Mechanical Models for Asphalt Behavior and Performance

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Today s Panelists and Moderator Eshan Dave, University of Minnesota, Duluth, evdave@d.umn.edu Amit Bhasin, University of Texas at Austin, a-bhasin@mail.utexas.edu Richard Kim, North Carolina State University, kim@ncsu.edu

About the webinar series The focus of these webinars is to: review the basic terms associated with mechanistic models so that the end users may effectively use these models, better appreciate the advantages of using mechanical models, develop a broader understanding of how mechanistic models work, and exemplify the use of models to solve specific problems faced by the asphalt materials and pavements community.

About the webinar series The webinars are not intended to: demonstrate the theoretical development of a model or models, or present the mathematical development of different models.

About the webinar series Current and future plan for webinar series Introduction to the building blocks of mechanistic models (this webinar) Overview of approaches to modeling damage in asphalt materials and pavements (December 2012) Exemplify the use of models to characterize cracking, rutting, moisture damage, etc. (future)

Mechanical Models for Asphalt Behavior and Performance Basics of Asphalt Modeling 1 Transportation Research Board webinar organized by:

Introduction Overall goal of this webinar is education Learning Objectives: Differences between modeling of asphalt concrete as compared to traditional civil engineering materials Stress-strain behavior of viscoelastic materials Differences between linear and non-linear viscoelastic behavior as well as elastic and plastic behavior Applications to solve boundary value problems Typical tests for viscoelastic characterization

Outline 1. Challenges in characterization of asphalt materials and pavements 2. Elastic versus viscoelastic response of materials 3. Linear viscoelasticity and superposition principle 4. Typical tests and terminology

1. Challenges Asphalt materials are visco-elasto-plastic heterogeneous composites with thermorheologically simple behavior Asphalt binders behave as: viscous fluids at high temperatures elastic solids at very low temperatures viscoelastic at intermediate temperatures Aggregates behave as elastic solids

1. Challenges +64 º C -22 º C New pavement 20 yr old pavement

1. Challenges The behavior of asphalt mix is quite complicated due to: viscoelastic and temperature dependent behavior of binder mismatch in properties of binder and aggregates effects of aggregate shape and texture macro and micro-scale interaction between particles and mastic In this webinar we will limit our discussions to the linear behavior, that is, no damage or cracking or failure

1. Challenges Standard strength of materials approach is often inadequate Need to account for: effects of and load history effects of temperatures effects of aging effects of heterogeneity effects of damage and failure (fatigue, fracture, etc.)

2. Elastic vs. Viscoelastic Response 2.1. Comparison of elastic and viscoelastic responses 2.2. Typical material models 2.3. Differences between and frequency domain based definitions 2.4. Differences between elastic and plastic response

2. Elastic vs. Viscoelastic Response 2.1. Comparison of Responses Elastic Viscoelastic Time, t = 0 s P P

2. Elastic vs. Viscoelastic Response 2.1. Comparison of Responses Elastic Viscoelastic Time, t = 10 s 0 s 10 s P P

2. Elastic vs. Viscoelastic Response 2.1. Comparison of Responses Elastic Viscoelastic Time, t = 100 s P 0 s 10 s 100 s P Under a constant prescribed load (stress), viscoelastic materials undergo creep deformation (strain)

2. Elastic vs. Viscoelastic Response 2.1. Comparison of Responses Load, P Excitation Load Deformation, δ Elastic Solids P Deformation, δ Deformation, δ Viscous Fluids Viscoelastic Response Deformation

2. Elastic vs. Viscoelastic Response 2.1. Comparison of Responses Deformation, δ Excitation Reaction, P Elastic Solids Deformation δ Reaction, P Viscous Fluids Viscoelastic Reaction, P Under a prescribed deformation (strain), viscoelastic materials undergo relaxation of reaction (stress) Response Reaction

2. Elastic vs. Viscoelastic Response 2.2. Material Models Constitutive behavior (or stress-strain relationship): Young s modulus for elastic solids Viscosity for viscous fluids Viscoelastic material constitutive behavior can be represented by: Creep compliance, D(t) strain = function of D(t) and stress Relaxation modulus, E(t) stress = function of E(t) and strain Various constitutive models are available for linear viscoelasticity: Spring-Dashpot Models (Prony series models) Mathematical Models (Power law is most common)

2. Elastic vs. Viscoelastic Response 2.2. Material Models Linear Spring Linear Dashpot (Elastic Solid) (Viscous Fluid) E λ E = Elastic Modulus ( lb / in 2 σ = E ε or N / m 2 ) λ = Coeff. of Viscosity ( Stokes or Poiseor N S σ = dε λ dt / m 2 )

2. Elastic vs. Viscoelastic Response 2.2. Material Models Basic Spring-Dashpot Models: Maxwell Model Voight-Kelvin Model Burger s Model 3-Parameter Solid Model 3-Parameter Liquid Model

2. Elastic vs. Viscoelastic Response 2.2. Material Models Response to Step Load: F Excitation Spring δ Dashpot δ Maxwell Response Model δ Voight-Kelvin Model δ

2. Elastic vs. Viscoelastic Response 2.2. Material Models Response to Step Load: Burger s F Excitation Model δ 3-Param. Solid Model δ Response 3-Param. Liquid Model δ

2. Elastic vs. Viscoelastic Response 2.2. Material Models Prony Series Models: Generalized Maxwell Model Generalized Voight-Kelvin Model (1) (1) (2) (2)

2. Elastic vs. Viscoelastic Response 2.2. Material Models 10 1 Behavior of Prony Series Stress 10 0 Time Creep Strain 10-1 P Initial Elastic Response, f(d 0 ) 10-2 10 0 10 1 10 2 10 3 10 4 10 5 10 6

2. Elastic vs. Viscoelastic Response 2.2. Material Models Power Law Model: Creep compliance or relaxation modulus is expressed by Power Law, D( t) = D Where, D 0, D 1 0 + D t 1 m and m are determined experimentally D(t) D 1 t m D 0 Time

2. Elastic vs. Viscoelastic Response 2.3. Time versus Frequency So far our discussion was limited to step or ramp type excitations Load, P Under cyclic or harmonic loading conditions we use slightly different terminology to define material response Deformation, δ Elastic Solids P

2. Elastic vs. Viscoelastic Response 2.3. Time versus Frequency 1 cycle = 1 Hz 1 cycle = 2π rad 1 cycle = 360 Deg + Applied Load, P (Stress) - t σ Complex modulus E* = σ/ε Deformation (Strain) + t Phase angle (ϕ) related to t t ε Phase angle, ϕ -

2. Elastic vs. Viscoelastic Response 2.3. Time versus Frequency Load, P δ P Elastic Solids Phase angle, ϕ = 0º δ Phase angle, ϕ = 90º Viscous Fluids δ Viscoelastic Phase angle, 0º < ϕ < 90º

2. Elastic vs. Viscoelastic Response 2.3. Time versus Frequency Time lag (phase angle) Stress, Strain Stress Strain σ ο ε o Complex Modulus E σ o * = ε o Phase Angle φ =ω t σ sin (ω t) o ε sin (ω t φ) o + Time

2. Elastic vs. Viscoelastic Response 2.4. Elastic versus Plastic Plastic behavior is when material does not make full recovery of strain on unloading Non-linear elastic Linear elastic Non-linear inelastic

2. Elastic vs. Viscoelastic Response 2.4. Elastic versus Plastic For materials exhibiting plastic behavior, Total Strain = Recoverable Strain + Plastic Strain Stress Total Strain Strain Plastic Strain Recov. Strain

Mechanical Models for Asphalt Behavior and Performance Basics of Asphalt Modeling 1

Goals The objective of this webinar is to learn about 1.Challenges in characterization of asphalt materials and pavements 2.Elastic versus viscoelastic response of materials 3.Linear viscoelasticity and superposition principle 4.Typical tests and terminology

3. Linear viscoelasticity & superposition 3.1. Linear viscoelastic response 3.2. Superposition principle and its applications 3.3. Inter-conversion 3.4. Example: Resilient modulus using creep-recovery

3. Linear viscoelasticity & superposition 1. Linear viscoelastic response Linear response is easily understood for elastic or independent materials We are just dealing with stress strain and does not play a role Simply put, for a linear elastic solid stress is proportional to strain (within certain limits) stress 1 E However, for a linear viscoelastic solid or -dependent elastic, the dimension also becomes important strain

3. Linear viscoelasticity & superposition 1. Linear viscoelastic response Let us first introduce the dimension for a linear elastic or independent material. σ Input σ 1 ε P ε=dσ 1 Response

3. Linear viscoelasticity & superposition 1. Linear viscoelastic response Let us first introduce the dimension for a linear elastic or independent material. σ Input ασ 1 σ 1 αp ε Response ε=dασ 1 ε=dσ 1

3. Linear viscoelasticity & superposition 1. Linear viscoelastic response What about a linear viscoelastic or dependent solid? σ Input σ 1 P ε Response ε t =D(t)σ 1 viscoelastic or dependent linear elastic or independent ε=dσ 1

3. Linear viscoelasticity & superposition 1. Linear viscoelastic response What about a linear viscoelastic or dependent solid? σ Input ασ 1 σ 1 The response is scaled BUT only when there is no history! ε t =D(t)ασ 1 αp ε Response ε t =D(t)σ 1 ε=dασ 1 viscoelastic or dependent linear elastic or independent ε=dσ 1

3. Linear viscoelasticity & superposition 1. Linear viscoelastic response What happens when there is a load or deformation history? Can we not just scale and add the response every the load increases or decreases? σ Input ασ 1 σ 1 ε t =D(t)ασ 1 ε Response ε t =D(t)σ 1 ε=dασ 1 viscoelastic or dependent linear elastic or independent ε=dσ 1

ε σ 3. Linear viscoelasticity & superposition 3.2. Superposition principle and its applications As before, let us first examine what happens to a linear elastic solid when you add (or subtract) new loads (or deformations) σ 2 σ 1 σ 1 + σ 2 ε=dσ 1 ε=d(σ 1 + σ 2 ) x seconds

3. Linear viscoelasticity & superposition 3.2. Superposition principle and its applications For a linear dependent or viscoelastic solid, we cannot just scale and add the strains σ 2 σ 2 σ σ σ 1 σ 1 + σ 2 σ 1 σ 1 + σ 2 ε ε=dσ 1 ε=d(σ 1 + σ 2 ) ε ε t =D(t)σ 1 ε t =D(t)(σ 1 +σ 2 ) x seconds x seconds

3. Linear viscoelasticity & superposition 3.2. Superposition principle and its applications For a linear dependent or viscoelastic solid, we cannot just scale and add the strains σ σ 2 σ σ 1 + σ 2 σ 1 σ 1 + σ 2 σ 1 σ 2 ε t =D(t)σ 1 +D(t-x)σ 2 ε ε ε=dσ 1 ε=d(σ 1 + σ 2 ) x seconds ε t =D(t)σ 1 ε t =D(t)σ 2 x seconds This is only when t>x seconds

3. Linear viscoelasticity & superposition 3.2. Superposition principle and its applications What about an arbitrary loading history? Recall ε t =D(t)σ 1 +D(t-x)σ 2 σ n σ n Simply put, the above idea is extended in the form of a summation of finite stresses: σ σ 4 σ 3 σ 2 σ 1 σ 3 σ 2 x 2 x 3 x 4 x n σ 1 A more common way to expressing the above is in its integral form:

3. Linear viscoelasticity & superposition 3.2. Superposition principle and its applications What about an arbitrary loading history? σ σ n σ 4 σ 3 σ 2 σ 1 σ n σ 3 σ 2 x 2 x 3 x 4 x n σ 1 It is important to remember that arbitrary load or deformation histories are accumulated using this superposition integral (also known as convolution integral or Boltzmann s integral)

3. Linear viscoelasticity & superposition 3.2. Superposition principle and its applications What about an arbitrary loading history? σ ε ε = Integrating {D(t), σ (t)} σ = Integrating {E(t), ε (t)}

3. Linear viscoelasticity & superposition 3.3. Inter-conversion Previously, we have seen that the material properties can be measured in domain (creep or relaxation) or frequency domain (complex modulus) These are properties of the material and hence it is mathematically possible to get one from the other ε σ ε = Integrating {D(t), σ (t)}

3. Linear viscoelasticity & superposition 3.3. Inter-conversion A note about compliance and relaxation In the case of elastic materials the inter-conversion or interrelationship is straightforward What about -dependent materials? There are methods to solve such integrals, for now it is sufficient to know that we can obtain one from the other

3. Linear viscoelasticity & superposition 3.4. Example: Resilient Modulus ε σ ε = Integrating {D(t), σ (t)}

Goals The objective of this webinar is to learn about 1.Challenges in characterization of asphalt materials and pavements 2.Elastic versus -dependent or viscoelastic response of materials 3.Linear viscoelasticity and superposition principle 4.Typical tests and terminology

4. Typical tests & terminology 4.1. Frequency sweep 4.2. Temperature sweep 4.3. Amplitude sweep 4.4. Time sweep

4. Typical tests & terminology 4.1. Frequency sweep Typically sinusoidal loading is used Frequency of loading is varied with Commonly used to determine the relationship between G* (or phase angle) and frequency and to determine creep compliance or relaxation Amplitude Time

4. Typical tests & terminology 4.2. Temperature sweep Also referred to as DMTA (Differential mechanical thermal analysis) Typically one stress amplitude is selected and temperature varies Commonly used with polymers to determine glass transition temperatures G δ γ δ T g Glass Transition Temperature

4. Typical tests & terminology 4.3. Amplitude sweep Typically sinusoidal loading is used Strain or stress amplitude is varied in linear or logarithmic scale Commonly used to determine limits for linear viscoelastic response Amplitude Time

4. Typical tests & terminology 4.4. Time sweep Typically sinusoidal loading is used A constant stress or strain amplitude is applied for several cycles Typically used to assess the fatigue damage evolution in materials; tests are typically carried out until specimen failure Amplitude Time