Mechanical Constitutive Equations Research Articles

Deriving fractional acoustic wave equations from mechanical and thermal constitutive equations

It is argued that fractional acoustic wave equations come in two kinds. The first kind is constructed ad hoc to have loss operators that fit power law measurements. The second kind is more fundamental as they in addition are based on underlying physical equations. Here that means constitutive equations. These equations are the fractional Kelvin–Voigt and the more general fractional Zener stress–strain relationships as well as a fractional version of the Fourier heat law. The properties of the wave equations are given in terms of attenuation, and phase/group velocities for low-, intermediate- and high-frequency regions. In the most general case, the attenuation exhibits power law behavior in all frequency ranges while the phase and group velocities increase sharply in the intermediate frequency range and converge to a constant, finite value for high frequencies. It is also shown that the fractional Zener wave equation is equivalent to the multiple relaxation model for attenuation.

Multiscale measurements of residual strains in a stabilized zirconia layer

Residual stresses in a polycrystalline material have been determined experimentally at different length scales using three different techniques, with the aim of obtaining quantitative values. The polycrystalline material used is the electrolyte of solid oxide fuel cells, made of yttria-stabilized zirconia and submitted to a high biaxial compression stress state. Macroscopic measurements were performed using traditional X-ray diffraction with the sin2ψ method. Residual stresses within the grains were determined by the X-ray microdiffraction technique using synchrotron radiation. The variation in the strain within each grain was analysed by high-resolution electron backscatter diffraction. The results are self-consistent and give further information on the relation between strain/stress values and grain orientation, and on intragranular strain variations. These results are very important for the validation of mechanical microscopic constitutive equations.

True intrinsic mechanical behaviour of semi-crystalline and amorphous polymers: Influences of volume deformation and cavities shape

A model enabling the determination of the intrinsic mechanical constitutive equations of uniaxially stretched polymers is presented. This model takes into account the cavitation-induced volume strain which can occur during the deformation of such materials. In particular, the true intrinsic axial stress and strain depends on the overall volume strain and a form factor depicting the evolution of the voids shape. Based on our model, the true intrinsic behaviour of high-density polyethylene (HDPE), polypropylene/ethylene-propylene rubber (PP/EPR), and polyethylene terephtalate (PET) was assessed in tension. Compared to the overall true behaviour, the intrinsic true behaviour of the materials did not exhibit anomalies at large strain levels with changing experimental parameters (strain rate and temperature), and can be accurately predicted by means of phenomenological constitutive equations as the one proposed by G’sell and Jonas (1979).

Multidimensional a priori hyper-reduction of mechanical models involving internal variables

Modeling large systems usually requires metamodels or response surfaces (RSs) of sub-systems. These metamodels are using sampling points in a parameter domain and related responses provided by the solution of parametric partial differential equations (PDEs). Between sampling points the responses are interpolated. We propose to incorporate the RS approximation into the weak form of parametric partial differential equations (PDEs). Hence a multidimensional model-reduction can be achieved. RSs provide very fast predictions. But their enrichment, by adding new sampling points, requires new response evaluations, and therefore new solutions of PDEs. Although it is off-line computations, their complexity does not facilitates the study of large-scale non-linear problems involving a large number of parameters. Reduced-order models can facilitate the solution of the PDEs used to enrich the RSs. We propose a multidimensional a priori model-reduction method to generate or to enrich RSs. It is coined multidimensional because the fields to forecast are defined over an augmented domain in terms of dimension. They are functions of both space variables and parameters that simultaneously evolve in time. This changes the functional space related to the weak form of the PDEs and the definition of the reduced-bases. It has a significant impact on the proposed model-reduction method. In particular, the variable interpolation in the framework of reduced-basis approximations has to be reconsidered. Moreover, a multidimensional reduced integration domain (MRID) is proposed to reduce the complexity of the reduced formulation. It is a subdomain of the full multidimensional domain. The multidimensional hyper-reduction method extracts from the MRID truncated equilibrium equations, truncated residuals and a truncated error indicator. This work is an extension of the a priori hyper-reduction (APHR) method to parametric PDEs coupled to a design of experiments (DOE) method. In this paper, the outputs of the metamodels are the elastic stiffness and the damping coefficient of non-linear mechanical sub-systems that could be incorporated in the model of aircrafts or cars subjected to vibrations. The proposed method has been designed to account for various recent and ongoing research on mathematical formulation of mechanical constitutive equations in material science. Here, we are using a constitutive model proposed by Qi and Boyce for polymers.The three following issues are addressed: Is the multidimensional APHR method more efficient than the APHR method applied individually on separated simulations? What is the efficiency of the proposed approach when adding sampling points in the current parameter domain? What is the efficiency of the method when adding a new dimension to the parameter domain, i.e. when adding a new parameter to enrich a former parametric study?

Analysis of thermo-hydrologic-mechanical impact of repository for high-level radioactive waste in clay host formation: an Indian reference disposal system

In our study, a coupled hydrologic-mechanical analysis is done of the excavation damaged zone, before the emplacement of nuclear waste. This is followed by a coupled thermal-hydrologic-mechanical analysis to evaluate the impact of nuclear waste repository in porous water containing rock mass. This analysis has been under taken in accordance with the Indian reference disposal system. The paper considers the changes in pore pressure and stresses due to excavation of disposal holes. A critical study of rock mass permeability, porosity, thermally induced stresses, strains, and temperature distribution after the emplacement of nuclear spent fuels in disposal holes has also been done. Coupling to the mechanical constitutive equations is done via “effective” normal strain rates.

Measuring Mechanical Behavior of Steel During Solidification: Modeling the SSCC Test

The submerged split chill contraction (SSCC) test can measure forces in a solidifying steel shell under controlled conditions that match those of commercial casting processes. A computational model of this test is developed and applied to increase understanding of the thermal-mechanical behavior during the initial solidification of steel. Determining the stress profile is difficult because of the complicated geometry of the experimental apparatus and the nonuniform temperature and strength across the shell. The two-dimensional axisymmetric elastic-viscoplastic finite-element model of the SSCC test features different mechanical properties and constitutive equations for delta-ferrite and austenite that are functions of both the temperature and the strain rate. The model successfully matches measurements of (1) temperature history, (2) shell thickness, (3) solidification force, and (4) failure location. In addition, the model reveals the stress and strain profiles through the shell and explains what the experiment is actually measuring. In addition to the strength of the shell, the measured force is governed by the strength of the junction between the upper and lower test pieces and depends on friction at the shell–cylinder interface. The SSCC test and the validated model together are a powerful analysis tool for mechanical behavior, hot tear crack formation, and other phenomena in solidification processes such as continuous casting.

A Hydromechanical Approach to Model Shrinkage of Air‐Dried Green Bodies

This paper presents a simple hydromechanical approach to model drying and shrinkage behavior of clay green bodies. Heat transfer and water vapor flow are neglected but the effect of air velocity, temperature, and relative humidity on evaporation rate can be taken into account through a boundary coefficient. Nonlinear hydraulic and mechanical constitutive functions are introduced and derived from experimental tests at the “elementary” volume scale. A state‐of‐the‐art tensiometer capable of measuring the tensile stress of water up to 2000 kPa was used to relate changes in volume and water content to the capillary suction, which is generated in the clay by the evaporation process. The proposed approach was validated against free dessication tests involving air‐drying of clay bars subjected to isotropic volume change and one‐dimensional water flow. Two different methods were used to solve the water flow equation. In the first method, nonlinear hydraulic and mechanical constitutive equations and nonlinear boundary conditions were implemented and the flow equation was solved numerically. The second method consisted in linearizing the flow equation and solving it using an analytical solution. The interest for this second method lies on the reduced number of clay parameters required, which can be obtained from simple routine tests.

Seismoelectric response of heavy oil reservoirs: theory and numerical modelling

We consider a linear poroelastic material filled with a linear viscoelastic solvent like wet heavy oils (oil as opposed to water or brines is wetting the surfaces of the pores). We extend the electrokinetic theory in the frequency domain accounting for the relaxation effects associated with resonance of the viscoelastic fluid. The fluid is described by a generalized Maxwell rheology with a distribution of relaxation times given by a Cole–Cole distribution. We use the assumption that the charges of the diffuse layer are uniformly distributed in the pore space (Donnan model). The macroscopic constitutive equations of transport for the seepage velocity and the current density have the form of coupled Darcy and Ohm equations with frequency-dependent material properties. These equations are combined with an extended Frenkel–Biot model describing the deformation of the poroelastic material filled with the viscoelastic fluid. In the mechanical constitutive equations, the effective shear modulus is frequency dependent. An amplification of the seismoelectric conversion is expected in the frequency band where resonance of the generalized Maxwell fluid occurs. The seismic and seismoelectric equations are modelled using a finite element code with PML boundary conditions. We found that the DC-value of the streaming potential coupling coefficient is also very high. These results have applications regarding the development of new non-intrusive methods to characterize shallow heavy oil reservoirs in tar sands and DNAPL contaminant plumes in shallow aquifers.

Soil–environment interactions in geotechnical engineering

The range of problems that geotechnical engineers must face is increasing in complexity and scope. Often, complexity arises from the interaction between the soil and the environment – the topic of this lecture. To deal with this type of problem, the classical soil mechanics formulation is progressively generalised in order to incorporate the effects of new phenomena and new variables on soil behaviour. Recent advances in unsaturated soil mechanics are presented first: it is shown that they provide a consistent framework for understanding the engineering behaviour of unsaturated soils, and the effects of suction and moisture changes. Building on those developments, soil behaviour is further explored by considering thermal effects for two opposite cases: high temperatures, associated with the problem of storage and disposal of high-level radioactive waste; and low temperatures in problems of freezing ground. Finally, the lecture examines some issues related to chemical effects on soils and rocks, focusing in part on the subject of tunnelling in sulphate-bearing rocks. In each case new environmental variables are identified, enhanced theoretical formulations are established, and new or extended constitutive laws are presented. Particular emphasis is placed on mechanical constitutive equations, as they are especially important in geotechnical engineering. The lecture includes summary accounts of a number of case histories that illustrate the relevance and implications of the developments described for geotechnical engineering practice.

Modeling of SMA superelastic behavior with nonlocal approach

Due to their thermomechanical properties (high mechanical work/volume ratio), shape memory alloys (SMA) are particularly interesting to be adopted in the design of micro-sensors and micro-actuators. Thus, various constitutive models have been developed and implemented in finite element codes in order to design such applications. If these ‘local’ models are well adapted to describe the behavior of the bulk material, they fail to satisfactorily describe phenomena such as transformation localization or size effects observed in small samples.A gradient constitutive model is presented in order to describe the localization of phase transformation in SMA structures. To achieve this development restricted to superelasticity, a non local variable (martensite volume fraction) is defined at a material point as the weighted average over the entire material domain of the local transformation variable. Using Taylor expansions, the non local definition is substituted by a gradient based equation, introducing a material length parameter which controls the size of the localization zone.The gradient and mechanical equilibrium constitutive equations have been numerically integrated by using the finite element method. Two kinds of finite elements have been developed (1D truss and 2D quadrilateral) and implemented in Abaqus®via a UEL subroutine. Several simulations have been performed which exhibit the localization phenomena of phase transformation in structures undergoing mechanical loading.

Statics of hybrid structures composed of interacting membranes and strings

The equations governing large deformations of hybrid structures are derived from basic principles of mechanics. The considered structures comprise membrane elements with strings arranged inside and possibly along boundaries of the entire structure. No special assumptions are imposed on the mechanical properties of membrane and string elements. The principle of virtual work is formulated under weak continuity hypotheses so that the derived theory takes into account general types of membrane-string attachments. The theory of mechanical constitutive equations for the considered structures is developed and the form of elastic constitutive equations is discussed. A number of specific aspects of string-membrane interactions are analyzed and possible applications of the developed theory are indicated.

Role of adsorption and swelling on the dynamics of gas injection in coal

An experimental technique is presented to perform gas injection experiments on coal cores confined by an external hydrostatic pressure, which makes use of the so‐called transient step method. The experiments are intended to improve the knowledge on the different mechanisms acting during CO2 storage in coal seams, in particular, those related to permeability. Helium, nitrogen, and carbon dioxide have been injected at pressure ranging from 10 to 80 bars and at confining pressures varying between 60 and 140 bars. The experiments with helium have been used to study the mechanical compliance of the coal core, whereas those with the adsorbing N2 and CO2 to study the effects of adsorption and swelling on the flow dynamics. The obtained experimental transient steps were successfully described using a mathematical model, consisting of mass balances accounting for gas flow and adsorption, and mechanical constitutive equations for the description of porosity and permeability changes during injection. A semiempirical relationship between permeability and operating pressures is validated, and the corresponding parameters have been evaluated. Results showed increase in permeability with decreasing effective pressure on the sample and, when an adsorbing gas was injected, a reduction in permeability caused by swelling, with CO2 having a stronger effect compared to N2.

CO2 storage through ECBM recovery: An experimental and modeling study

The permeability of the coal seam is the main petrophysical property controlling the performance of the ECBM operation, since it affects both the CO2 injection and CH4 recovery. In the present paper, coal swelling of intact coal samples is studied both under unconstrained and constrained conditions. Unconstrained swelling experiments are performed in a view cell under a static high pressure gas atmosphere, whereas gas injection experiments are carried out in a flow cell, where the sample is subjected to a given hydrostatic confinement. Both experiments are performed by using different gases, namely He, CO2, CH4 and N2, and under typical coal seam conditions, i.e. at high pressure and at 45 ∘C. The results of the unconstrained coal sample showed that swelling increases monotonically with pressure up to a few percents for adsorbing gases, with CO2 swelling coal more than CH4 that swells more than N2, whereas for helium, a non-adsorbing gas, volume changes are negligible. The results of the flow experiments were successfully described using a mathematical model consisting of mass balances accounting for gas flow and adsorption, and mechanical constitutive equations for the description of porosity and permeability changes during injection. Results showed increase in permeability with decreasing effective pressure on the sample. Moreover, when CO2 is used a permeability reduction was observed compared to Helium, which can be explained by taking into account the effects of swelling on the flow dynamics.

연속체 손상역학을 이용한 수치 피로시험 기법

Once assessment of material failure characteristics is captured precisely in a unified way, it can bedirectly incorporated into the structural failure assessment under various loading environments, based on the theoretical backgrounds so called Local Approach to Fracture. The aim of this study is to develop a numerical fatigue test method by continuum damage mechanics applicable for the assessment of structural integrity throughout crack initiation and structural failure based on the Local Approach to Fracture. The generalized elasto-visco-plastic constitutive equation, which can consider the internal damage evolution behavior, is developed and employed in the 3-D FEA code in order to numerically evaluate the material and/or structural responses. Explicit information of the relationships between the mechanical properties and material constants, which are required for the mechanical constitutive and damage evolution equations for each material, are implemented in numerical fatigue test method. The material constants selected from constitutive equations are used directly in the failure assessment of material and/or structures. The performance of the developed system has been evaluated with assessing the S-N diagram of stainless steel materials.

Development of Damage Assessment Method for Steel Structures Using Material-Structure Coupling Analysis

Once assessment of material failure characteristics is captured precisely in a unified way, it can be directly incorporated to the structural failure assessment under various loading environments, based on the theoretical backgrounds so called Local Approach to Fracture. The aim of this study is the development of an expert system applicable for the assessment of structural integrity throughout crack initiation and structural failure based on the Local Approach to Fracture. The generalized elasto-visco-plastic constitutive equation, which can consider the internal damage evolution behavior, was developed and employed in the 3-D FEA code in order to numerically evaluate the material and/or structural responses. Explicit information of the relationships between the mechanical properties and material constants, which are required for the mechanical constitutive and damage evolution equations for each material, was implemented in an automatic system using genetic algorithm based on an inference system. The material constants selected from genetic search and constitutive equations are used directly in the failure assessment of material and/or structures. The performance of the developed system has been evaluated for the S-N relationship assessment of several materials as well as the crack initiation assessment of various weldments in steel structures.

Ballistic Penetration of Conically Cylindrical Steel Projectile into Plain-woven Fabric Target – A Finite Element Simulation

This paper gives a finite element model to simulate the entire process of multilayered fabric target perforated by conically cylindrical steel projectile on the basis of the description of the actual structure of fabric considering crimps of warp and weft yarns. Commercially available finite element code Ls-Dyna is incorporated with the Weibull constitutive equations of filament yarns at high strain rate to simulate the ballistic impact response. The projectile and yarns in the fabric are meshed with a eight-node hexahedron element (Solid 164 in Ls-Dyna). From the comparisons of the residual velocities of the projectile, and deformation and damages of the fabric target after ballistic perforation between the experimental results and theoretical ones, it is proven that at the yarn-structural hierarchy, the finite element method with explicit algorithm and code Ls-Dyna could precisely simulate the ballistic impact between the steel projectile and the fabric target. It is also found from calculations that only when the mechanical properties and constitutive equations at a high strain rate are adopted in the finite element model, can the precision of simulation be improved. An interesting and nonintuitive result is that the existence of fabric interspaces between the interweaved yarns could not decrease the kinetic energy absorptions of the fabric target perforated by the projectile under the ballistic impact. The original ideas of this paper are the considerations of the actual structure of multilayered fabric target and the constitutive equations of yarns at a high strain rate in the finite element model. Compared with other papers in this field, the FEM model proposed in this paper can simulate the interaction between the projectile and the multilayered fabric target more precisely in fabric target deformation, kinetic energy absorption, and strain wave distribution. This model can also be extended to other fabric structures and materials under the ballistic impact.

Magnetoelastic behavior of ferromagnetic materials using stress dependent Preisach model based on continuum theory

The magnetic and the mechanical constitutive equations are proposed on the basis of the continuum theory of magnetoelastic materials. The rate type expression is derived from the moving Preisach model by taking into account the stress dependence as the magnetic constitutive equation. From the symmetry of the constitutive equations, it is shown that the stress magnetization effect can be derived from the stress-dependent magnetization and magnetostriction curves. It is experimentally confirmed that the constitutive equations can describe typical magnetoelastic behaviors of a ferromagnetic material such as the magnetization and the magnetostriction curves under the stress and the stress magnetization curve.

Systematic errors in flow stress measurement for the hot plane strain compression test

The hot plane strain compression test has been used for the determination of stress - strain curves at hot working temperatures. These curves are used for the establishment of the high temperature mechanical constitutive equations which are an essential part of numerical modelling tools. The test is particularly useful when the working process to be simulated is essentially plane strain (for instance, hot rolling). This work describes and quantifies the systematic errors that arise during stress determinations with the test. The paper uses analytical and numerical techniques (finite element methods) to explore the errors. Specimen geometry, friction, temperature, strain rate and strain are all important and test behaviour is different for cases where die breadth is less than specimen height compared with the reverse situation. Quantitative response surfaces for errors in stress have been determined. The paper describes how these surfaces might be used for optimising test conditions and also for the correction of experimental stress values to true effective stresses.

Response and failure of a double-shear beam subjected to mass impact

Abstract Experimental results are reported for the response and failure of a notched double-shear (DS) beam made from BS970/En24 medium carbon steel alloy when subjected to a mass impact loading. Mechanical properties and constitutive equations were obtained from quasi-static, dynamic and temperature tests on the beam material. Response and failure features were recorded for impact velocities between 30 and 110 m s −1 . A failure mode transition was observed between a ductile tensile rupture failure and an adiabatic shear banding failure. This test method could be used for studying shear stress–strain relations over a broad range of strain rates and for investigating different failure modes and their transitions. Finite-element numerical simulations were conducted for the response and failure in the notch section of the DS beam. Calibration factors were obtained, which may be used to calculate the actual stress and strain at the mid-point of a notch section in terms of the nominal values. Isothermal and adiabatic conditions obtained from a dimensional analysis were examined with the aid of FE simulations. An analytical model is used to predict the central block motions of DS beams and gives good agreement with experimental results. Various ductile fracture failure criteria were examined, among which the generalised plastic work density failure criterion leads to successful predictions for the ductile tensile failure within the notch section.

Analysis of localization in brittle materials through optical techniques

This paper is devoted to the experimental analysis of strain localization in rock like specimens subjected to uniaxial compression. Two different techniques are involved in this research: stereophotogrammetry and laser speckle photography. The strain fields obtained by these techniques exhibit a homogeneous response followed by a strain localization before the peak of the load verses displacement curve. These observations allow one to choose the appropriate mechanical constitutive equations to describe the behavior of such materials.