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Unformatted text preview: Biosolid Mechanics Chapter 2 Stress, strain and constitutive relation Introduction Influencing parameters for the failure of the structures External load: Weight W Geometry: cross section area Material Failure An inability to perform the intended mechanical function Break Fracture Tearing Rupture Deformation excessively Permanent Nonpermanent Figure 2.1 Concept of stress Hookes spring experiment: use different metallic springs ForceExtension curve: onetoone relationship F = k (xx ) k spring constant or stiffness Hooke original experiment did not consider the geometry Geometry: the thicker, the stiffer New parameter is needed to include cross area Intuition: F/A The force intensity: normal stress However, force effect is different when it act in different direction to a bodys surface Stress according to Cauchy: Force acting over an oriented area Depends on the vector of the force and surface Figure 2.3 Figure 2.2 Concept of stress Stress Type Normal stress: Tensile force: Measure of pulling (to elongate a material) action of an externally applied force in the direction perpendicular to the cut face Compressive force: Measure of pushing (to shorten a material) action of an externally applied force in the direction perpendicular to the cut face Shear stress: measure of resistance to the sliding action of an external force in the direction parallel to the cut face A force can always be decomposed into two: One normal to the surface: Normal stress One parallel to the surface: Shear stress Definition and components equ. (2.1) ij meaning: i (outward normal face direction) j (force direction) xy meaning Property Unit of stress Not a vector Depends on coordinate system A mathematic construct: tensor Can be described using positive sign convention Positive sign convention It is useful to represent the components of stress by arrows that act on the appropriate faces of a body in the appropriate direction For positive sign convention: Stress component is positive when tensile xx is directed in a positive direction on a positive face xx is directed in a negative direction on a negative face Stress component is negative when compress xy is assigned to positive direction on a positive face It is proved xy = yx under condition of equilibrium (no rotation) Figure 2.4 General prove: xy = yx Each point could be regarded as an infinitesimal cube The cube has six faces relative to each Cartesian coordinate system Under condition of equilibrium, no rotation Equ. (2.6) Thus, proved xy = yx (no rotation) Example General prove:...
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 Spring '12
 Mr.Wang

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