BioMech - Biological Materials ch. 9

External Forces impose loads

External forces on a body affect motion

External forces also load a body and affect internal structures

• Cartilage, ligaments, tendons, bones, muscle

Mechanical response of structures are related to injury

Mechanics of Materials

Considers segment to be a deformable body, not rigid

External forces imposed are resisted by internal forces and cause deformation

• external applied load: stress

• Deformation caused: strain

Mechanical Stress

How an applied force is distributed over the body it acts on

Mechanical stress equation

$$\sigma=\frac{F}{A}$$

$\sigma$ = stress

F = Internal force (N)

A = Cross-sectional area of the internal surface (m²)

• Units: N/m² (Pascal)

Internal force (F)

The force is the resultant of the many intermolecular bond forces across the imaginary cut plane

• F in the stress equation

Key feature of Mechanical stress

The amount of area the force is applied to:

Larger muscle harder to tear

Cross-Sectional Area

The cross-sectional area of the imaginary plane is the area over which the internal force is applied

• A in the stress equation

Types of Mechanical Stress

3 principal stresses depending on applied loading

• Tension

• Compression

• Shear

Axial Stress

external forces act perpendicular to the analysis plane

• Also call normal or longitudinal stress

Uniaxial load

external forces are colinear

Tensile stress

• an axial or normal stress

• occurs at the analysis plane as a result of a force or load pulling the molecules apart at that plane

• Tension causes body to deform by stretching (elongating)

Tension: Bone cross section

Bone cross-sectional area depends on plane of analysis

• Larger cross-sectional area: lower stress

• Small cross-sectional area: higher stress

Compression

Compressive stress

• an axial or normal stress

• occurs at the analysis plane as a result of a load squeezing the molecules together at that plane

• compression causes body to deform by shortening

Compression: Bone cross-section

Bone cross sectional area

• larger cross-sectional area: lower stress

• smaller cross-sectional area: higher stress

Shear

Shear stress

• a transverse stress

• acts parallel to the analysis plane as a result of non-colinear forces sliding molecules at that plane

Shear causes body to deform by skewing
- changes orientation of the sides of the object

shear stress represented with $\tau$

• $\tau$ =shear force / area

Tension Injury

Sprains (rupture) ligament and tendon

Strains (tear) muscle and cartilage

Compression injury

Bruises soft tissue

crushing fracture of bone

Shear injury

Blister on skin

Dislocation of a joint

Mechanical Loading

When external forces are not uniaxial, more complex loading occurs

• Bending

• Torsion

• Combined loads

Complexity of loading reflects

• Number, direction & location of external forces

• Shape of object

Bending

Bending load

• nonaxial loading

• Tension and compression produced at the analysis plane from 3 or more forces create force couples at ends of object

  • tends to rotate ends of body in opposite direction

• Bending loads cause body to deform by curve (bending)

  • One side squeezed

  • One side stretched

Torsion

Torsion load

• nonaxial loading

• shear stress acts parallel to the analysis plane as a result of opposing torques applied about the long axis of the body

• torsion causes body to deform by twisting

  • Shear stress from torsion load increases with greater distance from axis

Tubular and hollow cross-section of bones resists torsion

Combined Loading

Human bones under complex loading from gravity, tendons, ligaments, other bones, contact forces

• sustained loading a combination of

  • tension, compression, simple shear loads, bending, torsion

Strain

Quantifies the material’s deformation

Linear strain

change in an object’s length

• deformation produced by tensile or compressive stress

Shear Strain

change in orientation of an object’s molecules

• Deformation produced by shear stress

Reported as change in angle of perpendicular plane

$\Delta\theta$
Unit: Radians

Linear Strain mathematically

$$\varepsilon=\frac{\left(l-l_{o}\right)}{l_{o}}$$

$\varepsilon$ = linear strain

$l$ = stretched length

$l_{o}$ = original, undeformed length

$l-l_{o}$ = change in length

Units: dimensionless

Poisson’s Ratio

Width of an object also changes as it lengthens or shortens
When lengthened = wide ; When shortened = narrow

Ratio = (Axial strain) / (Transverse strain)

range: 0.1 to 0.5

Stress-Strain Relationship

Behavior of material under load

Elastic Behavior

Material deforms and returns to original length when unloaded

Plastic Behavior

Material deforms but doesn’t return to original length

• permanent deformation

Elastic Modulus (E)

Slope of the stress-strain curve

$$E=\frac{\Delta\sigma}{\Delta\varepsilon}$$

Also known as Young’s modulus

• Bulk modulus: slope describing compressive loading

• Shear modulus: Slope describing shear loading

Elastic Modulus & Materials

Stiff material: less strain per unit of stress (Large E)

Pilant material: more strain per unit of stress (low E)

• Bone is stiffer than ligaments or tendons

• Tendons and ligaments are more pilant than bone

Mechanical strength of a material

Maximum stress a material can withstand before failure

Elastic limit

The yield point for a material on the $\frac{\sigma}{\varepsilon}$ curve

• stress above this point causes permanent deformation of material

Yield strength

Stress at the elastic limit

• Disruption of internal organization, material does not regain shape

Ultimate strength

maximum stress material can withstand

• Material starts to give way

Failure strength

Stress level where total failure occurs

• Breakage or complete rupture of the material occurs

Failure strain

Strain exhibited when a total failure occurs

• Ductile material: exhibits large failure strain

  • Deforms a lot before giving way (ligament)

• Brittle material: exhibits a low failure strain

  • Deforms minimally before giving way (bone)

Toughness

The ability of a material to absorb energy before failure

• ligaments and tendons are tougher than bone

Connective Tissue

Components of the MS system exhibit different properties

• Muscle: active element of connective tissue

• passive elements of connective tissue

  • Bone, cartilage, ligament, tendon

Properties of collagen and elastin vary in connective tissue

Connective Tissue components

Collagen: Fibrous protein

• Very stiff

Elastin: Fibrous protein

• Very pilant

Connective tissue composition affects loading response

Isotropic

stiffness similar in different directions

Ansiotropic

Stiffness dependent on load direction

Mechanical properties of the MS: Bone

45% mineral, 30%—35% collagen, 20% water

Strongest and stiffest material in musculoskeletal system

• Compression > Tension > Shear

Mechanical properties vary with bone health

Bone strength affected by rate of loading

Mechanical properties of the MS: Cartilage

60%-80% water, 10%-30% collagen

• Hyaline (articular): covers bone ends at joints

• Fibrous: specialized in joints (invertebral disks, menisci)

Mechanical properties of the MS: Tendons & Ligaments

70% water, 25% collagen

Ligaments have more elastin than tendons

collagen fibers in bundles aligned with functional axis

• High stiffness to tensile loading

Mechanical properties of the MS: Muscle

Active component (sarcomere) determines muscle stiffness

• Passive component (connective tissue sheaths) create high failure strain