1. ELASTIC GROWTH IN TISSUES
Program in Applied Mathematics
ELASTIC GROWTH IN TISSUES
Rebecca Vandiver Program in Applied Mathematics
University of Arizona
Los Alamos Days, February 2007 Advisor: Alain Goriely

2. Outline Basic problem I. Introduction II. Theory and Methods
To study the mechanics of growing tissues
Obtain a general formulation of volumetric growth
Generate a description of cumulative growth
I. Introduction
Volumetric growth
Differential growth
Residual stress
II. Theory and Methods
Mechanics of soft tissues
Kinematics of growth
Cauchy stress
Equilibrium equations
III. Application
Growing cylinder
Future work

3. Part I I. Introduction Volumetric growth Differential growth
Residual growth

4. Growth Surface growth: Volumetric growth:
Occurs by accretion, deposition of new material on a preexisting surface
Volumetric growth:
Growth takes place in the bulk of the material

5. Bulk Growth: Plant Stems
Orange grows faster than green, so green is in tension. This stress is called residual stress.
Historical note: Tissue tension and differential growth first investigated by Hofmeister (1859) and Sachs (1865).

6. Residual stress Rhubarb Simple experiment reveals residual stress
Differential growth causes residual stress
Residual stress may increase rigidity

7. Bulk Growth: Arteries Stress in arteries
No Residual stress
Stress in arteries
Residual stress in unstressed arteries (Fung, 1984)
Create compressive and tensile layers
Believed to reduce stress gradient
Residual stress
Hozapfel-Ogden

8. Part II. II. Theory and Methods Mechanics of soft tissues
Kinematics of growth
Cauchy stress
Equilibrium equations

9. MECHANICAL PROPERTIES OF BIOLOGICAL TISSUES
BONE
AORTA
bone
aorta
σ (MPa)
σ (MPa) σs
1.75 70
Non-linear stress-strain curve ε ε
.004 6
Large deformation

10. KINEMATICS
Let X denote the position vectors in the reference configuration of an elastic body and x define the current configuration after growth.
x=c(X,t) X
The deformation tensor F = Grad(c) relates a material segment in the reference configuration to the same segment in the current configuration

11. KINEMATICS Decomposition: F= AG where A: pure elastic response
Rodriguez, et al. (1994)
Decomposition: F= AG
where A: pure elastic response
G: pure growth

12. Incremental Growth F(k) = FkFk-1…F2F1 G(k) = GkGk-1…G2G1
A(k)
F(k) = FkFk-1…F2F1
= AkGkAk-1Gk-1…A1G1
G(k) = GkGk-1…G2G1
A(k) = AkAk-1…A2A1

13. Cauchy Stress
We assume that the body is hyperelastic. That is, the material can be described by a strain energy function W=W(A) and the Cauchy stress tensor, T, can be written as
If there is one (or more) constraint such as incompressibility
then, we introduce a Lagrange multiplier p
where p is the hydrostatic pressure

14. EQUILIBRIUM EQUATIONS
The balance of linear momentum leads to Cauchy’s first law of motion:
The balance of angular momentum leads to Cauchy’s second law of motion:
If the body is at rest and body forces are absent,
With given boundary conditions: (e.g )

15. Part III
III. Applications
Growing cylinder
Future work

16. Growing cylinder
Consider the elastic growth of an incompressible cylindrical tube
ri = bi
ri = ai
Ri = B
Ri = A

17. Growing cylinder Growth along the z-axis: Gi = diag(1,1,ginc(ri))
 g(R) = ginc(ri)
Consider a linear incremental growth function
ginc = 1+μ(bi-ri) bi ai
Initial values:
A=1, B=2
Calculate μ at each iteration such that the volume increases by 1%

18. STRESSES IN THE CYLINDER
trr (radial stress)
tθθ (hoop stress)
tzz (axial stress)

19. Axial stress curves
Longitudinal stress plotted in the current configuration
Tensile
Compressive

20. Cumulative growth
Cumulative growth function plotted in the current configuration

21. Cumulative growth

22. Cumulative growth

23. Conclusions and perspectives
Elastic growth
Decomposition of growth
Formulation of incremental growth applied to a growing cylinder
Description of cumulative growth
Ongoing projects
Stress-dependent growth
Cavitation
Buckling formula for general 3-D case F

24. References
Ben Amar, M., and Goriely, A., Growth and instabilities in elastic tissues, J. Mech. Phys. Solids, 53: (2005).
Goriely, A., and Ben Amar, M. On the definition and modeling of incremental, cumulative, and continuous growth laws in morphoelasticity, Biomechanics and Modeling in Mechanobiology (2006).
Haughton, D. M., and Orr, A., On the eversion of incompresible elastic cylinders, Int. J. Non-Linear Mechanics 30:81-95 (1995).
Rodriguez, E. K., A. Hoger, and McCulloch, A. D., Stress-dependent finite growth in soft elastic tissues, Journal of Biomechanics, 27: (1994).

25. FUTURE WORK
Euler showed there was a critical load for buckling of a slender column
No comparable formula for a more general geometry F
Goal: a) Work on a general theory of buckling of a cylinder under a load in which we could relate parameters in a more general case
b) Explore how growth affects the buckling parameter

26. CUMULATIVE GROWTH Cumulative deformation gradient2: F(k) = FkFk-1…F2F1
= AkGkAk-1Gk1…A2G2A1G1
If growth and elastic tensors commute: A(k) = AkAk-1…A2A1,
G(k) = GkGk-1…G2G1
Stress in Bk:
where

27. CYLINDER: MODELING CUMULATIVE GROWTH
Incompressibility constraint
 det(A)=1  det(FG-1)=1  det(F)=det(G) 
Integrate to obtain

28. CYLINDER: MODELING CUMULATIVE GROWTH
Radial stress:
We want surface of cylinder to be free of any traction,
but t3=t3(R). Instead, impose a zero resultant load:
Solve for ai and λi at each iteration