In this paper, we consider the weakly damped wave equations with hereditary effects, and the nonlinearity f satisfies critical growth. The delay term g(t, u
t
) may be driven by a function with very weak assumptions, namely, just measurability. We analyze the well-posedness of solutions and verify the existence of the pullback D-attractor in
$${C_{H_0^1\left( \Omega \right)}} \times {C_{{L^2}\left( \Omega \right)}}$$
by constructing the energy functional and combining with the idea of the contractive function.
