StressStrain_03_Internal_Stress - Section 3.3 3.3 Internal...

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Section 3.3 Solid Mechanics Part I Kelly 40 3.3 Internal Stress The idea of stress considered in §3.1 is not difficult to conceptualise since objects interacting with other objects are encountered all around us. A more difficult concept is the idea of forces and stresses acting inside a material, “within the interior where neither eye nor experiment can reach” as Euler put it. It took many great minds working for centuries on this question to arrive at the concept of stress we use today, an idea finally brought to us by Augustin Cauchy, who presented a paper on the subject to the Academy of Sciences in Paris, in 1822. Augustin Cauchy 3.3.1 Cauchy’s Concept of Stress Uniform Internal Stress Consider first a long slender block of material subject to equilibrating forces F at its ends, Fig. 3.3.1a. If the complete block is in equilibrium, then any sub-division of the block must be in equilibrium also. By imagining the block to be cut in two, and considering free-body diagrams of each half, as in Fig. 3.3.1b, one can see that forces F must be acting within the block so that each half is in equilibrium. Thus external loads create internal forces ; internal forces represent the action of one part of a material on another part of the same material. If the material out of which the block is made is uniform over this cut, one can take it that a uniform stress A F / = σ acts over this interior surface, Fig. 3.3.1b. Figure 3.3.1: a slender block of material; (a) under the action of external forces F, (b) internal normal stress σ , (c) internal normal and shear stress F F FF ) a () b ( ) c ( F F F F A F = N S F imaginary cut F
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Section 3.3 Solid Mechanics Part I Kelly 41 Note that, if the internal forces were not acting over the internal surfaces, the two half- blocks of Fig. 3.3.1b would fly apart; one can thus regard the internal forces as those required to maintain material in an un-cut state. If the internal surface is at an incline, as in Fig. 3.3.1c, then the internal force required for equilibrium will not act normal to the surface. There will be components of the force normal and tangential to the surface, and thus both normal ( N σ ) and shear ( S ) stresses must arise. Thus, even though the material is subjected to a purely normal load, shear stresses develop. From Fig. 3.3.2a, the normal and shear stresses arising on an interior surface inclined at angle θ to the horizontal are { Problem 1} cos sin , cos 2 A F A F S N = = (3.3.1) Figure 3.3.2: stress on inclined surface; (a) decomposing the force into normal and shear forces, (b) stress at an internal point Although stress is associated with surfaces, one can speak of the stress “at a point”. For example, consider some point interior to the block, Fig 3.3.2b. The stress there evidently depends on which surface through that point is under consideration. From Eqn. 3.3.1a, the normal stress at the point is a maximum A F / when 0 = and a minimum of zero when o 90 = . The maximum normal stress arising at a point within a material is of special significance, for example it is this stress value which often determines whether a material will fail (“break”) there.
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StressStrain_03_Internal_Stress - Section 3.3 3.3 Internal...

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