Tham khảo tài liệu 'aircraft structures 1 2011 part 2', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 26 Basic elasticity For the case of plane stress they simplify to x x- y 1 y ơy vơx Zz - x- ry Txy lxy G 1-47 It may be seen from the third of Eqs that the conditions of plane stress and plane strain do not necessarily describe identical situations. Changes in the linear dimensions of a strained body may lead to a change in volume. Suppose that a small element of a body has dimensions 6x Sy and Sz. When subjected to a three-dimensional stress system the element will sustain a volumetric strain e change in volume unit volume equal to 1 Ex ổx l e 5y l Ez fe SxSySz e SxSySz Neglecting products of small quantities in the expansion of the right-hand side of the above equation yields r Ey z Substituting for ex Ey and z from Eqs we find for a linearly elastic isotropic body e E ơy ơz 2v ơx ơy or l-2iz . _ x e - o ơy Tz In the case of a uniform hydrostatic pressure ơx ơy ơz p and - P The constant 3 1 2v is known as the bulk modulus or modulus of volume expansion and is often given the symbol K. An examination of Eq. shows that V since a body cannot increase in volume under pressure. Also the lateral dimensions of a body subjected to uniaxial tension cannot increase so that V 0. Therefore for an isotropic material 0 V sg and for most isotropic materials V is in the range to below the elastic limit. Above the limit of proportionality V increases and approaches . Example A rectangular element in a linearly elastic isotropic material is subjected to tensile stresses of 83 N mm2 and 65 N mm2 on mutually perpendicular planes. Determine the strain in the direction of each stress and in the direction perpendicular to both Stress-strain relationships 27 stresses. Find also the principal strains the maximum shear stress the maximum shear strain and their directions at the point. Take E 200000N mm2 and V . If we assume that ơx 83N mm2 and ơy 65N mm2 then from Eqs 2Õ0ÕÕÕ 83- -3 x 65 3-175 x 10 200W 65- -3x83 M05xI 4 ez
