{"id":523,"date":"2020-04-19T15:59:50","date_gmt":"2020-04-19T19:59:50","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/humanbiomechanics\/chapter\/4-2-hookes-law-originally-section-5-3-elasticity-stress-and-strain-2\/"},"modified":"2020-09-03T15:05:18","modified_gmt":"2020-09-03T19:05:18","slug":"4-2-hookes-law-originally-section-5-3-elasticity-stress-and-strain-2","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/humanbiomechanics\/chapter\/4-2-hookes-law-originally-section-5-3-elasticity-stress-and-strain-2\/","title":{"raw":"10.1 Force-Deformation Curve","rendered":"10.1 Force-Deformation Curve"},"content":{"raw":"\n
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Forces can affect an object\u2019s shape. A change in shape due to the application of a force is a deformation<\/strong>. Even very small forces are known to cause some deformation. For small deformations, two important characteristics are observed. First, the object returns to its original shape when the force is removed\u2014that is, the deformation is elastic for small deformations. Second, the size of the deformation is proportional to the force\u2014that is, for small deformations, Hooke\u2019s law<\/strong> is obeyed. In equation form, Hooke\u2019s law is given by<\/p>\n\n

[latex]\\boldsymbol{F = k\\Delta{L},}[\/latex]<\/div>\n

where \u0394L<\/em><\/strong> is the amount of deformation (the change in length, for example) produced by the force F<\/strong><\/em>, and k<\/strong><\/em> is a proportionality constant that depends on the shape and composition of the object and the direction of the force. Note that this force is a function of the deformation \u0394L<\/em><\/strong> \u2014 it is not constant as a kinetic friction force is. Sometimes we use \u0394x<\/i><\/strong><\/b> instead of \u0394L.<\/i><\/strong><\/b>  The deformation can be along any axis.  Rearranging this to<\/p>\n\n

[latex]\\boldsymbol{\\Delta{L}\\: = }\\boldsymbol{\\frac{F}{k}}[\/latex]<\/div>\n

makes it clear that the deformation is proportional to the applied force. Figure 1<\/a> shows the Hooke\u2019s law relationship between the extension \u0394L<\/em><\/strong> of a spring or of a human bone. For metals or springs, the straight line region in which Hooke\u2019s law pertains is much larger. Bones are brittle and the elastic region is small and the fracture abrupt. Eventually a large enough stress to the material will cause it to break or fracture. Tensile strength<\/strong> is the breaking stress that will cause permanent deformation or fracture of a material.<\/p>\n\n

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HOOKE'S LAW<\/h3>\n
[latex]\\boldsymbol{F = k\\Delta{L},}[\/latex]<\/div>\n

where \u0394L<\/em><\/strong> is the amount of deformation (the change in length, for example) produced by the force F<\/strong><\/em>, and k<\/strong><\/em> is a proportionality constant that depends on the shape and composition of the object and the direction of the force.<\/p>\n\n

[latex]\\boldsymbol{\\Delta{L}\\:=}\\boldsymbol{\\frac{F}{k}}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n
<\/figcaption>\n\n[caption id=\"\" align=\"aligncenter\" width=\"225\"]\"Line Figure 1.<\/strong> A graph of deformation \u0394L<\/em><\/strong> versus applied force F<\/strong><\/em>. The straight segment is the linear region where Hooke\u2019s law is obeyed. The slope of the straight region is 1 \/ k<\/strong>. For larger forces, the graph is curved but the deformation is still elastic\u2014\u0394L<\/em><\/strong> will return to zero if the force is removed. Still greater forces permanently deform the object until it finally fractures. The shape of the curve near fracture depends on several factors, including how the force F<\/strong><\/em> is applied. Note that in this graph the slope increases just before fracture, indicating that a small increase in F<\/strong><\/em> is producing a large increase in L<\/strong><\/em> near the fracture.[\/caption]<\/figure>\n

The proportionality constant k<\/strong><\/em> depends upon a number of factors for the material.<\/p>\nWe now consider the type of deformation that will cause a change in length (tension and compression).   There are also sideways forces that cause shear (stress), and changes in volume, but we will not go into detail about them in this course.\n\n

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Changes in Length\u2014Tension and Compression: Elastic Modulus<\/h1>\n

A change in length \u0394L<\/em><\/strong> is produced when a force is applied to a muscle or tendon parallel to its length L<\/em>0<\/sub><\/strong>, either stretching it (a tension) or compressing it. (See Figure 3<\/a>.)<\/p>\n\n

\n\n[caption id=\"\" align=\"aligncenter\" width=\"175\"]\"Figure Figure 3.<\/strong> (a) Tension. The muscle\/tendon is stretched a length \u0394L<\/em><\/strong> when a force is applied parallel to its length. (b) Compression. The same rod is compressed by forces with the same magnitude in the opposite direction. For very small deformations and uniform materials, \u0394L<\/em><\/strong> is approximately the same for the same magnitude of tension or compression. For larger deformations, the cross-sectional area changes as the rod is compressed or stretched.[\/caption]<\/figure>\n

Experiments have shown that the change in length (\u0394L<\/em><\/strong>) depends on only a few variables. As already noted, \u0394L<\/em><\/strong> is proportional to the force F<\/strong><\/em> and depends on the substance from which the material is made. Additionally, the change in length is proportional to the original length L<\/em>0<\/sub><\/strong> and inversely proportional to the cross-sectional area of the wire or rod. For example, a long guitar string will stretch more than a short one, and a thick string will stretch less than a thin one. We can combine all these factors into one equation for \u0394L<\/em><\/strong>:<\/p>\n\n

[latex]\\boldsymbol{\\Delta{L}\\:=}\\boldsymbol{\\frac{1}{Y}\\frac{\\vec{\\textbf{F}}}{A}}\\boldsymbol{L_0},[\/latex]<\/div>\n

where \u0394L<\/em><\/strong> is the change in length, F<\/strong><\/em> the applied force, Y<\/strong><\/em> is a factor, called the elastic modulus or Young\u2019s modulus, that depends on the substance, A<\/strong><\/em> is the cross-sectional area, and L<\/em>0<\/sub><\/strong> is the original length. Table 3<\/a> lists values of Y<\/strong><\/em> for several materials\u2014those with a large Y<\/strong><\/em> are said to have a large tensile stiffness because they deform less for a given tension or compression.<\/p>\nBones, on the whole, do not fracture due to tension or compression. Rather they generally fracture due to sideways impact or bending, resulting in the bone shearing or snapping. The behaviour of bones under tension and compression is important because it determines the load the bones can carry. Bones are classified as weight-bearing structures such as columns in buildings and trees. Weight-bearing structures have special features; columns in building have steel-reinforcing rods while trees and bones are fibrous. The bones in different parts of the body serve different structural functions and are prone to different stresses. Thus the bone in the top of the femur is arranged in thin sheets separated by marrow while in other places the bones can be cylindrical and filled with marrow or just solid. Overweight people have a tendency toward bone damage due to sustained compressions in bone joints and tendons.\n\n<\/section>

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 Functionally, the tendon (the tissue connecting muscle to bone) must stretch easily at first when a force is applied, but offer a much greater restoring force for a greater strain. Figure 5<\/a> shows a stress-strain relationship for a human tendon. Some tendons have a high collagen content so there is relatively little strain, or length change; others, like support tendons (as in the leg) can change length up to 10%. Note that this stress-strain curve is nonlinear, since the slope of the line changes in different regions. In the first part of the stretch called the toe region, the fibers in the tendon begin to align in the direction of the stress\u2014this is called uncrimping<\/em>. In the linear region, the fibrils will be stretched, and in the failure region individual fibers begin to break. A simple model of this relationship can be illustrated by springs in parallel: different springs are activated at different lengths of stretch. Examples of this are given in the problems at end of this chapter. Ligaments (tissue connecting bone to bone) behave in a similar way.<\/p>\n\n

\n\n[caption id=\"\" align=\"aligncenter\" width=\"225\"]\"The Figure 5.<\/strong> Typical stress-strain curve for mammalian tendon. Three regions are shown: (1) toe region (2) linear region, and (3) failure region.[\/caption]<\/figure>\n

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\u2014the elastic behaviour 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.<\/p>\n\n

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Example 2: Calculating Deformation: How Much Does Your Leg Shorten When You Stand on It?<\/h3>\nCalculate 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.\n

Strategy<\/strong><\/p>\n

The force is equal to the weight supported, or<\/p>\n\n

[latex]\\boldsymbol{F = mg =(62.0\\textbf{ kg})(9.80\\textbf{ m\/s}^2) = 607.6\\textbf{ N}},[\/latex]<\/div>\n

and the cross-sectional area is  \u03c0r<\/em>2<\/sup>  =  1.257 \u00d7 10-3<\/sup> m2<\/sup><\/strong>. The equation [latex]\\boldsymbol{\\Delta{L} = \\frac{1}{Y}\\frac{F}{A}L_0}[\/latex] can be used to find the change in length.<\/p>\n

Solution<\/strong><\/p>\n

All quantities except \u0394L<\/em><\/strong> are known. Note that the compression value for Young\u2019s modulus for bone must be used here. Thus,<\/p>\n\n

[latex]\\boldsymbol{\\Delta{L}=(\\frac{1}{9\\times10^9\\textbf{ N\/m}^2})(\\frac{607.6\\textbf{ N}}{1.257\\times10^{-3}\\textbf{ m}^2})(0.400\\textbf{ m})}\\boldsymbol{=2\\times10^{-5}\\textbf{ m}}.[\/latex]<\/div>\n

Discussion<\/strong><\/p>\n

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 Table 3<\/a> have larger values of Young\u2019s modulus Y<\/strong><\/em>. In other words, they are more rigid.<\/p>\n\n<\/div>\n<\/div>\n

The equation for change in length is traditionally rearranged and written in the following form:<\/p>\n\n

[latex]\\boldsymbol{\\frac{F}{A}}\\boldsymbol{=Y}\\boldsymbol{\\frac{\\Delta{L}}{L_0}}.[\/latex]<\/div>\n

The ratio of force to area, [latex]\\boldsymbol{\\frac{F}{A}},[\/latex]  is defined as stress<\/span> <\/strong>(measured in N\/m2<\/sup><\/strong>), and the ratio of the change in length to length,[latex]\\boldsymbol{\\frac{\\Delta{L}}{L_0}},[\/latex]is defined as strain<\/span><\/strong> (a unitless quantity). In other words,<\/p>\n\n

[latex]\\boldsymbol{\\textbf{stress}=Y\\times\\textbf{strain}}.[\/latex]<\/div>\n

In this form, the equation is analogous to Hooke\u2019s law, with stress analogous to force and strain analogous to deformation. If we again rearrange this equation to the form<\/p>\n\n

[latex]\\boldsymbol{F=YA}\\boldsymbol{\\frac{\\Delta{L}}{L_0}},[\/latex]<\/div>\n

we see that it is the same as Hooke\u2019s law with a proportionality constant<\/p>\n\n

[latex]\\boldsymbol{k\\:=}\\boldsymbol{\\frac{YA}{L_0}}.[\/latex]<\/div>\n

This general idea\u2014that force and the deformation it causes are proportional for small deformations\u2014applies to changes in length, sideways bending, and changes in volume.<\/p>\n\n

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STRESS<\/h3>\n

The ratio of force to area, [latex]\\boldsymbol{\\frac{F}{A}},[\/latex] is defined as stress measured in N\/m2<\/sup>.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n

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STRAIN<\/h3>\n

The ratio of the change in length to length, [latex]\\boldsymbol{\\frac{\\Delta{L}}{L_0}},[\/latex] is defined as strain (a unitless quantity). In other words,<\/p>\n\n

[latex]\\boldsymbol{\\textbf{stress}=Y\\times\\textbf{strain}}.[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/section>
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Section Summary<\/h1>\n