Elasticity and Viscoelasticity
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Cohesive Crack with Rate-Dependent Opening and Viscoelasticity: I
International Journal of Fracture 86: 247±265, 1997. c 1997 Kluwer Academic Publishers. Printed in the Netherlands. Cohesive crack with rate-dependent opening and viscoelasticity: I. mathematical model and scaling ZDENEKÏ P. BAZANTÏ and YUAN-NENG LI Department of Civil Engineering, Northwestern University, Evanston, Illinois 60208, USA Received 14 October 1996; accepted in revised form 18 August 1997 Abstract. The time dependence of fracture has two sources: (1) the viscoelasticity of material behavior in the bulk of the structure, and (2) the rate process of the breakage of bonds in the fracture process zone which causes the softening law for the crack opening to be rate-dependent. The objective of this study is to clarify the differences between these two in¯uences and their role in the size effect on the nominal strength of stucture. Previously developed theories of time-dependent cohesive crack growth in a viscoelastic material with or without aging are extended to a general compliance formulation of the cohesive crack model applicable to structures such as concrete structures, in which the fracture process zone (cohesive zone) is large, i.e., cannot be neglected in comparison to the structure dimensions. To deal with a large process zone interacting with the structure boundaries, a boundary integral formulation of the cohesive crack model in terms of the compliance functions for loads applied anywhere on the crack surfaces is introduced. Since an unopened cohesive crack (crack of zero width) transmits stresses and is equivalent to no crack at all, it is assumed that at the outset there exists such a crack, extending along the entire future crack path (which must be known). -
An Investigation Into the Effects of Viscoelasticity on Cavitation Bubble Dynamics with Applications to Biomedicine
An Investigation into the Effects of Viscoelasticity on Cavitation Bubble Dynamics with Applications to Biomedicine Michael Jason Walters School of Mathematics Cardiff University A thesis submitted for the degree of Doctor of Philosophy 9th April 2015 Summary In this thesis, the dynamics of microbubbles in viscoelastic fluids are investigated nu- merically. By neglecting the bulk viscosity of the fluid, the viscoelastic effects can be introduced through a boundary condition at the bubble surface thus alleviating the need to calculate stresses within the fluid. Assuming the surrounding fluid is incompressible and irrotational, the Rayleigh-Plesset equation is solved to give the motion of a spherically symmetric bubble. For a freely oscillating spherical bubble, the fluid viscosity is shown to dampen oscillations for both a linear Jeffreys and an Oldroyd-B fluid. This model is also modified to consider a spherical encapsulated microbubble (EMB). The fluid rheology affects an EMB in a similar manner to a cavitation bubble, albeit on a smaller scale. To model a cavity near a rigid wall, a new, non-singular formulation of the boundary element method is presented. The non-singular formulation is shown to be significantly more stable than the standard formulation. It is found that the fluid rheology often inhibits the formation of a liquid jet but that the dynamics are governed by a compe- tition between viscous, elastic and inertial forces as well as surface tension. Interesting behaviour such as cusping is observed in some cases. The non-singular boundary element method is also extended to model the bubble tran- sitioning to a toroidal form. -
Analysis of Viscoelastic Flexible Pavements
Analysis of Viscoelastic Flexible Pavements KARL S. PISTER and CARL L. MONISMITH, Associate Professor and Assistant Professor of Civil Engineering, University of California, Berkeley To develop a better understandii^ of flexible pavement behavior it is believed that components of the pavement structure should be considered to be viscoelastic rather than elastic materials. In this paper, a step in this direction is taken by considering the asphalt concrete surface course as a viscoelastic plate on an elastic foundation. Assumptions underlying existing stress and de• formation analyses of pavements are examined. Rep• resentations of the flexible pavement structure, im• pressed wheel loads, and mechanical properties of as• phalt concrete and base materials are discussed. Re• cent results pointing toward the viscoelastic behavior of asphaltic mixtures are presented, including effects of strain rate and temperature. IJsii^ a simplified viscoelastic model for the asphalt concrete surface course, solutions for typical loading problems are given. • THE THEORY of viscoelasticity is concerned with the behavior of materials which exhibit time-dependent stress-strain characteristics. The principles of viscoelasticity have been successfully used to explain the mechanical behavior of high polymers and much basic work U) lias been developed in this area. In recent years the theory of viscoelasticity has been employed to explain the mechanical behavior of asphalts (2, 3, 4) and to a very limited degree the behavior of asphaltic mixtures (5, 6, 7) and soils (8). Because these materials have time-dependent stress-strain characteristics and because they comprise the flexible pavement section, it seems reasonable to analyze the flexible pavement structure using viscoelastic principles. -
Engineering Viscoelasticity
ENGINEERING VISCOELASTICITY David Roylance Department of Materials Science and Engineering Massachusetts Institute of Technology Cambridge, MA 02139 October 24, 2001 1 Introduction This document is intended to outline an important aspect of the mechanical response of polymers and polymer-matrix composites: the field of linear viscoelasticity. The topics included here are aimed at providing an instructional introduction to this large and elegant subject, and should not be taken as a thorough or comprehensive treatment. The references appearing either as footnotes to the text or listed separately at the end of the notes should be consulted for more thorough coverage. Viscoelastic response is often used as a probe in polymer science, since it is sensitive to the material’s chemistry and microstructure. The concepts and techniques presented here are important for this purpose, but the principal objective of this document is to demonstrate how linear viscoelasticity can be incorporated into the general theory of mechanics of materials, so that structures containing viscoelastic components can be designed and analyzed. While not all polymers are viscoelastic to any important practical extent, and even fewer are linearly viscoelastic1, this theory provides a usable engineering approximation for many applications in polymer and composites engineering. Even in instances requiring more elaborate treatments, the linear viscoelastic theory is a useful starting point. 2 Molecular Mechanisms When subjected to an applied stress, polymers may deform by either or both of two fundamen- tally different atomistic mechanisms. The lengths and angles of the chemical bonds connecting the atoms may distort, moving the atoms to new positions of greater internal energy. -
Viscoelasticity of Filled Elastomers: Determination of Surface-Immobilized Components and Their Role in the Reinforcement of SBR-Silica Nanocomposites
Viscoelasticity of Filled Elastomers: Determination of Surface-Immobilized Components and their Role in the Reinforcement of SBR-Silica Nanocomposites Dissertation zur Erlangung des akademischen Grades doctor rerum naturalium (Dr. rer. nat.) genehmigt durch die Naturwissenschaftliche Fakultät II Institut für Physik der Martin-Luther-Universität Halle-Wittenberg vorgelegt von M. Sc. Anas Mujtaba geboren am 03.04.1980 in Lahore, Pakistan Halle (Saale), January 15th, 2014 Gutachter: 1. Prof. Dr. Thomas Thurn-Albrecht 2. Prof. Dr. Manfred Klüppel 3. Prof. Dr. Rene Androsch Öffentliche Verteidigung: July 3rd, 2014 In loving memory of my beloved Sister Rabbia “She will be in my Heart ξ by my Side for the Rest of my Life” Contents 1 Introduction 1 2 Theoretical Background 5 2.1 Elastomers . .5 2.1.1 Fundamental Theories on Rubber Elasticity . .7 2.2 Fillers . 10 2.2.1 Carbon Black . 12 2.2.2 Silica . 12 2.3 Filled Rubber Reinforcement . 15 2.3.1 Occluded Rubber . 16 2.3.2 Payne Effect . 17 2.3.3 The Kraus Model for the Strain-Softening Effect . 18 2.3.4 Filler Network Reinforcement . 20 3 Experimental Methods 29 3.1 Dynamic Mechanical Analysis . 29 3.1.1 Temperature-dependent Measurement (Temperature Sweeps) . 31 3.1.2 Time-Temperature Superposition (Master Curves) . 31 3.1.3 Strain-dependent Measurement (Payne Effect) . 33 3.2 Low-field NMR . 34 3.2.1 Theoretical Concept . 34 3.2.2 Experimental Details . 39 4 Optimizing the Tire Tread 43 4.1 Relation Between Friction and the Mechanical Properties of Tire Rubbers 46 4.2 Usage of tan δ As Loss Parameter . -
Oversimplified Viscoelasticity
Oversimplified Viscoelasticity THE MAXWELL MODEL At time t = 0, suddenly deform to constant displacement Xo. The force F is the same in the spring and the dashpot. F = KeXe = Kv(dXv/dt) (1-20) Xe is the displacement of the spring Xv is the displacement of the dashpot Ke is the linear spring constant (ratio of force and displacement, units N/m) Kv is the linear dashpot constant (ratio of force and velocity, units Ns/m) The total displacement Xo is the sum of the two displacements (Xo is independent of time) Xo = Xe + Xv (1-21) 1 Oversimplified Viscoelasticity THE MAXWELL MODEL (p. 2) Thus: Ke(Xo − Xv) = Kv(dXv/dt) with B. C. Xv = 0 at t = 0 (1-22) (Ke/Kv)dt = dXv/(Xo − Xv) Integrate: (Ke/Kv)t = − ln(Xo − Xv) + C Apply B. C.: Xv = 0 at t = 0 means C = ln(Xo) −(Ke/Kv)t = ln[(Xo − Xv)/Xo] (Xo − Xv)/Xo = exp(−Ket/Kv) Thus: F (t) = KeXo exp(−Ket/Kv) (1-23) The force from our constant stretch experiment decays exponentially with time in the Maxwell Model. The relaxation time is λ ≡ Kv/Ke (units s) The force drops to 1/e of its initial value at the relaxation time λ. Initially the force is F (0) = KeXo, the force in the spring, but eventually the force decays to zero F (∞) = 0. 2 Oversimplified Viscoelasticity THE MAXWELL MODEL (p. 3) Constant Area A means stress σ(t) = F (t)/A σ(0) ≡ σ0 = KeXo/A Maxwell Model Stress Relaxation: σ(t) = σ0 exp(−t/λ) Figure 1: Stress Relaxation of a Maxwell Element 3 Oversimplified Viscoelasticity THE MAXWELL MODEL (p. -
A Bayesian Surrogate Constitutive Model to Estimate Failure Probability of Rubber-Like Materials
A Bayesian Surrogate Constitutive Model to Estimate Failure Probability of Rubber-Like Materials Aref Ghaderia,b, Vahid Morovatia,c, Roozbeh Dargazany1a,d aDepartment of Civil and Environmental Engineering, Michigan State University [email protected] [email protected] [email protected] Abstract In this study, a stochastic constitutive modeling approach for elastomeric materials is developed to consider uncer- tainty in material behavior and its prediction. This effort leads to a demonstration of the deterministic approaches error compared to probabilistic approaches in order to calculate the probability of failure. First, the Bayesian lin- ear regression model calibration approach is employed for the Carroll model representing a hyperelastic constitutive model. The developed model is calibrated based on the Maximum Likelihood Estimation (MLE) and Maximum a Priori (MAP) estimation. Next, a Gaussian process (GP) as a non-parametric approach is utilized to estimate the probabilistic behavior of elastomeric materials. In this approach, hyper-parameters of the radial basis kernel in GP are calculated using L-BFGS method. To demonstrate model calibration and uncertainty propagation, these approaches are conducted on two experimental data sets for silicon-based and polyurethane-based adhesives, with four samples from each material. These uncertainties stem from model, measurement, to name but a few. Finally, failure proba- bility calculation analysis is conducted with First Order Reliability Method (FORM) analysis and Crude Monte Carlo (CMC) simulation for these data sets by creating a limit state function based on the stochastic constitutive model at failure stretch. Furthermore, sensitivity analysis is used to show the importance of each parameter of the probability of failure. -
Viscoelastic Creep Crack Growth: a Review of Fracture Mechanical Analyses
Mechanics of Time-Dependent Materials 1: 241–268, 1998. 241 c 1998 Kluwer Academic Publishers. Printed in the Netherlands. Viscoelastic Creep Crack Growth: A Review of Fracture Mechanical Analyses W. BRADLEY1, W.J. CANTWELL2 and H.H. KAUSCH2 1Department of Mechanical Engineering, Texas A&M University, College Station, TX, U.S.A.; 2Materials Science Department, EPFL, DMX-LP, Swiss Federal Institute of Technology, 1015 Lausanne, Switzerland (Received 8 February 1997; accepted in revised form 2 October 1997) Abstract. The study of time dependent crack growth in polymers using a fracture mechanics approach has been reviewed. The time dependence of crack growth in polymers is seen to be the result of the viscoelastic deformation in the process zone, which causes the supply of energy to drive the crack to occur over time rather than instantaneously, as it does in metals. Additional time dependence in the crack growth process can be due to process zone behavior, where both the flow stress and the critical crack tip opening displacement may be dependent on the crack growth rate. Instability leading to slip-stick crack growth has been seen to be the consequence of a decrease in the critical energy release rate with increasing crack growth rate due to adiabatic heating causing a reduction in the process zone flow stress, a decrease in the crack tip opening displacement due to a ductile to brittle transition at higher crack growth rates, or an increase in the rate of fracture work due to more rapid viscoelastic deformation. Finally, various techniques to experimentally characterize the crack growth rate as a function of stress intensity have been critiqued. -
The Thermodynamic Properties of Elastomers: Equation of State and Molecular Structure
CH 351L Wet Lab 4 / p. 1 The Thermodynamic Properties of Elastomers: Equation of State and Molecular Structure Objective To determine the macroscopic thermodynamic equation of state of an elastomer, and relate it to its microscopic molecular properties. Introduction We are all familiar with the very useful properties of such objects as rubber bands, solid rubber balls, and tires. Materials such as these, which are capable of undergoing large reversible extensions and compressions, are called elastomers. An example of such a material is natural rubber, obtained from the plant Hevea brasiliensis. An elastomer has rather unusual physical properties; for example, an ordinary elastic band can be stretched up to 15 times its original length and then be restored to its original size. Although we might initially consider elastomers to be a solids, many have isothermal compressibilities comparable with that of liquids (e.g. toluene), about 10-4 atm-1 (compared with solids such as polystyrene or aluminum which have values of ~10–6 atm–1). Certain evidence suggests that an elastomer is a disordered "solid," i.e., a glass, that cannot flow as a result of internal, structural restrictions. The reversible deformability of an elastomer is reminiscent of a gas. In fact the term elastic was first used by Robert Boyle (1660) in describing a gas, "There is a spring or elastical power in the air in which we live." In this experiment, you will encounter certain formal thermodynamic similarities between an elastomer and a gas. One of the rather dramatic and anomalous properties of an elastomer is that once brought to an extended form, it contracts upon heating. -
Cross-Linked Polymers and Rubber Elasticity Chapter 9 (Sperling)
Cross-linked Polymers and Rubber Elasticity Chapter 9 (Sperling) • Definition of Rubber Elasticity and Requirements • Cross-links, Networks, Classes of Elastomers (sections 1-3, 16) • Simple Theory of Rubber Elasticity (sections 4-8) – Entropic Origin of Elastic Retractive Forces – The Ideal Rubber Behavior • Departures from the Ideal Rubber Behavior (sections 9-11) – Non-zero Energy Contribution to the Elastic Retractive Forces – Stress-induced Crystallization and Limited Extensibility of Chains (How to make better elastomers: High Strength and High Modulus) – Network Defects (dangling chains, loops, trapped entanglements, etc..) – Semi-empirical Mooney-Rivlin Treatment (Affine vs Non-Affine Deformation) Definition of Rubber Elasticity and Requirements • Definition of Rubber Elasticity: Very large deformability with complete recoverability. • Molecular Requirements: – Material must consist of polymer chains. Need to change conformation and extension under stress. – Polymer chains must be highly flexible. Need to access conformational changes (not w/ glassy, crystalline, stiff mat.) – Polymer chains must be joined in a network structure. Need to avoid irreversible chain slippage (permanent strain). One out of 100 monomers must connect two different chains. Connections (covalent bond, crystallite, glassy domain in block copolymer) Cross-links, Networks and Classes of Elastomers • Chemical Cross-linking Process: Sol-Gel or Percolation Transition • Gel Characteristics: – Infinite Viscosity – Non-zero Modulus – One giant Molecule – Solid -
Viscous-To-Viscoelastic Transition in Phononic Crystal and Metamaterial Band Structures
Viscous-to-viscoelastic transition in phononic crystal and metamaterial band structures Michael J. Frazier and Mahmoud I. Husseina) Department of Aerospace Engineering Sciences, University of Colorado Boulder, Boulder, Colorado 80309, USA (Received 3 March 2015; revised 8 September 2015; accepted 16 October 2015; published online 19 November 2015) The dispersive behavior of phononic crystals and locally resonant metamaterials is influenced by the type and degree of damping in the unit cell. Dissipation arising from viscoelastic damping is influenced by the past history of motion because the elastic component of the damping mechanism adds a storage capacity. Following a state-space framework, a Bloch eigenvalue problem incorpo- rating general viscoelastic damping based on the Zener model is constructed. In this approach, the conventional Kelvin–Voigt viscous-damping model is recovered as a special case. In a continuous fashion, the influence of the elastic component of the damping mechanism on the band structure of both a phononic crystal and a metamaterial is examined. While viscous damping generally narrows a band gap, the hereditary nature of the viscoelastic conditions reverses this behavior. In the limit of vanishing heredity, the transition between the two regimes is analyzed. The presented theory also allows increases in modal dissipation enhancement (metadamping) to be quantified as the type of damping transitions from viscoelastic to viscous. In conclusion, it is shown that engineering the dissipation allows one to control the dispersion (large versus small band gaps) and, conversely, engineering the dispersion affects the degree of dissipation (high or low metadamping). VC 2015 Acoustical Society of America.[http://dx.doi.org/10.1121/1.4934845] [ANN] Pages: 3169–3180 I. -
Elastomeric Materials
ELASTOMERIC MATERIALS TAMPERE UNIVERSITY OF TECHNOLOGY THE LABORATORY OF PLASTICS AND ELASTOMER TECHNOLOGY Kalle Hanhi, Minna Poikelispää, Hanna-Mari Tirilä Summary On this course the students will get the basic information on different grades of rubber and thermoelasts. The chapters focus on the following subjects: - Introduction - Rubber types - Rubber blends - Thermoplastic elastomers - Processing - Design of elastomeric products - Recycling and reuse of elastomeric materials The first chapter introduces shortly the history of rubbers. In addition, it cover definitions, manufacturing of rubbers and general properties of elastomers. In this chapter students get grounds to continue the studying. The second chapter focus on different grades of elastomers. It describes the structure, properties and application of the most common used rubbers. Some special rubbers are also covered. The most important rubber type is natural rubber; other generally used rubbers are polyisoprene rubber, which is synthetic version of NR, and styrene-butadiene rubber, which is the most important sort of synthetic rubber. Rubbers always contain some additives. The following chapter introduces the additives used in rubbers and some common receipts of rubber. The important chapter is Thermoplastic elastomers. Thermoplastic elastomers are a polymer group whose main properties are elasticity and easy processability. This chapter introduces the groups of thermoplastic elastomers and their properties. It also compares the properties of different thermoplastic elastomers. The chapter Processing give a short survey to a processing of rubbers and thermoplastic elastomers. The following chapter covers design of elastomeric products. It gives the most important criteria in choosing an elastomer. In addition, dimensioning and shaping of elastomeric product are discussed The last chapter Recycling and reuse of elastomeric materials introduces recycling methods.