Mathematics

Soft materials such as polymers and biological tissues have several engineering and biomechanical applications. These materials exhibit complex mechanical characteristics and the need to accurately predict their behavior has posed a tremendous challenge for scientists and engineers. Less than a half century old, continuum computational modeling of soft materials is undoubtedly still in its infancy and its projected outlook remains promising for improving humanity's quality of life. Continued advances in computers and computational methods are increasing our ability to handle large amounts of data and to model complex phenomena. Modeling of soft materials has a vital role to play in the development of the needed mathematical models and analyses. Because of the incredible complexity of the biochemophysical aspects of soft tissues and the chemophysical aspects of polymers, this type of modeling requires increased interdisciplinary and multidisciplinary research that brings scientists from various fields together in teams, both in research and education. In this talk, I shall present a seamless, fully variational constitutive model capable of capturing several complex mechanical characteristics exhibited by such materials. Application to traumatic brain injury (TBI) under impact/acceleration loading will be presented. Clinically observed brain damage is reproduced and a predictive capability of the distribution, intensity, and reversibility/irreversibility of brain tissue damage will be demonstrated. Another application to ballistic impact on a polyurea retrofitted DH36 steel plate is simulated and validated, and computational capability for assessing the blast performance of metal/elastomer composite shells will also be presented. Future directions of this work may lead to the formulation of head-injury criteria for medical, governmental, and industrial applications; addressing the definition of clinical-biomechanical injury thresholds and tolerances; the simulation of a wide range of injuries, including blast-induced TBI and the effects of growing tumors; neurosurgical simulations; and the design and the assessment of effective protective devices, such as helmets including honeycomb materials, polymers, or foam padding.