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This paper provides a formal connection between springs and continuum mechanics in the context of one-dimensional and two-dimensional elasticity. In the first stage, the equivalence between tensile springs and the finite element discretization of stretching energy of planar curves is established. Furthermore, when the strain is a quadratic function of stretch, this energy can be described with a new type of springs called tensile biquadratic springs. In the second stage, we extend this equivalence to nonlinear membranes (St Venant-Kirchhoff materials) on triangular meshes leading to triangular biquadratic and quadratic springs. Those tensile and angular springs produce isotropic deformations parameterized by Young modulus and Poisson ratios on unstructured meshes in an efficient and simple way. For a specific choice of the Poisson ratio, 1/3, we show that regular spring-mass models may be used realistically to simulate a membrane behavior. Finally, the different spring formulations are tested in pure traction and cloth simulation experiments.