Stress-strain relationships look similar to force-deformation relationships because:...stress is proportional to force...strain is proportional to deformation

1

Deformation, mm

Force, N

YLD PTFAIL PT

PROP LIMIT

Strain = Deform/Length mm/mm

Stress = Force Area

N/mm2

YLD PTFAIL PT

PROP LIMIT

Area

Length

Hooke's Law (1676): ut tensio, sic vis

= E

• The slope of the graph represents the stiffness of the material: E = /

• E is the elastic modulus, in N/m2

Stress & strain are linearly proportional in “linear elastic” materials:

2

Stiffness vs. Strength• Stiffness: resistance to deformation

– expressed as force required to produce a given amount of deformation

• Strength: resistance to failure– expressed as force required to produce

• a permanent deformation: yield strength• a fracture: ultimate or fracture strength

• stiffness and strength may apply to either– a structure (e.g. a femoral diaphysis)– a material (e.g. cortical or trabecular bone tissue)

3

Strength Depends on Mode of Loading

4

Torsion

CompressionTension

Shear

Bending

Three Basic Modes of Loading• Compression, tension, and shear• Each has an associated...

– anatomic “shape & size factor” = cross–sectional area, A

– material strength = failure stress, F = LF /A

NoteF is usually different in

compression, tension, and shear

• at failure: LF = F A

or F = LF /A

tension

compr.

shear

A

L

5

• Strength of a bone diaphysis depends on the product of– material (tissue) strength,

i.e., failure stress– and a "shape and size factor"

• Stiffness of a bone diaphysis depends on the product of– material (tissue) stiffness,

i.e., elastic modulus– and a "shape and size factor"

Strength & Stiffness of a Bone Diaphysiss

Figure: http://messenger.yahooligans.com/reference/gray/subjects/subject?id=59 6

Compressive strength of the femoral diaphysis

• Assume = circular cylinder– Periosteal, endosteal radii = 1.5 , 0.75 cm

– Failure stress is f = 105 x 106 N/m2

• Find diaphysis compressive failure load – Lfail = f A

– A = (rp2- re

2) = (1.52- 0.752) = 5.30 cm2

– A = 5.30 x 10-4 m2

– Lfail = f x A = (105 x 106) x (5.30 x 10-4)

– Lfail = 55,700 N = 12,500 lb

Strength in Simple Compression

7Figure: http://messenger.yahooligans.com/reference/gray/subjects/subject?id=59

(Note: 1 cm2 = 10-4 m2)

= 6+ tons !!!

• Primary stresses = tension & compression*

• Zero stress at central ‘neutral plane’ (NP)

• Tensile and compressive stresses occur on opposite sides of NP

compression

tensionNPfixed

end

8

Flexure (Bending) of a Bone Diaphysis

Cantilever Bending

3-point Bending

4-point Bending

F

* also shear stress; see slide 47

• Flexural stresses = tension + compression

• Stress increases linearly – on tensile and compressive sides of NP

• Stress decreases with distance from fixed end

compression

tensionNPfixed

end

c

magnified section9

Cantilever Bending Example

F

• Because the stress magnitude changes linearly with distance from the NP– from high tension to high compression...

• Bending produces shear stresses, too:

10

Bending Also Produces Shear Stress

Each layer of bone stretches a different amount...

...resulting in longitudinal shear stresses

c NP

Can you see that there are vertical shear stresses, too?

Cross-sectional Moment of Inertia, I

• “I” is the “anatomic shape and size factor" for bending

• Conceptually it is found by summing...– all the elements, A,

of cross–sectional area – each multiplied by y2,

the square of its distance from the “neutral plane”

– I = (A x y2)

Ayneutral

axis

For a circular cylinder, this works out to be I = ¼ (rp

4 – re4)

rpre

11

• Note that I is affected more by periosteal than endosteal bone formation

– because rp4 >> re

4

• Thus, the medullary canal is positioned to cause minimal loss of bending strength

Aneutral axis

rpre

12

Cross-sectional Moment of Inertia, I

e.g.: rp = 1.5 cm re = 0.75 cm

I = ¼ (1.54 - 0.754)I = ¼ (5.06 - 0.32) I = 4.74 cm4

M. canal

• During growth, bones expand radially– periosteal bone formation, endosteal resorption

• This continues throughout adulthood – but much more slowly, of course

• Periosteal expansion helps compensate for internal bone loss, particularly in postmenopausal women

Age-related Diaphyseal Expansion

13 I = ¼ (rp

4 – re4)

Periosteal Formation, Endosteal Resorption

• For a cylindrical structure

– = M c / I• M = bending moment (F x d)• c = distance from neutral axis• I = CSMI = ‘cross–sectional moment of inertia’

Bending Stress Equation

14

d F

Fixed end

Loaded end

• For a cylindrical structure

– = M c / I• M = bending moment (F x d)• c = distance from neutral axis

• I = CSMI = ¼ (rp4 – re

4)

– the failure moment is Mfail = (I/c) fail

• c = periosteal radius, where stress is greatest

Bending Stress Equation

15

anatomical strength factor

material strength factorNA

c

d F

Fixed end

Loaded end

Torsional loading produces shear stresses between adjacent bone segments, and...

16

...between circumferential bone layers because outer layers stretch more...

Resistance to torsion...

• ... depends on an anatomic strength factor called the “Polar Moment of Inertia” (IP)

– multiply each area element, A

– by the square of its distance from the center of rotation, r

– and sum these over the entire bone area: J = A x r2

– calculus shows that...

Ar

rpre

17

...for a cylindrical bone

IP = ½ (rp4 – re

4)

Angle of Twist• When a cylindrical bone of length L is subjected to a

twisting moment (torque) MT, the angle of twist between the ends, , is given by the equation:

where G = bone tissue shear modulus

IP = polar moment inertia = ¼ Rp4 - Re

4 )

Rp , Re = periosteal & endosteal radii 18

End View

L =

MT L G IP

Angle in radians (1 radian = 57.3o)

Rp

Re

Torsional Loading Example• A tibial diaphysis has the following properties

– shear modulus G = 8 x 108 Pascals (N/m2)

– length L = 0.30 m, radii: rp = 0.015 m , re = 0.005 m

• It is subjected to a torque of MT = 10 N-m in a skiing mishap. How much would it twist?

• Solution, assuming a circular cross-section: I = ¼ Rp

4– Re4) = ¼ 0.0154 - 0.0054) = 3.9 x 10-8 m4

= MT L/G I = 10 x 0.30 / (8 x 108) x (3.9 x 10-8)

19

L = 3.0 / 31.2

= 5.51 degrees

= 0.096 radians

• Local shear stress, torsional loading

– s = T r / J

• T = applied torque• r = distance from center• J = PMI, polar moment of inertia

– at failure, T = Tfail = (J/r) sf

• r = periosteal radius because that is where stress is greatest

• shape and size factor = J/r

• sf = shear failure stress

Torsional Stress Within a Cylindrical Bone

20

T

r

J =½ (rp4 - re

4)

Typical small child:

J =½ (1.004 – 0.304)

Jchild =cm4

Typical adult:

J =½ (1.804 – 0.604)

Jadult =cm4

Polar Moment of Inertia (J) Examples for Cylindrical Adult & Pediatric Bones

rp = outer radius, re = inner radius

Child: rp = 1.00 cm, re = 0.30 cm

Adult: rp = 1.80 cm, re = 0.60 cm

21

Cortical area: 3.2x greater

Polar M of I: 10x greater

Mechanically, periosteal bone “counts more” than medullary bone when resisting torsion (similar for bending)

• stiffness = shape factor x elastic modulus*

• strength = shape factor x material strength

• compression**: Lfail = A fail

• torsion: Tfail = (J/r) fail

• bending: Mfail = (I/c) fail

• bone tissue governs material strength (fail)

• bone size & shape govern structural properties (A, J, r, I, c)

Summary: Bone Stiffness & Strength

22* or shear modulus ** or tension

• the stress is– = L /A or L = A

• at failure, L = Lf , = f

– Lf = f A

– shape/size factor = A , area

– material strength = f , failure stress in compression, tension, or shear, as appropriate

tension

compr.

simple shear

For Tension, Compression & Simple Shear

23

For Bending and Torsion the “Shape and Size Factors” are Different...

• For a cylindrical bone these factors are:– Bending load:

Cross-sectional moment of inertia

I =(Rp4 - Re

4) / 4– Torsional load:

Polar moment of inertia

J =(Rp4 - Re

4) / 2• Because the radius is raised to the 4th power,

periosteal radius differences are important

24

Rp

Re

• Age-related changes in human long bone diameters

• at puberty– males add

periosteal bone– females add

endosteal (medullary) bone

• postmenopausally, women lose endosteal bone

AGE, years

0 20 40 60 80

DIA

ME

TE

R, m

m

2

4

6

8

10

MPMA

menwomen

Menarche Menopause

Gender, Age, Bone Anatomy & Strength

What are the mechanical implications of this?

Garn, The Earlier Gain and Later Loss of Bone, C.C. Thomas, Springfield, 1970.

In terms of cross-sectional moment of inertia, what is the advantage of endosteal vs. periosteal bone loss? 25

Periosteal

Endosteal

Examples of Strength Calculations for Cylindrical Bones

• Assume diaphysis is a circular cylinder– periosteal radius = 1.5 cm*– endosteal radius = 0.6 cm*– Calculate failure conditions for...

• Compression– A = (Rp

2- Re2) = (0.0152- 0.0062) = 5.94x10-4 m2

– Typical ultimate (i.e., failure) compressive stress, f = 195 MPa

– Failure load: Lf = Af = 1.16 x 105 N

26

Ro

Ri

* Convert to meters for calculations using standard stress units.

Examples of Strength Calculations for Cylindrical Bones

• Bending strength: Mfail = (I/c) fail

– again, Rp = 3 cm, Re = 1.2 cm

– I =(Rp4 - Re

4) / 4 – I = (81.0- 2.1)/4 = 62.0 cm4

– assume c = Rp = 2 cm– Typical tensile fx stress = 13,300 N/cm2

– Mfail = (I/c) fail = (62/2) x 13,300

– Mfail = 412,300 N-cm = 4123 N-m

27

Ro

Ri

• stiffness = shape factor x material stiffness

• strength = shape factor x material strength

• compression: Lfail = A fail

• torsion: Tfail = (J/r) fail

• bending: Mfail = (I/c) fail

• remodeling governs material properties

• modeling governs shape and size (A, J, I)

Determinants of Anatomic Bone Strength

28

Cortical Bone ‘Grain’• Micro-CT images show

Haversian canals (osteons) generally align with stress directions

• Consequently, bone has a ‘grain’ and properties are direction dependent, e.g....

• Elastic Modulus, GPa:

Cooper et al, Anatomical Record, 274B:169, 2003

Red dots: Haversian canal junctions

Longitudinal Transverse 18.2 + 0.9 11.7 + 1.0

Human Cortical Bone

29

Remodeling & Mechanical PropertiesLight areas are more mineralized:

Rho et al, J. Biomechs, 35:189, 2002

• newly formed bone mineralizes

– over time, more readily near blood supply (e.g., Haversian canal)

• osteons are less mineralized

– when more recently formed

– away from their Haversian canal

• interstitial bone more mineralized• average mineralization inversely

proportional to remodeling rate

– e.g. lower in children than adults

• elastic modulus tends to as remodeling rate

less mineralized

more mineralized

92 yr old

60 yr old 150 m

150 m30

Tension Compression Longitudinal Transverse Longitudinal

Transverse Elastic Modulus, GPa: Human 17.9 + 0.9 10.1 + 2.4 18.2 + 0.9 11.7 + 1.0 Bovine

Haversian 23.1 + 3.2 10.4 + 1.6 22.3 + 4.6 10.1 + 1.8 Primary 26.5 + 5.4 11.0 + 0.2 ----- -----

Failure Stress, MPa Human 136 + 16 53 + 11 105 + 17 131 + 21 Bovine Haversian 150 + 11* 49 + 7* 272 + 3 146 + 32 Primary 167 + 9 55 + 9* ----- -----

+ = Standard deviation, approximate due to averaging groups

Cortical bone mechanical properties vary with direction, remodeling, species, etc.

31Reilly et al, J. Biomechs, 7:271, 1974; J. Biomechs 8, 393, 1975

32

Proposed molecular basis for bone toughening: protein-based molecular “glue” on mineralized collagen fibers

Unloaded

Loaded:

Hansma, PK, J Musculoskelet Neuronal Interact 2005; 5(4):313-315

Ca++ ions hypothesized to link non-collagenous proteins, e.g., osteopontin

Loading stretches...

& separates glue-like fibers, absorbing energy.

Unloaded fibers “re-coil”

Strain Rate Also Affects Cortical Bone Strength &

Stiffness• Greatest energy absorption

occurs at 0.01-0.1/sec – i.e., impact at ~ 2mi/hr

• At increased rates of loading, bone is – stronger & stiffer– but more brittle– no. of fx fragments increases

with applied energy

0

Strain, mm/mm

Stress, MPa

0.020.00

200

.001/sec

.01/sec

.1/sec

1/sec

300/sec

1500/sec

McElhaney, J.H., J. Applied Physiology, 21:1231, 1966

100

33

Bone is more brittle at high loading rates: higher impact energy = more small fragments

+ greater total fracture surface areaL

iber

ated

Sur

face

Are

a, m

m2

Energy absorbed in fx (Joules)

Energy groups

Beardsley et al, J. Orthop. Res 23:686, 2005

• Bone stiffness and strength increase nonlinearly with– bone density, d (g/ml)– strain rate, = d/dt (sec-1)

• Elastic modulus (GPa)*

E = 3790 0.06 d3

• Compressive strength (MPa)*

= 68 0.06 d2

Combined Effects of Strain Rate & Density

35Carter and Hayes, J Bone & Joint Surg, 59A, 954, 1977.

dt

dstrain,

time, t

slope = .

Bone structure is infinitely variable, but just 2 basic types

• Bone tissue is usually – cortical – trabecular– rarely “in between”

• Both osteons & trabeculae tend to be aligned with stress directions

APPARENT DENSITY, gm/cm

1.01.21.41.61.82.0

POROSITY

0.0 0.2 0.4 0.6 0.8 1.0

ULT

IMA

TE

ST

RE

SS

, MP

a

0

50

100

150

200

250

Martin, R.B. et al, Skeletal Tissue Mechanics, Springer Verlag, New York, NY 1998.

cortical bone

trabecular bone

36

Trabecular Bone Compression Test

Courtesy of Prof. T. Keaveny, Univ. of California, Berkeley

Reduced diameter test region

Extensometer arms firmly attached with rubber bands

Bone ends embedded in metal end caps

37

Unlike cortical bone, trabecular bone fails in a progressive manner due to its porosity and fragility

Testing machine slowly applies force to specimenDumb-bell

shaped trabecular specimen

For

ce

Deformation

Trabecular Bone Failure ProfileA. Initial failure (yielding): region of collapsing (red) trabeculae tends to cascade across the specimen.

A

A

B. Progressive failure: band of collapsing trabeculae widens, another may form.

B

B

C. Progressive compaction: collapsing trabeculae are compacted, stiffness grows with compaction.C

C

For

ce

Deformation

initial specimen

Courtesy of Prof. T. Keaveny, Univ. of California, Berkeley 38

Compacted bone:

Trabecular vs. Cortical Bone Failure “Plateaus” representing crushing behavior.

Keaveny, TM, & Hayes, WC, in Hall, BK, Bone, CRC Press, Boca Raton, FL 1993. 39

Str

ess,

MP

a

0

50

100

200

150

0 5 10 15 20 25

0.30 g/cc

0.90 g/cc

1.85 = apparent density, g/cc

Strain, %

Cortical bone

Trabecular bone

In summary bone fractures are governed by...

http://www.nlm.nih.gov/medlineplus/ency/article/002966.htm

Nature of the bone Nature of the forces

Amount of energy40

The End

41Revised 3-8-07