End-grafted polymer chains exposed to strong shear solvent flow in the x-direction are investigated by a non-equilibrium Monte Carlo method using the bond-fluctuating model. The solvent flow is modelled by an enhanced jump rate of monomers in the Row direction. Under strong enough shear flow, the shear force is non-linear with the blob size xi depending on the shear force and not on the monomer volume fraction phi. For the case of a grafted single chain, our data on the end-to-end distance of the polymer for a wide range of shear rate gamma agree well with the proposed scaling form and compare reasonably well with the Langevin force law in the extreme strong shear case. Also the scaling for the blob size is derived with xi = aN(nu)K(gamma N2+nu) for some scaling function K, and verified by our simulation data. For the case of a polymer brush, we derive the crossover force scale F* separating the weak and strong shear regime with F*a/(kT) = phi(nu/(3 nu-1)) where nu is the usual self-avoiding walk exponent. For the case of a polymer brush under strong shear, the velocity profile, nu(z), inside and above the brush is calculated analytically by treating the flow as in a porous medium and solving the Brinkman equation. The solution of the velocity profile is then combined with non-equilibrium Monte Carlo simulation data which allow a self-consistent determination of the chain end-to-end length and the incline angle of the chain. Also we derive the scaling form for the positions x(n) of the nth monomer and verify it by our simulation data. We further obtain an analytical expression for x(n) in terms of the effective viscosity eta of the model, given by F = eta a nu(z). The effective viscosity is also found from our simulation to be eta a(2)/(kT) similar or equal to 0.02 +/- 0.005.
