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Background: In a previous work, the problem of identifying residual stresses through relaxation methods was demonstrated to be mathematically ill-posed. In practice, it means that the solution process is affected by a bias-variance tradeoff, where some theoretically uncomputable bias has to be introduced in order to obtain a solution with a manageable signal-to-noise ratio. Objective: As a consequence, an important question arises: how can the solution uncertainty be quantified if a part of it is inaccessible? Additional physical knowledge could—in theory—provide a characterization of bias, but this process is practically impossible with presently available techniques. Methods: A brief review of biases in established methods is provided, showing that ruling them out would require a piece of knowledge that is never available in practice. Then, the concept of average stresses over a distance is introduced, and it is shown that finding them generates a well-posed problem. A numerical example illustrates the theoretical discussion Results: Since finding average stresses is a well-posed problem, the bias-variance tradeoff disappears. The uncertainties of the results can be estimated with the usual methods, and exact confidence intervals can be obtained. Conclusions: On a broader scope, we argue that residual stresses and relaxation methods expose the limits of the concept of point-wise stress values, which instead works almost flawlessly when a natural unstressed state can be assumed, as in classical continuum mechanics (for instance, in the theory of elasticity). As a consequence, we are forced to focus on the effects of stress rather than on its point-wise evaluation. [ABSTRACT FROM AUTHOR]

This paper presents a forensic numerical study on hydro-fracturing at three locations within an excavation in a clayey till underlain by an aquifer in Ontario, Canada. The pioneering forensic studies were based on continuum mechanics using finite element method (FEM) combined with an effective medium approach, which did not explicitly simulate the physical process of fracture propagation. This has overestimated the size of the venting holes (i.e., model aperture approximately 8 m, observed approximately 0.1 m), hindering accurate assessment of the venting locations at the excavation base. For this reason, a coupled hydro-mechanical XFEM analysis, which explicitly models fracture propagation, was performed in this study to revisit the case history from the perspective of fracture mechanics. Parametric study is also performed to investigate the influence of flow rate on the direction of the fracture propagation, and thus location of venting holes. It is shown that the coupled hydro-mechanical XFEM analysis offers better predictions than previous FEM analysis for both, locations and sizes of the three observed venting holes. The parametric study reveals that an increase in the flow rate from 5 to 10 L/min (which is within the reported range) leads to a wider and less curved fracture, altering the predictions of the venting locations at the excavation base by around 8 m. This is because an increasing flow rate causes wider aperture and higher shear stress near the fracture tip, which intensifies the re-orientation of maximum principle stress that facilitates the re-direction of the propagating fractures. [ABSTRACT FROM AUTHOR]

After defining the fractional Λ -derivative, having all the prerequisites for corresponding to a differential, the fractional Λ -strain has already been established. Furthermore, only globally variational principles are allowed in the context of fractional analysis. Hence, balance laws, yielding the various field equations in Λ -fractional continuum mechanics, are derived, allowing corners in their fields. The basic balance laws of mass, linear and rotational momentum, and energy conservation with jump conditions are derived in the context of Λ -fractional analysis. [ABSTRACT FROM AUTHOR]

In this paper, we investigate stability and optimization of an axially loaded nonlocal beam that is simply supported at one end and elastically restrained against rotation on the other. Nonlocal continuum mechanics is a theoretical framework that extends classical continuum mechanics to account for the influence of small length scales, especially in nanostructures. We analyze elastically buckling nanobeam based on Eringen's nonlocal elasticity theory. The Euler method of adjacent equilibrium configuration is used to derive the nonlinear governing equations. The critical axial force and postbuckling shape are obtained for the beam with the unit cross-sectional area. The shape of the nonlocal beam stable against buckling that has minimal volume is determined by using Pontryagin's maximum principle. The optimality conditions are derived. The cross-sectional area for optimally designed beam is found from the solution of a nonlinear boundary value problem. New numerical results are obtained. The first integral (Hamiltonian) is used to monitor the accuracy of the integration. The numerical analysis includes the influence of the characteristic parameter of the small length scale on the critical load, the postbuckling shape and the optimal shape of the analyzed beams. It is shown that there are the saving materials for optimally designed nanobeam in all numerical examples. [ABSTRACT FROM AUTHOR]

A nonlinear phenomenological model of flexoelectricity in thermoelastic solids is presented within the frame of continuum mechanics and extended thermodynamics, incorporating the quasi-electrostatic approximation where the time derivative of the electric displacement vector can be neglected in Maxwell–Ampère's law. An expression for the energy flux is proposed, including many internal variables. An advantage of the proposed model is that it presents a unified approach to study several thermo-electromechanical couplings, including electrostriction, piezoelectricity, dependence of entropy and spontaneous polarization on flexoelectricity, among others. The model may be of interest in revealing the effects of such couplings on the properties of polarizable media, in particular ferroelectrics, when simple boundary conditions prevail. For a fully dynamic description of flexoelectricity, Hamilton's principle and the variational principle for external forces should be used instead. [ABSTRACT FROM AUTHOR]

The friction-assisted lateral extrusion process (FALEP) is gaining attention as a candidate for fabricating high-performance ultrafine grain alloys for potential industrial applications. It consists of extruding metal in bulk or powder form into a solid sheet in a single operation to obtain ultrafine-grained (UFG) structures. The sheet has high yield strength due to its UFG microstructure and a shear-type crystallographic texture that is fundamentally different from the textures of sheets obtained by rolling. Apart from its single-step feature, FALEP requires lower forces than in rolling, so less energy is required to achieve large reductions. The present work introduces analytical elastic/plastic continuum calculations for the mechanics of the FALEP process. The results of the calculations demonstrate the great advantages of FALEP with respect to rolling and equal/non-equal channel angular pressings. [ABSTRACT FROM AUTHOR]

In this paper, a novel analytical approach based on nonlocal strain gradient theory is proposed to investigate small-scale effects on the radial vibration of isotropic spherical nanoparticles interacting with a viscoelastic fluid. The viscoelastic fluid is assumed to be compressible and takes into account both compressional and shear relaxation processes. The frequency equation is derived by imposing the fluid-nanosphere interface continuity conditions. To demonstrate the validity and accuracy of the approach, a comparison is made with the literature results in some particular cases, which shows a good agreement. Numerical examples are finally conducted to reveal the significance of small-scale effects in the radial vibrations, which need to be included in the nonlocal strain gradient model of submerged spherical nanoparticles. It is found that the vibration behavior greatly depends on the nanosphere size, nonlocal parameter, strain gradient parameter and glycerol-water mixture. Particularly, the small-scale effects play a very pronounced role when the spherical gold nanoparticle radius is smaller than 3 nm. Thus, the obtained frequency equation is very useful to interpret the experimental measurements of vibrational characteristics of nanospheres submerged in a viscoelastic fluid. [ABSTRACT FROM AUTHOR]

The contribution at hand focuses on the introduction of a novel approach to model biological growth. The proposed formulations are chosen to represent plant like structures. Therefore, thermomechanically open systems are considered. The balance laws are presented for such systems. Furthermore, the proposed formulations are coupled with an adaptive meshing framework. Therefore, a so-called structural generator is presented and utilized in this work. Since no growth formulations within the framework of continuum mechanics exist so far for plant like systems, a novel set of constitutive equations is shown. The newly described principles are the phototropism and graviotropism. In the numerical examples, it is shown that the proposed formulation yields physically meaningful results. The combination of different growth principles results in plausible interactions of the aforementioned principles. Furthermore, results of numerical simulations are shown, which represent the growth process of plant like biological structures. [ABSTRACT FROM AUTHOR]

Martensitic steels are widely used as a structural material in critical components of fossil fuel and nuclear power plants, such as boilers, pipes, and fittings. Martensitic steels are known to have a hierarchical microstructure that follows the Kurdjumov–Sachs (K–S) orientation relationship, where a prior austenite grain is composed of packets separated by high angle grain boundaries or packet boundaries, which are, in turn, divided into blocks or variants segregated by high angle grain boundaries called block boundaries. Blocks themselves are an agglomeration of laths divided by low angle grain boundaries named lath boundaries which have precipitates scattered on them. This work seeks to examine, using a couple dislocation dynamics—continuum mechanics approach called multiscale dislocation dynamics plasticity (MDDP), the interactions between dislocations and packet, block, lath boundaries, and precipitates under uniaxial tension loading and their effect on the mechanical response of the material. The simulations are conducted at a strain rate of 105 s−1 at room temperature. The main crystallographic features that arise during the deformation process were extracted and analyzed in terms of their contribution to the mechanical response of the material. The orientation relationship governing the microstructure of martensitic steels, namely, the K–S orientation relationship, was incorporated in MDDP in an effort to accurately capture the deformation behavior of the material in question. The strength of lath martensitic steel was analyzed as a function of the lath width, block size, and packet size to determine the appropriate effective grain size. [ABSTRACT FROM AUTHOR]

For the multi-dimensional variational phase transition problem of continuum mechanics we propose a method for splitting the problem into two parts one of which is used to find a candidate for the volume fraction of one of the phases in the equilibrium state, whereas the other is used to find the corresponding equilibrium displacement field and equilibrium phase distribution. [ABSTRACT FROM AUTHOR]

The present article focuses on the Rayleigh surface wave propagation in homogeneous isotropic thermoelastic medium. This works aims to develop a new nonlocal elasticity model based on three-phase-lag model of hyperbolic thermoelasticity under double porosity structure. New constitutive relations and equations are derived using nonlocal continuum mechanics. State space approach is employed to solve the problem. Frequency equation is derived using appropriate boundary conditions. Path of surface particles is found elliptical at the time of Rayleigh surface wave propagation. Different characteristics of Rayleigh waves are calculated numerically and presented graphically for different wave number. Various characteristics of wave for different thermoelastic models are also presented graphically. The effect of nonlocal parameters and void parameters on phase velocity, attenuation coefficient, penetration depth, and specific loss of Rayleigh waves are shown graphically. Some special cases are deduced from this study which agree with the existing literatures. [ABSTRACT FROM AUTHOR]

A three-dimensional model for the mechanics of elastic/hyperelastic materials reinforced with bidirectional fibers is presented in finite elastostatics. This includes the constitutive formulation of matrix–fiber composite system and the derivation of the corresponding Euler equilibrium equation. The responses of the matrix material and reinforcing fibers are characterized, respectively, via the Neo-Hookean model and quadratic strain energy potential of the Green–Lagrange type. These are further refined by the Mooney–Rivlin strain energy model and the high-order polynomial energy potential of fibers to incorporate the nonlinear behaviors of the matrix material and fibers. Within the framework of differential geometry and strain-gradient elasticity, the general kinematics of bidirectional fibers, including the three-dimensional bending of a fiber and twist between the two adjoining fibers, are formulated, and subsequently integrated into the model of continuum deformation. The admissible boundary conditions are also derived by virtue of variational principles and virtual work statement. In particular, a dimension reduction process is applied to the resulting three-dimensional model through which a compatible two-dimensional model describing both the in-plane and out-of-plane deformations of thin elastic films reinforced with fiber mesh is obtained. To this end, model implementation and comparison with the experimental results are performed, indicating that the proposed model successfully predicts key design considerations of fiber mesh reinforced composite films including stress–strain responses, deformation profiles, shear strain distributions and local structure (a unit fiber mesh) deformations. The proposed model is unique in that it is formulated within the framework of differential geometry of surface to accommodate the three-dimensional kinematics of the composite, yet the resulting equations are reframed in the orthonormal basis for enhanced practical unitality and mathematical tractability. Hence, the resulting model may also serve as an alternative Cosserat theory of plates and shells arising in two-dimensional nonlinear elasticity. [ABSTRACT FROM AUTHOR]

We consider one-dimensional second-order partial differential equations describing waves in inhomogeneous and nonlinear media. Contact transformations and Euler differential substitution are used to construct general solutions. General and partial solutions of some nonstationary continuum mechanics models are found. [ABSTRACT FROM AUTHOR]

With the continuous research in seismology, the influence of the heterogeneity caused by microstructure interactions of the medium on seismic wave propagation has been paid more attention. Wang et al. (2020) proposed to describe the complex microstructures of the medium in the framework of the generalized continuum mechanics theory, and derived the asymmetric elastic wave equations containing the characteristic length scale parameter of the medium. In addition, scale effects of seismic wave propagation will appear when considering microstructures of the medium. In this work, in order to better analyze the scale effects and extract the new components of wave fields owing to the microstructure interactions, we propose an optimized finite-difference (FD) method based on the improved bat swarm optimization (BSO) algorithm. Then the optimized FD method is used to perform numerical modeling for the asymmetric elastic wave equations considering microstructures of the medium. Numerical dispersion analysis indicate that the optimized FD method has high accuracy. According to the numerical modeling results obtained by the optimized FD method, scale effects of seismic wave propagation can be clearly observed when considering microstructures of the medium. [ABSTRACT FROM AUTHOR]

Carbon nanotubes are one of the most influential constituents of advanced engineering systems. The classical continuum mechanics, however, ceases to hold in accurate description of the structural response of nanobars. The mixture unified gradient theory of elasticity is invoked for the nanoscopic study of the structural characteristics of nanobars. The associated boundary-value problem of equilibrium is established within a consistent stationary variational framework. The differential condition of equilibrium is properly equipped with extra non-standard boundary conditions. To examine the more realistic response of nano-sized bars, deformable boundary conditions are prescribed where the elastic deformation of the support is modeled by a translational restraint at the boundary. The idealized free and fixed end conditions are, accordingly, retrieved for ad hoc values of the translational elastic spring stiffness. The mechanics of the mixture unified gradient bar is analytically addressed and a closed-form solution of the axial deformation of the nanobar is detected. The axial structural features of the nanobar with elastic boundary conditions are numerically studied and the asymptotic structural behavior of the mixture unified gradient bar is analytically examined. A new benchmark is detected that can be efficaciously applied in the analysis and design of pioneering nano-systems. [ABSTRACT FROM AUTHOR]

We consider one-dimensional second-order partial differential equations describing waves in inhomogeneous and nonlinear media. Contact transformations and Euler differential substitution are used to construct general solutions. General and partial solutions of some nonstationary continuum mechanics models are found. [ABSTRACT FROM AUTHOR]

The development of reliable miniaturized devices capable of performing sensing, actuation, and computing functions in a robust manner strongly depends on the mechanical performance of arrays of micromachines subjected to alternating stresses during operation. Consequently, to enhance the functionality and longevity of these micromachines, it is essential to accurately determine the structural behavior under both static and dynamic loadings. However, this requires knowledge of the stress distribution and variation due to texture anisotropy effects in critical micromachine regions and insight into the material behavior at micrometer length scales. Accordingly, the objective of this study was to develop an effective computational mechanics framework for performing stress analysis of micromachines at resonance. Dynamic finite element analysis of resonating micromachines was performed to elucidate the effect of material damping on the dynamic response. A Poisson–Voronoi diagram was incorporated in the critical region of finite element models of polysilicon micromachines to mimic the local inhomogeneity and anisotropy of the polycrystalline microstructure. A computationally effective static finite element analysis of the steady-state resonant response, validated by the results of the dynamic analysis, was used to obtain the steady-state stress distribution in the critical region of polysilicon micromachines with different textures. Moreover, a probabilistic stress analysis of hundreds of simulations was carried out to examine the stochastic nature of the grain geometry, size, and orientation and the beam width on the maximum equivalent von Mises stress in the critical micromachine region. The present study provides an efficient computational methodology, which integrates local texture into continuum mechanics modeling, for performing stress analysis of dynamic microdevices exhibiting different polycrystalline microstructures. [ABSTRACT FROM AUTHOR]

The main aim of the present work is to investigate the role of the Maxwell stress tensor in the study of active materials. Despite the importance of this tensor in modeling mechatronic devices used in sophisticated applications, its non–symmetry still generates controversies in the literature, probably because classical continuum mechanics assumes a symmetric Cauchy stress, although the sum of Cauchy and Maxwell stresses is non–symmetric. In the framework of generalised continuum mechanics–a more advanced formalism than the classical one–, each material point has an associated microstructure so that the micro–rotations of the electric/magnetic dipoles present in real active materials may be simulated. To this end, a modified total stress formulation, including an angular momentum balance, is developed and implemented into a finite element research code using a complex–step formulation. It is concluded that generalised mechanics allows for incorporating both symmetric and non–symmetric contributions of the Maxwell tensor. Consequently, the rotations generated by the electromagnetic field may be analysed. The influence of the complete Maxwell tensor in a magnetostrictive actuator is studied by several magneto–mechanical numerical experiments of a Terfenol–D rod encircled by air, and several conclusions are highlighted. [ABSTRACT FROM AUTHOR]

Constitutive equations of continuum mechanics of the solid phase of anisotropic material is focused in the paper. First, a synoptic one-dimensional Maxwell model is explored, subjected to arbitrary deformation load. The explicit form is derived for stress on strain dependence. Further, the analogous explicit constitutive equation is taken in three spatial dimensions and treated mathematically. Later on, a simply supported straight concrete beam reinforced by the steel fibres is taken as an investigated domain. The reinforcement is considered and dealt as scattered within the beam. Material characteristics are determined in line with the theory of the reinforcement. Sinusoidal load is taken as the action, stress reaction function is observed. By exploitation of the Fourier transform within the stress-strain relation analysis, both time and frequency interpretation of the constitutive relation can be performed. [ABSTRACT FROM AUTHOR]

Inspired by biology and engineered soft active material systems, we propose a new constitutive formulation for a soft material consisting of soft contractile fibers embedded in a soft matrix. The mathematical implementation of the model is based on a multi-field invariant formulation within a nonlinear continuum mechanics framework. The coupled constitutive formulation highlights a new electromechanical coupling term that describes the intrinsic (or active) anisotropy due to the contractile units. The model demonstrates the relative role that intrinsic anisotropy plays in the overall stress response. The resulting formulation could be used to design and inspire the development of new soft material systems that seek to replicate three dimensional biological motion. [ABSTRACT FROM AUTHOR]

A higher-grade theory of non-ferromagnetic thermo-elastic dielectrics which incorporates the local mass displacement, the heat flux gradient, polarization inertia, and flexodynamic effects is developed. The process of local mass displacement is associated with changes in material microstructure. Using the fundamental principles of continuum mechanics, electrodynamics, and non-equilibrium thermodynamics, the gradient-type constitutive equations are derived. Due to accounting for the polarization inertia, the rheological constitutive equation for the polarization vector is obtained. In the balance equation of linear momentum, an additional term with the second time derivative of the polarization vector appears in comparison with the classical theory. This term controls the influence of the dynamic flexoelectric effect on the mechanical motion of dielectric solids. The propagation of a plane harmonic wave is analyzed within the context of the developed theory. It is shown that the theory allows for capturing the experimentally observed phenomenon of high-frequency dispersion of a longitudinal elastic wave. The theory may be useful for modeling coupled processes in nanodielectrics and heterogeneous polarized systems. [ABSTRACT FROM AUTHOR]

The study of energy transport in granular systems can involve a number of different angles to view the problem; for example, one can propagate sinusoidal waves within the granular assembly, which makes the particles vibrate; besides the large wavelength low-amplitude elastic limit, this can be at very large frequencies and medium-large amplitudes, thereby posing the particles in perturbations of different modes, like resulting in cyclic shear, which can be translational and/or rotational as well as oblique collisions between the particles to occur. If these particles are naturally occurring grains, they will have a far from classical "elastic" response and their morphologies will be evolving during these perturbations. If a viscous fluid is added, then the dynamics of these perturbations and the way the energy is transferred among the particles may be substantially different. One may wish to see this problem even at a smaller scale, examining only two perturbating particles in contact, or allowing them to impact each other in the presence of a fluid. If you load the granular system in a cyclic mode, but this time at a very low frequency, some mechanisms will be altered, and the way the energy will be dissipated may also be expected to be altered, thereby the interpretations made from such analysis. Of course, a granular assembly is often part of a larger system that we are interested in to study by stability analysis, as e.g. internal erosion, or the dynamics of a submarine landslide involving an extraordinarily large span of particle sizes and morphologies. Taking as example research works in soil dynamics, the rate of stiffness decrease in a granular system, caused by the nonlinearity of that system, is proportional to the rate of energy dissipation increase as macroscopically measured in medium-frequency torsional shear dynamic excitation. However, if the excitation amplitude is reduced enough to lead to measurements of elastic stiffness, some small dissipation of energy might still be observed, which generally contradicts principles of classic continuum mechanics. These, and many others are interesting and exciting, though challenging areas of research in granular matter, in which scientists from a wide span of expertise are working to provide answers, and perhaps raise more questions about what is happening in a granular system. The topical collection "Energy transport and dissipation in granular systems" aimed to provide a forum bringing together scientists and engineers from different disciplines to answer some simple, though challenging questions about what the involved mechanisms of energy transport and dissipation in granular systems are, and extending these towards understanding, how micromechanical-based features influence the macroscopic behavior of larger-scale systems involving particles or powders. Finally, we could see very proudly that a total of 18 high-quality articles were contributed and published under this topical collection in Granular Matter, which attempted to examine systems at various scales, from the scale of two interacting particles in contact, to element-size numerical and laboratory samples, to the behavior of a submarine landslide, or a system which reaches a fluidization state. The outcome is very rich and diverse, and we can be very confident that these works will comprise a state-of-the-art for future research in granular matter. We had a diversity of contributions in terms of demographics, from institutes in the U.S.A, Canada, and Europe, to Middle-East and East-Asia all the way to Australia-based researchers. The contributions have different perspectives, from physics, mechanical-aerospace engineering, civil and environmental engineering. The diversity was also present in the scientific methodology, from purely analytical works to experimental-based studies at various scales, with novel laboratory setups developed and presented which allow the examination of energy transport and dissipation analysis in granular systems. The majority of these works considered a granular system in which gravitational forces are dominant, i.e., frictional materials, but we also had contributions on adhesive-type granular micro-systems. Below is a summary of major outcomes from these contributions which are divided in different sub-sections based on some common characteristics of the different contributing works. [ABSTRACT FROM AUTHOR]

In this study, a numerical approach based upon the moving Kriging meshfree (MKM) formulations within the framework of the Gurtin–Murdoch continuum mechanics in conjunction with the three-dimensional shell theory is established. Thereafter, with the aid of proposed surface elastic-based MKM formulations incorporating proper polynomial basis function, the nonlinear three-dimensional large-amplitude vibration responses of cylindrical shells at nanoscale made of probabilistic nanocomposites and possessing different thicknesses are analyzed. It is revealed that by increasing the maximum lateral deflection associated with the nonlinear oscillation amplitude, the significant of the impact of surface stress tensor improves. Accordingly, by considering the maximum lateral deflection equal to the half of shell thickness, for the simply supported probabilistic inhomogeneous nanoshells having the thickness values of 5 nm , 10 nm , 20 nm , and 50 nm , the surface stress tensor causes to reduce the value of ω nl / ω l ratio about 6.31 % , 3.81 % , 2.10 % , and 0.89 % , respectively. While by assuming the maximum lateral deflection equal to the shell thickness, the surface stress tensor makes a reduction in the value of ω nl / ω l ratio about 16.16 % , 9.53 % , 5.19 % , and 2.19 % , respectively. [ABSTRACT FROM AUTHOR]

The peridynamic (PD) theory is a nonlocal reformulation of mechanics with various advantages over common approaches, mainly local continuum mechanics and molecular dynamics (MD). PD theory can capture phenomena at different dimensions, including nanoscale. However, limited studies have been performed by this theory in nanoscale, which have generally focused on the feasibility and accuracy of using PD in atomic-scale modeling. In the present study, based on the ordinary state-based peridynamic method, we investigate the fracture of pre-cracked single layer graphene sheets (SLGSs) under uniaxial tension. By simulating the exact atomic model of graphene, the failure strain and crack growth pattern in the zigzag and armchair directions in PD are compared with MD. We show that by considering some restrictions, these two methods have a good consistency with each other. Afterward, we study two different coarse-grained PD models and demonstrate this method can simulate the failure of graphene with acceptable accuracy. A significant reduction in simulation cost is an excellent point of the PD compared to the MD simulation model. Under these conditions, a massive atomic model with several million atoms can be easily simulated. [ABSTRACT FROM AUTHOR]

In this paper we study the dispersive properties related to a model of peridynamic evolution, governed by a non local initial value problem, in the cases of two and three spatial dimensions. The features of the wave propagation characterized by the nontrivial interactions between nonlocality and the regimes of low and high frequencies are studied and suitable numerical investigations are exposed. [ABSTRACT FROM AUTHOR]

Inspired by biology and engineered soft active material systems, we propose a new constitutive formulation for a soft material consisting of soft contractile fibers embedded in a soft matrix. The mathematical implementation of the model is based on a multi-field invariant formulation within a nonlinear continuum mechanics framework. The coupled constitutive formulation highlights a new electromechanical coupling term that describes the intrinsic (or active) anisotropy due to the contractile units. The model demonstrates the relative role that intrinsic anisotropy plays in the overall stress response. The resulting formulation could be used to design and inspire the development of new soft material systems that seek to replicate three dimensional biological motion. [ABSTRACT FROM AUTHOR]

In Continuum Mechanics a widely cited formula, due to Nanson (Mess Math 7:182–185, 1878), contributes the relation between the product of area and unit normal vectors relevant to corresponding surfaces in two configurations of a 3D body. A geometric treatment provides equivalent expressions of Nanson formula by direct elaborations on Euler–Jacobi volume change formula. Meaning and limits of Nanson formula are underlined. In the literature this formula has been improperly assumed to be expedient for attempts of imposing equilibrium in a reference shape of the body. A critical revision shows recourse to scaling and parallel transport of surface and bulk forces is impassable in assessing alleged referential equilibrium. A variational formulation of evolutive equilibrium in spacetime is developed and finite step elastic problems guided by control algorithms in computational procedures are illustrated. Step-by-step iterative methods of solution of nonlinear structural problems and simple counterexamples should help in convincing referential equilibria are not conceivable in Nonlinear Continuum Mechanics. [ABSTRACT FROM AUTHOR]

This paper analyzes the applicability of Eringen's Generalized Continuum Theories as a model for human blood in the microcirculation. The applied theory considers a fluid with a fully deformable substructure, namely a micromorphic fluid. This analysis is motivated by the fact that blood itself can be considered a suspension of deformable particles, i.e., red blood cells (RBCs), suspended in a Newtonian fluid, i.e., blood plasma. As a consequence, non-Newtonian phenomena such as shear-thinning are observed in blood. To test the micromorphic fluid as a model for blood, the solution for the velocity and the motion of substructure is determined for a cylindrical pipe flow and compared to experimental results of blood flow through narrow glass capillaries representing idealized blood vessels. A similar analysis was also conducted by Kang and Eringen in 1976, but it contains some misprints and minor errors regarding the mathematical expressions and subsequent discussion which are corrected in this paper. For certain material parameters, the micromorphic fluid models capture high-shear blood flow in narrow glass capillaries very well. This concerns both the velocity profiles and the shear-thinning behavior. Furthermore, a parameter study reveals that the flexibility of substructure governs the micromorphic shear-thinning. In this regard, parallels can be drawn to the shear-thinning of human blood, which is also induced by the deformability of RBCs. This makes the micromorphic fluid a complex but accurate model for human blood, at least for the considered experiments. [ABSTRACT FROM AUTHOR]

The second-order strain gradient wave equation is based on the generalized continuum mechanics theory, and it can enrich the content of classical continuum mechanics theory with the incorporation of the second-order spatial derivative term of displacement. Furthermore, this equation incorporates scale parameters of media characteristics to bridge the gap between micromodels and classical continuum mechanics to refl ect the microstructural characteristics in the media. Wang used a nonlocal theory to derive a constitutive equation of single-parameter secondorder strain gradient theory and provided a mathematical expression for the second-order strain gradient asymmetric elastic wave equation combined with geometric equation and differential equation of motion. The second-order strain gradient theory can reflect the small-scale effect of seismic wave propagation within the neighborhood with an average particle diameter of l of the medium. Given the relatively weak spatial-scale effect, the numerical dispersion generated when the difference operator approximates the differential operator can suppress the scale effect. Thus, the finite-difference operator must be optimized accurately to describe and analyze scale effects. In this paper, an improved black widow optimization algorithm was proposed to obtain optimized finite-difference coefficients, and numerical modeling based on the second-order strain gradient wave equation was performed. Numerical modeling results reveal the well-suppressed numerical dispersion, and seismograms display the clear extraction of small spatial-scale effects. [ABSTRACT FROM AUTHOR]

Due to the complex constitutive relationship of tread rubber and the strong non-linearity of the dynamic contact relationship between tread rubber and road surface, the torsional deformation characteristics and distribution mechanism are difficult to be modeled theoretically in the tire contact patch. In order to accurately express the nonlinear flexible large deformation of tire tread block under torsional condition, the constitutive theory modeling of tread rubber based on continuum mechanics and the constitutive model verification based on improved digital image correlation are studied. A novel constitutive model considering the hyperelastic, viscoelastic and frictional property of tread rubber is established. A full-field measuring equipment for torsional deformation of tread block is established based on machine vision technology. The digital image sequence of instantaneous torsion deformation is analyzed by the improved digital image correlation to obtain the strain rate distribution characteristics and distribution law of the marker points and the whole area of the tread block. The analysis results demonstrate that the proposed constitutive model is feasible to predict the full-field distribution characteristics and distribution trend of tread rubber blocks in the contact patch. [ABSTRACT FROM AUTHOR]

Tensor-nonlinear media are generalized models of continua having nonlinear relationship between the stress and strain rate, important for the descriptions of, for example, non-Newtonian fluids. We consider a wide class of such tensor-nonlinear, isotropic, incompressible media, which may possess a scalar stress potential in terms of velocity gradients, i.e., the analogue or Raleigh function. We also provide a setting of several linearized problems for these media in moving three-dimensional domains. Energy-based estimates and subsequent Lyapunov, asymptotic and exponential stability results are derived through an application of a sequence of integral inequalities. We also present particular cases of stability estimates for a variety of tensor-linear, or quasilinear, media that may or may not possess the dissipation potential, such as the Bingham and Saint-Venant media, or Newtonian viscous fluid. [ABSTRACT FROM AUTHOR]

The paper presents a rather general formulation of a porous medium deformation coupled with a fluid flowing through the pores within the framework of physical and geometric nonlinearity. The boundary-value problem is formulated in terms of the solid phase displacement increment, fluid pressure and porosity increments in the form of differential and variational equations. The equations were derived from the general conservation laws of Continuum Mechanics using spatial averaging over a representative volume element (RVE). The model takes into account the porosity and permeability evolutions during deformation process. The equations of filtration and porosity evolution are formulated in the material coordinate system related to the solid phase, according to the idea of Arbitrary Lagrangian–Eulerian (ALE) approach. The linearization of variational equations was done using Gateaux differentiation technique. The proper finite elements were used for the spatial discretization of the saddle system of equations to satisfy well-known Ladyzhenskaya–Babuška–Brezzi (LBB) correctness condition. A generalization of the implicit time integration scheme with internal iterations at each time step according to the Uzawa method is employed. The convergence of the iterative process is partly theoretically studied. The formulation is numerically implemented in the form of a self-made computer code. Examples of calculations are given. [ABSTRACT FROM AUTHOR]

Since bioparticles experience large deformations, it is imperative to develop an appropriate model for particle modeling based on its size to prevent Nano biological particle destruction such as Bacillus, which is modeled as a Nano-beam. In this paper, models with large deformation assumptions were suggested for a longitudinal particle that can be geometrically considered as a beam. By modeling a nano/microparticle using continuum mechanics theory and dynamic simulation based on FEM, the dynamic behavior and shape deformation caused by the manipulation forces were examined. Hence, the manipulation of bioparticles with a cylindrical shape, named Bacillus Subtilis, was investigated. Critical time and forces and the predominant mode of movement of biological nanoparticles in a manipulation process were obtained and the Euler–Bernoulli theory was utilized to investigate the dynamic of biological nanoparticles. After the discretization of governing equations of motion using the Galerkin method, mass and stiffness matrices were extracted and critical forces were calculated to examine the dynamic behavior of particles. The effect of the aspect ratio showed that the deformation of the particle increases with increasing aspect ratio. In addition, the manipulation force is an essential parameter when small and large deflection models are compared. It was evident that the difference between linear and nonlinear deflection models at L/D = 100 was decreased from 39.47% for the 10 µN force to 12.46% for the 5 µN force. It was deduced that the small and large deflection models tend to have similar results with reducing force. [ABSTRACT FROM AUTHOR]

A trend in the development of geophysics is to seek wave theory closer to the physical reality and derive corresponding wave equations to achieve highly accurate forward modeling, imaging, and inversion of complex structures. The generalized continuum mechanics (GCM) theory enriches the context of the conventional continuum mechanics theory by introducing the additional characteristic length scale parameters to represent the microstructural properties of the medium, and the asymmetric elastic wave equations derived from GCM theory can handle the influence of heterogeneity of a medium caused by the microstructural interactions on the propagation of seismic waves. To date, there are few studies on the numerical and analytical solutions of the elastic wave equations derived from the GCM theory, especially in the frequency band of seismic exploration. In addition, there are few studies in the existing literature that incorporate multiple theories and methods of the GCM theory into an integrated frame. In this paper, we introduce the concept of the multi-scale microstructural interactions and construct the quantitative relationship between the characteristic length scale parameter of the medium and the characteristic length scale parameter of the micro-pore reflecting the micro-pore structures, and then integrate the modified couple stress theory and the one-parameter second strain gradient theory into a unified framework for numerical modeling and analysis. [ABSTRACT FROM AUTHOR]

Material identification is critical for understanding the relationship between mechanical properties and the associated mechanical functions. However, material identification is a challenging task, especially when the characteristic of the material is highly nonlinear in nature, as is common in biological tissue. In this work, we identify unknown material properties in continuum solid mechanics via physics-informed neural networks (PINNs). To improve the accuracy and efficiency of PINNs, we develop efficient strategies to nonuniformly sample observational data. We also investigate different approaches to enforce Dirichlet-type boundary conditions (BCs) as soft or hard constraints. Finally, we apply the proposed methods to a diverse set of time-dependent and time-independent solid mechanic examples that span linear elastic and hyperelastic material space. The estimated material parameters achieve relative errors of less than 1%. As such, this work is relevant to diverse applications, including optimizing structural integrity and developing novel materials. [ABSTRACT FROM AUTHOR]

The computational efficiency has been largely hindering the application of the absolute nodal coordinate formulation (ANCF). To improve the computational efficiency of the absolute nodal coordinate formulation, the collocation strategy is employed to establish the ANCF beam element. A novel ANCF beam element based on multi-node collocation (ANCF_C) is proposed. Zero points of the second-order derivative of pth-order Legendre polynomial and the boundary points in the element domain are used as nodes to discretize the beam node-wisely. Accordingly, the (p - 1) th-order Lagrange interpolation is employed for the longitudinal displacement interpolation. The elastic force and stiffness matrix are deduced based on the enhanced continuum mechanics formulation (ECMF) to avoid the locking problem. By using p-point Lobatto quadrature for the numerical integration of the elastic force and the mass matrix, the quadrature points coincide with the discretized nodes in the element. The performance of the ANCF_C is verified by both static and dynamic examples. [ABSTRACT FROM AUTHOR]

Mechanical-diffusion coupling analysis at micro/nanotemporal and spatial scale has aroused great research interests with flourishing development of nanobattery system and fast rising of rapid charging technology, where the spatial nonlocal effects of mass transfer and elastic deformation as well as the influences of temporal nonlocal effects of mass transport (i.e., the phase laggings of diffusion flux vector and molar concentration gradient) will remarkably increase. In such cases, however, the accurate prediction of mechanical-diffusion responses is challenged: Firstly, the existing non-Fick diffusion–elasticity models are established by merely introducing mass diffusion model associated with the time rate of diffusion flux (i.e., phase laggings of diffusion flux); secondly, the spatial nonlocal effect of mass transfer is still not considered in the current on dual-phase-lag diffusion model. This work aims to develop a non-Fick diffusion–elasticity based on a new nonlocal dual-phase-lag diffusion model, which fully incorporates spatial and temporal nonlocal effects of mass transport. New constitutive and field equations are strictly derived via nonlocal continuum mechanics. To illustrate its application values, a one-dimensional isotropic homogeneous thin layer of finite thickness subjected to transient shock loadings of molar concentration is investigated. Dimensionless results are graphically presented to illustrate the effects of both nonlocal mass transfer and nonlocal elasticity on diffusive wave propagation and mechanical-diffusion responses. [ABSTRACT FROM AUTHOR]

The least action principle played a central role in the development of modern physics. A major drawback of the principle is that its applicability is limited to holonomic constraints. In the present work, we investigate the energy lost by particles as a result of the gravitational interaction in a homogeneous low-density medium subject to non-holonomic constraints. We perform the calculation for an arbitrary particle and outline the specific result for photons. The energy lost is calculated from first principles based on the principle of virtual work and the d'Alembert principle. Under the formalism mentioned above, the dissipative nature of the effect is established. Furthermore, we show that the results agree with an alternative derivation based on continuum mechanics and the Euler–Cauchy stress principle. [ABSTRACT FROM AUTHOR]

Soft materials that can demonstrate on demand reconfigurability and changing compliance are highly sought after as actuator materials in many fields such as soft robotics and biotechnology. Whilst there are numerous proof of concept materials and devices, rigorous predictive models of deformation have not been well-established or widely adopted. In this paper, we discuss programming complex three-dimensional deformations of a soft intrinsically anisotropic material by controlling the orientation of the contractile units and/or direction of the applied electric field. Programming is achieved by patterning contractile units and/or selectively activating spatial regions. A new constitutive model is derived to describe the soft intrinsic anisotropy of soft materials. The model is developed within a continuum mechanics framework using an invariant-based formulation. Computational implementation allows us to simulate the complex three-dimensional shape response when activated by electric field. Several examples of the achievable Gauss-curved surfaces are demonstrated. Our computational analysis introduces a mechanics-based framework for design when considering soft morphing materials with intrinsic anisotropy, and is meant to inspire the development of new soft active materials. [ABSTRACT FROM AUTHOR]

In the theory of partial differential equations, the fundamental solutions have undoubtedly occupied serious attention, as they have a critical role to play in a wide range of problems under mathematical physics and continuum mechanics. The present work explores the Galerkin-type representation of the field equations based on the recently developed Moore–Gibson–Thompson thermoelasticity theory. With the help of the Laplace transform technique, we then establish the short-time approximated fundamental solutions for two separate cases: concentrated body force and concentrated heat source. For the time domain, the fundamental solutions are obtained by taking inverse Laplace transforms of displacement and temperature components. Lastly, we obtain the fundamental solution for steady vibration in terms of elementary functions. [ABSTRACT FROM AUTHOR]

Abstract—A computer model of fracture in heterogeneous materials (including rocks) is proposed to study evolution of the defect structure (cracks) during deformation process. The model is based on the Discrete Element Method (DEM), which, in contrast to the methods relying on continuum mechanics, naturally simulates the formation and evolution of cracks. We used the bonded particle model (BPM), which is widely used in various modifications to study the fracture process. The material is modeled by a set of spherical particles (simulating polycrystalline grains), connected by bonds placed at particle contact points, which simulate grain boundaries. In the BPM model, crack initiation is determined by the breakage of bonds, and its propagation is provided by the coalescence of a numerous set of broken bonds. Computer experiments were performed with different material parameters (different mechanical properties of grains and grain boundaries, different grain sizes) to determine their effect on the local stress patterns, defect formation, and fracture source nucleation and development. The calculations were performed in the open-source DEM simulation software MUSEN. Samples were modeled by cylinders filled with spherical particles of the same or different radii packed up to reaching the void ratio 0.35–0.37. The materials simulating grains and bonds (grain boundaries) were specified with mechanical parameters corresponding to granite, quartz, orthoclase, oligoclase, and glass. The sample was placed in a virtual press in which a bottom plate was fixed and a top plate was moved towards the lower plate at a constant rate until the sample failed. The maximum local stress calculations have shown that homogeneous materials yield more heterogeneous spatial distributions of local stresses and vice versa—material heterogeneity leads to more uniform stress distributions. Comparison with the results of laboratory rock deformation experiments has shown that the proposed model of polycrystalline materials realistically describes some features of fracture in these materials when the main processes occur along the grain boundaries. These features include the brittle type of fracture in homogeneous materials and the presence of nonlinear elasticity (plasticity) revealed for more heterogeneous materials based on the stress-strain diagrams, as well as the time behavior of "acoustic emission rate" (or acoustic activity)—the number of bonds broken per unit time. For heterogeneous materials, the model shows a two-stage pattern of the fracture process with uniform defect accumulation throughout the sample at the first stage and the formation and growth of fracture nucleation sites at the second stage. [ABSTRACT FROM AUTHOR]

The deformation mechanism of fibrin fibers has been a long-standing challenge to uncover due to the fiber's complex structure and mechanical behaviors. In this paper, a phenomenological, bilinear, force-strain model is derived to accurately reproduce the fibrin fiber force-strain curve, and then, the phenomenological model is converted to a mechanistic model using empirical relationships developed from particle simulation data. The mechanistic model assumes that the initial linear fibrin fiber force-strain response is due to entropic extension of polypeptide chains, and the final linear response is due to enthalpic extension of protofibrils. This model is the first fibrin fiber tensile force-strain equation to simultaneously (1) reproduce the bilinear force-strain curve of fibrin fibers in tension; (2) explicitly include the number of protofibrils through the fibrin fiber cross section, persistence length of α C -regions, and stiffness of fibrin protofibrils; and (3) make demonstrably reasonable/accurate predictions of fibrin fiber mechanics when tempered against experimental results. The model predicted that the count of protofibrils through the cross section for the analyzed fibrin fibers is between 207 and 421, the persistence length of α C -regions is ∼ 0.36 nm , and the stiffness of protofibrils in a deforming fiber is ∼ 1.34 n N / strain . The predicted α C -region persistence length is within the range typical of amino acid residue lengths 0.34 - 0.4 nm , and the predicted protofibril stiffness is shown to correspond to half-staggered protofibrils of unfolded fibrin monomers. Our analysis supports the proposition that entropic extension of α C -regions could be responsible for fibrin fiber's initial force-strain stiffness and suggests a structural change in fibrin protofibrils during fibrin fiber deformation. The results from the model are compared to those from five candidate deformation mechanisms reported in the literature. Our work provides (1) strong quantitative support to a deformation mechanism that was previously supported by anecdote and qualitative argument, and (2) a model for rigorously analyzing fibrin fiber force-strain data and simulating fibrin fibers in tension. [ABSTRACT FROM AUTHOR]

Peridynamic (PD) theory can overcome the shortcomings of classical continuum mechanics (CCM) in simulating crack initiation and propagation. In this paper, a general anisotropic PD plane model is proposed based on the micro-beam bond. First, the general anisotropic expression of the micromoduli in different directions can be expressed through a certain transformation relation, which can make the bond exhibit general anisotropy. Then, with the use of the Allman interpolation method, the deformations of the bond can be obtained, and the PD strain energy density of the general anisotropic plane model can be expressed. Finally, the general anisotropic PD parameters in the micromoduli expression can be obtained by equating the strain energy densities of PD and CCM models. Numerical examples of static uniaxial tension, static shear, dynamic fracture of the compact tension test and dynamic fracture of the three-hole plate test prove the effectiveness of the proposed model in handling static in-plane problems and dynamic in-plane fracture problems. [ABSTRACT FROM AUTHOR]

As thermoset polymers find frequent implementation in engineering design, their application in structural engineering is rather limited. One key reason relies on the ongoing curing process in typical applications such as post-installed adhesive anchors, joints by structural elements or surface-mounted laminates glued by adhesive polymers. Mechanochemistry including curing and aging under thermal as well as mechanical loading causes a multiphysics problem to be discussed. For restricting the variety of material models based on empirical observations, we aim at a thermodynamically sound strategy for modeling thermosets. By providing a careful analysis and clearly identifying the assumptions and simplifications, we present the general framework for modeling and computational implementation of thermo-mechano-chemical processes by using open-source codes. [ABSTRACT FROM AUTHOR]

Nucleation and growth are phenomena that can be applied to several fields of science and technology. On the other hand, nucleation depends on the cooling rate, dislocating the equilibrium, as surface energy depends on the created and deformed surface area. The crystalline/glassy transition limit dependence on the thermal gradient is also analyzed. In this paper, under continuum mechanics, first and second-order nonequilibrium nucleation formulation models are derived, and a phase-change moving interface is considered in the thermal field. Important nucleation variables are plotted against the cooling rate for several nucleation angles. It is coupled with a theoretical model for the molar-specific heat capacity of solids to analyse its dependence on nucleation kinetics. [ABSTRACT FROM AUTHOR]

In the past, to model fibre stiffness of finite-radius fibres, previous finite-strain (nonlinear) models were mainly based on the theory of non-linear strain-gradient (second-gradient) theory or Kirchhoff rod theory. We note that these models characterize the mechanical behaviour of polar transversely isotropic solids with infinitely many purely flexible fibres with zero radius. To introduce the effect of fibre bending stiffness on purely flexible fibres with zero radius, these models assumed the existence of couple stresses (contact torques) and non-symmetric Cauchy stresses. However, these stresses are not present on deformations of actual non-polar elastic solids reinforced by finite-radius fibres. In addition to this, the implementation of boundary conditions for second gradient models is not straightforward and discussion on the effectiveness of strain gradient elasticity models to mechanically describe continuum solids is still ongoing. In this paper, we develop a constitutive equation for a non-linear non-polar elastic solid, reinforced by embedded fibers, in which elastic resistance of the fibers to bending is modelled via the classical branches of continuum mechanics, where the development of the theory of stresses is based on non-polar materials; that is, without using the second gradient theory, which is associated with couple stresses and non-symmetric Cauchy stresses. In view of this, the proposed model is simple and somewhat more realistic compared to previous second gradient models. [ABSTRACT FROM AUTHOR]

Evaluation of the stability of placement rooms and excavation damage in the surrounding geological formations of a deep geological repository (DGR) for high-level nuclear waste are essential components when assessing the overall repository performance. Assessment of the geomechanical performance of placement rooms for the duration of the regulatory period (typically 1 million years) usually involves simulations using numerical models capable of simulating coupled thermo-hydro-mechanical (THM) processes induced by different natural and repository-induced perturbations over different time and length scales. The numerical constitutive model used to represent the host medium should be capable of capturing highly nonlinear responses to stress changes, including development of damage and fracturing. In this paper, a novel approach for simulating the evolution of damage and fracturing around the repository rooms considering coupled THM processes is presented, including validation examples for the approach. The important and novel feature of this modeling approach is the use of the bonded-block model based on the distinct element method (DEM) to represent deformation, damage, and fracturing of the surrounding rock that could potentially result in rock fragmentation and collapse of the rock surrounding the placement rooms. As part of the study, the predictions of damage around placement rooms obtained using the new approach are compared with the conventional continuum mechanics model. Subsequently, the capability of the bonded-block model to simulate damage caused by in-situ and thermally induced stresses is validated by comparison with monitoring data and observations from field-scale experiments. Highlights: A novel numerical approach for modeling thermo-hydro-mechanical processes taking place around deep geological repositories is presented. The approach uses a coupling scheme that allows simulations of times relevant for performance assessment of the repositories. Formation of stress-induced microcracks and their effect on permeability changes and long-term stability of the rooms are explicitly simulated. The methodology is validated by comparison with field data and with a different numerical method. [ABSTRACT FROM AUTHOR]

Functional heterogeneity is a skeletal muscle's ability to generate diverse force vectors through localised motor unit (MU) recruitment. Existing 3D macroscopic continuum-mechanical finite element (FE) muscle models neglect MU anatomy and recruit muscle volume simultaneously, making them unsuitable for studying functional heterogeneity. Here, we develop a method to incorporate MU anatomy and information in 3D models. Virtual fibres in the muscle are grouped into MUs via a novel "virtual innervation" technique, which can control the units' size, shape, position, and overlap. The discrete MU anatomy is then mapped to the FE mesh via statistical averaging, resulting in a volumetric MU distribution. Mesh dependency is investigated using a 2D idealised model and revealed that the amount of MU overlap is inversely proportional to mesh dependency. Simultaneous recruitment of a MU's volume implies that action potentials (AP) propagate instantaneously. A 3D idealised model is used to verify this assumption, revealing that neglecting AP propagation results in a slightly less-steady force, advanced in time by approximately 20 ms, at the tendons. Lastly, the method is applied to a 3D, anatomically realistic model of the masticatory system to demonstrate the functional heterogeneity of masseter muscles in producing bite force. We found that the MU anatomy significantly affected bite force direction compared to bite force magnitude. MU position was much more efficacious in bringing about bite force changes than MU overlap. These results highlight the relevance of MU anatomy to muscle function and joint force, particularly for muscles with complex neuromuscular architecture. [ABSTRACT FROM AUTHOR]

The mechanics of fiber-reinforced composite materials is presented within the framework of the second strain gradient theory. As such, a continuum-based model is developed for the analysis of elastic materials reinforced with bidirectional fibers and subjected to finite plane deformations. The obtained model is subsequently applied to the cases of the unidirectional fiber composites for the purpose of model implementation. The Euler equilibrium equations and the associated boundary conditions are obtained via the variational principle and iterative integration by parts. In particular, we formulate the complete expressions of Piola-type triple stress and its coupled triple force arising in the third gradient of continuum deformations, which, in turn, yield the unique deformation maps in the presence of admissible boundary conditions of higher order. The solutions of the resulting systems of differential equations are obtained via the custom-built numerical scheme from which smooth and dilatational shear angle distributions are predicted throughout the entire domain of interest. It is also observed that the third gradient constitutive parameter is associated with the volume dilatation of third-gradient continua, which may be appeared in the form of shear band inclination angle. [ABSTRACT FROM AUTHOR]