Scaling Relation in Fracture of the Materials with Elastoplastic Response Inaccessible by Scaling Laws

Strong materials and a solid method of estimating their toughness are important for manufacturing materials valuable for human life, such as rubber and plastic. Such strong materials often exhibit a complex response to external force, which resists description by simple scaling laws. In addition, even for simple nonlinear materials, no theoretical or experimental reports have been available on a clear scaling law between fracture stress and crack size, which would, if available, provide a solid test for toughness. Here, we perform experiments on thin sheet samples to make important length scales well-separated. This practically suppresses all the finite-size effects so that we succeed in finding a clear scaling law for fracture (that between failure stress and crack size) by studying nonlinear polymer sheets. This leads to plausible estimates of the fracture toughness. Remarkably, we experimentally find the scaling law even though the nonlinearity in force response is not so simple as described by power laws. The fracture scaling can be explained by a theory developed here for simple nonlinear materials and we expect that this theory will be valid for many other materials with a complex nonlinearity, as demonstrated here. This clarifies the advantage of testing thin sheets. The scaling law established here can be regarded as a nonlinear extension of the Griffith's formula and holds also for thick samples: the nonlinear Griffith's formula is also applicable to three-dimensional bulk objects.