Automated constitutive modeling of isotropic hyperelasticity based on artificial neural networks

Herein, an artificial neural network (ANN)-based approach for the efficient automated modeling and simulation of isotropic hyperelastic solids is presented. Starting from a large data set comprising deformations and corresponding stresses, a simple, physically based reduction of the problem’s dimensionality is performed in a data processing step. More specifically, three deformation type invariants serve as the input instead of the deformation tensor itself. In the same way, three corresponding stress coefficients replace the stress tensor in the output layer. These initially unknown values are calculated from a linear least square optimization problem for each data tuple. Using the reduced data set, an ANN-based constitutive model is trained by using standard machine learning methods. Furthermore, in order to ensure thermodynamic consistency, the previously trained network is modified by constructing a pseudo-potential within an integration step and a subsequent derivation which leads to a further ANN-based model. In the second part of this work, the proposed method is exemplarily used for the description of a highly nonlinear Ogden type material. Thereby, the necessary data set is collected from virtual experiments of discs with holes in pure plane stress modes, where influences of different loading types and specimen geometries on the resulting data sets are investigated. Afterwards, the collected data are used for the ANN training within the reduced data space, whereby an excellent approximation quality could be achieved with only one hidden layer comprising a low number of neurons. Finally, the application of the trained constitutive ANN for the simulation of two three-dimensional samples is shown. Thereby, a rather high accuracy could be achieved, although the occurring stresses are fully three-dimensional whereas the training data are taken from pure two-dimensional plane stress states.

Orientational characteristics of magnetization precession excitation on the interference grating formed by femtosecond laser.

The subject of investigation in this work is the excitation of magnetization precession in magnetic film on the surface of which is formed the temperature relief which is formed by interference picture formatted by preliminary divided ray from femtosecond laser. It is mentioned the discovered in experiment the dependence of excitation efficiency from the orientation of magnetic field applied in the plane of film. The main aim of this work is the theoretical interpretation of observed orientation dependence. The realized in experiment scheme “pump-probe” is described. The whole geometry of task is proposed. This geometry includes in oneself the magnetic film with formed in its surface interference picture and applied in the plane of film the constant field. It is shown that by the thermal expansion in the film the elastic waves two types are excited; the surface Rayleigh waves and leaky longitudinal waves. The projections of wave-vectors of propagating waves to plane of film are normal to the strips of interference picture. The orientation of field may change from to longitudinal to transverse from the same strips. The components of deformation tensor of Rayleigh and leaky waves are determined. The precession of magnetization in the coordinate system connected with field is investigated. By using the apparatus of crossing matrixes it is found the components of deformation tensor are determined. In the frame of linear approach in this system the task about excitation of magnetization precession by elastic deformations by Rayleigh and leaky waves is solved. The dynamical component of magnetization precession which ensures the light polarization rotation which passes along the normal to the plane of film is found. It is shown that the angle of rotation is straight proportional to the tensor deformation components with the summarization with the resonance character of precession magnetization dependence from the value of magnetic field. The dependence of polarization rotation from orientation of field which is applied in the plane of film is found. It is shown that by the orientation of field along and across the interference stripes the rotation of polarization plane is absent. Between these extreme orientations the dependence has appearance as two maxima divided by deep minimum. The received results are compared with data of experiment. It is found the quality and in some cases quantity correlation. The recommendations for further development of work are proposed.

Global classical solutions to the elastodynamic equations with damping

In this paper, we show the global existence of classical solutions to the incompressible elastodynamics equations with a damping mechanism on the stress tensor in dimension three for sufficiently small initial data on periodic boxes, that is, with periodic boundary conditions. The approach is based on a time-weighted energy estimate, under the assumptions that the initial deformation tensor is a small perturbation around an equilibrium state and the initial data have some symmetry.

On a new class of non-Gaussian molecular-based constitutive models with limiting chain extensibility for incompressible rubber-like materials

In constitutive modelling of rubber-like materials, the strain-hardening effect at large deformations has traditionally been captured successfully by non-Gaussian statistical molecular-based models involving the inverse Langevin function, as well as the phenomenological limiting chain extensibility models. A new model proposed by Anssari-Benam and Bucchi ( Int. J. Non Linear Mech. 2021; 128; 103626. DOI: 10.1016/j.ijnonlinmec.2020.103626), however, has both a direct molecular structural basis and the functional simplicity of the limiting chain extensibility models. Therefore, this model enjoys the benefits of both approaches: mathematical versatility, structural objectivity of the model parameters, and preserving the physical features of the network deformation such as the singularity point. In this paper we present a systematic approach to constructing the general class of this type of model. It will be shown that the response function of this class of models is defined as the [1/1] rational function of [Formula: see text], the first principal invariant of the Cauchy–Green deformation tensor. It will be further demonstrated that the model by Anssari-Benam and Bucchi is a special case within this class as a rounded [3/2] Padé approximant in [Formula: see text] (the chain stretch) of the inverse Langevin function. A similar approach for devising a general [Formula: see text] term as an adjunct to the [Formula: see text] part of the model will also be presented, for applications where the addition of an [Formula: see text] term to the strain energy function improves the fits or is otherwise required. It is concluded that compared with the Gent model, which is a [0/1] rational approximation in [Formula: see text] and has no direct connection to Padé approximations of any order in [Formula: see text], the presented new class of the molecular-based limiting chain extensibility models in general, and the proposed model by Anssari-Benam and Bucchi in specific, are more accurate representations for modelling the strain-hardening behaviour of rubber-like materials in large deformations.

Evolving ultralight scalars into non-linearity with lagrangian perturbation theory

Many models of high energy physics suggest that the cosmological dark sector consists of not just one, but a spectrum of ultralight scalar particles with logarithmically distributed masses. To study the potential signatures of low concentrations of ultralight axion (also known as fuzzy) dark matter, we modify Lagrangian perturbation theory (LPT) by distinguishing between trajectories of different dark matter species. We further adapt LPT to include the effects of a quantum potential, which is necessary to generate correct initial conditions for ultralight axion simulations. Based on LPT, our modified scheme is extremely efficient on large scales and it can be extended to an arbitrary number of particle species at very little computational cost. This allows for computation of self-consistent initial conditions in mixed dark matter models. Additionally, we find that shell-crossing is delayed for ultralight particles and that the deformation tensor extracted from LPT can be used to identify the range of redshifts and scales for which the Madelung formalism of fuzzy dark matter can lead to divergences.

The mathematical foundations of anelasticity: existence of smooth global intermediate configurations

A central tool of nonlinear anelasticity is the multiplicative decomposition of the deformation tensor that assumes that the deformation gradient can be decomposed as a product of an elastic and an anelastic tensor. It is usually justified by the existence of an intermediate configuration. Yet, this configuration cannot exist in Euclidean space, in general, and the mathematical basis for this assumption is on unsatisfactory ground. Here, we derive a sufficient condition for the existence of global intermediate configurations, starting from a multiplicative decomposition of the deformation gradient. We show that these global configurations are unique up to isometry. We examine the result of isometrically embedding these configurations in higher-dimensional Euclidean space, and construct multiplicative decompositions of the deformation gradient reflecting these embeddings. As an example, for a family of radially symmetric deformations, we construct isometric embeddings of the resulting intermediate configurations, and compute the residual stress fields explicitly.

Contact of the edges of the interphase cut on the arc of the circle between the isotropic plate and the closed elastic rib

In the conditions of the general flat stress state created by uniformly distributed effects of tension (compression) at infinity, the mixed contact problem for an infinite isotropic plate with a circular hole, which contour reinforced by a closed elastic rib in the presence of a symmetrical interfacial section at the boundary of their connection and the edges of cut in the process of deformation is smoothly contacted, is considered. The components of the deformation tensor (unit elongation, the angle of rotation of the normal and the curvature) at the point of the contour of the hole of the plate are represented by integral dependences on the contact forces. By modeling the reinforcement of a closed elastic rod of a stable rectangular cross of large curvature and using the basic equations of linear theory of curvilinear rods the mathematical model of problems is constructed in the form of systems of three singular integral equations with Hilbert cores to find contact forces between plates and rib. To determine the initial parameters of a closed static indeterminate rod, the conditions of unambiguous displacement and angles of rotation at the point of its axis and the equilibrium conditions are used. The approximate solution of the problem is constructed by the method of mechanical quadrature and collocations, which investigated the influence on the stress state of the plate and the reinforcing rib and on the size of the area of smooth contact of stiffness factor of rib.

SHORTENED MAPPINGS OF SPACES WITH AFFINE CONNECTIVITY

The long history of theory of mappings was revived thanks to the tensor methods of inquiry. The notion of affine connectivity was introduced a hundred years ago. It enabled us to look at classic geometric problems from a different angle. Following the common tradition, this paper introduces a notion of a mapping for a space of affine connectivity. Modifying the method of A. P. Norden, we found the formulae for the main tensors: deformation tensor, Riemann tensor, Ricci tensor and their first and second covariant derivatives for spaces and , which are connected by a given mapping. These formulae contain both objects of and with covariant derivatives in respect to relevant connectivities. In order to simplify the expression, we introduced the notion of shortened mapping and its particular case: a half-mapping. The connectivity that appears in the case of a half-mapping is called a medium connectivity. The above mentioned formulae can be notably simplified in the case of transition to covariant derivatives in the medium connectivity. This fact permits us to obtain characteristics (the necessary conditions) for the estimates whether an object of inner character from the space of affine connectivity is preserved under a given type of mappings. Objects of the inner character are geometric objects implied by an affine connectivity. They include Riemann tensor, Ricci tensor, Weyl tensor. Every type of mapping received its own set of differential equations in covariant derivatives, which define a deformation tensor of connectivity with a necessity. The study of these equations can proceed by a research on integrability conditions. Integrability conditions are algebraic over-defined systems. That’s why there is a constant need in introduction of additionally specialized spaces or certain objects of these spaces. Applying the method of N. S. Sinyukov and J. Mikes, in the case of certain algebraic conditions, we obtained a form of a deformation tensor for a given mapping. Let us note that the medium connectivity was selected in order to simplify the calculations. Depending on the type of a model under consideration or on the physical limitations, we can construct any other connectivity (and mappings), which would be better suited for the given conditions. This approach is particularly fruitful when applied for invariant transformations connecting pairs of spaces of affine connectivity via their deformation tensor of connectivity.