Surface under Constant Normal Pressurep? For the particular case of aconstant pressure of magnitude[31-鲁2(1-x /0)E(1-v)α2[x2,t]=-p, t-x2 /c, ≥ 0e2. Evidently, a stress pulse equal in magnitude to the surfacepressure propagates vertically through the half space withspeed CL.: The velocity of the solid is constant in: O < x2 < tcL._ Ou [z,1] _ Cp (1+v)(1-2v) _ PV2[x2,t] =atE(1-v)pcL6
• For the particular case of a constant pressure of magnitude Surface under Constant Normal Pressure 6 2 2 2 2 2 2 1 1 2 , 1 , , L 0 L L c u x t t x c t x c E x t p p • Evidently, a stress pulse equal in magnitude to the surface pressure propagates vertically through the half space with speed cL . • The velocity of the solid is constant in: 0 < x2 < tcL . 2 2 2 2 , 1 1 2 , 1 L L u x t c p p v x t t E c p
Surface under Time-varying Shear Traction. The solid is at rest and stressq[]区区区区区区区区free at time t = 0etStrain-displacement relation:=二i+uiiIsotropic Hooke's Law:e21+vEVVOOi1EE(1+11a?uao1: Linear momentum balance equations:Cat?Ox? Symmetry condition indicatesu =u = O, u = us [x2,t]1 us The only nontrivial strain component: 8232 0x27
Surface under Time-varying Shear Traction 7 • The solid is at rest and stress free at time t = 0. • Strain-displacement relation: , , 1 2 ij i j j i u u 2 2 ij j i u x t 1 ; . 1 1 2 ij ij kk ij ij ij kk ij E E E • Linear momentum balance equations: • Isotropic Hooke’s Law: • Symmetry condition indicates u u u u x t 1 2 3 3 2 0, , • The only nontrivial strain component: 3 23 2 1 2 u x q t