Dynamics evolution of quantum correlation of two entangled qubits is investigated by using the techniques of weak measurement. The initial non-maximally entangled particles with entanglement sudden death happening are operated by pre-selection measurement, which not only improve the quantum correlations, but also can reduce or even eliminate entanglement sudden death region; As for the evolution of non-maximally entangled initial state without entanglement sudden death, the weak measurement will degrade the intensity of quantum correlation of qubits. The operation of weak measurement can not enhance entanglement between particles but can optimize quantum discord when particles are initially in maximally entangled state. Weak measurement has little effect on the quantum correlation transfer procession and can postpone the decay of entanglement, and even can make entanglement transfer happen earlier. In addition, the relation of system dissipation rate and coupling strength between the cavity field and qubits can also affect its dynamical evolution, showing Markov and non-Markov processes.