Applications of Porous Medium Theoriesfor Soil Mechanics and Bone Mechanics

Colby C. Swan, Assistant ProfessorCivil & Environmental EngineeringCenter for Computer−Aided Design

The University of Iowa

Joint ME Mech. Systems & CEE/SMT SeminarUniversity of Iowa, Iowa City, Iowa

23 April 1998

Acknowledgements:

Soil Mechanics Work: Young−Kyo Seo, CEE Graduate Student UI Old Gold Fellowship

Bone Mechanics Work:

Kristoffer Stewart, BME Graduate Student Elijah Garner, ME Graduate Student Professor Rod Lakes Professor Dick Brand The Whitaker Foundation

σ = σ’− p δ

σ22

σ21

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σ11

σ12

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σ11’

σ12’

σ22’

σ21’

Total Stresses, σ Effective Stresses, σ’ Pore Pressures, p

Stress Apportionment in Soils

Kinematics of Fluid and Solid Phases

* Motion of each phase is described by:

xα=xα(X,t) where α = s for solid α = f for fluid

* Dispacement, velocity and acceleration of the fluid and solid material points X

uα(X,t) = xα(X,t) − X

vα(X,t) =

aα(X,t) =

Dα

Dt(uα)

Dα

Dt (vα)

Both phases are assumed to be present at each Lagrangian point X.

Volume Fractions and Densities

* Volume fraction of α−phase

nα = dVα

dV

* Microscopic and Macroscopic densities

ρα = 1dV

∫ ρα dV = nα ρα

Basic Conservation Equations

* Conservation of Mass

∫∂ ρα

∂tdV + ∫ ρα vα . n dA = 0

* Conservation of Linear Momentum

DDt

∫ ρα vα dV = ∫ ρα bα dV + ∫ t α dA = 0

1) *** Incompressible soil grains; ***

2) On microscale, fluid has no shear viscosity;

4) Fluid pressure is independent of temperature;3) Fluid can be compressible/incompressible;

Constitutive Assumptions for Soils

5) Soil is either dry or fully saturated.

Constitutive Models

* Porous Solid Phase

* Fluid Phase

σw = −n w p w δ

(p w) = −DwDt

whereλw

nw[ ∇⋅ (n w vw) + ∇⋅ (n s vs) ]

.

σ’s = σ’s ( F, F, ζ)

Coupled Field Equations

ρs as = ∇ σ’s − n s ∇ pw − ξ ⋅ ( vs−vw) + ρsb

( vw)= ρw( vs−vw) ⋅ ∇ vw −n w ∇ pw +ξ ⋅ ( vs−vw) + ρwbDwDtρw

Time Integration

Matrix System of Coupled Equations

Mα = ∫ NA ρα NB dΩ

Z = ∫ NA ⋅ξ⋅ NB dΩ

+ [ Z −Z−Z Z][ vs

vw] +[ ns( ds, v )nw( v ) ] [ f s(ext)f w(ext) ]=[ Ms 0

0 Mw][ as

aw]

[ ns( ds, v )nw( v ) ] =[ ∫ BA∫ NAσ’ dΩ + ρw( vs−vw) ⋅ ∇ vw dΩ ]

∫ NAns pwdΩ− ∫ ∇NAnw pwdΩ

[ f s(ext)f w(ext) ] =[ ∫ NA ρs b dΩ+∫ NA hs dΓ

∫ NA ρw b dΩ+∫ NA hw dΓ]

Newmark’s family of implicit/explicit methods

vn+1 = vn +{(1− γ) an + γ an+1} ∆t

dn+1 = dn+ vn∆t + {(1−2 β) an + β an+1} ∆t 2/2

General Nature of Soils .....

Stiffness Characteristics

Ksoils < < K water : That is, bulk stiffness of water is much larger than that of soils;

Flow Characteristics

Coarse−grained soils: have low specific surface areas; this allows water to flow easily through such soils.

Fine−grained soils: have high specific surface areas; this means that pore fluid has extensive contact with soil grains; consequently, water does not flow easily in fine−grained soils.

f 3=0

f 1=0f 2=0

R( κ)

κΧ(κ)

ω

I 1J1c J1

t T

s

α

θ1

1−D Elastic Consolidation Problem

Porewater pressure at depth z=1.5m beneath the load

1m

∇

2−D Consolidation Problem (linear elastic media)

Porewater pressure at depth z=0.5B beneath the load

B

∇

DSCAL=1834

int ext

Can rn+1= 0 = fn+1(un+1) − f(tn+1) be solved?

n = n + 1 ∆t = ∆t/4

m = mmax?

Yes No

Yes

No

ext ext

flimit= f(tn) STOP

m = m + 1

n=0; m=0; t0=0;∆t = ∆tBaseline

tn+1= tn + ∆t

b) Limit state algorithm

slope displacement

g

glimit

equilibrium state does not exist

equilibrium state exists

a) gravity loading versus slope displacement

b) 0

30m

50m

80m 120m

fixed

rollers

a) Undeformed slope. b) Deformed slope at limit state.

a) Undeformed slope. b) Deformed slope at limit state.

30m

50m

80m

fixed

rollers

26m 94m

a) Undeformed slope. b) Deformed slope at limit state.

30m

50m

80m

fixed

rollers

26m 94m

a)Undeformed Configuration

50m

30m

fixed

68m52m80m

rollers

CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC

b)Deformed Configuration

a) Original state: clay over sand b)Limit state failure mechanism

c) Original state: sand over clay d)Limit state failure mechanism.

fixed

rollers

68m52m80m

50m

7.2m

22.8m

rollers

fixed

68m52m80m

50m

7.2m

22.8m

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b)Deformed limit state

fixed

80m

30m

50m

8m

54m30m9m

27m

rollers

a)Undeformed state

a) Undeformed slope. b) Deformed slope at limit state.

30m

50m

80m

fixed

rollers

26m 94m

c) Undeformed slope. d) Deformed slope at limit state.

30m

50m

80m

fixed

rollers

26m 94m

General Phenomenonology of Bone Adaptation

−> With increased mechanical stimulus, bones thicken and densify;

−> With lack of the same, the body resorbs bone mass; Bones become porous, and dimensions thin out; Consequently, bones lose their strength.

−> The fact that bones adapt/respond to mechanical stimulus has been recognized for over 100 years (Wolff’s Law).

−> However, the mechanisms by which bones adapt to mechanical stimulus remains very poorly understood!

−> Furthermore, the ability of orthopaedists/engineers to quantitatively predict bone adaptations is presently very poor.

Our Research Objectives:

−> To explore the potential role of fluid flow in bone adaptation;

−> To improve ability to quantitatively predict bone adaptation phenomena;

Experimental Observations.....

a) Bone adaptation depends upon the cyclic frequence of applied stimulus. [Hert et al, 1969, 1971, 1972; Chamay and Tschantz, 1972; Churches et al, 1979, 1982; Lanyon and Rubin, 1984; Turner and Forwood, 1993; McLeod et al, 1990, 1992;]

b) Bone adaptation shows selective dependence on the amplitude of mechanical stimulus. [Rubin and Lanyon, 1984, 1985; Frost, 1987]

c) Bone adaptation responds to stimulus in a trigger−like fashion. [Rubin and Lanyon, 1984, 1985; Turner et al, 1995; Brighton et al, 1992; Neidlinger−Wilke et al, 1995; Stanford et al, 1995]

Bone Adaptation Hypotheses

1) Damage in Bone: Under mechanical stimulus, bone accumulates damage. In repairing itself, bone adapts. [Burr et al, 1985; Martin and Burr, 1989; Prendergast and Taylor, 1994]

2) Strain Energy Density: Bone seeks to stay within a certain strain energy densities under applied loadings. [Beaupre et al, 1990; Cowin, 1993; Luo et al, 1995; Mullender et al, 1994; Jacobs et al, 1995;]

3) Mechanically Generated Electric Fields: a) piezoelectricity; [Spadaro, 1977; Binderman et al, 1985; Gjelsvik, 1973] b) streaming potentials; [Eriksson, 1974; Pollack et al, 1984]

4) Fluid Flow in Bone: Fluid flow provides stimulus to bone cells either by pressures, or shearing deformation that stimulates bone cells to initiate adaptation. [Weinbaum et al, 1994; Mak et al, 1996; Keanini et al, 1995; Turner et al, 1995]

1) Both bone matrix and fluid are compressible;

2) On microscale, fluid has no shear viscosity;

3) Both fluid and bone matrix are linear elastic;

Constitutive Assumptions for Cortical Bone

4) Bone is fully saturated.

Elastic Constitutive Model for Orthotropic Cortical Bone (following Biot 1956, 1957, 1962)

σ11

σ22

σ33

σ23

σ31

σ12

p

C11 C12 C13 0 0 0 C17

C21 C22 C23 0 0 0 C27

C31 C32 C33 0 0 0 C37

0 0 0 C44 0 0 0

C71 C72 C73 0 0 0 C77

0 0 0 0 C44 0 0

0 0 0 0 0 C66 0

ε11

ε22

ε33

ε23

ε31

γ12

ζ

=

σ values denote total stresses; ε values denote matrix strains;p denotes fluid pressure, pw; ζ denotes change of fluid content;

30% Haversian porosity

Undrained Test: Drained Test:Elong = 4.065GPa; Elong = 4.051GPa;Etrans= 3.391GPa; Etrans= 2.954GPa;

(pf/σtran)=0.332(pf/σlong)=0.0374

20% Haversian porosity

Undrained Test: Drained Test:Elong = 4.721GPa; Elong = 4.711GPa;Etrans= 4.067GPa; Etrans= 3.714GPa;

(pf/σtran)=0.317(pf/σlong)=0.0360

10% Haversian porosity

Undrained Test: Drained Test:Elong = 5.338GPa; Elong = 5.332GPa;Etrans= 4.849GPa; Etrans= 4.614GPa;

(pf/σtran)=0.309(pf/σlong)=0.0341

Drained/Undrained Response Characteristics of Haversian Bone

Properties : Ksolid=10GPa; νsolid= 0.40 Test Performed: Uniaxial stress (longitudinal) Kfluid=2GPa; νfluid= 0.50 Uniaxial stress (transverse)

λ1 λ2

Ωs

λ3

Next Step: Add Heterogeneity of bone matrix to osteon models.

Ongoing Research

a) Add hierarchy to bone models;

b) Perform 3D, dynamic bending & torsion tests on dry and saturated cortical bone specimens.

Differences in responses will be due, in part, to fluid flow effects.

c) Predict responses of saturated bone to experiments.

d) Verify and refine bone models.

e) Use bone models to calculate fluid stresses and flows in bone, at different length scales, during excitation.