This book provides readers with a deep understanding of the principles for generating formulations of compressible isotropic hyperelastic material models based on formulations of incompressible material models. The reference high-performance incompressible isotropic hyperelastic material model is Ogden's model, for which the elastic energy is generally represented as the sum of elemental energies based on strain tensors from the Doyle-Ericksen family. For the sake of transparency, the study is confined to considering the elastic energy only for one term of this sum based on the Finger strain tensor corresponding to the well-known neo-Hookean material model. The book presents a systematic study of the performance of two known types of compressible generalization of the incompressible neo-Hookean material model. The first type of generalization is based on the development of volumetric-isochoric neo-Hookean models and involves the additive decomposition of the elastic energy into volumetric and isochoric parts. The second, simpler type of generalization, is based on the development of mixed neo-Hookean models that do not use this decomposition. Theoretical studies of model performance and simulations of some homogeneous deformations have shown that when using "good" volumetric functions, mixed and volumetric-isochoric models show similar performance in applications and have physically reasonable responses in extreme states, which is convenient for theoretical studies. However, compared to volumetric-isochoric models, mixed models allow the use of a wider set of volumetric functions with physically reasonable responses in extreme states. Another feature of mixed models is that they allow for simpler expressions for stresses and tangent stiffness tensors. This book is useful both for novice researchers in developing hyperelastic equations for compressible materials and for experienced researchers by providing a brief overview of methods for generating compressible hyperelastic formulations based on available incompressible hyperelastic formulations.

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Dr. Sci. Sergey Korobeynikov has more than 50 years of experience in the field of Computational Nonlinear Solid Mechanics and Nano-mechanics and has worked in Novosibirsk State University as Professor. Currently he is working as Principal investigator in Lavrentiev Institute of Hydrodynamics of Siberian Branch of Russian Academy of Science. He has published 2 textbooks and one monograph (2000) in Russian, also he is the coauthor of one Springer brief book (2023). He has published 51 research papers in different International Journals and Proceedings of repute. He successfully supervised three PhD students and received numerous awards from Russian Academy of Science. He was the member of European Structural Integrity Society (ESIS) from 2006 up to 2019, and he is the member of Russian National Committee on Theoretical and Applied Mechanics (RNCTAM).

Dr. Aleksey Larichkin has more than 15 years of experience in the fields of Experimental Solid Mechanics and Computational Nonlinear Solid Mechanics and is working in Novosibirsk State University as Senior Lecturer. Currently, he is working as Senior researcher in Lavrentiev Institute of Hydrodynamics of Siberian Branch of Russian Academy of Science. He is the coauthor of one Springer brief book (2023) and textbook in Russian (2022). He has published 15 research papers in different International Journals and Proceedings of repute. Furthermore, he successfully supervised one PhD student.

Professor Patrizio Neff graduated with a degree in mathematics from the TU Darmstadt in 1995, where he subsequently completed his Ph.D in 2000 concerning multiplicative plasticity. In collaboration with Professor Jörg Schröder he extended the Ball's polyconvexity requirement to anisotropic materials. From 2001-2002 he served as a Visiting Associate Professor at the California Institute of Technology in Pasadena, USA. In 2004, he successfully completed his habilitation on the subject of finite strain Cosserat theory and Cosserat shells at the TU Darmstadt. Since 2009, he has been a full professor of Nonlinear Analysis and Modelling at the University of Duisburg-Essen. He is the author of over 240 research papers published in various prestigious international journals and proceedings. Throughout his academic career, he has supervised numerous doctoral students, five of whom now hold professorships. His research is in the field of applied mathematics and partial differential equations, including notably generalized Korn's inequalities and generalized continua (micromorphic, Cosserat) as well as nonlinear elasticity. The primary focus of his research endeavours has recently been directed towards isotropic nonlinear elasticity theory.