solutions manual for biofluid mechanics the human ... · biofluid mechanics: the human circulation...

SOLUTIONS MANUAL FOR BIOFLUID MECHANICS THE HUMAN CIRCULATION 2ND EDITION CHANDRAN

SOLUTIONS


https://testbankplus.com/shop/solutions-biofluid-mechanics-human-circulation-2nd-edition-chandran/



Biofluid Mechanics: The Human Circulation

Solutions to problems 7/11/2011 6

Chapter 2

2.2 Following a uniaxial extension test, to calculate the stress-strain data from load-extension

data, engineering stress and strain are given as: Stress, 0

F

A where, F is force and A0 is the

original cross-sectional area; and Strain, 0

0

l

ll where, l0 is the original length and l is the

current length. However, the cross-sectional area actually diminishes with extension (or strain).

Therefore, it is only appropriate that the true stress be defined as, true

stress, true

F

A ,where A is the ‘true’ diminished cross-sectional area

which would be dependent on extension (and therefore strain) and the

compressibility of the tissue. Assuming that the arterial tissue is

perfectly incompressible (i.e. the volume of the specimen is a constant

throughout the test) and that the specimen shape is always rectangular,

derive a relationship for ‘A’ as a function of A0 and ε, and consequently

show that 0

1true

F

A

Let v0 be the volume of the tissue at zero-extension v the volume of the tissue at any point during extension

By assumption of incompressibility,

v0 = v l0 xA0 = l x A 0 0 00

l A lA A

l l

But,0

0

l

ll 1

0l

l

1

10

l

l

A

A0

l

l0





Therefore, 0

1

AA

Since, true

F

A

0

1true

F

A

2.3 A circular aluminum tube 40 cm in length is subject to a tensile load of 2 kN. The

outside and inside diameters of the tube are 4.2 and 4.0 cm respectively. What is the

amount of tensile stress on the bar? Aluminum has a elastic modulus of 73.1 GPa.

Determine the axial strain and the increase in length of the bar for the given load.

Assuming a typical elastic modulus of about 0.1MPa for a typical artery, what would be

the corresponding tensile load on an arterial specimen of the same dimension that would

result in the same axial strain?

L = 40 cm =0.4m; P = 20 x 103 N

Tensile stress

37

2 2

2 101.55 10 15.5

.021 0.02a

P xx Pa MPa

A

Axial strain

7

49

1.55 102.12 10 / 212

73.1 10a

a

x mx m m mE x

Increase in length 4 52.12 10 0.4 8.5 10 0.085axial a L x x x m mm

For the artery with the same axial strain, the axial stress is 6 40.1 10 2.12 10 21.2a art aE x x x Pa

The corresponding axial load is 2 221.2 (0.021) (0.02) 0.0027art aP xA x x N





2.4 A high strength steel rod (E = 200 GPa and ν = 0.32) with a diameter of 5 cm is being

subjected to a compressive load of 10 kN. Determine the increase in diameter of the tube after

the load is applied.

Axial stress

3

2

10 105.09

0.025a

P xMPa

A

Axial strain

6

59

5.09 102.55 10 25.5

200 10a

a

x mmx m mE x

Lateral strain

6 60.32 25.5 10 8.15 10

8.15

lat amx x x m

mm

Increase in diameter 6 68.15 10 0.5 4.07 10 0.00407dia lat D x x x m mm

2.5 A brass specimen 10 mm in diameter and a length of 50 mm is loaded with a 20 kN force

in tension. If the length increases by 0.12 mm, determine the elastic modulus of the brass. If the

diameter of the bar decreases by 0.0083 mm, calculate the Poisson’s ratio of the material.

Axial stress

3

23 2

20 10255

5 10a

F x NMPa

A x m





0.120.0024 /

50a

mmmm mm

mm

Elastic modulus for brass –

106.1a

brass

a

E GPa

0.00830.00083 /

10lat

mmmm mm

mm

Poisson’s ratio for brass –

0.000830.35

0.0024lat

a

2.6 An aluminum soda can (E = 73.1 Gpa; 0.35 ) has a radius to thickness ratio of 200:1 and holds

soda under pressure. When the lid is opened to release the pressure, the strain in the longitudinal

direction is measured as 170 μm/m. What was the internal pressure in the can? Express your answer

in the units of mm Hg. Compare this pressure magnitude with the typical mean blood pressure in an

artery.

For thin-walled tubes;

2a

pr

t

and

pr

t

Considering only the axial strain:





9 6 3

2

73.1 10 170 10 2 / 200 124.3 10 934.4

a

a

a a

a

E

E

prE

t

p x x x x x Pa mmHg

Including the effect of circumferential strain in the analysis, Hooke’s law relationship gives

1 1

2

1

2

a a

pr pr

E E t t

pr

Et

Hence,

0.5a

Ep

rt

Substituting the values for axial strain and the elastic modulus and Poisson’s ratio for aluminum,

54.14 10 3,115p x Pa mmHg

Compare this magnitude to an average mean arterial pressure of 100 mmHg.



Courtesy of CRC Press/Taylor & Francis Group

Strain-hardening

Strain (ε) Strain (ε)(a) (b)

Rupture

Stre

ss (σ

)

Stre

ss (σ

)

σy

σb

σb

σσult

Yield Necking

FIGURE 2.2 Stress–strain diagrams for typical ductile (a) and brittle (b) material.

002x002.eps




Pδ

Stre

ss (σ

)Strain (ε)(a) (b)

ℓℓ0

FIGURE 2.1 Stress–strain behavior of Hookean elastic material: (a) a specimen with an initial length ℓ0 is shown to deform to the final length ℓ with an axial load P; (b) the resulting linear stress–strain plot for a material that follows the Hooke’s law.

002x001.eps




P

σx =PA

z

A

x

y

δ

FIGURE 2.4 An axially loaded slender bar.

002x004.eps




Strain (ε)

Stre

ss (σ

)

dε dσ

Einc = dσ/dε

FIGURE 2.3 Schematic of a nonlinear stress versus strain relationship for biological soft tissue.

002x003.eps




z x

yσy

σzσx

τyzτzy

τzx

τyxτxy

τxz

FIGURE 2.6 General stress conditions on a cubic element.

002x006.eps




y y π2 + γxy

π2

– γxy

τxy

τyx

τyx

τxy

x x

FIGURE 2.5 An element subjected to shearing stresses.

002x005.eps




σz

σz

z

tσθ

σθ

FIGURE 2.7 Stresses on a thin-walled cylindrical vessel.

002x007.eps




(a) (b)

σθ

σθ

σz

p

p

RR

Rz

∆z

z

t

tt

x

FIGURE 2.8 Forces on segments of the thin-walled cylindrical pressure vessel: (a) schematic of the forces in the diametrical plane for hoop stress, σθ; (b) forces in the axial direction for axial stress, σz.

002x008.eps




p2

p10

R2

R1

dz

p2

p1dθ

σr

σ

σr + dσr

dr

σ+–

(a) (b)

z

FIGURE 2.9 Stresses on a thick-walled open-ended vessel: (a) schematic of the thick-walled cylindri-cal vessel subjected to internal and external pressure; (b) forces acting on a small element of the thick-walled vessel.

002x009.eps




t= 0Time

Time

ε0

σ0

Stre

ss (σ

)St

rain

(ε)

ε

t=

FIGURE 2.11 Creep test for a viscoelastic material.

002x011.eps




σ

σ σ

σ

σ

(a)

(c)

(b)

1 1

(d)

σ

σ σσσ

ε

ε

µKs

µ

µ

µ

Ks

Ks

Ks

ε̇

ε̇

FIGURE 2.10 A typical viscoelastic material schematically represented by (a) a linear spring repre-senting the elastic component with the corresponding stress–strain relationship and (b) the viscous com-ponent represented by a dashpot with the corresponding stress–strain rate relationship. The combined viscoelastic behavior of a solid represented by (c) a Maxwell model with the spring and dashpot in series and (d) a Kelvin model with spring and dashpot in parallel.

002x010.eps




Dynamic loading

Φ

Static loadingPr

essu

reRa

dius

Time

FIGURE 2.13 Response to time-dependent pressure loading of a viscoelastic structure.

002x013.eps




t= 0Time

Time

σ

σ0

ε0

Stra

in (ε

)St

ress

(σ)

t=

FIGURE 2.12 Stress relaxation experiment for a viscoelastic material.

002x012.eps



solutions manual for biofluid mechanics the human ... · biofluid mechanics: the human circulation...

Documents