Showing papers on "Constitutive equation published in 2018"

A constitutive model for simple shear of dense frictional suspensions

Abstract: Discrete particle simulations are used to study the shear rheology of dense, stabilized, frictional particulate suspensions in a viscous liquid, toward development of a constitutive model for steady shear flows at arbitrary stress. These suspensions undergo increasingly strong continuous shear thickening (CST) as the solid volume fraction ϕ increases above a critical volume fraction, and discontinuous shear thickening (DST) is observed in a range of ϕ. When studied at controlled stress, the DST behavior is associated with nonmonotonic flow curves of the steady-state shear rate as a function of stress. Recent studies have related shear thickening to a transition from mostly lubricated to predominantly frictional contacts with the increase in stress. In this study, the behavior is simulated over wide ranges of concentration, dimensionless shear stress, and coefficient of interparticle friction. The simulation data have been used to populate the lubricated-to-frictional rheology model of Wyart and Cates [Phy...

Tensiometry and rheology of complex interfaces

Abstract: Complex fluid-fluid interfaces play an important role in a variety of application domains, from emulsion and foam stability, to thin films in biomedical applications, to coating flow phenomena. The current work reviews progress made in particular for interfaces where rheological stresses, peculiar to the interface, play an important role. The developments made in the area of constitutive modeling are briefly reviewed to clarify which material functions can be measured. For shear rheometry, progress in analyzing the flow field in the measurement device has been key, combined with advances in control over surface concentration and microstructural evaluation. For dilation/compressional rheometry much work has been done on separating changes in the surface tension from the extra rheological stresses. Finally, we discuss how “simple complex flows”, such as thin film and drainage flows, offer a first step up in complexity and seem to present a good benchmark problem for testing constitutive equations and the interplay between transport phenomena, interfacial rheology and the changes in state variables.

Continuum Mechanics: Elasticity, Plasticity, Viscoelasticity

Abstract: FUNDAMENTALS OF CONTINUUM MECHANICS Material Models Classical Space-Time Material Bodies Strain Rate of Strain Curvilinear Coordinate Systems Conservation of Mass Balance of Momentum Balance of Energy Constitutive Equations Thermodynamic Dissipation Objectivity: Invariance for Rigid Motions Coleman-Mizel Model Fluid Mechanics Problems for Chapter 1 Bibliography NONLINEAR ELASTICITY Thermoelasticity Material Symmetries Isotropic Materials Incompressible Materials Conjugate Measures of Stress and Strain Some Symmetry Groups Rate Formulations for Elastic Materials Energy Principles Geometry of Small Deformations Linear Elasticity Special Constitutive Models for Isotropic Materials Mechanical Restrictions on the Constitutive Relations Problems for Chapter 2 Bibliography LINEAR ELASTICITY Basic Equations Plane Strain Plane Stress Properties of Solutions Potential Energy Special Matrix Notation The Finite Element Method of Solution General Equations for an Assembly of Elements Finite Element Analysis for Large Deformations Problems for Chapter 3 Bibliography PLASTICITY Classical Theory of Plasticity Work Principle von Mises-Type Yield Criterion Hill Yield Criterion for Orthotropic Materials Isotropic Hardening Kinematic Hardening Combined Hardening laws General Equations of Plasticity Strain Formulation of Plasticity Finite Element Analysis Large Deformations Thermodynamics of Elastic-Plastic Materials Problems for Chapter 4 Bibliography VISCOELASTICITY Linear Viscoelasticity Effect of Temperature Nonlinear Viscoelasticity Thermodynamics of Materials with Fading Memory Problems for Chapter 5 Bibliography FRACTURE AND FATIGUE Fracture Criterion Plane Crack through a Sheet Fracture Modes Calculation of the Stress Intensity Factor Crack Growth Problems for Chapter 6 Bibliography MATHEMATICAL TOOLS FOR CONTINUUM MECHANICS Sets of Real Numbers Matrices Vector Analysis Tensors Isotropic Functions Abstract Derivatives Some Basic Mathematical Definitions and Theorems Problems for Chapter 7 Bibliography INDEX

A general constitutive model of soft elastomers

Abstract: It is a long-standing challenge to predict the general constitutive behaviors of soft elastomers under finite deformation, since most of the constitutive relations calibrated from uniaxial tension tests cannot accurately characterize the responses under complex deformation states. In this paper, we develop a general constitutive model of soft elastomers based on a new microscopic picture, in which the free energy is decomposed into two parts: one comes from the cross-linked network and the other from the entangled network. To calibrate and verify the proposed constitutive model, we test several kinds of materials including vulcanized rubber, natural rubber, Entec Enflex S4035A TPE, silicone rubber, and Tera-PEG gel. The results are compared with those from other similar constitutive models (extended tube model and nonaffine network model). With only three material parameters, our model not only captures the softening with a stress plateau and hardening with a sharp rise in stress, but also accurately characterizes the constitutive behaviors of soft rubberlike materials under various deformation states. As an example, we show that the model can predict well the inhomogeneous deformation of an inflated balloon diagram. This constitutive model possesses considerable advantages in the applications of soft robots, soft electronics, elastomeric transducers, etc.

Artificial neural networks and intelligent finite elements in non-linear structural mechanics

Abstract: In recent years, artificial neural networks were included in the prediction of deformations of structural elements, such as pipes or tensile specimens. Following this method, classical mechanical calculations were replaced by a set of matrix multiplications by means of artificial intelligence. This was also continued in finite element approaches, wherein constitutive equations were substituted by an artificial neural network (ANN). However, little is known about predicting complex non-linear structural deformations with artificial intelligence. The aim of the present study is to make ANN accessible to complicated structural deformations. Here, shock-wave loaded plates are chosen, which lead to a boundary value problem taking geometrical and physical non-linearities into account. A wide range of strain-rates and highly dynamic deformations are covered in this type of deformation. One ANN is proposed for the entire structural model and another ANN is developed for replacing viscoplastic constitutive equations, integrated into a finite element code, leading to an intelligent finite element. All calculated results are verified by experiments with a shock tube and short-time measurement techniques.

A Study on the Mechanical Behavior and Statistical Damage Constitutive Model of Sandstone

Abstract: Triaxial compression test results of sandstone indicate that the peak point strain, elastic modulus, peak deviatoric stress and residual deviatoric stress of the tested sandstone increase with increasing confining pressure, and the variations in them with the confining pressure can be described with a linear function, a logistic function, the generalized Hoek–Brown criterion and the linear Mohr–Coulomb criterion, respectively. Supposing that the rock material can be divided into an elastic part and a damaged part in the rock failure process, the deviatoric stress–strain relationship of the elastic part satisfies Hooke’s law, while the damaged part provides residual deviatoric stress. On this basis, it was assumed the rock meso-element strength follows a composite power function distribution. Then, the damage evolution equation was deduced using a statistical method, and a new damage model, which can reflect the rock residual deviatoric stress, was proposed. The reasonability of the new model was verified using the test results of the sandstone. A comparison of the predicted and test results shows that this damage model can well simulate the deviatoric stress–strain response in the failure process of the tested sandstone. In particular, it can reflect the residual deviatoric stress after rock failure.

Similarity solution for cavity expansion in thermoplastic soil

Abstract: Summary This paper presents an analytical solution for cavity expansion in thermoplastic soil considering non-isothermal conditions. The constitutive relationship of thermoplasticity is described by Laloui's advanced and unified constitutive model for environmental geomechanical thermal effect (ACMEG-T), which is based on multi-mechanism plasticity and bounding surface theory. The problem is formulated by incorporating ACMEG-T into the theoretical framework of cavity expansion, yielding a series of partial differential equations (PDEs). Subsequently, the PDEs are transformed into a system of first-order ordinary differential equations (ODEs) using a similarity solution technique. Solutions to the response parameters of cavity expansion (stress, excess pore pressure, and displacement) can then be obtained by solving the ODEs numerically using mathematical software. The results suggest that soil temperature has a significant influence on the pressure-expansion relationships and distributions of stress and excess pore pressure around the cavity wall. The proposed solution quantifies the influence of temperature on cavity expansion for the first time and provides a theoretical framework for predicting thermoplastic soil behavior around the cavity wall. The solution found in this paper can be used as a theoretical tool that can potentially be employed in geotechnical engineering problems, such as thermal cone penetration tests, and nuclear waste disposal problems.

The importance of Thermo-Hydro-Mechanical couplings and microstructure to strain localization in 3D continua with application to seismic faults. Part I: Theory and Linear Stability Analysis

Abstract: A Thermo-Hydro-Mechanical (THM) model for Cosserat continua is developed to explore the influence of frictional heating and thermal pore fluid pressurization on the strain localization phenomenon. A general framework is presented to conduct a bifurcation analysis for elasto-plastic Cosserat continua with THM couplings and predict the onset of instability. The presence of internal lengths in Cosserat continua enables to estimate the thickness of the localization zone. This is done by performing a linear stability analysis of the system and looking for the selected wavelength corresponding to the instability mode with fastest finite growth coefficient. These concepts are applied to the study of fault zones under fast shearing. For doing so, we consider a model of a sheared saturated infinite granular layer. The influence of THM couplings on the bifurcation state and the shear band width is investigated. Taking representative parameters for a centroidal fault gouge, the evolution of the thickness of the localized zone under continuous shear is studied. Furthermore, the effect of grain crushing inside the shear band is explored by varying the internal length of the constitutive law.

A micro–macro constitutive model for finite-deformation viscoelasticity of elastomers with nonlinear viscosity

Abstract: Elastomers are known to exhibit viscoelastic behavior under deformation, which is linked to the diffusion processes of the highly mobile and flexible polymer chains. Inspired by the theories of polymer dynamics, a micro–macro constitutive model is developed to study the viscoelastic behaviors and the relaxation process of elastomeric materials under large deformation, in which the material parameters all have a microscopic foundation or a microstructural justification. The proposed model incorporates the nonlinear material viscosity into the continuum finite-deformation viscoelasticity theories which represent the polymer networks of elastomers with an elastic ground network and a few viscous subnetworks. The developed modeling framework is capable of adopting most of strain energy density functions for hyperelastic materials and thermodynamics evolution laws of viscoelastic solids. The modeling capacity of the framework is outlined by comparing the simulation results with the experimental data of three commonly used elastomeric materials, namely, VHB4910, HNBR 50 and carbon black (CB) filled elastomers. The comparison shows that the stress responses and some typical behaviors of filled and unfilled elastomers can be quantitatively predicted by the model with suitable strain energy density functions. Particularly, the strain-softening effect of elastomers could be explained by the deformation-dependent (nonlinear) viscosity of the polymer chains. The presented modeling framework is expected to be useful as a modeling platform for further study on the performance of different type of elastomeric materials.

Mechanical Behavior and Damage Constitutive Model of Granite Under Coupling of Temperature and Dynamic Loading

Abstract: Dynamic compression tests of Huashan granite were conducted using an improved split–Hopkinson pressure bar. The effects of the treatment temperature and strain rate on the mechanical behaviors (for example, the stress–strain curve, dynamic strength, elastic modulus, energy absorption, and failure mode) of the granite samples were explored. In addition, a statistical damage constitutive model for the rock was developed based on a Weibull distribution, and the influencing factors of the model parameters were analyzed. The results show that the enhancement effect of the strain rate on dynamic compressive strength under high temperatures still exists. However, the strain rate has no significant effect on the elastic modulus. The influences of the treatment temperature on the dynamic strength and elastic modulus are complex. There is a positive linear correlation between the energy absorbed by the sample and the incident energy. As the strain rate or incident energy increases, the failure modes of heat-treated samples change from axial splitting to pulverization. Under the same dynamic loading, an increase in the temperature can exacerbate the fragmentation degree of the sample. The proposed statistical damage constitutive model can accurately describe the effects of the treatment temperature and strain rate on the stress–strain responses of rock, and its parameters have definite physical meanings. Thus, the model is a very good tool for the analysis of thermo-mechanical coupling problems involved in deep rock mass engineering.

Fully resolved numerical simulations of fused deposition modeling. Part II – solidification, residual stresses and modeling of the nozzle

Abstract: The purpose of this paper is to continue to describe the development of a comprehensive methodology for fully resolved numerical simulations of fused deposition modeling.,A front-tracking/finite volume method introduced in Part I to simulate the heat transfer and fluid dynamics of the deposition of a polymer filament on a fixed bed is extended by adding an improved model for the injection nozzle, including the shrinkage of the polymer as it cools down, and accounting for stresses in the solid.,The accuracy and convergence properties of the new method are tested by grid refinement, and the method is shown to produce convergent solutions for the shape of the filament, the temperature distribution, the shrinkage and the solid stresses.,The method presented in the paper focuses on modeling the fluid flow, the cooling and solidification and volume changes and residual stresses, using a relatively simple viscoelastic constitutive model. More complex material models, depending, for example, on the evolution of the conformation tensor, are not included.,The ability to carry out fully resolved numerical simulations of the fused deposition process is expected to be critical for the validation of mathematical models for the material behavior, to help explore new deposition strategies and to provide the “ground truth” for the development of reduced-order models.,The paper completes describing the development of the first numerical method for fully resolved simulation of fused filament modeling.

Modified porosity model in analysis of functionally graded porous nanobeams

Abstract: This work studies the mechanical bending and vibration of functionally graded nanobeams using finite elements according to Euler beam theory. We implement a modified porosity model that represents porosity and Young’s modulus in an implicit form, where the density is assumed as a function of the porosity parameter, while Young’s modulus is assumed as a ratio of the mass density with porosity to that without porosity. The effect of nano-scale is described by the nonlocal continuum theory by adding the length scale into the constitutive equations as a material parameter comprising information about nanoscopic forces and its interactions. The material gradation of constituents is described by a power function through the thickness of nanobeam. The beam is simply supported and is assumed to be thin, and hence, the kinematic assumptions of Euler–Bernoulli beam theory are held. The mathematical model is solved numerically using the finite-element method. Numerical results show that the increase of porosity, material graduation, and nano-scale parameters tend to decrease the bending resistance as well as the fundamental frequencies of the nanobeam.