.4 Incompressibility . . . . . . . . . . . . . . . . . . . . . 745.5 Comparison of formulations . . . . . . . . . . . . . . . 75

1 Appendix – Piola Identity . . . . . . . . . . . . . . . . . . . . 79

Paper III 831 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 852 General theory . . . . . . . . . . . . . . . . . . . . . . . . . . 86

2.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . 862.2 Constitutive law . . . . . . . . . . . . . . . . . . . . . 892.3 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . 892.4 General identification problem . . . . . . . . . . . . . . 90

3 The rotationally symmetric case . . . . . . . . . . . . . . . . . 913.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . 913.2 Equilibrium and boundary conditions . . . . . . . . . . 933.3 Identification problem . . . . . . . . . . . . . . . . . . 96

4 A numerical example . . . . . . . . . . . . . . . . . . . . . . . 984.1 A specific constitutive law . . . . . . . . . . . . . . . . 984.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

x

CONTENTS

6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

Paper IV 1151 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1172 General Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 118

2.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . 1182.2 Balance Equations . . . . . . . . . . . . . . . . . . . . 1202.3 Constitutive Equations . . . . . . . . . . . . . . . . . . 121

3 Arterial Application . . . . . . . . . . . . . . . . . . . . . . . 1233.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . 1233.2 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . 1243.3 Growth and Remodeling Equations . . . . . . . . . . . 1253.4 Strain Energy . . . . . . . . . . . . . . . . . . . . . . . 1263.5 Remodeling of the Collagen Fibers . . . . . . . . . . . 128

4 Numerical Solution . . . . . . . . . . . . . . . . . . . . . . . . 1295 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1316 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1347 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

Paper V 1431 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1452 Mathematical Framework . . . . . . . . . . . . . . . . . . . . 146

2.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . 1462.2 Constitutive Relations . . . . . . . . . . . . . . . . . . 1462.3 Damage Evolution . . . . . . . . . . . . . . . . . . . . 1502.4 The non–Equilibrium Stresses . . . . . . . . . . . . . . 152

3 The Elasticity Stiffness Tensor . . . . . . . . . . . . . . . . . . 1533.1 The Derivative of the 2nd Piola–Kirchhoff Stress . . . . 155

4 Illustrative Examples . . . . . . . . . . . . . . . . . . . . . . . 1565 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . 157

xi

Part I

Introduction

1

Background

Biomechanics can be defined as the development and application of mechanicsto solve problems in biology. The main point is that we first observe thebiology and then try to develop a model that represents that behavior. Itmay not be possible to determine the beginning of biomechanics; one of thepioneers was Leonardo da Vinci (1452–1519). He discovered how the valves inveins made the blood flow in only one direction, from the veins to the heartand to the arteries. That, together with the assumption of conservationof mass, led to the conclusion that blood must return from the arteries tothe veins, and therefore circulate within the body. But the developmentof biomechanics was slow, due to that the particle mechanics derived byGalileo and Newton was not suitable to describe the continuous blood flow.The development of a theory appropriate to the mechanics of a continuousmedia, continuum mechanics, began in the early 18th century with Euler andlater with Navier, Cauchy and others. In the late 19th century, experimentson different specimens indicated that soft tissues do not obey Hooke’s law,that is their constitutive behavior is not linear. With the non–linearity ofsoft tissues, further understanding was delayed until the middle of the 20thcentury, when the theory of finite deformations was developed.

More than twenty years ago, laboratory work began to develop and moreand more realistic models began to appear. Even today there are a greatmany challenging problems in soft tissue modeling that need more study, forexample residual stress, growth, remodeling and inelasticity. Residual stressis the stress that is contained in an unloaded tissue. Most tissues grow withinthe body and during a lifetime the mechanical properties may change due tothe wellness of that particular person. It may be that some parameters thatdescribe the tissue are dependent on the person’s age and former illnesses.Remodeling means that the tissue can change its constitutive behavior dueto some deficiency, for example the heart may remodel itself after a cardiacinfarction and arteries can develop aneurysms due to degradation of the stiff-

1

CHAPTER 1. BACKGROUND

ness in the arterial wall. Inelastic effects such as damage and viscosity arealso important for understanding the total behavior of the tissue, in partic-ular damage propagation during hypertension.

Cardiovascular diseases are a major health problem. In the United Statesof America, for example, they accounted for almost 39% of all deaths in2001 (American Heart Association, 2003). That is almost 3 times as many asthose caused by cancer, and the British Heart Foundation Health PromotionResearch Group (2000) reports that the percentage is the same in Europe.

In this dissertation different models that include residual stress, growth,remodeling and damage are developed. We give examples of how to findthe residual stress by optimizing specific material parameters against realexperimental data. We develop a model that lets the tissue grow under load.If we define a stress free unloaded tissue and then apply pressure and let thetissue grow and remodel for some time, and then remove the pressure, wewill see that the tissue is no longer stress free when unloaded. In that waywe can say that the residual stress develops during growth and remodeling.When an artery grows it can also develop artifacts such as aneurysms and it isbelieved that they grow during a phase when the wall is weakening due to thedegradation of some constituents. This type of degradation of the elasticityof the wall is called remodeling, and a theory for this is also presented. If ananeurysm grows in such a way that it can rupture, it is very critical for thesurvival of the patient. Therefore, it is of importance to understand how thedamage propagates through the wall, due to some disease or hypertension,and where an aneurysm may appear.

2

2

Soft Tissues

Soft tissues are complex materials. Although each type of tissue has uniquebehaviors, there are many properties that are common. All organs live in apressurized environment which is complex to simulate and most tissues arecomprised of different layers, were each layer has specific properties. For ex-ample, large elastic arteries consist mainly of three layers each with differentmaterial properties. Furthermore, visco–elastic and anisotropic behaviors arenot uncommon. If the environment changes, for example due to some kindof disease, the tissue can grow and remodel itself to optimize its function inthis new environment. All tissues are under pressure in their normal state.Therefore, is it difficult to obtain accurate data from experiments done invitro (in a laboratory rather than in the tissue’s natural setting).

From a mechanical point of view, it is also difficult to find a suitableconfiguration that is stress free. It is important to be able to conclude thatstress in the reference set is known or zero. In standard continuum mechanicsit is often assumed that the reference configuration is stress free, but in softtissue mechanics it is not generally true that an unloaded configuration isstress free; we call the stress within an unloaded body residual stress. To beable to make accurate stress estimations it is important to find this residualstress or to find a configuration where the body is stress free. One way tofind a stress free configuration is to cut the tissue into many pieces whereeach part is assumed to be stress free. This procedure can be describedlocally by a deformation, but will in general end up with a non–compatibleconfiguration, see for example Klarbring et al. (2007).

2.1 Elastic Arteries

Most elastic arteries consist of three layers; the intima (tunica intima), themedia (tunica media) and the adventitia (tunica adventia), see Figure 1. In

3

CHAPTER 2. SOFT TISSUES

healthy young people the innermost layer, the intima, consists of mainlyone layer of endothelial cells. Due to the thinness of the innermost layer,it is often assumed that the intima does not contribute to the mechanicalproperties of the wall. However, the endothelial cells have an indirect effecton the mechanical properties since they are sensitive to shear stress and canstimulate the tissue to grow. Even though the intima has no mechanicalproperties in young people, it thickens with age and may contribute to themechanical properties in older people.

The media is the largest part of the wall (about 67%), and consists ofa complex three dimensional network of smooth muscle cells and a mix ofcollagen and elastin fibers (Sonneson et al., 1994). The smooth muscle cellsare concentrically arranged through the arterial wall and its active propertieshelp to regulate the stiffness of the artery and the the blood flow. It isoften assumed that the media is responsible for the elastic properties of theartery. The post–natal generation of elastin is mainly developed during thefirst two weeks of life and the turnover time is very long. Elastin’s half–lifeis best measured in years (Dubick et al., 1981; Humphrey, 2001) whereasfor collagen it is best measured in days (Reinhart et al., 1978; Humphrey,2001). Due to the slow adaptation of elastin it is believed that it may beresponsible for the development of residual stresses in arteries during post–natal growth. The media is separated from the intima and adventitia by two

Figure 1: Schematic of the different layers in an elastic artery. The innermostlayer (black) is the intima, the middle layer (grey) is the media, and theoutermost layer (light grey) is the adventitia.

elastic membranes. The mix of smooth muscle cells and the fibers constitutea helix with a small twist (pitch) (Holzapfel et al., 2000). This arrangementgives the media great strength and the ability to resist loads in both axialand circumferential directions.

The outermost layer, the adventitia (about 33%), consists mainly of cellsthat produce collagen and elastin. The collagen fibers are arranged in ahelical structure and reinforce the wall. The adventitia is not as stiff at

4

2.2. RESIDUAL STRESS

low pressures as it is at high pressures. This is due to the fact that at lowpressures the collagen fibers are undulated and do not reach its full length,but when the pressure increases the collagen fibers straighten and begin tocarry load. When all collagen fibers are stretched the adventitia becomesalmost rigid and prevents the artery from rupture.

2.2 Residual Stress

As already mentioned, residual stress is the stress that is left within thebody when all external forces are removed. One of the first discoveries thatunloaded arteries are not stress free was made by Bergel (1960). He wrote:

“When an artery is split open longitudinally it will unroll itself. . .This surely indicates some degree of stress even when there is nodistending pressure.”

Twenty years later, Chuong and Fung (1986) performed experiments witharteries from rabbits. They found that when the arteries were cut alongtheir symmetry axis the arteries opened up. Later, they performed the sametype of experiments with left ventricles and they found the same type ofbehavior as with the arteries. This indicates that stress exists in unloadedsoft tissues (not only arteries). It is believed that this residual stress reducesthe stress gradients in the pressurized environment and gives the tissue a morehomogenized stress distribution. In arteries it mainly reduces the tangentialstress at the inner wall, which otherwise would be high. The residual stressis believed to develop during growth in a pressurized environment. Figure 2shows how the tangential residual stress is develops from a stress free state.The left picture shows a hypothetical embryo that is by definition stress free.That embryo is then pressurized with a luminal pressure at 13.3 kPa and tohomogenize the stresses the growth process begins. After some time, whenthe growth has stabilized, the pressure is removed and the picture to theright with the residual stress is obtained.

2.3 Growth and Remodeling

The mass of a living tissue both increases and decreases with time. Thechange of mass is often referred to as growth. Since the tissue occupies apart of a pressurized body, the growth must take place under the influence ofpressure. During growth the tissue develops residual stresses, which meansthat the residual stress field is dependent on time. It may be possible to model

5

CHAPTER 2. SOFT TISSUES

Figure 2: Schematic of the initial (unloaded) stress free configuration (left)and the corresponding grown configuration (unloaded) with the tangentialcomponent of the residual stress (right).

the residual stress due to growth by assuming a stress free configuration atsome reference time and then letting a strain–like tensor (growth tensor)evolve with time. This tensor should be defined locally at every materialpoint, so each part of the tissue can grow independently of other parts. Thisunfortunately also implies that a grown unstressed body is not necessarilycompatible. The residual stress is then believed to occur as a result of theelastic deformation required from the incompatible grown configuration to acompatible physical configuration, see Figure 4 for a simple sketch and Klischet al. (2001), Rodriguez et al. (1994), Skalak et al. (1996), and Taber andEggers (1996).

When using continuum mechanics to describe growth one must add thatthe mass changes with time, instead of remaining constant. Mass change inliving tissues arises primarily from a change of volume whereas the densityis almost conserved.

The remodeling of soft tissues is also important for their behavior. Asexamples of remodeling we have: changes in the heart wall after an infarc-tion, remodeling of the arterial wall stiffness due to illness or changes inthe surrounding environment, changes of the angle between the fibers, andchanges in material constants. Since growth and remodeling are both time–dependent, we need equations or differential equations (evolution laws) de-scribing the evolution.

As more and more sophisticated models appear that predict the residual

6

2.3. GROWTH AND REMODELING

stress, growth and remodeling, the ability to make a correct stress calculationwill increase. The possibility to estimate the stress may help in decisionsinvolving surgery and increase the ability to predict ruptures, diseases etc.

7

CHAPTER 2. SOFT TISSUES

8

3

Mechanics

The first step in mechanical modeling is to define a set that represents thebody in a particular configuration. We define a set B0, as a sub–domainof the physical space. Further, let χ define a time–dependent deformationfrom B0 onto a deformed configuration B that represents the body under aparticular load. We write this as

χ : B0 × R → B.

Let X = (X1, X2, X3) denote the coordinates in the reference configurationB0 and x = (x1, x2, x3) the coordinates in the physical configuration B. Thedeformation in then given by

x = χ(X, t).

Given a deformation, the deformation gradient is calculated as the derivativeof the deformation χ:

F =∂χ(X , t)

∂X=

∂x

∂X.

The deformation gradient maps tangent vectors on B0 to tangent vectors onB, this is written

F : TB0 → TB,

where TB and TB0 are the union of all tangent spaces to B and B0, re-spectively. For an illustration, see Figure 3.

Now introduce a normal vector n(x) defined on the boundary of physicalconfiguration B. From Cauchy’s theorem (see, for example Gurtin, 1981)we know that there exists a tensor σ such that σ times the normal vector is

9

CHAPTER 3. MECHANICS

F

TB0 TB

Figure 3: The deformation gradient and the corresponding tangent spaces.

equal to the traction t applied on the surface at the point x. This is usuallywritten as

t(n) = σn. (1)

The tensor σ is more well known as the Cauchy stress tensor. This stressmeasure is represented by force per deformed area. Sometimes it is more con-venient to define a stress tensor that represent the force per undeformed area.To define such a tensor we use the deformation gradient and its determinant:

P = (det F )σF−T , (2)

where P is known as the first Piola–Kirchhoff stress tensor and the trans-formation to the undeformed configuration is called a Piola transformation.For later use we also define the second Piola–Kirchhoff stress tensor as

S = (det F )F−1σF−T = F−1P . (3)

The physical meaning of the second Piola–Kirchhoff stress is more vaguethan, for example the Cauchy or the first Piola–Kirchhoff stress respectively.Sometimes, for example when modeling viscosity, it can be more advanta-geous to use this kind of stress measure.

Now assume the body is in equilibrium. The forces acting on the bodyare of two kinds: traction forces t on the body surface and body forces ρb,where ρ is the density of the physical body B. The total force acting on thebody in equilibrium is equal to zero and is written

∫

∂B

t ds +

∫

B

ρb dv = 0.

Writing the surface integral as a volume integral by Stoke’s theorem (di-vergence theorem) using (1) and localizing, we obtain the local form of theequilibrium equation:

div σ + ρb = 0. (4)

10

3.1. GROWTH AND REMODELING

Using the Piola transformation (2) we can show that the equilibrium equationtransforms into

Div P + ρ0b = 0,

where Div is the divergence with respect to the reference coordinates X andρ0 = (det F )−1ρ is the reference density.

If we assume the existence of a strain energy function W dependent on theelastic deformation gradient F , we can show from the dissipation inequality(see equation (7) below) that the stress P can be calculated as the derivativeof the strain energy. That is,

P =∂W∂F

,

and with (2) we obtain the Cauchy stress as

σ = (det F )−1∂W∂F

F T .

A material where a strain energy can be defined is called hyper–elastic.Hyper–elastic materials are a very large class of materials and this is themost common way (by a large margin) to model elastic phenomena.

3.1 Growth and Remodeling

To determine what kind of equations may drive growth and remodeling weagain use the dissipation inequality. This law says, in words, that the sumof the internal power and the change of energy is always less than or equalto zero. We take the total deformation gradient F from TB0 onto the TB

to be the composition of an elastic part F e and a growth part G. That is,F = F eG. An illustration is presented in Figure 4. Note that the growthtensor maps vectors in TB0 to vectors in TB0 .

We also introduce a strain energy W as an isotropic scalar valued functiondependent only on the elastic deformation F e and remodeling (material)parameters mα. The energy per unit volume in the reference configurationB0 is defined as

Ψ =

∫

B0

(det G)W(F e, mα) dV0. (5)

To define the internal power we need to choose what kind of velocity fieldswe want to use. Of course we have the deformation rate F , but we are also

11

CHAPTER 3. MECHANICS

F

TB0

TB0 TB

G F e

Figure 4: The deformation, growth, and the corresponding tangent spaces.

interested in growth and remodeling. Introducing the growth rate G and theremodeling rates mα we can define the internal power as

Pi = −∫

B0

(P : F + Y : G + Mαmα

)dV0, (6)

where P , Y and Mα are forces to balance the corresponding rates. Note thatY and Mα are material forces (configurational forces), for example they arerelated to the change in the structure or properties of the body. The colon( : ) should be interpreted as a double contraction, and, as is customary, arepeated index means summation over that index. Now, if we require thatthe dissipation inequality,

Ψ + Pi ≤ 0, (7)

must hold for all subsets of B0, it can be localized. Standard arguments nowgive that the force P can be explicitly expressed as

P = (det G)∂W∂F

(8)

and we obtain the reduced dissipation inequality(

(det G)∂W∂mα

− Mα

)mα +

((det G)G−1W − Y − F T

e P)G ≤ 0. (9)

12

3.2. DAMAGE

We note that the force P in (8) is in fact the first Piola–Kirchhoff stresstensor. To be able to satisfy the reduced dissipation inequality (9) the ma-terial is treated as a generalized standard material (Halphen and Nguyen,1975; Moreau, 1974) and a convex potential function ϕ is introduced. Theresulting evolution equations are given by the following system of ordinarydifferential equations

∂ϕ

∂G= F T

e P − (det G)G−TW + Y (10)

∂ϕ

∂mα= Mα − (det G)

∂W∂mα

, (11)

The simplest possible potential that can be chosen and at the same timeguarantees that the thermodynamics are satisfied is the quadratic function

ϕ =c0

2G : G +

c1

2mαmα, c0, c1 > 0. (12)

In arteries it is believed that the transmural stress distribution is almostconstant. In Figure 5 it is shown how we can use the growth evolution todrive the stress to a more homogenized state. The growth is initiated after aluminal pressure at 13.3 kPa is applied. The tissue is then allowed to grow.The almost constant stress distribution is obtained by choosing a drivingforce Y as

Y = (det G)G−1W − F eP∗, (13)

where P ∗ is a given homeostatic stress. The equations (10) and (11) togetherwith (12) and (13) become

c0G = F Te (P − P ∗) (14)

c1mα = Mα − (det G)

∂W∂mα

, (15)

The evolution law (14) is such that the tissue grows (increases and decreasesits mass) until the stress P is equal to the given homeostatic stress P ∗.

3.2 Damage

As discussed in previous sections, the zero stress state for soft tissues is notthe same as the unloaded state and residual stresses are important for thetissue to function in its natural environment. But when the tissue is stretchedway beyond its normal working range, the residual stress is small compared

13

CHAPTER 3. MECHANICS

Figure 5: Illustration of how growth can influence the stress distribution.The tissue is initiated with a inner pressure at 13.3 kPa and the tissue isthen allowed to change its mass to homogenizes the stress distribution.

with the stress within this overstretched tissue, see for example Fung (1993).In this section we neglect the residual stresses and model the degradation ofa tissue due solely to very large strains.

When modeling damage one usually multiplies the strain energy by asmooth function g with a range in the interval 1 to 0, where 1 signifiesundamaged material and 0 total failure or rupture, see Lemaitre (1992). Thedamaged strain energy is denoted Wd and is written

Wd = g(δ∗)W (16)

where δ∗ is some kind of strain measure and W is the undamaged strainenergy. For example, when modeling damage of fibers, δ∗ may be the stretchof the fibers. If we use the damaged strain energy in the dissipation inequality,we see by using (5) and (7) that if we require that

∂g

∂δ∗δ∗ ≤ 0, (17)

together with the evolution of the growth and the remodeling, the dissipa-tion inequality is always satisfied. This means that if the strain measure δ∗

increases, the damage function g(δ∗) must decrease and vice versa.The damage function g(δ∗) can be described by an evolution law in the

same way as growth and remodeling, see equations (14) and (15). How-ever, we will not use that approach, instead we explicitly define the damage

14

3.2. DAMAGE

function as a monotonically decreasing function with a few parameters. InNatali et al. (2003) and Natali et al. (2005) it is suggested and shown that fora transversely isotropic material, e.g., tendons and ligaments, the followingfunctional form gives satisfactory results

g(δ∗) =1 − eγ(δ∗−δ+)

1 − eγ(δ−−δ+)(18)

where γ is a parameter that sets the characteristic of the damage and, δ+ andδ− describe at what strain the tissue fails and when the damage is initiated.This means that for strains less than δ−, the tissue is undamaged (g = 1),and for strains equal to δ+, the tissue ruptures (g = 0). Figure 6 showns howthe damage function depends on the parameter γ. For small absolute valuesthe damage is close to linear and for large positive values the damage is veryslow at the beginning, but as the strain δ∗ tends to the limit δ+, the damagefunction goes rapidly to zero and the tissue fails. For negative values of γthe damage propagates rapidly close to the initial strain δ− and slows downwhen the strain gets close to the rupture state δ+.

1.1 1.12 1.14 1.16 1.18 1.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

δ*

g(δ*

)

γ >0

γ <0

γ=0

Figure 6: A plot of how the damage function g(δ∗) depends on the parameterγ. The initial strain is δ− = 1.1 and the maximum strain is taken to beδ+ = 1.2

3.2.1 Validation of the Model

This section is taken from an upcoming conference paper. It is a continuationof Paper V and for simplicity the notation is as therein, and may differ fromthat is used in the rest of this introduction.

15

CHAPTER 3. MECHANICS

To test the damage theory against experimental data, an extension testwas performed on a porcine ligament and the resulting stress–strain curve wasused to find the material and damage parameters. Ligaments are treated asa transversely isotropic material since their anisotropic behavior is mainlydue to one fiber family, aligned along the ligament.

In Paper V we have chosen to use a strain energy that is a slight modifi-cation of the model presented in Natali et al. (2005) and Natali et al. (2003),it is composed of an isotropic part and another that is anisotropic. Theisotropic contribution is given by a standard Mooney–Rivlin material withtwo parameters, c1 and c2. The fibrous part is constructed as an exponentialfunction with two parameters, c3 and c4. Most tissues include a high per-centage of water and can therefore be treated as almost incompressible. Theincompressibility condition is obtained by penalization. To do this we adda volumetric part to the strain energy, and it is customary to use one thattends to infinity (+∞) as the volume ratio J = det F approaches zero (seePaper V). The total strain energy is given by

Ψ = g1(δ∗

1)(Ψ0(J) + Ψ1(I1, I2)) + g2(δ∗

2)Ψ2(I4) (19)

where I1, I2 and I4 are deformation invariants (see Paper V), and g1 andg2 are damage functions with the corresponding damage variables δ∗1 andδ∗2, respectively. The material parameters are included in the volumetric,isotropic, and anisotropic parts and the explicit expression for those are

Ψ0 = c0(J2 − 1

2− ln J) (20)

Ψ1 = c1(I1 − 3) + c2(I2 − 3) (21)

Ψ2 =c3

c4(ec4(I4−1) + c4(I4 − 1) − 1) (22)

where c0 is a penalization constant, c1 and c2 describe the elasticity of thenon fibrous part of the tissue and c3 is the resistance in the undulated fibers,and c4 represents the strength of the stretched fibers. The parameters areobtained by using a Nelder–Mead method (using the fminsearch function inMATLAB) for solving a non–linear minimization problem in the least squaresense.

The damage is here modeled by two damage functions one for the strainenergies Ψ0 and Ψ1 and another for Ψ2, see equation (19). This meansthat we need to determine 6 damage parameters (10 parameters in total).For simplicity we have taken the initial damage values δ−1 and δ−2 from theexperimental data set. This gives a total of 8 parameters. The result of theparameter fit is given in Figure 7 and it shows that the model is capable ofcapturing the behavior of this kind of tissue (ligament).

16

3.3. VISCOSITY

1 1.1 1.2 1.3 1.4 1.5 1.60

1

2

3

4

5

6

Strain [−]

Nom

inal

str

ess

[MP

a]

Elastic Parameters:c

1 = 6.877e+4

c2 = 1.209e+4

c3 = 6.078e+4

c4 = 4.355

Damage Paramters:γ1 = 7.300e−2

γ2 = −4.704e−1

δ1− = 1.316 (exp.)

δ1+ = 1.631

δ2− = 1.250 (exp.)

δ2+ = 1.631

Figure 7: Result after optimization of the damage model used on a porcineligament. Two damage functions are used: one for the isotropic part and onefor the anisotropic part respectively.

3.3 Viscosity

A viscous material is such that the current stress depends on the evolutionof the deformation. The stress is rate–dependent. The total stress is usuallydivided into a volumetric part, an elastic part, and a viscous part. Whenmodeling viscosity it is preferable to use the second Piola–Kirchhoff stressmeasure. The total stress is written:

S = Svol + Se + H , (23)

where Svol is the stress solely due to volumetric changes, Se is the stress fromthe pure elastic process, and H is a second Piola–Kirchhoff–like stress tensordescribing the viscous stress (non–equilibrium stress). The viscous effectsare assumed to follow the behavior of a generalized Maxwell element (see forexample Simo, 1987; Holzapfel and Gasser, 2001) and the non–elastic stressH is described by the following ordinary differential equation

H +1

τH = βSe (24)

where τ is the relaxation time and β is a free energy factor associated withthe relaxation time (see, Holzapfel and Gasser, 2001). Equation (24) is easily

17

CHAPTER 3. MECHANICS

integrated and the result can be written using convolution:

H = H0e−T/τ +

∫ T

0

Seβe(t−T )/τ dt (25)

where H0 is the non–elastic stress at time t = 0, here taken to be zero(no initial viscosity). To solve (25) we use the following recursive algorithm,derived from the midpoint rule (Simo, 1987), that has proved to convergerapidly:

Hn+1 = Hne−∆t/τ + βτ1 − e−∆t/τ

∆t

(Sn+1

e − Sne

), (26)

where n + 1 is the current iteration and ∆t is the time increment.To obtain the true stress (Cauchy stress) we solve equation (7) for σ and

the Cauchy stress is given by

σ = (det F )F−1SF−T .

18

4

Future Work

The development of realistic models for soft tissues has just begun and muchmore work needs to be done in this field. So far, most models treat the tissuesas consisting of passive materials, but recent research has shown that thecells that make up the tissue are highly active and can regulate growth andremodeling of the tissue. For example, the endothelial cells in the intima reacton the concentration of Ca2+ ions and can send signals through the tissueto stimulate growth. If we can find out how the cells react on mechanicaland chemical stimuli, we may be able to understand the complex behaviorof cardiovascular diseases, for example aneurysms. This challenging field is aunion of many different fields. To be successful, a close collaboration betweenbiologists, chemists, physicians and mechanicans is needed.

Another interesting area is to build a finite element model of a portionof an artery and simulate the blood flow. By using the blood shear stress asboundary conditions on the endothelial cells, we may be able to simulate arealistic growth and remodeling response.

The damage theory presented in this thesis is not stable when the stressreaches the point where the tangent stiffness is zero (the maximum stressor turning point on the curve). Obtaining a more stable method that cansimulate rupture should be of interest.

19

CHAPTER 4. FUTURE WORK

20

5

Abstract of Appended Papers

I

On Compatible Strain with Reference to

Biomechanics

In previous studies, residual stresses and strains in soft tissues have beenexperimentally investigated by cutting the material into pieces that are as-sumed to become stress free. The present paper gives a theoretical basisfor such a procedure, based on a classical theorem of continuum mechanics.As applications of the theory we study rotationally symmetric cylinders andspheres. A computer algebra system is used to state and solve differentialequations that define compatible strain distributions. A mapping previouslyused in constructing a mathematical theory for the mechanical behavior ofarteries 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 canbe cut from pole to pole to form a stress and strain free configuration.

II

Theory of Residual Stresses with Application

to an Arterial Geometry

This paper presents a theory of residual stresses, with applications to biome-chanics, especially to arteries. For a hyper–elastic material, we use an initiallocal deformation tensor K as a descriptor of residual strain. This tensor, ingeneral, is not the gradient of a global deformation, and a stress free referenceconfiguration, denoted B, therefore, becomes incompatible. Any compatiblereference configuration B0 will, in general, be residually stressed. However,

21

CHAPTER 5. ABSTRACT OF APPENDED PAPERS

when a certain curvature tensor vanishes, it does actually exist a compatibleand stress free configuration, and we show that the traditional treatment ofresidual stresses in arteries, using the opening–angle method, relates to sucha situation.

Boundary value problems of non-linear elasticity are preferably formu-lated on a fixed integration domain. For residually stressed bodies, threesuch formulations naturally appear: (i) a formulation relating to B0 with anon-Euclidean metric structure; (ii) a formulation relating to B0 with a Eu-clidean metric structure; and (iii) a formulation relating to the incompatibleconfiguration B. We state these formulations, show that (i) and (ii) coincidein the incompressible case, and that an extra term appears in a formulationon B, due to the incompatibility.

III

Modeling Initial Strain Distribution in Soft

Tissues with Application to Arteries

A general theory for computing and identifying the stress field in a residu-ally stressed tissue is presented in this paper. The theory is based on theassumption that a stress free state is obtained by letting each point deformindependently of its adjacent points. This local unloading represents an ini-tial strain, and can be described by a tangent map. When experimental datais at hand in a specific situation the initial strain field may be identified bystating a non linear minimization problem where this data is fitted to itscorresponding model response. To illustrate the potential of such a methodfor identifying initial strain fields, the application to an in vivo pressure–radius measurement for a human aorta is presented. The result shows thatthe initial strain is inconsistent with the strain obtained with the opening–angle–method. This indicates in this case that the opening-angle-method hasa too restrictive residual strain parametrization.

IV

Residual Stresses in Soft Tissue as a Conse-

quence of Growth and Remodeling

We develop a thermodynamically consistent model for growth and remodelingin elastic arteries. The model is specialized to a cylindrical geometry, strain

22

energy of the Holzapfel–Gasser–Ogden type and remodeling of the collagenfiber angle. A numerical method for calculating the evolution of the adaptionprocess is developed. For a particular choice of the thermodynamic forces ofgrowth and remodeling (configurational forces), it is shown that an almosthomogeneous transmural axial and tangential stress distribution is obtained.Residual stresses develop during this adaption process and these stressesresemble what is found by the widely used opening-angle model.

V

Modeling of Passive Behavior of Soft

Tissues Including Viscosity and Damage

This article describes a continuum damage model for anisotropic soft tissues.The model is developed with the underlying framework of hyper-elasticity.As usual, the corresponding strain energy is additively split into a volumetricpart and a volume–preserving part. the damage of the tissue involves bothisotropic and anisotropic contributions. The viscous properties of the tissueare modeled by a generalized linear standard solid with a finite number ofMaxwell elements, which allows for the approximation of frequency indepen-dent responses. The results are obtained with the commercial FE softwareABAQUS and are in agreement with other studies done by different authorsin the field.

23

CHAPTER 5. ABSTRACT OF APPENDED PAPERS

24

Bibliography

American Heart Association, Heart, Disease and Stroke Statistics–2004 Up-date, American Heart Association, Dallas, Texas.

Bergel D. A., The Visco–Elastic Properties of the Arterial Wall, Ph.D. The-sis, University of London, UK, 1960.

British Heart Foundation Health Promotion Research Group, European Car-diovascular Disease Statistics–2000 Edition, Institute of Health and Sci-ence, University of Oxford, Oxford.

Dubick M. A., Rucker R. B., Cross C. E. and Last J.A., Elastin Metabolismin Rodent Lung, Biochimica et Biophysica Acta, 672, 303–306, 1981.

Fung Y. C., Biomechanics: Mechanical Properties of Living Tissues,Springer, New York, 1993.

Gurtin M. E., An Introduction to Continuum Mechanics, Academic Press,San Diego, California, 1981.

Sonneson B., Lanne T., Vernersson E., Hansen F., Sex Difference in theMechanical Properties of the Abdominal Aorta in Human Beings, Journalof Vascular Surgery, 20, 959–969, 1994.

Hokanson J. and Yazdani S., A Constitutive Model of the Artery with Dam-age, Mechanics Research Communications, 24, 151–159, 1997.

Holzapfel G. A., Gasser T. C. and Ogden R. W., A New Constitutive Frame-work for Arterial Wall Mechanics and a Comparative Study of MaterialModels, Journal of Elasticity, 61, 1–48, 2000.

Holzapfel G. A. and Gasser T. C., A Viscoelastic Model for Fiber–ReinforcedComposites at Finite Strains: Continuum Basis, Computational Aspects

25

BIBLIOGRAPHY

and Applications, Computer Methods in Applied Mechanics and Engineer-ing, 190, 4379–4430, 2001.

Choung C. J. and Fung Y. C., Residual Stress in Arteries. In: Schmidt–Schonbien G. W., Woo S. L.–Y., Zweifach BW (Eds.) Frontiers in Biome-chanics, Springer, Berlin Heidelberg New York, pp 117–129, 1986.

Halphen B. and Nguyen Q. S., Sur les Mereriaux Standards Generalises,Journal de Mecanique, 14, 39–62, 1975.

Humphrey J. D., Cardiovascular Solid Mechanics; Cells, Tissues, and Or-gans, Springer, New York, 2001.

Klarbring A., Olsson T. and Stalhand J., Theory of Residual Stresses withApplication to an Arterial Geometry, Archives of Mechancis, Submitted,2007.

Klisch A. M., Van Dyke T. J. and Hoger A., A Theory of Volumetric Growthfor Compressible Elastic Biological Materials, Mathematics and Mechanicsof Solids, 6, 551–575, 2001.

Lemaitre J., A Course on Damage Mechanics, Springer, New York, 1992.

Moreau J. J., On Unilateral Constraints, Friction and Plasticity, In: G.Capriz and G. Stampacchia (Eds.), New Variational Techniques in Matem-atical Physics, Edizione Cremonese, Rome, 1974.

Natali A. N., Pavan P. G., Carniel E. L. and Dorow C., A TransverselyIsotropic Elasto–Damage Constitutive Model for the Periodontal Liga-ment, Computer Methods in Biomechanical and Biomedical Engineering,6, 329–338, 2003.

Natali A. N., Pavan P. G., Carniel E. L., Luciasano M. E. and TaglialavoroG., Anisotropic Elasto–Damage Constitutive Model for the BiomechanicalAnalysis of Tendons, Medical Engineering and Physics, 27, 209–214, 2005.

Reinhart N., Cardinale G. J. and Udenfriend S., Increased Turnover of Arte-rial Collagen in Hypertensive Rats, Proceedings of the Natianal Academyof Sciences, 75, 451–453, 1978.

Rodriguez E. K., Hoger A. and McCulloch A. D., Stress–Dependent FiniteGrowth in Soft Elastic Tissues, Journal of Biological Mechanics, 27, 455–467, 1994.

26

BIBLIOGRAPHY

Simo J. C., On a Fully Three–Dimensional Finite–Strain Viscoelastic DamageModel: Formulation and Computational Aspects, Computer Methods inApplied Mechanics and Engineering, 60, 153–173, 1987.

Skalak R., Zargaryan S., Jain R., Netti P. and Hoger A., Compatibility andthe Genesis of Residual Stress by Volumetric Growth, Journal of Mathe-matical Biology, 34, 889–914, 1996.

Taber L. A. and Eggers D. W., Theoretical Study of Stress–ModulatedGrowth in the Aorta, Journal of Theoretical Biology, 180, 343–357, 1996.

27