Mechanically instructive biomaterials: a synergy of mechanics, materials and biology
Zhenwei Ma, Jianyu Li
Department of Mechanical Engineering, McGill University, Montreal, Canada
Prof. Yu Deng of the University of Chicago Department of Mathematics won the Fields Medal in Mathematics for solving Hilbert's Sixth Problem. How does this differ from the Unified Mechanics Theory? I raised this question to Gemini 3.1 Pro Extended Thinking. The answer is attached. I hope you enjoy reading it.
Maximiliano Larrain Silva Amit Acharya
A solution procedure is formulated and solved for treating the nonlinear Dissipation Inequality as a constraint equation within continuum mechanics, and allowing for incomplete knowledge of constitutive behavior. The scheme is demonstrated in the context of the rate-dependent, elastoplastic response of a bar, resulting in a nonlinear problem of constrained optimization. Both closed form and computational results are developed. The computational solutions utilize a sequence of convex optimization problems, and are shown to be accurate. In the example considered, the approach is shown to automatically correct an (intentionally) faulty constitutive specification, resulting in the solution to be in accord with the fundamental postulates of continuum mechanics.
Arash Yavari Amit Acharya
Curl forces are position-dependent, non-conservative, and non-dissipative forces that, in general, cannot be derived from an ordinary potential energy. Consequently, their equations of motion do not, in general, follow from a standard variational principle. In this paper, we present a dual variational formulation for particle dynamics under curl forces. By introducing variables dual to position and velocity and an auxiliary function, we construct a pre-dual action in which the equations of motion act as constraints. Stationarity with respect to the primal variables defines a dual-to-primal mapping, whose substitution into the pre-dual action gives an action expressed entirely in terms of the dual variables. The Euler–Lagrange equations of the dual action recover both the original equations of motion and their prescribed initial conditions. We also introduce an auxiliary dual Hamiltonian that is conserved along stationary dual trajectories, although it does not represent the physical energy. The formulation is illustrated using two nonlinear curl force fields in two and three dimensions and the classical Ziegler column. These examples demonstrate that non-conservative curl-force dynamics can admit variational descriptions even in the absence of an ordinary potential energy or a conventional Lagrangian.
Our latest paper is now freely accessible for the next 50 days from this link: https://authors.elsevier.com/a/1nUCG3In-v8hqF
Someone asked how "Thermodynamic state index in the Unified Mechanics Theory relates, conceptually or mathematically, to conditional phase-space entropy in finite Hamiltonian systems." I posed the question to Gemini AI. Here is its answer in the attached file.
The lemma of Hill (or of Hill-Mandel) is crucial to the consistent treatment of effective properties in the theory of composites. Building on various classical results (notably by Maxwell and Airy), I've recently applied the lemma sort of "out of context" to characterize (count really) the (infinitesimal) isometric deformations of periodic surfaces.
Dear colleagues,
I am pleased to invite you to the next webinar in the Extreme Mechanics Letters (EML) Webinar Series.
Our upcoming seminar will be delivered by Prof. Miguel Bessa (Brown University):
"Disentangling Uncertainty in AI for Engineering"
Date: Wednesday, 15 July 2026
Time: 10:00 am Boston / 3:00 pm London / 4:00 pm Paris / 10:00 pm Beijing
