## Solid Mechanics

### Abstract

1 Continuum Description of Forces within the Lung: Stress
2 Boundary Conditions
3 Nonuniform Deformation: Displacement and Strain
4 Constitutive Equations
5 Methods of Problem Solving
6 Linear Elasticity
7 Two Elementary Deformations
 Figure 1. Top: typical transmission electron micrograph of tissue from an air‐filled, perfusion‐fixed lung. In the intact lung, forces are transmitted across the plane of the cut by tissue elements and surface tension. Squares A and B denote expanding area in which stress is measured.Courtesy of H. Bachofen. Bottom: force per unit area plotted as a function of the area of expanding square. Limiting value of force per unit area is the stress. At the scale at which this limiting value is reached, parenchyma can be treated as a continuum Figure 2. A: stress components that act on faces of any small cube of parenchyma. B: thin slice of tissue at surface of lung, including pleural membrane. In the uniformly expanded lung, shear stresses equal zero and normal stresses (τ) equal transpulmonary pressure (PL). Figure 3. Shear strain: deformation in which material changes shape with no change in volume. Solid lines, undeformed shapes; dashed lines, deformed shapes. Stresses (τ) that accompany this deformation may include shear and normal stresses, depending on the orientation of the plane on which they act. γ, Change of angle. Figure 4. Model of cylindrical hole cut in uniformly expanded lung. Normal stress equal to transpulmonary pressure (PL) has to be applied at the inner boundary of hole to prevent distortion of the surrounding parenchyma. Vessel that fits perfectly in this hole is subjected to both alveolar gas pressure and material stress equal to and opposite that shown. If hole expands, the parenchyma distorts with no change in volume and the material stress is reduced by 2μ times the fractional change in hole radius (δ/a). μ, Shear modulus; a, radius of hole in uniformly expanded lung; δ, change of hole radius. Figure 5. Gravitational distortion of cylindrical solid in a rigid cylindrical container. This distortion produces local volume and shape changes (dashed lines). Because of the resistance of the solid to shear, the vertical gradient of the surface stress and the vertical gradient of the volume per unit mass are less than they would be for a liquid of the same specific weight and compressibility. l, Height; r, radius; z, vertical coordinate.
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