Isotropic Invariants Research Articles

Response to Shariff’s comments to my paper on his isotropic invariants (Shariff, 2023)

In the paper by Shariff (Q. J. Mech. Appl. Math. 76, 143–161, 2023) a functional basis of vectors and symmetric tensors based on eigenvectors of the first tensor is proposed. In the paper by Itskov (Mech. Soft Mater. 6(4), 1–3, 2024) a counterexample is reported demonstrating that the proposed terms do not generally represent isotropic invariants and cannot thus serve as a functional basis of symmetric tensors. In the latest response (Mech. Soft Mater. 6, 4, 2024) Shariff shows many examples of symmetric tensors for which his functional basis is valid but he does not really oppose the counterexample.

A Research of Analytical Solution on Tubular Structure which is Subjected to an Axis Expansion in Different Contributions

In this research paper, we proposed a classical mechanical study of the behavior of a tubular structure which is subjected to an expansion following the axis of the material. The first part, is focused on kinematic and dynamic aspects, Gradient tensor, left Cauchy-Green tensor ant isotropic invariants are used to proceed to find an analytical exact logarithmic solution function of the radius, of the problem at the limits through a system of partial differential equations in equilibrium condition. The simulations show that in the case of a sinusoidal, logarithmic and exponential function of time, the solution has a strong dependence to the radius. As a results, sinusoidal function of time has a big influence in the shape of the solution graphic which is negative or positive depending to the value of the time and radius. But logarithmic and exponential contribution has not a big influence in the arc shape of the solution graphic but a great influence in the values and it remains positive whatever the value of the radius.

On the functional basis of isotropic vector and tensor functions by Shariff (2023)

In the paper Shariff (Q. J. Mech. Appl. Math. 76:143–161, 2023) a functional basis of a system of vectors and symmetric tensors is proposed. The functional basis is expressed in terms of eigenvalues and eigenvectors of the first tensor and includes a smaller number of terms in comparison to the classical irreducible representation (see, e.g., Boehler, J. Appl. Math. Mech. 57:323–327, 1977; Pennisi and Trovato, Int. J. Eng. Sci. 25:1059–1065, 1987). In the present contribution, we show that elements of the functional basis by Shariff (Q. J. Mech. Appl. Math. 76:143–161, 2023) do not represent isotropic invariants of the vector and tensor arguments and cannot thus be referred to as the functional basis. To this end, a counterexample with two symmetric tensors is considered. Under an arbitrary orthogonal transformation the functional basis (Shariff, Q. J. Mech. Appl. Math. 76:143–161, 2023) of these two tensors should remain constant but it does change in contrast to the classical representation.

Role of anisotropic invariants in numerically modeling soft biological tissues as transversely isotropic hyperelastic materials: A comparative study

Fiber-reinforced biological soft tissues are often modeled as anisotropic hyperelastic materials. Four strain invariants can be used to define a strain energy function for soft tissues reinforced with a single fiber family: two isotropic invariants I1,I2 and two anisotropic invariants I4,I5. Invariant I5 is often omitted in the strain energy functions to simplify the problem mathematically. In this study, the implications of using only I4 or the use of both anisotropic invariants I4,I5 in the numerical modeling of soft tissues were analyzed. A simple modification to the Holzapfel–Gasser–Ogden (HGO) model is proposed by adding a term that contains invariant I5. Material parameters were calculated by fitting the models with experimental data of uniaxial traction in the tibialis anterior tendon tissue of rats. General analytical solutions for the simple load scenarios were obtained. Such solutions are taken as a reference point to measure the precision of the numerical results obtained in the finite element simulations. The main differences between the models were observed in the shear behavior. The proposed model predicts three different shear responses (two responses with fiber reinforcement and one isotropic response), while the HGO model predicts two equal isotropic responses and only one with fiber reinforcement. Two sets of simple shear experimental data on a fiber-reinforced elastomer material were used to verify the shear stress prediction of the models. The experimental data show that the three shear behaviors are different; thus, the HGO model offers a limited description of the shear behavior. Furthermore, analytical solutions and experimental data suggest that invariant I5 is related to the elastic energy of the fibers when the material is subjected to shear in a direction parallel to the fibers. The models for non-homogeneous deformations were also compared. For this comparison, an irregular geometry was implemented in ABAQUS, and the traction and shear conditions were simulated by changing the fiber inclination. The most critical difference was found when the shear was parallel to the direction of the fiber and the minor difference when the fiber was at 45° to the load force.

Quadratic-stretch elasticity

A nonlinear small-strain elastic theory is constructed from a systematic expansion in Biot strains, truncated at quadratic order. The primary motivation is the desire for a clean separation between stretching and bending energies for shells, which appears to arise only from reduction of a bulk energy of this type. An approximation of isotropic invariants, bypassing the solution of a quartic equation or computation of tensor square roots, allows stretches, rotations, stresses, and balance laws to be written in terms of derivatives of position. Two-field formulations are also presented. Extensions to anisotropic theories are briefly discussed.

Effect of Fibers Configuration on Nonlinear Vibration of Anisotropic Dielectric Elastomer Membrane

To describe the specific behavior of soft biological tissues that can be regarded as incompressible, a strain energy function for fiber-reinforcement materials based on two isotropic and anisotropic deformation invariants is developed and investigated. The anisotropic properties of simple composite materials were initially studied and now extensive hyperelastic modeling is used to analyze the dynamic characteristics of a dielectric elastomer membrane that experiences finite strains and is reinforced by one family of fiber. It should be noted that the presence of fibers in the rubber matrix will strengthen the structure, but here the presence of fibers will greatly reduce the electric field required to activate the dielectric elastomer and thus be stimulated with a lower voltage. It even helps to reduce the issue of instability that is a common thing in dielectric elastomers. Therefore, the effect of fiber angle orientation in an anisotropic dielectric elastomer under combined loads of biaxial tensile forces and an electric field with a focus on the damping effects is completely studied for different cases. Based on the Gent model, phase diagrams, dynamic response diagrams, and resonance frequency diagrams are also plotted for several values of fibers directions and damping coefficients.

A dynamic constitutive model for fiber-reinforced composite under impact loading

A constitutive model was proposed to predict the deformation and characterize the failure features of fiber-reinforced composite structures under dynamic loading. Primarily, quasi-static and dynamic mechanical tests were carried out on composite laminates with designed plying sequences. The material modulus was found rising as the increase of strain rate, while the ultimate strain exhibits no strain rate dependency. Nonlinear behavior was observed under transverse compression and in-plane shear, and was attributed to the progressive damage of material. Based on the results of uniaxial tests, constitutive relations were established for plane stress case and three-dimensional case respectively. Two damage factors were introduced to describe the nonlinear behavior, and failure criteria based on transversely isotropic invariants were proposed. Additionally, the constitutive model was implemented in Abaqus through user-defined material subroutine and validated by the simulation of uniaxial tests and bird impact tests. Comparing to the traditional model, the constitutive relations presented in this paper was proved to have better accuracy and efficiency in predicting dynamic response and ultimate failure patterns of composite structures under dynamic loading.

An irreducible polynomial functional basis of two-dimensional Eshelby tensors

The two-dimensional (2D) Eshelby tensors are discussed. Based upon the complex variable method, an integrity basis of ten isotropic invariants of the 2D Eshelby tensors is obtained. Since an integrity basis is always a polynomial functional basis, these ten isotropic invariants are further proven to form an irreducible polynomial functional basis of the 2D Eshelby tensors.

Two irreducible functional bases of isotropic invariants of a fourth-order three-dimensional symmetric and traceless tensor

The elasticity tensor is one of the most important fourth-order tensors in mechanics. Fourth-order three-dimensional symmetric and traceless tensors play a crucial role in the study of the elasticity tensor. In this paper, we present two isotropic irreducible functional bases for a fourth-order three-dimensional symmetric and traceless tensor. One of them is exactly the minimal integrity basis introduced by Smith and Bao in 1997. It has nine homogeneous polynomial invariants of degrees two, three, four, five, six, seven, eight, nine and ten, respectively. We prove that it is also an irreducible functional basis. The second irreducible functional basis also has nine homogeneous polynomial invariants. It has no quartic invariant but has two sextic invariants. The other seven invariants are the same as those of the Smith–Bao basis. Hence, the second irreducible functional basis is not contained in any minimal integrity basis.

Irreducible function bases of isotropic invariants of a third order three-dimensional symmetric and traceless tensor

Third order three-dimensional symmetric and traceless tensors play an important role in physics and tensor representation theory. A minimal integrity basis of a third order three-dimensional symmetric and traceless tensor has four invariants with degrees two, four, six, and ten, respectively. In this paper, we show that any minimal integrity basis of a third order three-dimensional symmetric and traceless tensor is also an irreducible function basis of that tensor, and there is no syzygy relation among the four invariants of that basis, i.e., these four invariants are algebraically independent.

Isotropic polynomial invariants of Hall tensor

The Hall tensor emerges from the study of the Hall effect, an important magnetic effect observed in electric conductors and semiconductors. The Hall tensor is third order and three dimensional, whose first two indices are skew-symmetric. In this paper, we investigate the isotropic polynomial invariants of the Hall tensor by connecting it with a second order tensor via the third order Levi-Civita tensor. We propose a minimal isotropic integrity basis with 10 invariants for the Hall tensor. Furthermore, we prove that this minimal integrity basis is also an irreducible isotropic function basis of the Hall tensor.

An irreducible function basis of isotropic invariants of a third order three-dimensional symmetric tensor

In this paper, we present an eleven invariant isotropic irreducible function basis of a third order three-dimensional symmetric tensor. This irreducible function basis is a proper subset of the Olive-Auffray minimal isotropic integrity basis of that tensor. The octic invariant and a sextic invariant in the Olive-Auffray integrity basis are dropped out. This result is of significance to the further research of irreducible function bases of higher order tensors.

Current density tensors.

It is shown that nonsymmetric second-rank current density tensors, related to the current densities induced by magnetic fields and nuclear magnetic dipole moments, are fundamental properties of a molecule. Together with magnetizability, nuclear magnetic shielding, and nuclear spin-spin coupling, they completely characterize its response to magnetic perturbations. Gauge invariance, resolution into isotropic, deviatoric, and antisymmetric parts, and contributions of current density tensors to magnetic properties are discussed. The components of the second-rank tensor properties are rationalized via relationships explicitly connecting them to the direction of the induced current density vectors and to the components of the current density tensors. The contribution of the deviatoric part to the average value of magnetizability, nuclear shielding, and nuclear spin-spin coupling, uniquely determined by the antisymmetric part of current density tensors, vanishes identically. The physical meaning of isotropic and anisotropic invariants of current density tensors has been investigated, and the connection between anisotropy magnitude and electron delocalization has been discussed.

Limit analysis and homogenization of nanoporous materials with a general isotropic plastic matrix

In this paper, a closed-form expression of a macroscopic strength criterion for ductile nanoporous materials is established, by considering the local plastic behavior as dependent on all the three isotropic stress invariants and by referring to the case of axisymmetric strain-rate boundary conditions. The proposed criterion also predicts void-size effects on macroscopic strength properties. A homogenization procedure based on a kinematic limit-analysis is performed by addressing a hollow-sphere model comprising a rigid-ideal-plastic solid matrix. Void-size effects are accounted for by introducing an imperfect-coherent interface at the cavity boundary. Both the interface and the solid matrix are assumed to obey to a general isotropic yield function, whose parametric form allows for a significant flexibility in describing effects induced by both stress triaxiality and stress Lode angle. Taking advantage of analytical expressions recently provided by Brach et al. [Int J Plasticity 2017; 89: 1–28] for the corresponding support function and for the exact velocity field under isotropic loadings, a parametric closed-form relationship for the macroscopic strength criterion is obtained as the solution of an inequality-constrained minimization problem, the latter being faced via the Lagrangian method combined with Karush-Kuhn-Tucker conditions. Finally, several comparative illustrations are provided, showing the influence of local-yield-function parameters on the established criterion, as well as the model capability to describe the macroscopic strengthening, typical of nanoporous materials, induced by a void-size reduction for a fixed porosity level.

Efficient evaluation of the material response of tissues reinforced by statistically oriented fibres

For several classes of soft biological tissues, modelling complexity is in part due to the arrangement of the collagen fibres. In general, the arrangement of the fibres can be described by defining, at each point in the tissue, the structure tensor (i.e. the tensor product of the unit vector of the local fibre arrangement by itself) and a probability distribution of orientation. In this approach, assuming that the fibres do not interact with each other, the overall contribution of the collagen fibres to a given mechanical property of the tissue can be estimated by means of an averaging integral of the constitutive function describing the mechanical property at study over the set of all possible directions in space. Except for the particular case of fibre constitutive functions that are polynomial in the transversely isotropic invariants of the deformation, the averaging integral cannot be evaluated directly, in a single calculation because, in general, the integrand depends both on deformation and on fibre orientation in a non-separable way. The problem is thus, in a sense, analogous to that of solving the integral of a function of two variables, which cannot be split up into the product of two functions, each depending only on one of the variables. Although numerical schemes can be used to evaluate the integral at each deformation increment, this is computationally expensive. With the purpose of containing computational costs, this work proposes approximation methods that are based on the direct integrability of polynomial functions and that do not require the step-by-step evaluation of the averaging integrals. Three different methods are proposed: (a) a Taylor expansion of the fibre constitutive function in the transversely isotropic invariants of the deformation; (b) a Taylor expansion of the fibre constitutive function in the structure tensor; (c) for the case of a fibre constitutive function having a polynomial argument, an approximation in which the directional average of the constitutive function is replaced by the constitutive function evaluated at the directional average of the argument. Each of the proposed methods approximates the averaged constitutive function in such a way that it is multiplicatively decomposed into the product of a function of the deformation only and a function of the structure tensors only. In order to assess the accuracy of these methods, we evaluate the constitutive functions of the elastic potential and the Cauchy stress, for a biaxial test, under different conditions, i.e. different fibre distributions and different ratios of the nominal strains in the two directions. The results are then compared against those obtained for an averaging method available in the literature, as well as against the integration made at each increment of deformation.

Extension or Compression Induced Twisting in Fiber-Reinforced Nonlinearly Elastic Circular Cylinders

The problem of pure extension or compression of a solid circular fiber-reinforced cylinder is considered in the context of the theory of nonlinear elasticity. Here we are interested in a particular feature of this problem which only occurs for certain classes of incompressible anisotropic hyperelastic materials and is completely absent in the isotropic case. We show that pure extension or compression of a transversely isotropic fiber-reinforced cylinder can induce twisting of the cylinder. The transverse isotropy is due to a single family of helically wound extensible fibers distributed throughout the cylinder. For general pitch angles, it is shown that in the absence of an applied torsional moment, the cylinder subjected to an applied tensile or compressive force exhibits an induced twisting due to mechanical anisotropy. Such an induced twist does not occur when the fibers are oriented longitudinally or azimuthally. It is also shown that at a special pitch angle (a “magic” angle), the initial slope of the axial force and induced moment with respect to stretch is independent of the Young’s modulus of the fibers. The results on induced twist are described in detail for three specific strain-energy densities. These models are quadratic in the anisotropic invariants, linear in the isotropic strain invariants and are consistent with the linearized theory of elasticity for fiber-reinforced materials. The results are illustrated on using elastic moduli for muscles based on experimental findings of other authors.

Kinking Instability in the Torsion of Stretched Anisotropic Elastomeric Filaments

The problem of torsion superimposed on extension of incompressible nonlinearly elastic solid circular cylinders is a classic problem of nonlinear elasticity that has been considered by many authors in the context of rubber elasticity particularly for isotropic materials. An interesting aspect of such problems is the instability that arises with sufficiently large twist where a kink or knot suddenly appears. An energy approach to examining this instability when the extension and twist are prescribed was given by Gent and Hua (Int. J. Nonlinear Mech. 39:483–489, 2004) and illustrated there for a neo-Hookean isotropic elastic material. The theoretical results were compared with experimental observations on natural rubber rods. In a recent paper Murphy (Int. J. Nonlinear Mech. 68:96–100, 2015) has shown that the approach in Gent and Hua for isotropic materials can be simplified when the rods are assumed to be thin. Motivated by potential applications to the biomechanics of soft tissues, we are concerned here with the implications of this simplified thin rod instability theory for transversely isotropic materials. The results are illustrated on using three specific strain-energy density functions. These models are quadratic in the anisotropic invariants, linear in the isotropic strain invariants and are consistent with the linear theory. In contrast with the case for isotropic materials, it is shown that the kinking instability arises even in the absence of stretch that is in the case of pure torsion. The results are illustrated on using elastic moduli for muscles based on experimental findings of other authors.

Origin invariance in vibrational resonance Raman optical activity

A theoretical investigation on the origin dependence of the vibronic polarizabilities, isotropic and anisotropic rotational invariants, and scattering cross sections in Resonance Raman Optical Activity (RROA) spectroscopy is presented. Expressions showing the origin dependence of these polarizabilities were written in the resonance regime using the Franck-Condon (FC) and Herzberg-Teller (HT) approximations for the electronic transition moments. Differently from the far-from-resonance scattering regime, where the origin dependent terms cancel out when the rotational invariants are calculated, RROA spectrum can exhibit some origin dependence even for eigenfunctions of the electronic Hamiltonian. At the FC level, the RROA spectrum is completely origin invariant if the polarizabilities are calculated using a single excited state or for a set of degenerate states. Otherwise, some origin effects can be observed in the spectrum. At the HT level, RROA spectrum is origin dependent even when the polarizabilities are evaluated from a single excited state but the origin effect is expected to be small in this case. Numerical calculations performed for (S)-methyloxirane, (2R,3R)-dimethyloxirane, and (R)-4-F-2-azetidinone at both FC and HT levels using the velocity representation of the electric dipole and quadrupole transition moments confirm the predictions of the theory and show the extent of origin effects and the effectiveness of suggested ways to remove them.

Isotropic invariants of a completely symmetric third-order tensor

In both theoretical and applied mechanics, the modeling of nonlinear constitutive relations of materials is a topic of prime importance. To properly formulate consistent constitutive laws some restrictions need to be imposed on tensor functions. To that aim, representations theorems for both isotropic and anisotropic functions have been extensively investigated since the middle of the 20th century. Nevertheless, in three-dimensional physical space, most of the results are restricted to sets of tensors up to second-order. The purpose of the present paper is thus to get one step further and to provide an integrity basis for isotropic polynomial functions of a completely symmetric third-order tensor. We exploit the link between the O(3)-action on harmonic tensors and the \documentclass[12pt]{minimal}\begin{document}$\mathrm{SL}(2,\mathbb {C})$\end{document} SL (2,C)-action on the space of binary forms to explicitly construct this basis. We believe that such an integrity basis may found interesting applications both in continuum mechanics and in other fields of theoretical physics.

Dominant negative Poynting effect in simple shearing of soft tissues

We identify three distinct shearing modes for simple shear deformations of transversely isotropic soft tissue which allow for both positive and negative Poynting effects (that is, they require compressive and tensile lateral normal stresses, respectively, in order to maintain simple shear). The positive Poynting effect is that usually found for isotropic rubber. Here, specialisation of the general results to three strain-energy functions which are quadratic in the anisotropic invariants, linear in the isotropic strain invariants and consistent with the linear theory suggests that there are two Poynting effects which can accompany the shearing of soft tissue: a dominant negative effect in one mode of shear and a relatively small positive effect in the other two modes. We propose that the relative inextensibility of the fibres relative to the matrix is the primary mechanism behind this large negative Poynting effect.