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Linear_Elasticity_02_Elastic_Model - Section 4.2 4.2 The...

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Section 4.2 Solid Mechanics Part I Kelly 97 4.2 The Linear Elastic Model 4.2.1 The Response of Real Materials The Tension Test Consider the following key experiment, the tensile test , in which a small, usually cylindrical, specimen is gripped and stretched, usually at some given rate of stretching. The force required to hold the specimen at a given displacement/stretch is recorded, Fig. 4.2.1. For many engineering materials, for example steel, rocks and concrete, it is found that the force is proportional to displacement as with the portion OA in Fig. 4.2.1. The following observations will also be made: (1) if the material is unloaded, the force/displacement curve will trace back along the line OA down to zero force and zero displacement; further loading and unloading will again be up and down OA (2) the force-displacement curve will be more or less the same regardless of the rate at which the specimen is stretched (at least at moderate temperatures). (3) the loading curve remains linear up to a certain force level, the yield point or elastic limit of the material (point A ). Beyond this point, permanent deformations are induced; on unloading to zero force, the specimen will have a permanent elongation. (4) the strains up to the elastic limit are small Figure 4.2.1: force/displacement curve for the tension test Stress-Strain Curve There are two definitions of stress used to describe the tension test. First, there is the force divided by the original cross sectional area of the specimen 0 A ; this is the nominal stress or engineering stress , 0 A F n = σ (4.2.1) 0 A Force Displacement elastic limit
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Section 4.2 Solid Mechanics Part I Kelly 98 Alternatively, one can evaluate the force divided by the (smaller) current cross-sectional area A , leading to the true stress A F = σ (4.2.2) in which F and A are both changing with time. For small elongations, within the linear range OA , the cross-sectional area of the material undergoes negligible change and both definitions of stress are more or less equivalent. Similarly, one can describe the deformation in two alternative ways. Denoting the original specimen length by 0 l and the current length by l , one has the engineering strain 0 0 l l l = ε (4.2.3) Alternatively, the true strain accounts for the fact that the “original length” is continually changing; a small change in length dl leads to a strain increment l dl d / = ε and the true strain is defined as the accumulation of these increments: = = 0 ln 0 l l l dl l l t ε (4.2.4) The true strain is also called the logarithmic strain . Again, at small deformations, the difference between these two strain measures is negligible. The stress-strain diagram for a tension test can now be described using the true stress/strain or nominal stress/strain definitions, as in Fig. 4.2.2. The stress value at the elastic limit is called the yield stress Y .
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