This paper proposes to solve and simulate various Kirchhoff models of nonlinear strings thanks to Volterra series. Two nonlinearities are studied : the string tension is supposed to depend either on the global elongation of the string, either on the local strain located at x. For each model, a Volterra series is used to represent the displacement as a functional of excitation forces. The Volterra kernels are solved in the Laplace domain thanks to a modal decomposition. As a last step, systematic identifications on these kernels lead to a structure in the time domain, which is composed of ”linear filters”, instantaneous sums and instantaneous products. Such a structure allows for sound synthesis thanks to standard signal processing techniques. The nonlinear dynamics introduced by this simulation is significant and perceptible on sounds for sufficiently large excitations.