Constitutive Quantity Research Articles

Extended micropolar approach within the framework of 3M theories and variations thereof

As part of his groundbreaking work on generalized continuum mechanics, Eringen proposed what he called 3M theories, namely the concept of micromorphic, microstretch, and micropolar materials modeling. The micromorphic approach provides the most general framework for a continuum with translational and (internal) rotational degrees of freedom (DOF), whilst the rotational DOFs of micromorphic and micropolar continua are subjected to more and more constraints. More recently, an “extended” micropolar theory has been presented by one of the authors: Eringen’s 3M theories were children of solid mechanics based on the concept of the indestructible material particle. Extended micropolar theory was formulated both ways for material systems as well as in spatial description, which is useful when describing fluid matter. The latter opens the possibility to model situations and materials with a continuum point that on the microscale consists no longer of the same elementary units during a physical process. The difference culminates in an equation for the microinertia tensor, which is no longer a kinematic identity. Rather it contains a new continuum field, namely an independent production term and, consequently, establishes a new constitutive quantity. This makes it possible to describe processes of structural change, which are difficult if not impossible to be captured within the material particle model. This paper compares the various theories and points out their communalities as well as their differences.

An Instrumented High-Speed Rotor With Embedded Telemetry for the Continuous Spatial Pressure Profile Measurement in Gas Lubricated Bearings: A Proof of Concept

Abstract Pressure is the constitutive quantity governing the flow field in gas lubricated bearings. Knowledge of the pressure is of principal importance in the fundamental understanding of such bearings as well as for the validation of their models. Pressure measurements can be done from the bearing side using pressure taps, yet, several details will not be captured. In order to acquire a continuous scan of the pressure field inside the bearing, it is necessary to measure from the rotor side. This paper presents an instrumented measurement high-speed rotor with embedded pressure probes and a wireless telemetry that is capable of the continuous pressure field measurement within the gas film of journal bearings. The rotor was tested on externally pressurized gas journal bearings (EPGJBs) (D = 40 mm and L/D = 1), and pressure profile measurements were acquired up to 37.5 krpm (DN 1.5 M). Measurements at discrete points using pressure taps inside the test bearing were also performed for comparison. The measurements from both sides (bearing and rotor) were in good agreement at quasi-static conditions. At higher operational speeds, it was necessary to perform an in situ system identification and calibration for the embedded pressure probes using the bearing side measurements as a reference. The in situ system identification technique was successful to reconstruct the attenuated pressure signals for a wide range of supply pressures (amplitudes) and rotor speeds (excitation frequencies). The instrumented rotor was proven qualified to perform time-resolved pressure measurements within the gas film of journal bearings up to 37.5 krpm.

A novel two-stage discrete crack method based on the screened Poisson equation and local mesh refinement

We propose an alternative crack propagation algorithm which effectively circumvents the variable transfer procedure adopted with classical mesh adaptation algorithms. The present alternative consists of two stages: a mesh-creation stage where a local damage model is employed with the objective of defining a crack-conforming mesh and a subsequent analysis stage with a localization limiter in the form of a modified screened Poisson equation which is exempt of crack path calculations. In the second stage, the crack naturally occurs within the refined region. A staggered scheme for standard equilibrium and screened Poisson equations is used in this second stage. Element subdivision is based on edge split operations using a constitutive quantity (damage). To assess the robustness and accuracy of this algorithm, we use five quasi-brittle benchmarks, all successfully solved.

Damage and fracture algorithm using the screened Poisson equation and local remeshing

We propose a crack propagation algorithm which is independent of particular constitutive laws and specific element technology. It consists of a localization limiter in the form of the screened Poisson equation with local mesh refinement. This combination allows the capturing of strain localization with good resolution, even in the absence of a sufficiently fine initial mesh. In addition, crack paths are implicitly defined from the localized region, circumventing the need for a specific direction criterion. Observed phenomena such as multiple crack growth and shielding emerge naturally from the algorithm. In contrast with alternative regularization algorithms, curved cracks are correctly represented. A staggered scheme for standard equilibrium and screened equations is used. Element subdivision is based on edge split operations using a given constitutive quantity (either damage or void fraction). To assess the robustness and accuracy of this algorithm, we use both quasi-brittle benchmarks and ductile tests.

A constitutive-based element-by-element crack propagation algorithm with local mesh refinement

In the context of multiple constitutive models, multiple finite element formulations and crack nucleation and propagation hypotheses, we propose a simple yet effective algorithm to initiate and propagate cracks in 2D models which is independent of the constitutive and element specific technology. Observed phenomena such as multiple crack growth and shielding emerge naturally, without specialized algorithms for calculating the crack growth direction. The algorithm consists of a sequence of mesh subdivision, mesh smoothing and element erosion steps. Element subdivision is based on the classical edge split operations using a given constitutive quantity (either damage or void fraction). Mesh smoothing makes use of edge contraction as function of a given constitutive quantity (such as void fraction or principal stress). To assess the robustness and accuracy of this algorithm, we use classical quasi-brittle benchmarks and ductile tests.

Scalar, vectorial, and tensorial damage parameters from the mesoscopic background

In the mesoscopic theory a distribution of different crack sizes and crack orientations is introduced. A scalar damage parameter, a second order damage tensor and a vectorial damage parameter are defined in terms of this distribution function. As an example of a constitutive quantity the free energy density is given as a function of the damage tensor. This equation is reduced in the uniaxial case to a function of the damage vector and in case of a special geometry to a function of the scalar damage parameter.

A unified evolution equation for the Cauchy stress tensor of an isotropic elasto-visco-plastic material

In the present study an evolution equation for the Cauchy stress tensor is proposed for an isotropic elasto-visco-plastic continuum. The proposed stress model takes effects of elasticity, viscosity and plasticity of the material simultaneously into account. It is ascribed with some scalar coefficient functions and, in particular, with an unspecified tensor-valued function N, which is handled as an independent constitutive quantity. It is demonstrated that by varying the values and the specific functional forms of these coefficients and N, different known models in non-Newtonian rheology can be reproduced. A thermodynamic analysis, based on the MA¼ller-Liu entropy principle, is performed. The results show that these coefficients and N are not allowed to vary arbitrarily, but should satisfy certain restrictions. Simple postulates are made to further simplify the deduced general results of the thermodynamic analysis. They yield justification and thermodynamic consistency of the existing models for a class of materials embracing thermoelasticity, hypoelasticity and in particular hypoplasticity, of which the thermodynamic foundation is established successively for the first time in literature. The study points at the wide applicability and practical usefulness of the present model in different fields from non-Newtonian fluid to solid mechanics. In this paper the thermodynamic analysis of the proposed evolution-type stress model is discussed, its applications are reported later.

A unified evolution equation for the Cauchy stress tensor of an isotropic elasto-visco-plastic material

In the present study an evolution equation for the Cauchy stress tensor is proposed for an isotropic elasto-visco-plastic continuum. The proposed stress model takes effects of elasticity, viscosity and plasticity of the material simultaneously into account. It is ascribed with some scalar coefficient functions and, in particular, with an unspecified tensor-valued function N, which is handled as an independent constitutive quantity. It is demonstrated that by varying the values and the specific functional forms of these coefficients and N, different known models in non-Newtonian rheology can be reproduced. A thermodynamic analysis, based on the Muller–Liu entropy principle, is performed. The results show that these coefficients and N are not allowed to vary arbitrarily, but should satisfy certain restrictions. Simple postulates are made to further simplify the deduced general results of the thermodynamic analysis. They yield justification and thermodynamic consistency of the existing models for a class of materials embracing thermoelasticity, hypoelasticity and in particular hypoplasticity, of which the thermodynamic foundation is established successively for the first time in literature. The study points at the wide applicability and practical usefulness of the present model in different fields from non-Newtonian fluid to solid mechanics. In this paper the thermodynamic analysis of the proposed evolution-type stress model is discussed, its applications are reported later.

Texture simulation based on tensorial Fourier coefficients

The main result of the paper is the derivation of the evolution equation of the tensorial texture coefficients of the crystallite orientation distribution function (codf). The evolution equation of each coefficient depends on the complete codf and the lattice spin, which is a constitutive quantity. For the solution of the differential equation based on a finite number of coefficients, the codf has to be estimated. This estimate is obtained here by the maximum entropy method. By this approach the texture evolution can be described by modeling some low-order Fourier coefficients. It will be shown that such a low-dimensional approach yields a reasonable description of the texture evolution and of mechanical properties like the Taylor factor.

A constitutive equation for the porosity field

This paper deals with the derivation of an evolution equation for the field here denoted by Δ n, which represents the difference between the porosity in an arbitrary state of a deformable porous medium and its equilibrium value. The equation is obtained assuming that Δ n is a constitutive quantity depending on the history of the fields describing macroscopically a porous material. Moreover, at least in the vicinity of the state of stable equilibrium, the equation obtained for Δ n can be regarded as a balance equation liable to various degrees of approximation, so including various models with additional balance laws known in the literature.

On Thermodynamic Consistency of Turbulent Closures

The problem of the implementation of the second law of thermodynamics for the determination of the thermodynamic consistency of solutions determined by turbulent closures is considered for incompressible fluids. The possibility of the application of the methods of thermodynamics to constraining constitutive laws describing turbulent flow features, but not material behaviour, is discussed. It is shown that the ordinary realizability conditions requiring non-negative values of the averaged squared fluctuations are necessary and sufficient conditions determining the thermodynamic consistency of a process governed by a closure model. Because turbulent closures are not universal, using the second law of thermodynamics to constrain them can impose unnecessary restrictions on the models, when the turbulent entropy is considered as a constitutive quantity. The notion and validity of different forms of the turbulent entropy is discussed. It is found that the form of the turbulent entropy originating from the analogy between the turbulent kinetic energy and absolute temperature contradicts the principle of irreversibility. In a particular case of small temperature fluctuations, the second law yields correct constraints, if the turbulent entropy is assumed not to be a constitutive quantity, but a variable governed by an evolution equation of special form generated by the balance equation for internal energy.

A sketch of continuum thermodynamics

Different formulations of non-equilibrium continuum thermodynamics are discussed: Thermodynamics of Irreversible Processes (TIP), Rational Thermodynamics (RAT), Extended Thermodynamics (ET), Mesoscopic Continuum Thermodynamics (MT), and the GENERIC version of thermodynamics. Concepts as constitutive quantity, state space, material frame indifference, exploiting dissipation inequality, mesoscopic variables, and GENERIC balance equations are taken into consideration.

An extended field theory of viscoelastic materials

A theory of viscoelastic materials is presented here as an extended field theory in which the viscous stress tensor is regarded as an independent field variable rather than a constitutive quantity as usual. In addition to the usual conservation laws, an extra balance law for energy tensor is postulated similar to the equation of second moment in the kinetic theory of gases. The consequences of material frame indifference, the entropy principle, and the requirement of hyperbolicity on the constitutive equations are exploited. The paper offers a field theory of solids parallel to the extended thermodynamics of gases proposed recently by Liu and Müller.