The upper triangular decomposition provides an alternative method to multiplicatively decompose the deformation gradient tensor into a product of a rotation tensor and an upper triangular distortion tensor. The six components of the distortion tensor can be directly related to pure stretch and simple shear deformations, which are physically measurable. In this study, four new constitutive models for hyperelastic materials are developed by using strain energy density functions in terms of the distortion tensor. An untangled cross-linked network model and a tube-like constraint model are employed along with four different inverse Langevin function approximations to construct four micromechanically motivated strain energy density functions, each of which involves the first and second invariants and contains three material parameters. The Cauchy stress components, derived directly as partial derivatives of the strain energy density function with respect to the distortion tensor components, have simpler expressions than those based on the invariants of the right Cauchy–Green deformation tensor. Four fundamental deformation modes—uniaxial tension/compression, pure shear, equi-biaxial tension, and simple shear—together with the general biaxial stretch case are analyzed by directly applying the newly proposed constitutive models. The four new analytical models are validated by comparing the predicted stress-deformation curves with those obtained experimentally for brain tissue and rubber under various loading conditions. The numerical results reveal that the four models can effectively capture the mechanical behavior of hyperelastic materials, thereby providing a new approach based on the upper triangular decomposition and offering physically interpretable constitutive equations.

We analyze the evolution of red and blue galaxies in different cosmic web environments from redshift z = 3 to z = 0 using the IllustrisTNG simulation. We use Otsu's method to classify the red or blue galaxies at each redshift and determine their geometric environments from the eigenvalues of the deformation tensor. Our analysis shows that initially, blue galaxies are more common in clusters followed by filaments, sheets and voids. However, this trend reverses at lower redshifts, with red fractions rising earlier in denser environments. At z < 1, most massive galaxies (log(M */M ⊙) > 10.5) are quenched across all environments. In contrast, low-mass galaxies (log(M */M ⊙) < 10.5) are more influenced by their environment, with clusters hosting the highest red galaxy fractions at low redshifts. We observe a slower mass growth for low-mass galaxies in clusters at z < 1. Filaments show relative red fractions (RRF) comparable to clusters at low masses, but host nearly 60% of low-mass blue galaxies, representing a diverse galaxy population. It implies that less intense environmental quenching in filaments allows galaxies to experience a broader range of evolutionary stages. Despite being the densest environment, clusters display the highest relative blue fraction (RBF) for high-mass galaxies, likely due to interactions or mergers that can temporarily rejuvenate star formation in some of them. The (u-r) colour distribution transitions from unimodal to bimodal by redshift z = 2 across all environments. At z < 1, clusters exhibit the highest median colour, with stellar mass being the primary driver of colour evolution in massive galaxies. The suppression of star formation rate (SFR) and specific SFR (sSFR) is also most pronounced in clusters during this period. Our study suggests that stellar mass governs quenching in high-mass galaxies, while a complex interplay of mass and environment shapes the evolution of low-mass galaxies.

In classical mechanics, stress and strain are defined within real vector spaces using real-valued second-order tensors. Recent developments in quantum field theory, holography, and complex spacetime geometry indicate that such real-valued frameworks may be insufficient to describe phenomena like quantum entanglement, nonlocal curvature, and holographic effects. This paper proposes an extension of the classical stress-strain theory into complex vector spaces by introducing complex displacement fields. From these fields, complex strain and stress tensors are derived. The imaginary components of these tensors are interpreted as internal curvature, holographic tension, or phase-related deformations, which may represent hidden quantum degrees of freedom. The proposed formalism accommodates dissipative processes, non-Hermitian behavior, and stress effects induced by quantum entanglement. Applications of this framework are discussed for quantum materials, holographic analog systems, and cosmological models where classical stress-energy tensors might require complex-valued generalizations. This approach provides a unified tensorial representation for exploring deformation across classical and quantum regimes, potentially offering insights into the interplay between geometry and quantum fields.

We consider the Hookean dumbbell model, a system of nonlinear PDEs arising in the kinetic theory of homogeneous dilute polymeric fluids. It consists of the unsteady incompressible Navier–Stokes equations in a bounded Lipschitz domain, coupled to a Fokker–Planck-type parabolic equation with a centre-of-mass diffusion term, for the probability density function, modelling the evolution of the configuration of noninteracting polymer molecules in the solvent. The micro–macro interaction is reflected by the presence of a drag term in the Fokker–Planck equation and the divergence of a polymeric extra-stress tensor in the Navier–Stokes balance of momentum equation. We introduce the concept of generalised dissipative solution—a relaxation of the usual notion of weak solution, allowing for the presence of a, possibly nonzero, defect measure in the momentum equation. This defect measure accounts for the lack of compactness in the polymeric extra-stress tensor. We prove global existence of generalised dissipative solutions satisfying additionally an energy inequality for the macroscopic deformation tensor. Using this inequality, we establish a conditional regularity result: any generalised dissipative solution with a sufficiently regular velocity field is a weak solution to the Hookean dumbbell model. Additionally, in two space dimensions we provide a rigorous derivation of the macroscopic closure of the Hookean model and discuss its relationship with the Oldroyd-B model with stress diffusion. Finally, we improve a result by Barrett and Süli (Nonlinear Anal. Real World Appl. 39:362–395, 2018) by establishing the global existence of weak solutions for a larger class of initial data.

Abstract In this paper we present a collection of general identities relating the deformation tensor K = L η g of an arbitrary vector field η with the tensor Σ = L η ∇ on an abstract hypersurface H of any causal character. As an application we establish necessary conditions on H for the existence of a homothetic Killing vector on the spacetime where H is embedded. The sufficiency of these conditions is then analysed in three specific settings. For spacelike hypersurfaces, we recover the well-known homothetic KID equations in the language of hypersurface data. For two intersecting null hypersurfaces, we generalize a previous result by Chruściel and Paetz, valid for Killings, to the homothetic case and, moreover, demonstrate that the equations can be formulated solely in terms of the initial data for the characteristic Cauchy problem, i.e. without involving a priori spacetime quantities. This puts the characteristic KID problem on equal footing with the spacelike KID problem. Furthermore, we highlight the versatility of the formalism by addressing the homothetic KID problem for smooth spacelike-characteristic initial data. Other initial value problems, such as the spacelike-characteristic with corners, can be approached similarly.

This paper simulates the blood clot extraction process inside an idealized cylindrical blood vessel model using the aspiration-based thrombectomy technique. A fully Eulerian technique is used within the finite volume method where incompressible Navier–Stokes equations are solved in the fluid region. In contrast, the Cauchy stress equation is solved in the clot region. Blood is assumed to be a Newtonian fluid, while the clot is either hyperelastic or viscoelastic material. In the hyperelastic formulation, the clot deformation is calculated based on the left Cauchy–Green deformation tensor, while the stresses are based on the linear Mooney–Rivlin model. In the viscoelastic formulation, the Oldroyd B model is used within the log conformation approach to calculate the viscoelastic stresses in the clot. The interface between the blood and the clot is tracked with the help of the geometric volume-of-fluid method. We focus on the role of flow variables like the pressure, velocity, and proximity between the clot and the catheter tip to successfully capture the clot under catheter suction. We observe that, once the clot is attracted to the catheter port due to pressure forces, the viscous stresses try to drag it inside the catheter. On the other hand, if the clot is not initially attracted, it is carried downstream by the viscous stresses. If the suction velocity is low (∼0.2 m/s), the clot cannot be sucked inside the catheter, even if it is touching the catheter. At a higher suction velocity of 0.4 m/s, the suction effect is strong enough to capture the clot despite the larger initial distance from the catheter. Hence, the pressure distribution and viscous stresses play essential roles in the suction or escape of the clot during the thrombectomy process. Also, the viscoelastic model predicts the rupture of the clot inside the catheter during suction.

ABSTRACT Cosmic filaments are the transport channels of matter in the Megaparsec Universe, and represent the most prominent structural feature in the matter and galaxy distribution. Here, we describe and define the dynamical nature of cosmic filaments, based on the realization that the complex spatial pattern and connectivity of the cosmic web are already visible in the primordial random density field, in the spatial pattern of the primordial tidal and deformation eigenvalue field. The filaments emerging from this are multistream features and structural singularities in phase-space. The caustic skeleton formalism allows a fully analytical classification, identification, and treatment of the nonlinear cosmic web. The caustic conditions yield the mathematical specification of web-like structures in terms of the primordial deformation tensor eigenvalue and eigenvector fields, in which filaments are identified – in two dimensions – with the so-called cusp caustics. These are centred around points that are maximally stretched. The resulting mathematical conditions represent a complete characterization of filaments in terms of their formation history, dynamics, and orientation. We illustrate the workings of the formalism based on a set of constrained N-body simulations of proto-filament realizations. These realizations are analysed in terms of spatial structure, density profiles, and multistream structure and compared to simpler density or potential field saddle point specifications. The presented formalism, and its three-dimensional generalization, will facilitate the mining of the rich cosmological information contained in the observed web-like galaxy distribution, and be of key significance for the analysis of cosmological surveys such as Sloan Digital Sky Survey, Dark Energy Spectroscopic Instrument (DESI), and Euclid.

Abstract Non-crystalline solid materials have significant applications in the domains of science and engineering. This study focuses on the use of fundamental concepts of molecular interactions to develop a constitutive equation that can predict the stress-stretch behaviors of these materials. The strain energy density function of the material is derived using Helmholtz free energy of van der Waals potential. It is obtained in terms of excluded volume and number density of the molecules. To make consistent with continuum approximation, the excluded volume and number density are defined in terms of strain invariants of right Cauchy-Green deformation tensor. Finally, the constitutive model is represented in the form of Cauchy stress tensor. The current model is used for predicting finite deformation of non-crystalline solid phase of material. The results derived from the current model are also compared with the experimental results of polyurethane foam and poly vinyl alcohol gel materials. The current constitutive model can also be used for predicting the deformation characteristics of micro/nano components used in engineering systems. This study can provide a basis for the future scope of the constitutive modelling for the non-crystalline solid materials considering their complex molecular structures.

Abstract In this article, we consider the incompressible limit of global-in-time strong solutions with arbitrary large initial velocity for the compressible viscoelastic fluids in the sense of critical Besov framework. We decouple our compressible system into two coupling sub-systems by introducing a skew symmetric matrix, which is related to the deformation tensor. This work generalizes the similar result obtained by Hu et al. (Incompressible limit for compressible viscoelastic flows with large velocity, Advances in Nonlinear Analysis 12 (2023), 20220324) to the critical functional space with respective to the natural scaling of the system. The proof relies on the dispersive property of the linear system on the high-frequency regime and the parabolic property on the low-frequency regime. The dispersion tends to disappear when λ \lambda tends to infinite, but having large λ \lambda provides strong dissipation on the potential part of the velocity and thus makes the flow almost incompressible. In addition, by exploiting the intrinsic structure of the viscoelastic system, we obtain the global uniform estimates of the solutions near equilibrium.

Abstract Hydration is a major process that controls defect equilibria in oxides through the exchange of oxygen and hydrogen species between the solid and its gaseous environment. For yttria-stabilized zirconia (YSZ), the presence of intrinsic oxygen vacancies that provide charge compensation to the acceptor dopants and the inherent structural disorder pose significant problems towards an understanding of how hydration operates at the atomistic level. First-principles calculations and ab-initio thermodynamics are employed in order to study the hydration reaction in cubic YSZ and the two types of defects appearing therein as reactants and products, the oxygen vacancies and protons, respectively, yielding the defect-formation energies, defect-induced deformation tensors and chemical expansion coefficients. The calculations are based on density-functional theory using a semilocal density functional and a screened-exchange functional approach and take into account the intrinsic structural disorder of the YSZ lattice. The various terms to the free energy of the hydration reaction are determined as a function of temperature and water-vapor partial pressure. The calculations provide estimates of the enthalpy and entropy of hydration in cubic YSZ examining how the solid-state and gas-phase contributions affect the free-energy balance. The final results are discussed in connection with experimental observations of hydration effects in YSZ and other oxides.

We consider the singularity formation of strong solutions to the three-dimensional incompress-ible fractional Navier-Stokes equations in the whole space. By making use of the Bony decomposition technique, we prove that a unique local strong solution does not blow-up at time T if deformation tensor belongs to nonhomogeneous Besov spaces. As a bi-product, the result improves some well-known results on regularity for the particular case of classical Navier-Stokes equations.

The paper presents an analytical solution to the problem of a circular pipe turned inside out in a rigid gasket. Formulas were obtained for the magnitude of the radial stress, which is responsible for the adhesion between the pipe and the gasket. The solution is obtained for an arbitrary incompressible hyperelastic material with a hyperelastic potential that depends only on the first invariant of the left Cauchy – Green deformation tensor (various generalizations of the neo-Hookean solid) or on the second invariant of the logarithmic Hencky strain tensor (various generalizations of the incompressible Hencky material). The solution takes into account the occurrence of plastic flow in areas adjacent to the lateral surfaces of the pipe. Both ideally plastic and isotropically hardening materials of a general type are considered. For the latter, a solution scheme is given; in the particular case of a linearly hardening material, a closed-form solution is obtained. For the perfect plasticity model, a closed-form solution was obtained for the neo-Hookean solid, for an incompressible Hencky material, and for the Gent material.

The purpose of the research is differentiation of recent geodynamic processes within the Carpathian Mountains on the basis of freely available GNSS data. Methodology. The methodology included GNSS data collection, processing and analysis. An algorithm for processing was proposed, which consisted of 5 main stages: transformation of data into an internal format, verification of time series for compliance with requirements, determination of horizontal velocities, division of the GNSS network into triangles, and determination of deformation parameters. Results. This study presents a comprehensive analysis of recent geodynamic processes based on GNSS data freely available from the Nevada Geological Survey. Taking into account the requirements for time series, 50 GNSS stations were selected and processed. In general, absolute and regional velocities were obtained and analysed during 2000–2023. Regional velocities of horizontal movements were used to calculate the deformation tensor and deformation parameters. The results of the study are consistent and correlate well with the studies of other scientists. The obtained results confirm the presence of active geodynamic processes within the Carpathians. Originality. The proposed approach made it possible to estimate the main deformation parameters (value and direction of deformation axes, total shear and dilation) within the Carpathian Mountains. This makes it possible to analyse and predict recent geodynamic processes in the region. Practical significance. On the basis of the calculated values, maps of the distribution of vectors of absolute and regional horizontal velocities, total shear rates, dilatation rates, and rotation rates were constructed.

Abstract We develop a blow-up criterion in this paper for the strong solution of the three-dimensional magnetohydrodynamic (MHD) flows Dirichlet problem. The velocity field’s deformation tensor’s norm is the only factor influencing this blow-up criterion. Specifically, it is not affected by temperature or magnetic field. These findings also suggest that the strong solutions of compressible MHD flows maintain global regularity as long as the velocity tensor remains bounded.

Nonlinear free vibrations of a structurally tailored anisotropic pre-twisted thin-walled beam subjected to large deformations

The torsional deformation behavior of an elastic bar with a circular cross-section was investigated by applying invariant dyadic analysis, where the small finite displacement functions advocated by Saint-Venant (1855) were fully employed. It was found that the previously overlooked circumferential shear force field generated by pure torsion on the side walls of a bar produces an unusual torque term induced by the skew-symmetric part of the deformation tensor and exhibits quadratic length dependence along the z-axis of the bar. The adaptation of this torque term for a helical conformation of α-peptides creates moments acting on the circular cross-sections and is directed along the surface normal of circular cross-sections, which coincides with the tangent vector of the helix. The projection of this torque along the z-axis of the helix varies quadratically with the azimuthal angle. The radial component of the unusual torque, which also lies along the principal normal vector of the helix, starts to perform a precession motion by tracking a spiral orbit around the z-axis, whereas its apex angle decreases asymptotically with the azimuthal angle and finally reaches a finite value depending on the height of the helix along the z-axis. The ordinary torque terms, which are also deduced from the self- and anti-self-conjugate parts of the deformation tensor, have magnitudes half that of the full torque term reported in the literature. The present results were applied to the helical conformation of α-peptides designated by {3.611} to show that the mechanical stability of strained open-ended helical conformations can be successfully achieved by spontaneous readjustments of the surface and bulk Helmholtz free energies under isothermal isochoric conditions. It has been demonstrated that the main contribution to the mechanical stability of α-peptide 3.611 cannot come alone from the electrostatic dipole-dipole interaction potential of the anti-align excess dipole pairs but also from the surface Helmholtz free energy, which is characterized by a binding free energy of -15.5 eV/molecule (-32.56 Kcal/mole) for an alpha-peptide composed of 11 amino acid residues with a critical arc length of approximately 10 nm, assuming that the shear modulus is G = 1GPa and the surface Helmholtz specific free energy density is fs = 800 erg/cm2. This result was in excellent agreement with the experimental observations of the AH-1 conformation of (Glu)n Cys at pH 8. The present theory indicates that only two excess permanent anti-align dipole pairs for one α-Helical peptide molecule is requirement to stabilize the whole secondary structure of the protein that is exposed to heavy torsional deformation during the folding processes which amounts to 7.75 eV/molecule stored electrostatic energy compared to the interfacial Helmholtz free energy of -23.25 eV/molecule, which is exposed to hydrophobic environments.

Abstract Non-crystalline molecular solid materials have many scientific and engineering applications. This study develops a constitutive equation for understanding stress-stretch behaviour of non-crystalline molecular solid using Lennard-Jones (LJ) intermolecular interaction. The strain energy derived from Lennard-Jones interactions between molecules. Based on the excluded volume (spherical volume occupied by the molecules maintaining centre to centre distance with a reference molecule) and density of the molecules, strain energy density is developed. In order to relate the molecular approach with continuum approximation, the excluded volume and density are expressed as a function of strain invariants of right Cauchy-Green deformation tensor. Finally, the constitutive equation in the form of Cauchy stress tensor is developed using the present strain energy density function. The present constitutive model is used to study finite deformations of the molecular solid like uniaxial extension. We compare our theoretical results with the experimental data of flexible polyurethane foams and obtain very good agreements. The current constitutive model can predict the deformation of micro/nano engineering system components.

Dense communities of bacteria, also known as biofilms, are ubiquitous in all of our everyday life. They are not only always surrounding us, but are also active inside our bodies, for example in the oral cavity. While some biofilms are beneficial or even necessary for human life, others can be harmful. Therefore, it is highly important to gain an in-depth understanding of biofilms which can be achieved by in vitro or in vivo experiments. Since these experiments are often time-consuming or expensive, in silico models have proven themselves to be a viable tool in assisting the description and analysis of these complicated processes. Current biofilm growth simulations are using mainly two approaches for describing the underlying models. The volumetric approach splits the deformation tensor into a growth and an elastic part. In this approach, the mass never changes, unless some additional constraints are enforced. The density-based approach, on the other hand, uses an evolution equation to update the growing tissue by adding mass. Here, the density stays constant, and no pressure is exerted. The in silico model presented in this work combines the two approaches. Thus, it is possible to capture stresses inside of the biofilm while adding mass. Since this approach is directly derived from Hamilton’s principle, it fulfills the first and second law of thermodynamics automatically, which other models need to be checked for separately. In this work, we show the derivation of the model as well as some selected numerical experiments. The numerical experiments show a good phenomenological agreement with what is to be expected from a growing biofilm. The numerical behavior is stable, and we are thus capable of solving complicated boundary value problems. In addition, the model is very reactive to different input parameters, thereby different behavior of various biofilms can be captured without modifying the model.

Abstract In response to the unanswered relevant questions surrounding atherosclerosis, it becomes imperative to investigate arterioles using sophisticated mathematical modelling techniques to shed light on critical stress and strain patterns influenced by gravity. The primary objective of this study is to scrutinize flow characteristics and probe stress and strain distributions experienced by the intima layer of arterioles, encompassing coronary, renal, cerebral, mesenteric, and pulmonary arteries, under gravitational forces. This investigation employs a fluid-structure interaction methodology utilizing arbitrary Eulerian–Lagrangian formulation. The study delves into blood flow characteristics within coronary, renal, cerebral, mesenteric, and pulmonary arterioles using the fluid-structure interaction technique, employing an arbitrary Eulerian–Lagrangian formulation. It thoroughly examines various biomechanical parameters such as the Cauchy–Green stress tensor, Principal strain, Piola–Kirchoff stress tensor, deformation tensor, and volume strain along the intima layer under the gravitational influence, elucidating vulnerable regions prone to endothelial dysfunction. Higher values of δV are found at the left shoulder and in the intima’s post stenosis area due to the pressure gradient along the flow channel, whereas other intima regions show a null volume strain. A thorough understanding of stress distribution is essential to create focused therapies to lessen vascular health problems. The stress in the post-stenosis region seems to affect the endothelial layer to a significant extent.

We perform the numerical simulation of primordial black hole formation from a nonspherical profile of the initial curvature perturbation ζ. We consider the background expanding universe filled with the perfect fluid with the linear equation of state p = wρ (w = 1/3 or 1/5), where p and ρ are the pressure and the energy density, respectively. The initial condition is set in a way such that the principal directions of the second derivatives of ζ and △ζ at the central peak are misaligned, where △ is the Laplacian. In this setting, since the linearized density is proportional to △ζ, the inertia tensor and deformation tensor ∂ i ∂ j ζ are misaligned. Thus tidal torque may act and the spin of a resultant primordial black hole would be non-zero in general, although it is estimated to be very small from previous perturbative analyses. As a result, we do not find a finite value of the spin within our numerical precision, giving support for the negligibly small value of the black hole spin for 1/5 ≲ w ≲ 1/3. More specifically, our results suggest that the dimensionless PBH spin s is typically so small that s ≪ 0.1 for w ≳ 0.2.