CHAPTER 14 COMPLEX STRAIN AND THE ELASTIC CONSTANTS Summary The relationships between the elastic constants are E=2G(1+v)and E=3K(1-2v) Poisson's ratio v being defined as the ratio of lateral strain to longitudinal strain and bulk modulus K as the ratio of volumetric stress to volumetric strain. The strain in the x direction in a material subjected to three mutually perpendicular stresses in the x,y and z directions is given by -2=Ea-o,-o,) Similar equations apply for E,and e. Thus the principal strain in a given direction can be found in terms of the principal stresses, since 6=是-管2-，-0-o For a two-dimensional stress system (i.e.a3 =0),principal stresses can be found from known principal strains,since -(e+veE and 02 eveE 01 (1-2) (1-2) When the linear strains in two perpendicular directions are known,together with the associated shear strain,or when three linear strains are known,the principal strains are easily determined by the use of Mohr's strain circle. 14.1.Linear strain for tri-axial stress state Consider an element subjected to three mutually perpendicular tensile stresses ao,and o. as shown in Fig.14.1. If o,ando were not present the strain in the x direction would,from the basic definition of Young's modulus E,be Ex= E 361CHAPTER 14 COMPLEX STRAIN AND THE ELASTIC CONSTANTS Summary The relationships between the elastic constants are E = 2G(1 +v) and E = 3K(1-2v) Poisson's ratio v being defined as the ratio of lateral strain to longitudinal strain and bulk modulus K as the ratio of volumetric stress to volumetric strain. The strain in the x direction in a material subjected to three mutually perpendicular stresses in the x, y and z directions is given by Qx 0 0, 1 & =--v-L,-v-=-((a -VQ -VQ) "E E EE" " Similar equations apply for E, and E,. since Thus the principal strain in a given direction can be found in terms of the principal stresses, For a two-dimensional stress system (i.e. u3 = 0), principal stresses can be found from known principal strains, since E (-52 + VEl) (1 - v2) E and o2 = (81 + YE21 (1 - v2) 61 = When the linear strains in two perpendicular directions are known, together with the associated shear strain, or when three linear strains are known, the principal strains are easily determined by the use of Mohr's strain circle. 14.1. Linear strain for tri-axial stress state Consider an element subjected to three mutually perpendicular tensile stresses Q,, Q, and Q, If Q, and Q, were not present the strain in the x direction would, from the basic definition of as shown in Fig. 14.1. Young's modulus E, be QX E, = - E 36 1