. According to numerical simulation and field testing, Zhang determined that the vehicle load on the top surface of expressway subgrade is approximately 30~60kPa, and the vehicle load waveform on the top of the subgrade could be approximated as a half-sine pulse waveform .
where [mathematical expression not reproducible]; [bar.p] and [bar.q] are the effective hydrostatic pressure and the effective Mises equivalent deviatoric stress
, respectively; [[??].sub.max] is the algebraically maximum eigenvalue of [bar.[sigma]]; the brackets <x> are used in Macaulay sense; [[bar.[sigma]].sub.c]([[??].sup.p.sub.c]) is the uniaxial compressive effective strength; [alpha] and [gamma] are dimensionless material constants, which can be determined by comparing the initial equibiaxial and uniaxial compressive yield stress and by comparing the yield conditions along the tensile and compressive meridians, respectively; and [beta]([[??].sup.p]) can be calculated by
13 and 21 and ignoring elastic deformations, a pseudo-linear relationship between the microscopic deviatoric stress
and deformation rate can be further expressed as
The test surface of the crack of the maximum tensile stress criterion in the deviatoric stress
plane, meridian plan, and state of plane stress is shown in Figures 4-6.
where v is the velocity vector, [rho] the density, f the body force, [sigma] the total stress tensor, p the pressure, [tau] the deviatoric stress
tensor, [C.sub.p] the heat capacity of polymer, T the temperature of polymer, k the thermal conductivity of polymer, [[eta].sub.E] the Newtonian viscosity, and D the rate of strain tensor.
where [s.sub.T] is the second-order deviatoric stress
tensor, which represents the deviatoric stress
tensor of the intersection point of the yield surfaces [f.sub.m+1] and [f.sub.m], and [[alpha].sub.m] and (p' + [p'.sub.0])[[alpha].sub.m+1] are the center of the yield surfaces [f.sub.m] and [f.sub.m+1], respectively.
The resulting stress-strain curves show that the axial deviatoric stress
continues to increase until the strain exceeds 20%, which can be chosen as the maximum permissible strain [[gamma].sub.f].
where [rho] is density of the shell, t is the time coordinate, P = P(r,t) is the pressure in the shell, [[tau].sub.rr] = [[tau].sub.rr] (r,t) is the deviatoric stress
tensor in the shell fluid in the r direction that is due to applied stress in the r direction, and [[tau].sub.[theta][theta]] = [[tau].sub.[theta][theta]] (r,t) is the deviatoric stress
tensor in the shell fluid in the [theta] direction that is due to applied stress in the [theta] direction.
[S.sub.ij] is the deviatoric stress
, and [[sigma].sub.kk] is the spherical stress.
where [[sigma]'.sub.ij] = [[sigma].sub.ij] - [1/3][[delta].sub.ij][[sigma].sub.ij] is the deviatoric stress
, [~.[sigma]] = [square root of ([3/2][[sigma]'.sub.ij][[sigma]'.sub.ij])] is the equivalent stress or flow stress, [~.[epsilon]] = [square root of ([2/3][[epsilon].sub.ij][[epsilon].sub.ij])] is equivalent strain, and [[??].[epsilon]] = [square root of ([2/3][dot.[epsilon].sub.ij][dot.[epsilon].sub.ij])] is the equivalent strain rate.
Caption: FIGURE 5: Relationship between deviatoric stress
and axial strain at different applied confining pressures: (a) 2.5W+C100+P40, (b) 3.0W+ C100+P70, (c) 3.0W+C200, and (d) 3.0W+C200+P50.
which is adopted from the Stassi equation, where [I.sub.1] is the first stress invariant, [J.sub.2] is the second deviatoric stress
invariant, and R is the ratio of the "yield" stress in compression to that in tension.