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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 • Hooke’s 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 body’s 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) • σ ij meaning: σ i (outward normal face direction) j (force direction) • σ 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 • σ xx is directed in a positive direction on a positive face • σ xx is directed in a negative direction on a negative face – Stress component is negative when compress – σ xy is assigned to positive direction on a positive face – It is proved σ xy = σ yx under condition of equilibrium (no rotation) Figure 2.4 General prove: σ xy = σ 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 σ xy = σ yx (no rotation) • Example General prove:...
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 Spring '12
 Mr.Wang
 Force, Shear Stress, Stress, Polar coordinate system

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