In the laboratory study of extreme conditions of pressure, temperature, and timescale, the irradiation of matter by high intensity sources has been of central importance.Here we investigate the performance of layered targets in experiments involving very high intensity hard x-ray irradiation, motivated by the advent of extremely high brightness x-ray sources such as free electron lasers. These sources enable massive (µm to mm) scale targets with confinement of the directly irradiated regions by thick, x-ray transparent tampers. The thermal and mechanical response of these layers, from femtosecond-picosecond energy delivery, to mechanical relaxation on nanosecond timescales and heat flow and dissipation on microseconds, is controlled by the differential energy deposition, shock and release wave production, and thermal transport properties. Sample survival over one or more exposures can depend on damage arising from heating or mechanical stresses induced by heating, depending on the materials used and target geometry. This study also doubles as an investigation into the properties of diamond anvil high pressure cells, long used in conjunction with x-ray synchrotron sources, at new high intensity x-ray beamlines at free electron lasers and upgraded synchrotrons. Configuring a tamped target as a high pressure cell confers certain advantages, such as resistance to expansion and thermal stresses, and sample confinement, in conjunction with efficient control of heat. This study suggests new routes to studying thermodynamic equilibrium states of high energy density matter via the capacity to confine extreme states for particularly long durations.