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4.8 Elasticity: stress and strain  (Page 4/15)

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Unlike bones and tendons, which need to be strong as well as elastic, the arteries and lungs need to be very stretchable. The elastic properties of the arteries are essential for blood flow. The pressure in the arteries increases and arterial walls stretch when the blood is pumped out of the heart. When the aortic valve shuts, the pressure in the arteries drops and the arterial walls relax to maintain the blood flow. When you feel your pulse, you are feeling exactly this—the elastic behavior of the arteries as the blood gushes through with each pump of the heart. If the arteries were rigid, you would not feel a pulse. The heart is also an organ with special elastic properties. The lungs expand with muscular effort when we breathe in but relax freely and elastically when we breathe out. Our skins are particularly elastic, especially for the young. A young person can go from 100 kg to 60 kg with no visible sag in their skins. The elasticity of all organs reduces with age. Gradual physiological aging through reduction in elasticity starts in the early 20s.

Calculating deformation: how much does your leg shorten when you stand on it?

Calculate the change in length of the upper leg bone (the femur) when a 70.0 kg man supports 62.0 kg of his mass on it, assuming the bone to be equivalent to a uniform rod that is 40.0 cm long and 2.00 cm in radius.

Strategy

The force is equal to the weight supported, or

F = mg = 62 . 0 kg 9 . 80 m /s 2 = 607 . 6 N , size 12{F= ital "mg"= left ("62" "." 0`"kg" right ) left (9 "." "80"`"m/s" rSup { size 8{2} } right )="607" "." 6``N} {}

and the cross-sectional area is πr 2 = 1 . 257 × 10 − 3 m 2 size 12{πr rSup { size 8{2} } =1 "." "257"` times "10" rSup { size 8{ - 3} } m rSup { size 8{2} } } {} . The equation Δ L = 1 Y F A L 0 size 12{ΔL= { {1} over {Y} } { {F} over {A} } L rSub { size 8{0} } } {} can be used to find the change in length.

Solution

All quantities except Δ L size 12{ΔL} {} are known. Note that the compression value for Young’s modulus for bone must be used here. Thus,

Δ L = 1 9 × 10 9 N/m 2 607 . 6 N 1. 257 × 10 − 3 m 2 ( 0 . 400 m ) = 2 × 10 −5 m. alignl { stack { size 12{ΔL= { {1} over {9 times "10" rSup { size 8{9} } " N/m" rSup { size 8{2} } } } times { {"607" "." "6 N"} over {1 "." "257" times "10" rSup { size 8{ - 3} } " m" rSup { size 8{2} } } } times 0 "." "400 m"} {} #=0 "." "002" times "10" rSup { size 8{ - 3} } " m" {} } } {}

Discussion

This small change in length seems reasonable, consistent with our experience that bones are rigid. In fact, even the rather large forces encountered during strenuous physical activity do not compress or bend bones by large amounts. Although bone is rigid compared with fat or muscle, several of the substances listed in [link] have larger values of Young’s modulus Y size 12{Y} {} . In other words, they are more rigid and have greater tensile strength.

The equation for change in length is traditionally rearranged and written in the following form:

F A = Y Δ L L 0 . size 12{ { {F} over {A} } =Y { {ΔL} over {L rSub { size 8{0} } } } } {}

The ratio of force to area, F A size 12{ { {F} over {A} } } {} , is defined as stress    (measured in N/m 2 ), and the ratio of the change in length to length, Δ L L 0 size 12{ { {ΔL} over {L rSub { size 8{0} } } } } {} , is defined as strain    (a unitless quantity). In other words,

stress = Y × strain . size 12{"stress"=Y times "strain"} {}

In this form, the equation is analogous to Hooke’s law, with stress analogous to force and strain analogous to deformation. If we again rearrange this equation to the form

F = YA Δ L L 0 , size 12{F= ital "YA" { {ΔL} over {L rSub { size 8{0} } } } } {}

we see that it is the same as Hooke’s law with a proportionality constant

k = YA L 0 . size 12{k= { { ital "YA"} over {L rSub { size 8{0} } } } } {}

This general idea—that force and the deformation it causes are proportional for small deformations—applies to changes in length, sideways bending, and changes in volume.

Stress

The ratio of force to area, F A size 12{ { {F} over {A} } } {} , is defined as stress measured in N/m 2 .

Strain

The ratio of the change in length to length, Δ L L 0 size 12{ { {ΔL} over {L rSub { size 8{0} } } } } {} , is defined as strain (a unitless quantity). In other words,

stress = Y × strain . size 12{"stress"=Y times "strain"} {}
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Source:  OpenStax, Introduction to applied math and physics. OpenStax CNX. Oct 04, 2012 Download for free at http://cnx.org/content/col11426/1.3
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