Physical reliability of mathematical models in continuum mechanics

This paper presents ordered rate constitutive theories in Lagrangian description for compressible as well as incompressible homogeneous, isotropic thermoviscoelastic solid matter with memory in which the material derivative of order m of the deviatoric stress tensor and heat vector are functions of temperature, temperature gradient, time derivatives of the conjugate strain tensor up to any desired order n, and the material derivatives of up to order m−1 of the stress tensor. The thermoviscoelastic solids described by these theories are called ordered thermoviscoelastic solids with memory due to the fact that the constitutive theories are dependent on orders m and n of the material derivatives of the conjugate stress and strain tensors. The highest orders of the material derivative of the conjugate stress and strain tensors define the order of the thermoviscoelastic solid. The constitutive theories derived here show that the material for which these theories are applicable have fading memory. As is well known, the second law of thermodynamics must form the basis for deriving constitutive theories for all deforming matter (to ensure thermodynamic equilibrium during evolution), since the other conservation and balance laws are independent of the constitution of the matter. The entropy inequality expressed in terms of Helmholtz free energy density \({\Phi}\) does not provide a mechanism to derive a constitutive theory for the stress tensor when its argument tensors are stress and strain rates in addition to others. With the decomposition of the stress tensor into equilibrium and deviatoric stress tensors, the constitutive theory for the equilibrium stress tensor is deterministic from the entropy inequality. However, for the deviatoric stress tensor, the entropy inequality requires a set of inequalities to be satisfied but does not provide a mechanism for deriving a constitutive theory. In the present work, we utilize the theory of generators and invariants to derive rate constitutive theories for thermoviscoelastic solids with memory. This is based on axioms and principles of continuum mechanics. However, we keep in mind that these constitutive theories must satisfy the inequalities resulting from the second law of thermodynamics. The constitutive theories for heat vector q are derived: (i) strictly using conditions resulting from the entropy inequality; (ii) using the theory of generators and invariants with admissible argument tensors that are consistent with the stress tensor as well as the theories in which simplifying assumptions are employed which yield much simplified theories. It is shown that the rate theories presented here describe thermoviscoelastic solids with memory. Mechanisms of dissipation and memory are demonstrated and discussed, and the derivation of memory modulus is presented. It is shown that simplified forms of the general theories presented here result in constitutive models that may resemble currently used constitutive models but are not the same. The work presented here is not to be viewed as extension of the current constitutive models; rather, it is a general framework for rate constitutive theories for thermoviscoelastic solids with memory based on the physics and derivations that are consistent within the framework of continuum mechanics and thermodynamics. The purpose of the simplified theories presented in the paper is to illustrate possible simplest theories within the consistent framework presented here.

Using geologic structures to constrain constitutive laws not accessible in the laboratory

In the field of solid mechanics it is often necessary to use constitutive (physical) relations in the rate form, for example, for formulation of the boundary value problem in the rate form with contact conditions when a contact area is a priori unknown and changes in a deformation process.The paper considers some questions of formulating geometrically nonlinear constitutive relations of elastic material in the rate form and the relationship of these equations and the constitutive relations in the finite form.In many existing elastic and elasto-plastic models Hooke's law (written in terms of the actual configuration) is used as the constitutive relation.As a rule, the rate measure of the strain is deformation rate tensor (the symmetrical part of velocity gradient) and the stress rate measure is some objective derivative (convective or corotational) of weighted Kirchhoff stress tensor.The use of the convective derivative leads to certain difficulties, for example, analysis of evolution stress state in a deformable basis is complicated; consequently, convective derivatives are excluded from consideration.Using a stress tensor corotational derivative instead of the material derivative (derivative by time) allows satisfying the principle of material indifference (the principle of independence of constitutive relation from the choice of reference frame), however a choice of the derivative type can be implemented by many ways.An arbitrary selection of stress corotational derivative leads to undesirable effects: stress oscillations for the simple shear monotonic loading (for example, for Jaumann derivative), "not closed" stress trajectories and nonzero stress work for a closed deformation trajectory.In papers of A. Meyers, H. Xiao, O. Bruhns corotational derivative (with logarithmic spin or logspin) was proposed.The derivative of the right Hencky's strain tensor is exactly equal to deformation rate tensor.When using this derivative in the constitutive relation the described effects are missing, on the basis of this remarked logspin authors state the logspin exclusivity and recommend it only for using in rate form constitutive relations.The article shows full compliance between Hooke's law in the rate and finite form under conditions: existence of the material basis in which properties of the body (in this case -elastic) remain unchanged and the use of the same type corotational derivative for stress and strain measures.Computational experiments for illustration of the proved assertion about the equivalence of different Hooke's law forms were conducted: various

We would like to know whether year-to-year changes, or month-tomonth changes in crop yields or prospects are consistent with observed weather data. Generally, historical weather records extend back farther than records of crop yields. We wish to make use of weather data for the entire period of record even though yield data may be available for a much shorter period. This paper reports on an exploratory inverted matrix approach used in one phase of a crop-weather study. The application of multiple regression methods in the study of relationships between crop yields and weather factors is, of course, not new, but the large amount of computational labor involved has discouraged many workers and our people from attempting correlations studies on a very extensive scale. As pointed out by R. A. Fisher, the use of the inverse matrix of a set of normal equations greatly reduces the amount of computations when the same set of independent variables is used repeatedly; in addition, it serves to simplify the calculation of sampling errors of the regression coefficients. However, a large amount of computational work is still required when the various dependent variables are available for only relatively few years, and these periods vary from crop to crop because of the fact that the data or series were started at different points in time. We would like some way of utilizing all the weather and crop yield data available. Therefore, we would like to devise what might be called inverse matrix solution for a given State or area which could be used whenever the given set of weather factors were appropriate. However, the sampling errors of the regression coefficients cannot be computed using the elements of this generalized where the dependent variables are used for only a subperiod. The inverse matrix is obtained for a given set of independent variables (i.e., weather factors) for the entire period of the weather

Enjoyment Mediates Associations of the Physical and Family Environment With Adolescent Physical Activity: A Structural Equation Modeling Approach

In this chapter we tabulate the elements of the integrity basis which are invariant under the magnetic symmetry group of the electromagnetic crystalline solids. For the sake of simplicity, the electromagnetic solids considered here are assumed to have the constitutive equations $$ W = W\left( {{E_{KL}},{P_K},{M_k},\theta } \right) $$ (7.1) for the internal energy. We have similar expressions for the entropy, the electric field, the magnetic induction, the heat flux, and the stress tensor. The arguments E KL , P K , M K , and θ appearing in (7.1) are, respectively, the material description of the strain tensor (a true, symmetric, and i-tensor), the polarization (a polar and time-symmetric vector), the magnetization (an axial, time-asymmetric vector), and the temperature (a true time-symmetric scalar). The dependent quantities such as the entropy, the electric field, the magnetic induction, and the stress tensor are derived from the internal energy, W, since these materials are conservative (cf. eq. (1.40)).

Smart materials are active and multifunctional materials, which play an important part for sensor and actuator applications. These materials have the potential to transform passive structures into adaptive systems. However, a prerequisite for the design and the optimization of these materials is, that reliable models exist, which incorporate the interaction between the different combinations of thermal, electrical, magnetic, optical and mechanical effects. Polymeric electroelastic materials, so-called (EAP), own the characteristic to deform if an electric field is applied. EAP's possesses the benefit that they share the characteristic of polymers, these are lightweight, inexpensive, fracture tolerant, elastic, and the chemical and physical structure is well understood. However, the description electroactive polymer is a generic term for many kinds of different microscopic mechanisms and polymeric materials. Based on the laws of electromagnetism and elasticity, a visco-electroelastic model is developed and implemented into the finite element method (FEM). The presented three-dimensional solid element has eight nodes and trilinear interpolation functions for the displacement and the electric potential. The continuum mechanics model contains finite deformations, the time dependency and the nearly incompressible behavior of the material. To describe the possible, large time dependent deformations, a finite viscoelastic model with a split of the deformation gradient is used. Thereby the time dependent characteristic of polymeric materials is incorporated through the free energy function. The electromechanical interactions are considered by the electrostatic forces and inside the energy function.

While time-series studies have consistently provided evidence for an effect of particulate air pollution on mortality, uncertainty remains as to the extent of the life-shortening implied by those associations. In this paper, the authors estimate the association between air pollution and mortality using different timescales of variation in the air pollution time series to gain further insight into this question. The authors' method is based on a Fourier decomposition of air pollution time series into a set of independent exposure variables, each representing a different timescale. The authors then use this set of variables as predictors in a Poisson regression model to estimate a separate relative rate of mortality for each exposure timescale. The method is applied to a database containing information on daily mortality, particulate air pollution, and weather in four US cities (Pittsburgh, Pennsylvania; Minneapolis, Minnesota; Seattle, Washington; and Chicago, Illinois) from the period 1987-1994. The authors found larger relative rates of mortality associated with particulate air pollution at longer timescale variations (14 days-2 months) than at shorter timescales (1-4 days). These analyses provide additional evidence that associations between particle indexes and mortality do not imply only an advance in the timing of death by a few days for frail individuals.

In Part 1 of this paper, a machining force model was developed based on an enhanced version of Zheng’s Continuum Mechanics model that incorporates microstructural effects. Machining experiments identified Secondary Dendrite Arm Spacing (SDAS) as a significant microstructure feature of 319 aluminum in terms of machinability. A new material constitutive relationship that incorporates SDAS microstructure effects on the flow stress was proposed. In this part of the paper, disk turning tests are performed to simulate the orthogonal cutting process. The cutting forces obtained from some of these tests are used in concert with an inverse form of the continuum mechanics machining model to estimate the parameters in the material constitutive equation. The enhanced continuum mechanics orthogonal cutting model is then applied to predict cutting forces when machining Al319. Comparison of the model predicted and experimentally acquired cutting forces is demonstrated to show good agreement.

Fully implicit compositional simulator for modeling of asphaltene deposition during natural depletion

Questions and comments are provided for (1). I find Yulii Shikhmurzaev's opinion paper on the contact line problem interesting to read. It is helpful as it has allowed me to better identify the points where I disagree. First, he defines microscopic and macroscopic levels of continuum description (that may be applied to the contact line problem) as mutually exclusive. The definition is done in a way that excludes the mesoscopic models for the contact line problem used in the literature. Then he introduces his own model that in my view is also a mesoscale model. In contrast, I think that already his definition of levels of continuum description is not a good representation of reality, and I will below offer a slightly different one. If one accepts my classification, the widely employed mesoscale models become admitted again and Yulii Shikhmurzaev's model becomes one of them, and competes on a common ground. My first major concern is his classification of continuum descriptions. I do not un- derstand the way Yulii Shikhmurzaev draws the line between models in Continuum Mechanics and Statistical Physics. If statistical physics deals with "field variables describing statistical distributions of molecular properties" (1) then continuum me- chanics deals with field variables describing mean values of molecular properties (i.e., the lowest order moments of statistical distributions). Continuum mechanics is a part of statistical physics 1 . To obtain a valid description based on mean values (as in

In connection with the increase in requirements for details in studying the geological cross section, it seems natural to proceed from models of continuum mechanics (CM) in mathematical modeling of direct problems of seismic exploration. A complete system of CM equations in Lagrange variables is set forth, including equations of state in a very general form. It is shown that artificial viscosity needs to be introduced in the stage of constructing a difference scheme for more adequate calculation of elastic waves. A model calculation of diffraction of a plane wave on a thin inclusion is given. A problem is solved for which results of physical modeling are available: the direct problem of seismics for a medium with a layer of permafrost. Comparison of results obtained in the form of wave fields and common-shot-point (CSP) seismograms makes it possible to draw a positive conclusion about the applicability of this concept for solving complex direct problems of seismic exploration.

This paper proposes a hybrid peridynamic and classical continuum mechanical model for the high-temperature damage and fracture analysis of concrete structures. In this model, we introduce the thermal expansion into peridynamics and then couple it with the thermoelasticity based on the Morphing method. In addition, a thermomechanical constitutive model of peridynamic bond is presented inspired by the classic Mazars model for the quasi-brittle damage evolution of concrete structures under high-temperature conditions. The validity and effectiveness of the proposed model are verified through two-dimensional numerical examples, in which the influence of temperature on the damage behavior of concrete structures is investigated. Furthermore, the thermal effects on the fracture path of concrete structures are analyzed by numerical results.

Partial least squares (PLS) regression (a.k.a. projection on latent structures) is a recent technique that combines features from and generalizes principal component analysis (PCA) and multiple linear regression. Its goal is to predict a set of dependent variables from a set of independent variables or predictors. This prediction is achieved by extracting from the predictors a set of orthogonal factors called latent variables which have the best predictive power. These latent variables can be used to create displays akin to PCA displays. The quality of the prediction obtained from a PLS regression model is evaluated with cross‐validation techniques such as the bootstrap and jackknife. There are two main variants of PLS regression: The most common one separates the roles of dependent and independent variables; the second one—used mostly to analyze brain imaging data—gives the same roles to dependent and independent variables. Copyright © 2010 John Wiley & Sons, Inc.This article is categorized under: Statistical Models > Linear Models Algorithms and Computational Methods > Least Squares

Chapter 29 - An Empirical Analysis of Short-Biased Hedge Funds’ Risk-Adjusted Performance: A Panel Approach