Measurements of Stress and Strain

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Relationship between force and body
Stress
Strain
Application of Stress-Strain Curve
Measurements of Stress and Strain

1. Frankel V.H. & Nordin M (2001): Biomechanics of Bone. In Nordin M. & Frankel VH (eds): Basic Biomechanics of the Musculoskeletal System. Philadelphia, PS, USA: Lippincott Williams & Wilkins. pp.26-58.
2. Chaffin & Andersson, 1999: 101-124, 146-158, 167-170

Relationship Between Force and Body

Force

• definition： an action that changes the state of rest or motion to which it is applied
• external force vs. internal force
• strength： maximum force that a body can be loaded
• stress： load per unit

Body

• an object that may be real or imaginary but represents a definite quantity of matter (mass), with certain dimensions, occupying a definite position in space
• rigid vs. deformable body
  • rigid body： no relative displacement can occur between the particles when forces are applied to the body
  • deformable body： the adjacent particles can be displaced relative to one another when the forces are applied to the body

Effect of forces on a body

• in dynamic sense
  • linear motion (translation) in the direction of net force
  • rotary motion (rotation) in the direction of net moment
• in static sense
  • static equilibrium if the body is rigid or if the stress is low or if the duration is short
  • deformation (shape and size changes) if the body is deformable
• long-term biological changes
  • growth
  • injuries
  • degeneration

Mechanics of Materials

• a branch of applied mechanics that develops relationship between the external loads applied to a deformable body and the internal forces acting within the body
  • deformation of the body
  • body's stability when it is subjected to external loads

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Stress

Definition of stress

definition： the intensity of force per unit area of the tissue normal stress： the intensity of internal force acting perpendicular to a plane
s = DF / DA assumptions： the material is homogeneous the cross-sectional area at each point is the same the strain is even the resultant load is passing through its centroid
shear stress： the intensity of internal force acting tangent to a plane t = DV / DA
SI unit：  Pa (Pascal) = N/m2 USCS unit： psi = lb/in2 tensile stress is positive while compressive stress is negative

Types of stress

• tensile stress (tension)
  • one kind of normal stress that is applied perpendicular to the body and takes it apart
  • the body tends to be elongated in the direction of the applied forces

• compressive stress (compression)
  • one kind of normal stress that is applied perpendicular to the body and puts it together
  • the body tends to be shrink in the direction of the applied forces

• shear stress
  • the force acting in directions tangent to the area resisting the force
  • also named as tangential force
  

• bending stress
  
  • failure under bending stress
    • three point bending: failure at the point of the middle force
    • four point bending: failure at the weakest point

• torsion stress： loads parallel to the surface of the structure and in the same direction, resulting in the tensile stresses and strains at one side and compressive stresses and strains at the other side of the structure; there are no stresses and strains along the neutral axis
  

• combined stress
  

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Strain

Definition of stain

the extent of deformation relative to its initial condition normal strain： the ratio of the change in length to the original length e = DL / L shear strain： g = d / h
unit： normal strain = % (dimensionless quantity) or   mm/m            shear strain = rad tensile strain is positive while compressive strain is negative

Factors affecting the extent of deformation

• mechanical properties
• size of the body
• shape of the body
• temperature
• humidity
• magnitude, direction, and duration of applied forces

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Application of Stress-Strain Curve

Stress-Strain curve

• elastic region： When the magnitude of the stress is small, the elastic force can be represented by the relation for an ideal spring (Hooke's law), i.e., the elastic force exerted by the viscoelastic material is proportional to the amount of deformation
  • F = k x
    where F = elastic force
                k = spring stiffness which is a constant
                x = amount of deformation
• plastic region
• yield point 　
• failure point

Strength

• maximum stress that a body can be loaded (ultimate stress)
• maximum strain that a body can be deformed (ultimate strain)
• maximum energy stored

Stiffness

• modulus of elasticity (Young's modulus)： for both tensile and compression stress
  • the ratio of the stress to strain in the elastic region of the stress-strain curve

    E = s / e

  • named after Thomas Young (1773-1829, English scientist)
  • SI unit： Pascal
  • USCS： psi

• Hooke's law ： only for tensile stress
  • for an elastic material, the strain is a linear function of the stress applied
  • named after Robert Hooke (1653-1703, English scientist)

• modulus of rigidity (shear modulus of elasticity)

  G = t / g      where g = d / h

• SI unit： Pascal
• USCS： psi

Poisson's ratio

• When a material is under a tensile stress, the tensile strain and the lateral contraction is proportional.

  n = e_lateral / e_longitudinal

  • assumptions：
    • the material is homogeneous
    • the material is isotropic
  • named after Simeon Denis Poisson (1781-1840)

  

• unit： dimensionless
• 0 £ n £ ½
• relationship between the modulus of elasticity and that of rigidity

  G = E / 2(1 + n)

Brittle vs. Ductile materials

Creep Phenomenon (潛變現象)

Load Relaxation Phenomenon (鬆弛現象)

Hysteresis (遲滯現象)

Elastic vs. Plastic materials

elasticity： the ability of a body to resume its original size and shape on removal of the applied loads
NOTE： the elastic material is not necessary to have a linear relationship on the stress-strain curve

plasticity： When a tissue is stretched to the plastic region and then released, the tissue will assume a new resting length that is longer than the initial length because of plastic changes in its structure.

clinical application： flexibility exercise or joint mobilization 

Allowable stress

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Measurements of Stress and Strain

Tension test

Compression test

ASTM

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