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Continuum-type constitutive relations of odd matter need to be formulated according to the second law of thermodynamics. Based on the primitive thermodynamics of Edelen, a procedure admitting most general relations, is outlined for heat-conducting fluids and solids. For viscous responses of odd matter, the theory accounts for the irreversible and non-dissipative forces, besides the hyperdissipative ones. For quasi-static responses, the theory grasps the elastic and non-conservative forces besides those derivable from the free energy, this being the realm of Cauchy elasticity beyond hyperelasticity. In Cosserat-type odd matter, primitive thermodynamics also accounts for curvature-torsion and couple-stress besides, respectively, deformation and force-stress tensors. In both, classical and micropolar cases, the theory grasps all possible couplings between the thermodynamic velocity and force vectors, along with a full range of anisotropies of hyperdissipative and hyperelastic responses (both linear and nonlinear).

We generalize the odd elasticity of planar materials to thermoelasticity, admitting spatially inhomogeneous properties. First, we show that for active systems breaking Onsager relations thermal evolution is given by an odd generalization of the Maxwell-Cattaneo relation. Next three different heat conduction models of odd solids are considered leading, respectively, to a classical coupled thermoelasticity with Fourier law, thermoelasticity with relaxation times of the Maxwell-Cattaneo type, and thermoelasticity with two relaxation times. Governing equations are established in terms of either displacement-temperature pair, stress-heat flux pair, or stress-temperature pair. Next, we establish a form of the stiffness tensor, ensuring its inversion to a compatibility tensor, and write equations of elasticity in the presence of eigenstrains, such as thermal strains, where we find that the stress field remains unchanged for a specific additive change of the compliance tensor field. This so-called stress invariance gives an equivalence class of a wide range of odd materials with different values of material properties. Effectively, within each class, the elastic compliances may be modified by a field linear in the plane without affecting the stress field. Finally, we study hydrodynamic modes in an odd thermoelastic solid with Fourier heat conduction and argue that contrary to even elastic solids, the temperature can affect both dilatational and shear waves. We present odd corrections to sound attenuation and diffusion coefficients., Comment: 7 pages

We present a novel thermodynamic parameter estimation framework for energy-based surgery on live tissue, with direct applications to tissue characterization during electrosurgery. This framework addresses the problem of estimating tissue-specific thermodynamics in real-time, which would enable accurate prediction of thermal damage impact to the tissue and damage-conscious planning of electrosurgical procedures. Our approach provides basic thermodynamic information such as thermal diffusivity, and also allows for obtaining the thermal relaxation time and a model of the heat source, yielding in real-time a controlled hyperbolic thermodynamics model. The latter accounts for the finite thermal propagation time necessary for modeling of the electrosurgical action, in which the probe motion speed often surpasses the speed of thermal propagation in the tissue operated on. Our approach relies solely on thermographer feedback and a knowledge of the power level and position of the electrosurgical pencil, imposing only very minor adjustments to normal electrosurgery to obtain a high-fidelity model of the tissue-probe interaction. Our method is minimally invasive and can be performed in situ. We apply our method first to simulated data based on porcine muscle tissue to verify its accuracy and then to in vivo liver tissue, and compare the results with those from the literature. This comparison shows that parameterizing the Maxwell--Cattaneo model through the framework proposed yields a noticeably higher fidelity real-time adaptable representation of the thermodynamic tissue response to the electrosurgical impact than currently available. A discussion on the differences between the live and the dead tissue thermodynamics is also provided., Comment: Accepted for publication in the IEEE Transactions on Biomedical Engineering. Research reported in this publication was supported by the National Institute of Biomedical Imaging and Bioengineering of the National Institutes of Health under award number R01EB029766

We present a novel 3D adaptive observer framework for use in the determination of subsurface organic tissue temperatures in electrosurgery. The observer structure leverages pointwise 2D surface temperature readings obtained from a real-time infrared thermographer for both parameter estimation and temperature field observation. We introduce a novel approach to decoupled parameter adaptation and estimation, wherein the parameter estimation can run in real-time, while the observer loop runs on a slower time scale. To achieve this, we introduce a novel parameter estimation method known as attention-based noise-robust averaging, in which surface thermography time series are used to directly estimate the tissue's diffusivity. Our observer contains a real-time parameter adaptation component based on this diffusivity adaptation law, as well as a Luenberger-type corrector based on the sensed surface temperature. In this work, we also present a novel model structure adapted to the setting of robotic surgery, wherein we model the electrosurgical heat distribution as a compactly supported magnitude- and velocity-controlled heat source involving a new nonlinear input mapping. We demonstrate satisfactory performance of the adaptive observer in simulation, using real-life experimental ex vivo porcine tissue data., Comment: Paper accepted to the 2022 IEEE Conference on Decision and Control (CDC 2022)

In this paper, we review the history, current state-of-art, and physical applications of the spectral theory of two classes of random functions. One class consists of homogeneous and isotropic random fields defined on a Euclidean space and taking values in a real finite-dimensional linear space. In applications to continuum physics, such a field describes physical properties of a homogeneous and isotropic continuous medium in the situation, when a microstructure is attached to all medium points. The range of the field is the fixed point set of a symmetry class, where two compact Lie groups act by orthogonal representations. The material symmetry group of a homogeneous medium is the same at each point and acts trivially, while the group of physical symmetries may act nontrivially. In an isotropic random medium, the rank 1 (resp. rank 2) correlation tensors of the field transform under the action of the group of physical symmetries according to the above representation (resp. its tensor square), making the field isotropic. Another class consists of isotropic random cross-sections of homogeneous vector bundles over a coset space of a compact Lie group. In applications to cosmology, the coset space models the sky sphere, while the random cross-section models a cosmic background. The Cosmological Principle ensures that the cross-section is isotropic. For convenience of the reader, a necessary material from multilinear algebra, representation theory, and differential geometry is reviewed in Appendix., Comment: 2 figures

A generalization of virial theorems and virial stresses to micropolar continuum mechanics is explored. The linear momentum balance in dyadic product with translation leads to (i) the first virial theorem of micropolar continuum mechanics involving the infinite-time limit of the kinetic translational energy and (ii) a classical formula for computing the virial force-stress known in molecular dynamics. The angular momentum balance in dyadic product with rotation leads to (i) the second virial theorem of micropolar continuum mechanics involving the infinite-time limit of the kinetic rotational energy and (ii) a virial couple-stress along with a formula for its computation. The latter stress is also uncovered in the dyadic product of linear momentum balance with rotation. The virial force-stress and virial couple-stress contain, respectively, the Reynolds force-stress and turbulent couple-stress.

Self-affine morphology of random interfaces governs their functionalities across tribological, geological, (opto-)electrical and biological applications. However, the knowledge of how energy carriers or generally classical/quantum waves interact with structural irregularity is still incomplete. In this work, we study vibrational energy transport through random interfaces exhibiting different correlation functions on the two-dimensional hexagonal lattice. We show that random interfaces at the atomic scale are Cantor composites populated on geometrical fractals, thus multifractals, and calculate their quantized conductance using atomistic approaches. We obtain a universal scaling law, which contains self-similarity for mass perturbation, and exponential scaling of structural irregularity quantified by fractal dimension. The multifractal nature and Cantor-composite picture may also be extendable to charge and photon transport across random interfaces.

This study is motivated by the observation, based on photographs from the Cassini mission, that Saturn's rings have a fractal structure in radial direction. Accordingly, two questions are considered: (1) What Newtonian mechanics argument in support of that fractal structure is possible? (2) What kinematics model of such fractal rings can be formulated? Both challenges are based on taking Saturn's rings' spatial structure as being statistically stationarity in time and statistically isotropic in space, but statistically non-stationary in space. An answer to the first challenge is given through the calculus in non-integer dimensional spaces and basic mechanics arguments (Tarasov (2006) \textit{Celest. Mech. Dyn. Astron.} \textbf{94}). The second issue is approached in Section~3 by taking the random field of angular velocity vector of a rotating particle of the ring as a random section of a special vector bundle. Using the theory of group representations, we prove that such a field is completely determined by a sequence of continuous positive-definite matrix-valued functions defined on the Cartesian square $F^{2}$ of the radial cross-section $F$ of the rings, where $F$ is a fat fractal., Comment: 15 pages, 1 figure

For each of the $8$ isotropy classes of elastic materials, we consider a homogeneous random field taking values in the fixed point set $\mathsf{V}$ of the corresponding class, that is isotropic with respect to the natural orthogonal representation of a group lying between the isotropy group of the class and its normaliser. We find the general form of the correlation tensors of orders $1$ and $2$ of such a field, and the field's spectral expansion., Comment: Modified version, typos corrected, 100 pages, 1 figure

We establish spectral expansions of homogeneous and isotropic random fields taking values in the $3$-dimensional Euclidean space $E^3$ and in the space $\mathsf{S}^2(E^3)$ of symmetric rank $2$ tensors over $E^3$. The former is a model of turbulent fluid velocity, while the latter is a model for the random stress tensor or the random conductivity tensor. We found a link between the theory of random fields and the theory of finite-dimensional convex compacta., Comment: 24 pages

Over the past few decades, various conjectures were advanced that Saturn's rings are Cantor-like sets, although no convincing fractal analysis of actual images has ever appeared. We focus on the images sent by the Cassini spacecraft mission: slide #42 "Mapping Clumps in Saturn's Rings" and slide #54 "Scattered Sunshine". Using the box-counting method, we determine the fractal dimension of rings seen here (and in several other images from the same source) to be consistently about 1.6~1.7. This supports many conjectures put forth over several decades that Saturn's rings are indeed fractal.

We derive basic equations of electromagnetic fields in fractal media which are specified by three indepedent fractal dimensions {\alpha}_{i} in the respective directions x_{i} (i=1,2,3) of the Cartesian space in which the fractal is embedded. To grasp the generally anisotropic structure of a fractal, we employ the product measure, so that the global forms of governing equations may be cast in forms involving conventional (integer-order) integrals, while the local forms are expressed through partial differential equations with derivatives of integer order but containing coefficients involving the {\alpha}_{i}'s. First, a formulation based on product measures is shown to satisfy the four basic identities of vector calculus. This allows a generalization of the Green-Gauss and Stokes theorems as well as the charge conservation equation on anisotropic fractals. Then, pursuing the conceptual approach, we derive the Faraday and Amp\`ere laws for such fractal media, which, along with two auxiliary null-divergence conditions, effectively give the modified Maxwell equations. Proceeding on a separate track, we employ a variational principle for electromagnetic fields, appropriately adapted to fractal media, to independently derive the same forms of these two laws. It is next found that the parabolic (for a conducting medium) and the hyperbolic (for a dielectric medium) equations involve modified gradient operators, while the Poynting vector has the same form as in the non-fractal case. Finally, Maxwell's electromagnetic stress tensor is reformulated for fractal systems. In all the cases, the derived equations for fractal media depend explicitly on fractal dimensions and reduce to conventional forms for continuous media with Euclidean geometries upon setting the dimensions to integers.

When an electric field is applied to an electrolyte-saturated polymer gel embedded with charged colloidal particles, the force that must be exerted by the hydrogel on each particle reflects a delicate balance of electrical, hydrodynamic and elastic stresses. This paper examines the displacement of a single charged spherical inclusion embedded in an uncharged hydrogel. We present numerically exact solutions of coupled electrokinetic transport and elastic-deformation equations, where the gel is treated as an incompressible, elastic Brinkman medium. This model problem demonstrates how the displacement depends on the particle size and charge, the electrolyte ionic strength, and Young's modulus of the polymer skeleton. The numerics are verified, in part, with an analytical (boundary-layer) theory valid when the Debye length is much smaller than the particle radius. Further, we identify a close connection between the displacement when a colloid is immobilized in a gel and its velocity when dispersed in a Newtonian electrolyte. Finally, we describe an experiment where nanometer-scale displacements might be accurately measured using back-focal-plane interferometry. The purpose of such an experiment is to probe physicochemical and rheological characteristics of hydrogel composites, possibly during gelation.