Nonlinear Elastic Constitutive Equation Research Articles

Variational inequalities of multilayer elastic contact systems with interlayer friction: Existence and uniqueness of solution and convergence of numerical solution

Inspired by the layered structure models in pavement mechanics research, in this study, a class of multilayer elastic contact systems with interlayer frictional contact conditions and deformable supporting frictional contact conditions on the foundation has been constructed. Based on the nonlinear elastic constitutive equations, the corresponding system of partial differential equations and variational inequalities are respectively introduced. Under the framework of variational inequalities, the existence and uniqueness of solutions for such models, along with the approximation properties of finite element numerical solutions, are proven and analyzed. The aforementioned conclusions provide fundamental and broadly applicable theoretical support for addressing mechanical problems in multilayer elastic contact systems within the framework of variational inequalities. Finally, the numerical experimental results based on the mixed finite element method also substantiate our theoretical conclusions.

Advances on kinematically exact rod models for thin-walled open-section members: Consistent warping function and nonlinear constitutive equation

This work proposes two advances on kinematically exact rod models for thin-walled open section members: (i) an explicit analytical expression for the cross-section warping function, including both primary and secondary warping for arbitrary section geometries, and (ii) a nonlinear elastic constitutive equation with all higher-order strain terms included, suitable for truly finite strains. By not neglecting any of the strain terms at the constitutive level, full bending, compression and torsional strain couplings are enabled. This improves the model performance in capturing buckling phenomena and describing the developed post-critical configurations. Since the stress-resultants end up with elaborate expressions, numerical rather than analytical integration over the cross-section is preferred for their computation, which explains the need for knowing a priori the warping function (and its directional derivatives) in the entire cross-section, including in the thickness direction. The advances are implemented in a 7-DOF (3 displacements, 3 rotations and one warping intensity) rod model available in an in-house finite element program. Validation is performed using existing results from the literature as well as solutions obtained with shell models from a commercial software. The proposed advances proved to fix pathological behavior of previous rod models in both torsional and lateral–torsional buckling problems, as well as numerical convergence issues in developed post-critical configurations.

Failure Analysis of Cement Sheath Mechanical Integrity Based on the Statistical Damage Variable.

Maintaining the cement sheath mechanical integrity is the key to ensuring the benefit and safety of oil and gas well drilling and production. The main function of the cement sheath is to isolate oil and gas from water layers, which prevents the formation fluid from channeling to other layers. At present, how to effectively evaluate the cement sheath sealing performance is the fundamental problem to be solved urgently. This paper first carried out the cement sheath annulus isolation experiment and analyzed the main forms of cement sheath seal failure. Then, the interaction of the cement sheath, casing, and surrounding rock in the initial state and subsequent wellbore operations was explained. An analysis model for the cement sheath mechanical integrity incorporating the nonlinear elastic constitutive equation was proposed. Based on the statistical damage variable in continuum mechanics theory, a damage variable was established. The results show that the main form of cement sheath integrity failure is tensile crack damage and micro-annulus caused by plastic yielding. The damage variable can quantitatively describe the cement sheath mechanical integrity. The field case analysis shows that the damage variable is highly correlated with wellbore pressure and also verifies the applicability of the variable. Reducing wellbore pressure will help maintain the mechanical integrity of the cement sheath, providing sealing performance. This research can provide a reference for designing the mechanical properties of the cement sheath and improving the sealing ability of the cement sheath.

Buoyancy-driven mixed convection flow of FENE-P fluids over a flat plate

The primary purpose of this study is to investigate the buoyancy mixed convection flow of non-Newtonian fluid over a flat plate. The addition of a small amount of polymers into a Newtonian solvent raises the viscosity and generates elastic properties in the resulting solution. To study the behavior of these viscoelastic fluids, finite extensible nonlinear elastic constitutive equations along with Peterlin’s closure (FENE-P model) are used. Along with mass, momentum and energy equations, viscoelastic constitutive equations are also used to examine the rheology of the resulting polymer solution. Similarity transformations are introduced to convert the governing equations into nondimensional forms. The nondimensional equations are solved using the fourth-order boundary value solver in MATLAB. The distribution of the velocity and temperature fields is displayed graphically under the impact of various involved parameters like Eckert number (Ec), Richardson number (Ri), Prandtl number (Pr). The addition of polymers increases the friction among the different fluid layers, leading to viscous dissipation in the fluid. The presented model’s validation is done with the Newtonian fluid to verify the results. The Nusselt number is also computed and analyzed to study the heat transfer rate. The effects of viscoelastic parameters like Weissenberg number (W[Formula: see text]), polymer viscosity ratio ([Formula: see text]) and polymer extensibility parameter ([Formula: see text]) on heat transfer rate are also shown graphically. Buoyancy parameter (Richardson number, [Formula: see text]) represents the dominance of natural convection relative to that of forced convection. The temperature of the resulting fluid falls with the increase in the value of Ri. The Nusselt number tends to decrease with increasing Richardson number when viscous dissipation effects are active.

Thermal boundary layer of laminar flow of dilute polymer solution

The thermal boundary layer flow is a canonical flow with characteristics that are present in most natural and industrial convection flows. An approximate self-similar solution is proposed for the first time for the thermal boundary layer of steady laminar flow of viscoelastic fluids, described by the finitely extensible nonlinear elastic constitutive equation with Peterlin's closure (FENE-P model). This semi-analytical thermal solution is obtained by performing an order of magnitude analysis and ensuing simplifications of the governing equations by assuming that the fluid properties are independent of temperature therefore decoupling the flow governing equations from the energy equation. The effects of viscoelasticity quantified with the Weissenberg number based on the streamwise coordinate (x) (Wix) up to Wix=1 and viscous dissipation (results are presented for Brinkman numbers between -40 and +40) on thermal boundary layer characteristics are investigated comprehensively for both constant wall temperature and constant wall heat flux. At low elasticity levels (Wix<0.01) the solution exhibits a global self-similar behavior in which flow and thermal quantities collapse on the corresponding Newtonian curves, and the polymer characteristics show a unique behavior if adequately normalized. However, by increasing flow elasticity the unique self-similar behavior of the approximate solution is lost, with the elasticity dependent results exhibiting local variations. In addition, the effects of elasticity are intensified by viscous dissipation. For the present study cases, it is observed that elasticity may change Nusselt numbers by more than 8%, and the thermal boundary layer thickens by up to 10%.

Impact of high-temperature-water cooling damage on the mechanical properties of concrete

Abstract A series of impact tests were conducted on split Hopkinson pressure bar equipment to study the influence of high-temperature-cooling damage on the mechanical properties of concrete. The experimental investigation showed the influence on the peak stress, peak strain, and fracture mode. The results indicate that the strain rate is directly related to the loading rate and heating temperature. The test results of natural cooling and water immersion cooling treatment were compared, which showed that the strength of water-cooled specimens is higher than that of naturally cooled specimens before the heating temperature reaches 400 °C and becomes much lower when the heating temperature is higher than 400 °C. Based on the experimental study, a dynamic non-linear elastic constitutive equation was established using a particle swarm optimization algorithm and an inversion analysis method. These results could provide theoretical references for the design of the ultimate strength of concrete materials for practical applications, such as fire and explosion prevention.

Thermo-elasto-plastic analysis of geosynthetic clay liners exposed to thermal dehydration

Geosynthetic clay liners (GCLs) are designed to minimise contaminant migration into subsoils in landfills and other waste containment systems. The so-called mixture theory has typically been chosen as a basis for numerical models describing the complicated behaviour of GCLs subjected to thermo-hydro-mechanical changes. Despite progress in this area in the past decade, some aspects of the response of GCLs have not been captured by these simulations. For example, a typical assumption of these analyses is that the mechanical response to thermal loads of GCLs can be described by linear or non-linear elastic constitutive equations. However, there is evidence that in many instances plastic deformation cannot be avoided. The primary aim of this study is to analyse the hydration and dehydration behaviour of GCLs within a thermo-elasto-plastic framework. First, a thermo-elasto-plastic constitutive model is introduced and numerically implemented. Next, the results of numerical simulations of the performance of GCLs are presented and discussed and the effects of deviations from elastic behaviour are identified. Finally, a parametric study is conducted to identify the influence of key parameters in the constitutive model on the horizontal stress in the GCL’s bentonite during restrained shrinkage driven by thermal dehydration.

Large-eddy simulations of forced isotropic turbulence with viscoelastic fluids described by the FENE-P model

A new subgrid-scale (SGS) model developed for large-eddy simulations (LES) of dilute polymer solutions, described by the finitely extensible nonlinear elastic constitutive equation closed with the Peterlin approximation, is presented. In this distortion similarity model (DSIM) the filtered conformation tensor evolution equation is based on the self-similarity of the polymer stretching terms, and on a global equilibrium of the trace of the conformation tensor, which is proportional to the elastic energy stored in the polymer molecules, while the SGS stresses are modelled with the classical Smagorinsky model. The DSIM closure is assessed in direct numerical simulations (DNS) of forced isotropic turbulence using classical a priori tests, and in a posteriori (LES) showing very good agreement with all the exact (filtered DNS) results. The DSIM model is simple to implement and computationally inexpensive and represents a major step forward in the numerical simulation of turbulent flows of Newtonian fluids with polymer additives.

Study on compressive mechanical capabilities of concrete after high temperature exposure and thermo-damage constitutive model

Deterioration of mechanical properties of concrete structure, caused by the high temperature of a fire, would seriously threaten people’s lives and their properties. This paper aims to study the physical and mechanical properties such as strength, deformation, and elastic modulus and the changing regularity of the failure mode of concrete, with the uniaxial compression tests on C35 concrete being separately carried out under normal temperature and high temperature (300–1200°C). A nonlinear elastic constitutive equation of concrete on thermo-damage is established on the basis of experiment. And its characteristic parameter is identified by the differential coefficient algorithm. Its reliability is testified by comparing and analyzing the calculated curve and testing curve.

Nonlinear elastic behavior of two-dimensional molybdenum disulfide

This research explores the nonlinear elastic properties of two-dimensional molybdenum disulfide. We derive a thermodynamically rigorous nonlinear elastic constitutive equation and then calculate the nonlinear elastic response of two-dimensional MoS${}_{2}$ with first-principles density functional theory (DFT) calculations. The nonlinear elastic properties are used to predict the behavior of suspended monolayer MoS${}_{2}$ subjected to a spherical indenter load at finite strains in a multiple-length-scale finite element analysis model. The model is validated experimentally by indenting suspended circular MoS${}_{2}$ membranes with an atomic force microscope. We find that the two-dimensional Young's modulus and intrinsic strength of monolayer MoS${}_{2}$ are 130 and 16.5 N/m, respectively. The results approach Griffith's predicted intrinsic strength limit of ${\ensuremath{\sigma}}_{\mathrm{int}}\ensuremath{\sim}\frac{E}{9}$, where $E$ is the Young's modulus. This study reveals the predictive power of first-principles density functional theory in the derivation of nonlinear elastic properties of two-dimensional MoS${}_{2}$. Furthermore, the study bridges three main gaps that hinder understanding of material properties: DFT to finite element analysis, experimental results to DFT, and the nanoscale to the microscale. In bridging these three gaps, the experimental results validate the DFT calculations and the multiscale constitutive model.

1040 第一原理計算に基づく非線形弾性構成式を用いた単結晶の大変形挙動の有限要素解析(OS9,OS10 合同ポスターセッション,OS・一般セッション講演)

1040 第一原理計算に基づく非線形弾性構成式を用いた単結晶の大変形挙動の有限要素解析(OS9,OS10 合同ポスターセッション,OS・一般セッション講演)

Finite element modeling for soft tissue surgery based on linear and nonlinear elasticity behavior

New surgical techniques require fine control from the surgeon's point of view. Until recently, the necessary experience was only obtainable through traditional training protocols (using cadavers, animals, etc.). However, numerous training simulators have now been developed for use in this area. We present a new approach based on a three-dimensional finite element software and on different kinds of linear and nonlinear elastic constitutive equations that is able to predict realistic results. To classify these equations in terms of accuracy, we performed ex-vivo experimental measurements on lamb kidneys. The software has been applied to soft tissue deformation, namely lamb kidney and human uterus, and the numerical results have been compared to experimental ones.

Finite-element analysis of an axisymmetric, thin-walled, nonlinear elastic pressurized torus

A finite-element methodology for the static analysis of an axisymmetric, toroidal membrane of initially circular cross section under internal uniform pressure is developed. The torus experiences large elastic deformations and its material behavior is described by a nonlinear elastic constitutive equation. Both incompressible and compressible material cases are considered. Axisymmetric finite elements are employed in the formulation, which is established with the aid of a variational principle. The resulting nonlinear stiffness equations are solved by a highly accurate and efficient least squares method. The accuracy of the finite element method is assessed. Numerical results for the deformational configuration and the developed stresses of a torus as functions of the applied internal pressure are presented in order to illustrate the method and demonstrate its accuracy by comparison with analytical results whenever this is possible.

A comparison of second-order constitutive theories for hyperelastic materials

To develop a finite elastic constitutive equation for a real material, it is necessary to experimentally determine a number of material functions. This is much more difficult than determining the material constants needed for construction of a second-order constitutive equation for the same material. So it can be advantageous to use a second-order constitutive equation if it provides an adequate description of a material’s mechanical behavior over the deformation regimes to be considered. Of course, the second-order expansion of a constitutive relation always provides a good description of the mechanical response for sufficiently small strains and rotations. But by neglecting terms in these expansions which are higher than second order, we can construct a St Venant–Kirchhoff-type material model which may be applied for any deformation. In this paper we investigate the circumstances under which such St Venant–Kirchhoff-type second-order constitutive equations can successfully be used in lieu of the fully nonlinear elastic constitutive equation to solve boundary value problems. Two nonlinear elastic materials are considered: the generalized Blatz–Ko and the harmonic materials. For these materials we examine the performance of constitutive equations which are second order in the displacement gradient, the Biot strain, and the Green strain. A variety of boundary value problems are solved with these second-order constitutive equations, and the resulting solutions are compared to the solutions obtained with the corresponding fully nonlinear constitutive equation. We find that both the nature of the material and the nature of the deformation have a large impact on the performance of these three second-order constitutive equations. Overall, the constitutive equation which is second order in the Biot strain provides the most accurate solutions over the largest range of strains and rotations.

State Surfaces of Partially Saturated Soils: An Effective Pressure Approach

State surfaces may be given by interpolation of experimental data or may represent the integration of linear or nonlinear elastic constitutive equations. Anelastic behavior may also be considered if only loading is taken into account to ensure the uniqueness of the surface. State surfaces are usually defined in the (p, q, s) space, where p, q, and s represent, respectively, the excess of mean stress over air pressure, the deviatoric stresses, and suction. An alternative approach assumes similar constitutive equations but defined in the space (p′, q), where p′ represents the effective pressure. This latter approach is extended to the space (p′, q, s) and improved to carefully deal with the role of suction, in conjunction with the confining pressure, in controling soil expansion (swelling).

Cam-Clay plasticity, Part II: Implicit integration of constitutive equation based on a nonlinear elastic stress predictor

An improved stress integration algorithm for the modified Cam-Clay plasticity model capable of accounting for nonlinear elasticity is proposed. The algorithm entails the use of secant elastic moduli for integrating the nonlinear elastic constitutive equation in conjunction with a fully implicit scheme for calculating plastic strains. Numerical experiments demonstrate the algorithm's good accuracy and stability under a wide variety of imposed stress and strain paths. This algorithm should be useful in situations dominated by combined nonlinear elastic unloading and plasticity such as would occur in tunneling and excavations.

Inflation Characteristics of Unvulcanized Gum and Compounded Rubber Sheets

Abstract The inflation of rubber compound membranes to fill molds has a long history in rubber processing. There have, however, been few basic experimental studies of this topic. This free inflation of vulcanized rubber sheets was studied by Treloar and Rivlin and Saunders during the 1940's and 50's where it was used to investigate the form of the stress-strain relationship. Rivlin and Saunders sought to fit nonlinear elastic constitutive equations to data obtained at small inflation levels. In the early 1970's Denson and his coworkers utilized the inflation of unvulcanized polyisobutylene membranes to apply this as a procedure to evaluate biaxial and planar extensional viscosities. Similar studies were made by Maerker and Schowalter. These studies all involve free inflation of membranes. These studies of inflation of raw gum polyisobutylene sheets only looked at relatively small deformation levels where a hemispherical shape or less was achieved. Failure mechanisms in similarly inflated vulcanized SBR 1502 rubber sheets were described by Dickie and Smith. These authors did not describe the shapes of the inflated sheets. There seem to be no studies of inflation of rubber membranes into molds or useful equivalent studies of molten plastics such as thermoforming. It is the purpose of this paper to present a basic study of the inflation of membranes of various gum and compounded elastomers. We observe the manner in which the membranes freely inflate into air and also the manner in which they fill molds of various shapes. We begin with qualitative considerations but will also present quantitative measurements. Biaxial elongational viscosities have been estimated.

The Effects of Molecular Weight on the Single Lap Shear Creep and Constant Strain Rate Behavior of Thermoplastic Polyimidesulfone Adhesive

The bonded shear creep and constant strain rate behaviors of zero, one, and three percent endcapped thermoplastic polyimidesulfone adhesive were examined at room and elevated temperatures. Endcapping was accomplished by the addition of phthalic anhydrides. The primary objective was to determine the effects of molecular weight on the mechanical properties of the adhesive. Viscoelastic and nonlinear elastic constitutive equations were utilized to model the adhesive. Ludwik's and Crochet's relations were used to describe the experimental failure data. The effects of molecular weight changes on the above mentioned mechanical behavior were assessed. The viscoelastic Chase-Goldsmith and elastic nonlinear relations gave a good fit to the experimental stress strain behavior. Crochet's relations based on Maxwell and Chase-Goldsmith models were fit to delayed failure data. Ludwik's equations revealed negligible rate dependence. Ultimate stress levels and the safe levels for creep stresses were found to decrease as molecular weight was reduced.

Material characterization of structural adhesives in the lap shear mode

A general method for characterizing structural adhesives in the bonded lap shear mode is proposed. Two approaches — semi-empirical and theoretical — are assessed. The semi-empirical approach includes Ludwik's and Zhurkov's equations to describe, respectively, the failure stresses in the constant strain-rate and constant stress loading modes with the inclusion of temperature effects. The theoretical approach is used to describe adhesive shear stress/strain behaviour by utilizing viscoelastic or non-linear elastic constitutive equations. On the basis of this approach an empirical extension of Crochet's delayed failure equation is proposed to account for the effects of high temperature.

Non-linear elastic constitutive equations

The non-linear elastic constitutive equations proposed by Kauderer, [1], are a particular case of a class of quasi-linear equations. They cannot give an account of some specific non-linear effects of the Poynting and the Kelvin types. It is more useful in two and three dimensional problems to start with general non-linear forms proposed initially by Brillouin, [2].