Axial stress and strain

Here is the elastic modulus tensor. It has 3 = 81 elements, however since the stress and strain are represented by symmetric matrices with six independent elements each, the number of independent modulus tensor elements is reduced to 36. An additional reduction to 21 is achieved by considering elastic materials for which a strain energy function exists. Physically, C2323 represents a shear modulus since it couples a shear stress with a shear strain. Cim couples axial stress and strain in the 1 or x direction,... 

Fig. 4.4. Axial stress and strain in the notched cross section for two different materials Linear-elastic (dashed line) and elastic-plastic (solid line). The figure shows the nominal stresses and strains (Tnss, nss) and the maximum values in the linear-elastic (o-max,el, Smax.el) and elastic-plastic case (Umax, Smax)...

Figure 2.6 Comparison of axial stress and strain (left) with shear stress and strain (right). (Reprinted with permission from Prof. Hiroshi Toshiyoshi, Institute of Industrial Science (IIS), The University of Tokyo, Japan.)...

It can be seen that there is a correspondence between the axial stress and strain and the shear stress and strain ... 

Partially Plastic Thick-Walled Cylinders. As the internal pressure is increased above the yield pressure, P, plastic deformation penetrates the wad of the cylinder so that the inner layers are stressed plasticady while the outer ones remain elastic. A rigorous analysis of the stresses and strains in a partiady plastic thick-waded cylinder made of a material which work hardens is very compHcated. However, if it is assumed that the material yields at a constant value of the yield shear stress (Fig. 4a), that the elastic—plastic boundary is cylindrical and concentric with the bore of the cylinder (Fig. 4b), and that the axial stress is the mean of the tangential and radial stresses, then it may be shown (10) that the internal pressure, needed to take the boundary to any radius r such that is given by... 

The axial load (compression) gives a uniform stress and strain in the absence of a bending moment. If a bending moment exists, then one side is extended while the other is compressed. 

In the context of elastic deformation two parameters, known as stress and strain respectively, are very relevant. Stress is an internal distributed force which is the resultant of all the interatomic forces that come into play during deformation. In the case of the solid bar loaded axially in tension, let the cross sectional area normal to the axial direction be A0. From a macroscopic point of view the stress may be considered to be uniformly distributed on any plane normal to the axis and to be given by o A0 where o is known as the normal stress. The stress has to balance the applied load, F, and one must, therefore have o Aq = F or o = F/Aq. The units of stress are those of force per unit area, i.e., newtons per square... 

Acoustic studies were performed on specimens of the Bridgman type, but fortunately lower levels of axial compressive stress than Bridgman had used were employed in order to more closely simulate the concentrated shear process of metal cutting. The apparatus used that was capable of measuring stresses and strains as well as acoustic signals arising from plastic flow is described in the dissertation of Walker [16]. Two important results were obtained ... 

The axial strain increased and maximum axial strain was 0.67% when we load axial stress, and the sample was compressed in axial direction. And the axial strain continued going up to 1.71% in the process of waiting for pore water pressure to be stable. The axial strain went down with unloading and to 1.19% when the deviate stress was 0. In the following process, the axial strain continued going down in a nearly same rate to 1.14% finally. 

With these two data, one of the best-known mechanical data for materials, the so-called elastic modulus, F, is calculated. It describes the relation between stress and strain when the strain and stress are both axially aligned ... 

The governing idea in this approach is to delimit the state variables to stress and strain in a finite number of sections of a rod of the magnetostrictive material. In a radial-axial model [69] an example of such sections is shown in Fig. 6.35. The constitutive equation for the field distribution inside the rod... 

There is a continuous decrease as the angles 0 and a deviate from 0°. There is no rise in axial strength, as indicated by the maximum stress and strain theories. 

A fundamental difference between the maximum work and the other theories lies in the question of interaction among the failure modes. The maximum stress and strain theories assume that there is no interaction among the three failure modes (axial, transverse, and shear failures). 

Hibbeler, R. C. 2011. Mechanics of Materials, 8th ed. Upper Saddle River, NJ Prentice Hall. Clear and comprehensive, this text includes stress, strain, mechanical properties of materials, axial load, torsion, bending, stress and strain transformations, design and deflection of beams and shafts, buckling of columns, and energy methods. Includes a photorealistic art program that helps students visualize concepts. 

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