Beer & Johnston Vector Mechanics for Engineers Statics 9th nvrehs.info Mazen Nawareg. nvrehs.info Page 1 11/27/08 PM user-s Eleventh Edition Vector Mechanics For Engineers Statics and Dynamics Ferdinand P. Beer Late of Lehigh University E. Russell Johnston, Jr. Late of University. Vector Mechanics for. Engineers, Statics. Ninth Edition. Volume 1, Chapters 2–5. Ferdinand P. Beer. Late of Lehigh University. E. Russell Johnston, Jr. University.
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The deformation band which formed at the upper yield point will propagate along the gauge length at the lower yield point.
The band occupies the whole of the gauge at the luders strain. Beyond this point, work hardening commences. The appearance of the yield point is associated with pinning of dislocations in the system.
For example, solid solution interacts with dislocations and acts as pin and prevent dislocation from moving. Therefore, the stress needed to initiate the movement will be large. As long as the dislocation escape from the pinning, stress needed to continue it is less. After the yield point, the curve typically decreases slightly because of dislocations escaping from Cottrell atmospheres.
As deformation continues, the stress increases on account of strain hardening until it reaches the ultimate tensile stress. Until this point, the cross-sectional area decreases uniformly because of Poisson contractions. Then it starts necking and finally fractures.
The appearance of necking in ductile materials is associated with geometrical instability in the system. Due to the natural inhomogeneity of the material, it is common to find some regions with small inclusions or porosity within it or surface, where strain will concentrate, leading to a locally smaller area than other regions.
For strain less than the ultimate tensile strain, the increase of work-hardening rate in this region will be greater than the area reduction rate, thereby make this region harder to be further deform than others, so that the instability will be removed, i. However, as the strain become larger, the work hardening rate will decreases, so that for now the region with smaller area is weaker than other region, therefore reduction in area will concentrate in this region and the neck becomes more and more pronounced until fracture.
After the neck has formed in the materials, further plastic deformation is concentrated in the neck while the remainder of the material undergoes elastic contraction owing to the decrease in tensile force. The stress-strain curve for a ductile material can be approximated using the Ramberg-Osgood equation. Brittle materials[ edit ] Brittle materials, which includes cast iron, glass, and stone, are characterized by the fact that rupture occurs without any noticeable prior change in the rate of elongation,  sometimes they fracture before yielding.
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