Formulation of a Basic Constitutive Model for Fine - Grained Soils Using the Hypoplastic Framework

A hypoplastic approach to constitutive modelling was developed by Kolymbas 1996 considering a non-linear tensor function in the form of strain and stress rate. However, the implicit formulation of the hypoplastic model with indirect material parameters severely limits its applicability to real-world geotechnical problems. In many cases, the numerical analysis of geotechnical problems relies on simple elastoplastic constitutive models that cannot model a wide range of soil response aspects. One promising paradigm of constitutive modelling in geotechnics is hypoplasticity, but many of the hypoplastic models belong to advanced models. In the article, we present the simple hypoplastic model as an alternative to the widely used Mohr-Coulomb elastoplastic model.

Contribution estimation of singular points in solving problems of radiation of HF VHF wire antennas arbitrarily located above the earth

The radiation solving the problem features of the wire antennas placed above the earth are considered. Based on the method of integral equations, the singular points contribution in solving radiation problems of the HF VHF wire antennas arbitrarily located above the earth is researched. The contribution of singular points is estimated depending on the frequency and parameters of the earth. It is shown that in the HF and VHF waves, their contribution of singular points is significant. The radiation solving the problem features of the wire antennas placed above the earth are considered. The value of elements of the Green’s tensor function is usually expressed in the Sommerfeld’s integrals form. The aim of this work is to estimate the contribution of singular points to the value of Sommerfeld’s integrals. Based on the method of integral equations, the singular points contribution in solving radiation problems of the HF VHF wire antennas arbitrarily located above the earth is researched. The contribution of singular points is estimated depending on the frequency and parameters of the earth. It is shown that in the HF and VHF waves, their contribution of singular points is significant.

Non-Linear Constitutive Equations for Transversely Isotropic Materials Belonging to the С∞ and С∞h Symmetry Groups

Transversely isotropic materials feature infinite-order symmetry axes. Depending on which other symmetry elements are found in the material structure, five symmetry groups may be distinguished among transversely isotropic materials. We consider constitutive equations for these materials. These equations connect two symmetric second-order tensors. Two types of constitutive equations describe the properties of these five material groups. We derived constitutive equations for materials belonging to the C∞ and C∞h symmetry groups in the tensor function form. To do this, we used corollaries of Curie's Symmetry Principle. This makes it possible to obtain a fully irreducible form of the tensor function.

On the existence of elastic minimizers for initially stressed materials

A soft solid is said to be initially stressed if it is subjected to a state of internal stress in its unloaded reference configuration. In physical terms, its stored elastic energy may not vanish in the absence of an elastic deformation, being also dependent on the spatial distribution of the underlying material inhomogeneities. Developing a sound mathematical framework to model initially stressed solids in nonlinear elasticity is key for many applications in engineering and biology. This work investigates the links between the existence of elastic minimizers and the constitutive restrictions for initially stressed materials subjected to finite deformations. In particular, we consider a subclass of constitutive responses in which the strain energy density is taken as a scalar-valued function of both the deformation gradient and the initial stress tensor. The main advantage of this approach is that the initial stress tensor belongs to the group of divergence-free symmetric tensors satisfying the boundary conditions in any given reference configuration. However, it is still unclear which physical restrictions must be imposed for the well-posedness of this elastic problem. Assuming that the constitutive response depends on the choice of the reference configuration only through the initial stress tensor, under given conditions we prove the local existence of a relaxed state given by an implicit tensor function of the initial stress distribution. This tensor function is generally not unique, and can be transformed according to the symmetry group of the material at fixed initial stresses. These results allow one to extend Ball's existence theorem of elastic minimizers for the proposed constitutive choice of initially stressed materials. This article is part of the theme issue ‘Rivlin's legacy in continuum mechanics and applied mathematics’.

Discrete estimators of covariance components of vectorial periodically nonstationary random processes

The properties of estimators for invariants of covariance tensor-function of vectorial periodically correlated random processes, calculated on the base of discrete data, are analyzed. It is shown that aliasing effect of the first kind leads to incorrect estimation of the mean function Fourier coefficients and the second kind leads to decreasing a convergence of covariance components estimator. The conditions of avoidance of the aliasing effect of the first and the second kinds are obtained. Formulas for the estimator variance and bias, which allow comparing efficiency of the discrete and the continuous estimators, are derived. The consistency of estimators is proved. Dependences of the estimators variances and biases on realization length and signal parameters are found.

Thermal Elastic Constitutive Equation of Orthotropic Materials

In the finite deformation range, the numbers of orthotropic 2n order elastic constants are studied on the basis of tensor function and of its representation theorem. On the basis of elastic constant research, the elastic orthotropic constitutive equation is derived by using the tensor method. Based on orthotropic elastic constitutive equations an in-depth study on the constitutive theory of orthotropic nonlinear thermal elasticity is carried out, and by considering the deformation produced by the coupling of temperature and load, nonlinear orthotropic thermoelastic constitutive equation is further derived with representation of the tensor invariant and scalar invariant. The constitutive equations could be used very convenient to the application in reality.

Dunford-Taylor Integral and the Isotropic Tensor Valued Functions Having the Commutative Property with their Tensor Argument

Isotropic tensor valued functions of tensor arguments play an important role in the formulation of the equations governing the behavior of solid materials in the field of continuum mechanics. When the tensor argument is non-symmetric, the complexity and the difficulty in dealing with the tensor functions are high. In this work, the issue of expressing an isotropic tensor valued tensor function of a non-symmetric tensor argument is attention by utilizing the Dunford-Taylor integral. An important subclass of the isotropic tensor functions is considered with the commutative property.