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1. An introduction to the Ogden model in biomechanics: benefits, implementation tools and limitations

   https://doi.org/10.1098/rsta.2021.0365  

   Lohr, Matthew J.; Sugerman, Gabriella P.; Kakaletsis, Sotirios; Lejeune, Emma; Rausch, Manuel K. (October 2022, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences)

   Constitutive models are important to biomechanics for two key reasons. First, constitutive modelling is an essential component of characterizing tissues’ mechanical properties for informing theoretical and computational models of biomechanical systems. Second, constitutive models can be used as a theoretical framework for extracting and comparing key quantities of interest from material characterization experiments. Over the past five decades, the Ogden model has emerged as a popular constitutive model in soft tissue biomechanics with relevance to both informing theoretical and computational models and to comparing material characterization experiments. The goal of this short review is threefold. First, we will discuss the broad relevance of the Ogden model to soft tissue biomechanics and the general characteristics of soft tissues that are suitable for approximating with the Ogden model. Second, we will highlight exemplary uses of the Ogden model in brain tissue, blood clot and other tissues. Finally, we offer a tutorial on fitting the one-term Ogden model to pure shear experimental data via both an analytical approximation of homogeneous deformation and a finite-element model of the tissue domain. Overall, we anticipate that this short review will serve as a practical introduction to the use of the Ogden model in biomechanics. This article is part of the theme issue ‘The Ogden model of rubber mechanics: Fifty years of impact on nonlinear elasticity’. 

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2. A mathematical justification for nonlinear constitutive relations between stress and linearized strain

   Rajagopal, KR; Rodriguez, C (May 2024, 24_RajRod_Preprint)

   We present an asymptotic framework that rigorously generates nonlinear constitutive relations between stress and linearized strain for elastic bodies. Each of these relations arises as the leading order relationship satisfied by a one-parameter family of nonlinear constitutive relations between stress and nonlinear strain. The asymptotic parameter limits the overall range of strains that satisfy the corresponding constitutive relation in the one parameter family while the stresses can remain large (relative to a fixed stress scale). This differs from classical linearized elasticity where a fixed constitutive relation is assumed, and the magnitude of the displacement gradient serves as the asymptotic parameter. Also unlike classical approaches, the constitutive relations in our framework are expressed as implicit relationships between stress and strain rather than requiring stress explicitly expressed as a function of strain, adding conceptual simplicity and versatility. We demonstrate that our framework rigorously justifies nonlinear constitutive relations between stress and linearized strain including those with density-dependent Young’s moduli or derived from strain energies beyond quadratic forms. 

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3. A mathematical justification for nonlinear constitutive relations between stress and linearized strain

   Rajagopal, K; Rodriguez, C (October 2024, Zeitschrift für angewandte Mathematik und Physik)

   We present an asymptotic framework that rigorously generates nonlinear constitutive relations between stress and linearized strain for elastic bodies. Each of these relations arises as the leading-order relationship satisfied by a oneparameter family of nonlinear constitutive relations between stress and nonlinear strain. The asymptotic parameter limits the overall range of strains that satisfy the corresponding constitutive relation in the one-parameter family, while the stresses can remain large (relative to a fixed stress scale). This differs from classical linearized elasticity where a fixed constitutive relation is assumed, and the magnitude of the displacement gradient serves as the asymptotic parameter. Also unlike classical approaches, the constitutive relations in our framework are expressed as implicit relationships between stress and strain rather than requiring stress explicitly expressed as a function of strain, adding conceptual simplicity and versatility. We demonstrate that our framework rigorously justifies nonlinear constitutive relations between stress and linearized strain including those with density-dependent Young’s moduli or derived from strain energies beyond quadratic forms. 

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4. Well-posedness of the governing equations for a quasi-linear viscoelastic model with pressure-dependent moduli in which both stress and strain appear linearly

   https://doi.org/10.1007/s00033-023-02160-0  

   Itou, Hiromichi; Kovtunenko, Victor A; Rajagopal, Kumbakonam R (February 2024, Zeitschrift für angewandte Mathematik und Physik)

   The response of a body described by a quasi-linear viscoelastic constitutive relation, whose material moduli depend on the mechanical pressure (that is one-third the trace of stress) is studied. The constitutive relation stems from a class of implicit relations between the histories of the stress and the relative deformation gradient. A-priori thresholding is enforced through the pressure that ensures that the displacement gradient remains small. The resulting mixed variational problem consists of an evolutionary equation with the Volterra convolution operator; this equation is studied for well-posedness within the theory of maximal monotone graphs. For isotropic extension or compression, a semi-analytic solution of the quasi-linear viscoelastic problem is constructed under stress control. The equations are studied numerically with respect to monotone loading both with and without thresholding. In the example, the thresholding procedure ensures that the solution does not blow-up in finite time. 

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5. Unbiased construction of constitutive relations for soft materials from experiments via rheology-informed neural networks

   https://doi.org/10.1073/pnas.2313658121  

   Mahmoudabadbozchelou, Mohammadamin; Kamani, Krutarth M; Rogers, Simon A; Jamali, Safa (January 2024, Proceedings of the National Academy of Sciences)

   The ability to concisely describe the dynamical behavior of soft materials through closed-form constitutive relations holds the key to accelerated and informed design of materials and processes. The conventional approach is to construct constitutive relations through simplifying assumptions and approximating the time- and rate-dependent stress response of a complex fluid to an imposed deformation. While traditional frameworks have been foundational to our current understanding of soft materials, they often face a twofold existential limitation: i) Constructed on ideal and generalized assumptions, precise recovery of material-specific details is usually serendipitous, if possible, and ii) inherent biases that are involved by making those assumptions commonly come at the cost of new physical insight. This work introduces an approach by leveraging recent advances in scientific machine learning methodologies to discover the governing constitutive equation from experimental data for complex fluids. Our rheology-informed neural network framework is found capable of learning the hidden rheology of a complex fluid through a limited number of experiments. This is followed by construction of an unbiased material-specific constitutive relation that accurately describes a wide range of bulk dynamical behavior of the material. While extremely efficient in closed-form model discovery for a real-world complex system, the model also provides insight into the underpinning physics of the material. 

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