Mechanical Models for Asphalt Behavior and Performance Introduction and Review of Linear Viscoelastic Behaviors
About the webinar series Past, current, and future plan for webinar series Introduction to the building blocks of mechanistic models (October 18, 212) http://www.trb.org/electronicsessions/blurbs/167848.aspx Overview of approaches to modeling damage in asphalt materials and pavements (December 11, 212) http://www.trb.org/main/blurbs/16883.aspx Viscoelastic models to unify asphalt stiffness measures (this webinar) Exemplify the use of models to characterize cracking, rutting, moisture damage, etc. (future)
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. AFK5(1) fall workshop series 213 Viscoplasticity (Dr. Charles Schwartz (UMD) and Dr. Regis Carvalho, Oaken Consult) 214 Viscoelastic Continuum Damage Model (Dr. Richard Kim, NCSU and Dr. Shane Underwood, ASU)
About this webinar 1. Review of linear viscoelastic behaviors 2. Application of linear viscoelastic models for laboratory evaluations 3. Application of linear viscoelastic models for field evaluations Shane Underwood, Ph.D. ASU Emin Kutay, Ph.D. MSU
Application of Linear Viscoelastic Models for Field Evaluations
Learning Objectives Understand the role of viscoelastic relaxation on in-situ stress-strain response of asphalt pavements. Explain how Falling Weight Deflectometer should be run and what data should be collected in order to be able to backcalculate in-situ E* mastercurve.
Strain response in the field ε y ε z z x y ε z ε y ε x Layered Viscoelastic Moving Load Model (LAVAM)
Elastic vs. Viscoelastic assumption in the field Transverse Microstrain 5 4 3 2 1 z=3.99", y=1",x=", (-): Compr., (+): Tens. Viscoelastic relaxation Measured-1 Measured-2 Predicted - LAVA Predicted - Layered elastic -1.2.4.6.8 1 1.2 1.4 1.6 1.8 2 time
Elastic vs. Viscoelastic assumption in the field Longitudinal Microstrain 4 2 z=3.99", x=", (-): Compr., (+): Tens. Elastic Viscoelastic -2.2.4.6.8 1 time Transverse Microstrain Vertical Microstrain 8 6 4 2 z=3.99", x=", (-): Compr., (+): Tens. Elastic Viscoelastic.2.4.6.8 1 time z=3.99", x=", (-): Compr., (+): Tens. 2-2 -4 Elastic Viscoelastic -6.2.4.6.8 1 time
What is different? Elastic vs Viscoelastic assumption. Elastic stress-strain relationship: σ() t = Eε() t Viscoelastic stress-strain relationship: t dε σ() t = Et ( τ) dτ dτ
Viscoelastic properties E* or D* E(t) D(t) 1 5 1 1 5 1 1 4 1-1 1 4 1-1 1 3 1-2 1 3 1-2 E* D* E(t) D(t) 1 2 1-3 1 2 1-3 1 1 1-4 1 1 1-4 1 1-1 1-5 1 1 5 1 1 Freq(Hz) 1-5 1 1-1 1-5 1 1 5 1 1 Time(s) 1-5 Sigmoid functions: ( ) = c 1 + log E(t) c 2 1+ exp( c 3 c 4 log(t R )) 7
E* Master curve (a) log( E* ) (psi) T = -1 o C T = 4 o C T = 21 o C T = 37 o C log( f ) (Hz)
E* Master curve Time Temperature Superposition Principle Shift data horizontally to develop a single master curve (b) log( E* ) (psi) T = -1 o C T = 4 o C T = 21 o C T = 37 o C log ( E* ) = b 1 + 1 + exp(-b 3 b 2 b 4 log( f R )) f R = f a(t ) Reduced Frequency, log(f R ) (Hz)
In-situ E* mastercurve Falling Weight Deflectometer (FWD) test FWD test? E* Mastercurve
E* backcalculation approach General Steps: Deflection (in).35.3.25.2.15.1.5 Measured u(t) field -.5.1.2.3.4.5.6.7 Time(seconds) Deflection (in).25.2.15.1.5 Predicted u(t) from FWD simulation -.5.1.2.3.4.5.6.7 Time(seconds) Minimize Error = m 1 n ( k k u ) i uo ( k u ) k = 1 i, o= 1 i max 11
Layered Viscoelastic (forward) Algorithm (LAVA) Response of linear viscoelastic material (Shapery 1974) R ve (t) = t τ = R ve H (t τ) di(τ) For a layered system with a circular load: ve u vertical (t,z,r) = t τ = ve u H vertical (t τ,z,r) (Boltzmann s superposition integral) dσ(τ)
Layered Viscoelastic Algorithm (LAVA) Shapery s quasi-elastic approximation (1965; 1974) u H ve (t,r,z) u H e (t,r,z) Use layered elastic theory to get Use Boltzman s superposition integral to obtain VE response ve u vertical (t,z,r) = t τ = e u H vertical (t τ,z,r) dσ(τ) u H e (t,r,z) Unit load
Forward Algorithms: LAVA 1-2 Step 1: TT ii h 1 h 2 h 3 =Inf AC Base σ(t) Subgrade E subgrade, ν 3 1 E(t), ν 1 E base, ν 2 r=5.9 2 3 4 5 6 Step 3: (t), (inches) u ve H =(ue H 1-3 1-4 rc = in rc = 8 in rc = 12 in rc = 18 in rc = 24 in rc = 36 in rc = 48 in rc = 6 in Relaxation modulus, E(t) psi 1 7 1 6 1 5 1 4 Step 2: tt RRii = Number of time interval NE=1 tt ii aa TT (TTTT) t 1 3 Ri 1-1 1-5 1 1 5 1 1 Reduced time, t R at 19 o C (sec) EE(tt RRii ) Step 4: Stress (psi) t Ri 1-1 1-5 1 1 5 1 1 Reduced time t R, (sec) 1-5 12 1 8 6 4 2.2.4.6.8.1 Time (sec) i AFK 5(1): Sub committee ve on advanced e models to understand behavior and performance of asphalt mixtures AFK 5: Committee u ( on ticharacteristics ) = u Hof ( asphalt ti paving τ j ) dmixtures σ ( τ j to ) meet structural whererequirements i = 1,214,... N s j=
Backcalculation algorithms Known variables: Number of layers Thickness of each layer Poisson s ratio of each layer Unknowns (iteratively backcalculated) E(t) sigmoid constants (4 unknowns) log( E(t) ) = c 1 + c 2 1+ exp( c 3 c 4 log(t R )) E i of unbound layers (typically 2-3 unknowns) Optimization algorithms: fminsearch, ga:genetic algorithm
Backcalculation algorithm o Once the E(t) sigmoid constants are known o o log( E(t) ) = c 1 + c 2 1+ exp( c 3 c 4 log(t R )) Prony series is fit to the E(t) master curve E(t) = E + N i=1 E i e E* is calculated using the following: t R ρ i E'= E * cos φ E"= E * sin φ N ( ) 2 ( ) 2 ( ) = E + E i 2πfρ i N i=1 ( ) = E i 2πfρ i i=1 1+ 2πfρ i ( ) 2 1+ 2πfρ i E* = E'+iE" E* = E' 2 +E" 2 φ = tan 1 (E"/ E')
Stress (psi) 1 8 6 4 2 In order to get E* AND shift factor (a(t)) u VE (t) in.4.3.2.1 rc = in rc = 7.99 in rc = 12.1 in rc = 17.99 in rc = 24.2 in rc = 35.98 in.1.2.3.4 t(s) AC BASE SUBGRADE -.1.1.2.3.4 t(s) FWD 1 @ T 1o C FWD 2 @ T 2o C FWD 3 @ T 3o C FWD n @ T no C Backcalculation (BACKLAVA) Relaxation Modulus (psi) 1E+7 1E+6 1E+5 1E+4 T= C T=2C 1E+3 T=3C T=4C 1E+2 1E-8 1E-5 1E-2 1E+1 1E+4 1E+7 17 Time (sec)
Stress (psi) 1 8 6 4 2 In order to get E* AND shift factor (a(t)).1.2.3.4 t(s) u VE (t) in.4.3.2.1 rc = in rc = 7.99 in rc = 12.1 in rc = 17.99 in rc = 24.2 in rc = 35.98 in -.1.1.2.3.4 t(s) AC 1E+7 BASE SUBGRADE AC sublayer 1 T 1 = 2 o C AC sublayer 2 T 2 = 15 o C AC sublayer 3 T 3 = 1 o C 1 FWD Backcalculation (BACKLAVA) Relaxation Modulus (psi) 1E+6 1E+5 1E+4 T= C T=2C 1E+3 T=3C T=4C 1E+2 1E-8 1E-5 1E-2 1E+1 1E+4 1E+7 18 Time (sec) 18
Backcalculation example 6.9" 12" LTPP Section 46-84 Asphalt Concrete (Linear Viscoelastic) ν=.35 Granular Base (Ebase) ν=.4 Subgrade (Esubgrade) ν=.45 31.18 o C 28.17 o C 25.48 o C Deflection (in).4.35.3.25.2.15.1.5 Measured S1 Measured S3 Backcalculated S1 Backcalculated S3.3.6.9.12.15 Time (sec) Relaxation modulus, E(t) (psi) 1 7 1 6 1 5 1 4 Backcalculated-1 Backcalculated-2 1 2 Computed from measured D(t) 1-5 1-4 1-3 1-2 1-1 1 1 1 1 2 1 3 1 3 Reduced time at 19 o C (sec)
Challenges Dynamics/wave propagation cannot be simulated with the method presented herein. Dynamic + viscoelastic forward solutions were recently developed, but computational speed of these algorithms make them impractical. Often the measured deflection time history is unreliable.
Summary o Consideration of viscoelasticity of AC is important o In-situ E* may be backcalculated with o o Single FWD run on an AC with large temperature gradient FWDs run at different temperatures o Accurate measurement of deflection time history is crucial
Mechanical Models for Asphalt Behavior and Performance Application of Linear Viscoelastic Models for Laboratory Evaluations
Learning Objective and Topics Explain the relationship between dynamic modulus and resilient modulus of asphalt concrete. Topics 1. Characterization of linear viscoelastic functions 2. Relationship between linear viscoelastic functions 3. Relationship between resilient modulus, M R, and dynamic modulus, E*
1. LVE Characterization Determine the value of coefficients used in mathematical models that define mechanical properties. Relevant material properties Stress response to unit input of strain = Relaxation modulus, E(t) Strain response to unit input of stress = Creep compliance, D(t) Sinusoidal response to sinusoidal input = Complex modulus/compliance, E*/D* Others: Shear modulus/compliance, bulk modulus/compliance, Poisson s ratio
1. LVE Characterization Review from Webinar 1 σ σ n σ n Simply put, the response of an arbitrary loading history is interpreted based on the summation of responses to a series of unit step inputs, e.g., a summation of finite stresses: time 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: t dσ ε() t = Dt ( τ) dτ dτ
1. LVE Characterization Time Domain Creep Compliance, D(t) Test Constant load applied for prescribed period of time. Strain response is monitored Principle advantage Creep load is relatively easy to apply Principle concern Can easily exceed the LVE limits σ Input ε Response linear elastic or time independent σ 1 time D(t)=ε t (t)/σ 1 viscoelastic or time dependent ε=dσ 1 time
1. LVE Characterization Generalized Voight-Kelvin Model Generalized Maxwell Model (1) (2) (1) (2) N t t i D( t) = D + + Di 1 e τ η i= 1 ( ) N = + i= 1 t i E t E Ee ρ i October 212 Webinar
1. LVE Characterization Frequency Domain Complex Modulus, Test E*(ω) Constant sinusoidal stress or strain applied for prescribed temperature and frequency. Stress or strain response is monitored for amplitude and delay. Principle advantage Can ensure LVE strain limits are not exceeded. Principle concern Ensuring loading retains sinusoidal shape. σ Input ε Response time time Reported with respect to frequency and angular frequency ω = 2π f
2. LVE Functional Relationships Time-Domain Relationships In the case of elastic materials the inter-conversion or inter-relationship is straightforward D x E =1 What about time-dependent materials? t E ( t τ ) D( τ ) dτ = t E( τ ) D( t τ ) dτ = t 1. Exact analytical solution 2. Numerical solution 3. Semi-analytical solution t October 212 Webinar
2. LVE Functional Relationships Time-to-Frequency Relationships Exact relationships ( ) N = + i= 1 t i E t E Ee ρ i ( ω) E = E + E ωρ N 2 2 i i 2 2 i= 1 ωρi + 1 N i i= 1 ωρi + 1 i ( ω) 2 2 E = E ωρ Approximate relationships R = R( t) 2.1 R = λ R( t) 1 t= = πω f t= ω
3. M R and E* Relationship M R = engineering parameter to relate recoverable strain in AC material under simulated, field-like loading pulse. Stress 2 14 16 σ.1 s.9 s Strain.1 s.9 s Resilient modulus is not a LVE response function ε r M R = σ ε ASTM D4123 AASHTO T322 NCHRP 1-28 r
3. M R and E* Exact Relationship Measurement E *, ϕ LVE Interconversion Theory ( ω) E = E + N 2 2 i i 2 2 i= 1 ωρi + 1 N i= 1 E ωρ t i E( t) = E Ee ρ + i t ( τ) ( ) E t D t dτ = t Stress σ ( t) 2 ωt σ sin t.1 = 2.1 < t.9 2 14 16 σ.1 s.9 s t dσ ε() t = Dt ( τ) dτ dτ M R N t i D( t) D Di 1 e τ = + i= 1 = σ ε r Strain.1 s.9 s ε r
3. M R and E* Exact Relationship Lacroix et al. 27 3, S12.5C 3, S12.5CM Predicted M R, MPa 2, 1, Predicted M R, MPa 2, 1, 1, 2, 3, Measured M R, MPa 1, 2, 3, Measured M R, MPa 3, B25.C 3, S12.5FE Predicted M R, MPa 2, 1, Predicted M R, MPa 2, 1, AFK 5(1): Sub committee on advanced models to understand behavior and performance of asphalt mixtures 1, 2, 3, 1, 2, 3, Measured M R, MPa Measured M R, MPa
3. M R and E* Approx. Relationship E* @ 5 Hz versus M R Xiao 29 Lacroix 27 3 Dynamic Modulus ( E*, MPa 2 1 y = 1.134x R² =.992 1 2 3 Resilient Modulus (M R ), MPa Can models help us explain the root cause of this correlation?
3. M R and E* Approx. Relationship Superposition allows decomposition of input load into separate components. Total response is summation of responses from each component. Stress 2 14 16 σ.1 s.9 s = Stress 2 14 16 + Stress ε = ε ε r.5 s 1 s = Strain Strain +
3. M R and E* Approx. Relationship Sinusoidal Part Creep Part Stress Stress 2 14 16 Assume steady state loading σ 2 D ε = ( ) ε = ( ) ( ).5 s.5 σ 1 s 1.9 2 D D Approximate Interconversion ε.5 s R = R t t 1 1 = 1πω = f 1 1 εr ε.5 s ε1 s σ = = + 2 E* ( 1 ) E* ( 2 ) E* (.1 ) E* (.11) σ 1 σ 1 σ ε 1 1 ε 2 E * f 1 Hz ( = ) ( ) 2.1.5 s 2 E * 1 s ( 2 σ σ ) 2 E* (.1 ) E* (.11) ε = = M E* ( f=4.8 Hz) AFK 5(1): Sub committee on advanced r models to understand behavior and performance of asphalt mixtures R AFK 5: Committee on characteristics of asphalt M paving mixtures * ( 4.8 to meet ) structural requirements R E
Summary Major uniaxial LVE response functions include the Creep Compliance, Relaxation Modulus, Complex Modulus, and Complex Compliance. The M R is an engineering response function based on a characteristic loading history similar to what occurs in a pavement system. LVE theory can be used to mathematically predict the resilient modulus from know response functions. Characterization Interconversion Integration (Exact) M R E* (f = 5 Hz) (Approximate)
Review of Linear Viscoelastic Behaviors
Linear Viscoelastic Behaviors Elastic P
Linear Viscoelastic Behaviors Elastic Viscoelastic Time, t = s P P
Linear Viscoelastic Behaviors Elastic Viscoelastic Time, t = 1 s s 1 s P P
Linear Viscoelastic Behaviors Elastic Viscoelastic Time, t = 1 s P s 1 s 1 s P Under a constant prescribed load (stress), viscoelastic materials undergo creep deformation (strain)
Linear Viscoelastic Behaviors Deformation, δ Excitation Reaction, P Elastic Solids time time Deformation δ Reaction, P Viscous Fluids Viscoelastic time Reaction, P time Under a prescribed deformation (strain), viscoelastic materials undergo relaxation of reaction (stress) Response Reaction
Linear Viscoelastic Behaviors Strain Input Stress Response ε 1 ε 2 ε 1 ε 2 ε 3 ε 3 ε 4 ε 4 Time Time
Linear Viscoelastic Behaviors Strain Input Stress Response T > 1 > T2 > T3 T4 T 4 T 3 T 2 T 1 Time Time
Linear Viscoelastic Behaviors Under cyclic or harmonic loading conditions we use slightly different terminology to define material response Load, P Deformation, δ time Relate the magnitude of excitation and the delay between excitation and response. Elastic Solids time P
Linear Viscoelastic Behaviors Time lag (phase angle) Stress, Strain Stress Strain 2πf σ ο ε o Complex Modulus σ o E * = ε o Phase Angle ϕ = ω t ϕ = 2π f t σ sin (ω t) o ε sin (ω t φ) o + Time
Linear Viscoelastic Behaviors When stress is input, R H (t) = D(t) and use: t dσ ε() t = Dt ( τ) dτ dτ When strain is input, R H (t) = E(t) and use: t dε σ() t = Et ( τ) dτ dτ When stress or strain are steady state sinusoidal use: σ = ( E *) ε ε = ( E *) σ
Linear Viscoelastic Behaviors Summary Viscoelastic responses exhibit a combination of elastic and viscous behaviors. This characteristic behavior means that relationships between stress and strain must include time-, frequency-, temperature- dependent functionality. In the next presentation we will briefly talk about how these relationships are characterized and how the relationships between these characteristics can be exploited for useful material prediction.