A variable-stiffness module includes a rigid structure having a first stiffness, an intermediate substrate having a second stiffness less than the first stiffness, and a flexible substrate having a third stiffness less than the second stiffness. The rigid structure is disposed on the intermediate substrate and the intermediate substrate is disposed on the flexible substrate. A conductor is disposed partially on the intermediate substrate and partially on the flexible substrate and is connected to the rigid structure. The conductor extends from the rigid structure to the intermediate substrate to the flexible substrate. In some embodiments, a variable-stiffness module includes any combination of multiple rigid structures, multiple intermediate substrates, and multiple conductors. The conductor can be an optical conductor or an electrical conductor and can be disposed over the rigid structure or between the rigid structure and the intermediate substrate.