Epitaxial layers about 1 mu m thick of GaAs, InP, CdTe, EuS or SrS on Si or GaAs substrates exhibit a strain -10(-3) < epsilon(0) < 10(-3) at the temperature of deposition. The strain remains on cooling until a critical temperature T-c reached. The layers adhere to the substrates below T-c and adopt different strains epsilon" and epsilon(perpendicular to) parallel and perpendicular respectively to the substrate. The T-c and epsilon(0) values often vary on annealing above 160-400 degrees C. The ratio -(epsilon(perpendicular to) -epsilon(0))/(epsilon" -epsilon(0)) remains independent of temperature and annealing. Stable epitaxial layers with constant epsilon(0) and T-c values can be obtained in some cases by deposition on buffer layers or stepped substrates.