By G. W Housner
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Additional info for The analysis of stress and deformation
A surface of a stressed, isotropic body lies in the xy-plane. At a point on the surface where there are no applied loads acting, the strains e,, e,, and y,, are measured. Write the arrays for the state of stress and strain at the point in terms of the measured strains. 46 Strain-Measurements. If a material is isotropic, its elastic properties are described by the two constants E and G. These physical constants can be measured easily in the laboratory. If a bar is pulled in tension in a testing machine, its elongation per inch of length, when correlated with the applied force, will determine the value of E.
The stress-strain diagrams for some real materials are indicated in Fig. 32. The precise shapes of these curves depend upon the composition of the material, its prior thermal and strain treatment and the characteristics of the loading system. It is seen that for small strains steel can be considered to be linearly elastic. For strains somewhat beyond the yield point steel is a reasonable approximation to an elastic-plastic material. The point of initial yielding is called the upper yield point to distinguish it from the lower yield point which corresponds to the horizontal portion of the curve in Fig.
This state of stress is called pure biaxial compression or tension. If the principal stresses are such that U , = -a,, then at 45' from the principal plane the maximum planar shear stress has the value a,, and the stresses a,. and 0,. are zero. This state of stress is called pure shear. Mohr's circle may be used for the transformation of the general state of stress at a point when the stress element is rotated about the x, y, or z axis. A Mohr's circle for rotation of the stress element about each of the three principal axes is drawn in Fig.
The analysis of stress and deformation by G. W Housner