Speaker
Description
Bone material can be seen as porous and fiberous, exhibiting the quasi-brittle type mechanical response, with some degree of plasticity. Softening plasticity and fracture mechanics lead to ill-posed mathematical problems due to the loss of monotonicity. Multiple co-existing solutions are possible when softening elements are coupled together, and solutions cannot be continued beyond the point of complete failure of a material. Moreover, spatially continuous models with softening suffer from localization of strains and stresses to measure-zero submanifolds.
We formulate a problem of quasistatic evolution of elasto-plastic spring networks (Lattice Spring Models, suitable for fiberous materials) with a plastic flow rule that allows for softening. The fundamental kinematic and static characteristics of the network are described by the rigidity theory and structural mechanics. The assumptions on the network itself allow for inhomogeneous, disordered or porous structures, such as bone material.
To solve the evolution problem we convert it to a type of a differential quasi-variational inequality known as the state-dependent sweeping process. The existence of solution to the associated time-stepping problem (implicit catch-up algorithm) can be shown, and the associated estimates imply the existence of a solution to the (time-continuous) sweeping process.
Using numerical simulations of regular grid-shaped networks with softening we demonstrate the development of non-symmetric plastic slips. At the same time, in toy examples it is easy to analytically derive multiple co-existing solutions, appearing in a bifurcation which happens when the parameters of the networks continuously change from hardening through perfect plasticity to softening.
