In this study, we extended the distributed-Lagrange-multiplier/fictitious-domain (DLM/FD) formulation of Glowinski et al. [Int. J. Multiphase Flow 25 (1999) 755] for the fluid/rigid-body interactions to deal with the fluid/flexible-body interactions by replacing Newton’s equations of motion for the rigid body with the continuum equations for the general solid material. Similar to the rigid-body case where the DLM is introduced as a pseudo body force to enforce the constraint of rigid-body motion of the fictitious fluid in the solid domain, the Lagrange multiplier in our formulation is to enforce the fictitious fluid to move at the same velocity as the solid. For our computational scheme, a first-order accurate fractional step scheme is employed to decouple the entire system into three sub-systems: a fluid problem, a solid problem and a Lagrange multiplier problem; the flow problem is solved with the projection method on half-staggered grids; the solid problem is solved with the Lagrangian finite element method and the Newton iterative method; and the incompressibility of the material is implemented with the penalty function method. The proposed method is applied to two typical fluid–structure interaction problems: the flow-driven oscillation of a flexible plate along the flow direction and the self-sustained oscillation across the flow direction. Both results compare favorably with previously reported numerical and experimental results, and show that our method is suited to the simulation of the motion of an incompressible non-linear elastic material in a fluid.