Part 7 Nonlinearity
Linear System Superposition, Convolution re ( ) re ( ) = r 1 1 = r re ( 1 + e) = r1 + r e excitation r = r() e response In the time domain: t rt () = et () ht () = e( τ) ht ( τ) dτ et () excitation rt () response ht () impulse function of the system R( ω ) = E( ω) H( ω ) R( ω ) Fourier spectrum of the response E( ω ) Fourier spectrum of the excitation H ( ω ) transfer function of the system
Fourier Transform H ( ω ) = F { ht ( )} and ht ( ) = F -1 { H( ω)} F -1 F {()} ht = ht () e ω i t dt 1 { H ( ω )} = H( ) eiωt d π ω ω
Nonlinear System re ( + e) re ( ) + re ( ) 1 1 Higher-Order Elastic Coefficients 1 1 c ij ij ijk ij k 3 ijk mn ij k mn E = E + gc ε + C ε ε + C ε ε ε +... E c static energy density g is a (bias) constant ε ij are the complete strain components C ij first-order elastic coefficients Cijk second-order elastic coefficients (81/) Cijk mn third-order elastic coefficients (79/3) u ½( i uj uk uk u j ) ½( i u ε ij = + + + ) x j xi xi xj xj xi Without the lower-order terms: 1 1 ijk ij k 3 ijk mn ij k mn E = C ε ε + C ε ε ε +...
Nonlinear Elastic Behavior of Materials I material (stress-strain relationship) Stress [a. u.] Elastic Limit Linear Limit Ultimate Failure Strain [a. u.] II geometrical (strain-displacement relationship) k k F F ε F ε ε = a + u a u, F a ε = k ε, F a uf = = a 3 k u 3 a
Lattice Nonlinearity Potential Energy [a. u.] parabolic potential well typical 0 1 Normalized Lattice Distance parabolic potential function Elastic Stiffness [a. u.] unstrained typical 0.9 0.95 1 1.05 1.1 Normalized Lattice Distance (Strain)
Harmonics Generation Harmonic (sinusoidal) excitation: Nonlinear response: () = sin( ω ) et e0 t r r ret r 0 et e t e e ( ( )) = + ( ) + ½ ( ) +... r 0 is assumed to be zero (since it is independent of the excitation) h1 r = is the linear transfer coefficient e h = ½ r e is the quadratic transfer coefficient rt he 1 0 ω t + he0 ω t () sin( ) sin ( ) sin ( ξ ) = ½ - ½cos( ξ ) rt h e0 he 1 0 t he0 t () ½ + sin( ω ) ½ cos( ω ) +... Higher-order terms: rt he 3 3 1 0 t he0 t he 3 0 t ( ) sin( ω ) + sin ( ω ) + sin ( ω ) +... rt he 3 3 0 + he 1 0 + he 3 0 ωt he 0 ωt he 3 0 ω t ( ) ½ ( ¾ )sin( ) ½ cos( ) ¼ cos(3 ) For weak nonlinearity: rn e0 n
Nonlinear Distortion Continuous build-up of the harmonic distortion: Basic Harmonic nonlinear material Higher-Harmonics Localized harmonic distortion: Basic Harmonic imperfect interface Higher-Harmonics
Harmonics Generation 11 ( β σ = C ε + ε )
Nonlinear distortion build-up d = 0 1 Instantaneous Pressure [a. u.] d > d 1 d > d 3 Time [a. u.] quasi-static pulse formation Oscilloscope Transmitter Receiver High-Pass Filter & Amplifier Ch 1 Ch Specimen Low-Pass Filter & Amplifier
Acousto-Elasticity Linear regression: c( σ e) = c0 + η1σ e +ησ e +... (4.14) η 1 = c/ σe is the first-order acousto-elastic coefficients ½ c/ e is the second-order acousto-elastic coefficients η = σ c o is the sound velocity in the undeformed material σ e is the external stress applied to the specimen c( σe) c0 + η1σ e (4.15) c s,np c d,n c s,nn c d,p c s,p tension σ Five independent combinations of wave and polarization directions:
Wave Velocities in the Principal Directions σ λ + μ ρcd, p = λ + μ + [ + λ + ( 4m + 4λ + 10μ)] 3λ + μ μ σ λ ρcdn, = λ + μ + [ ( m + λ + μ)] 3λ + μ μ σ λ n ρcsp, = μ + ( m 4λ 4μ) 3λ + μ 4μ (4.16b) + + + (4.16c) σ λ n ρcsnp, = μ + ( m λ μ) 3λ + μ 4μ + + + (4.16d) σ λ + μ ρcsnn, = μ + ( m n λ) 3λ + μ μ, m, (4.16e) and n Murnaghan coefficients ρ σ density tensile stress material λ μ m n [109 Pa] [109 Pa] [109 Pa] [109 Pa] [109 Pa] Aluminum 7064 59.3 7.4-34 -397-403 Armco iron 110 8-348 -1030 1100 Polystyrene.9 1.4-18.9-13.3-10 Pyrex 13.5 7.5 14 9 40
Normalized Wave Velocities in the Principal Directions λ +μ +λ+ (4m + 4λ+ 10 μ) λ+ μ μ cd, p [1 + σ ] = cd0 (1 + Nd, pσ ) ρ ( λ + μ)(3λ + μ) λ ( m + λ + μ) λ+ μ μ cdn, [1 + σ ] = cd0 (1 + Ndn, σ ) ρ ( λ + μ)(3λ + μ) λ n m + + 4λ + 4μ μ 4μ csp, [1 + σ ] = cs0 (1 + Nsp, σ) ρ μ(3λ + μ) λ n m + + λ + μ μ 4μ csnp, [1 + σ ] = cs0 (1 + Nsnp, σ) ρ μ(3λ + μ) λ +μ m n λ μ μ csnn, [1 + σ ] = cs0 (1 + Nsnn, σ) ρ μ(3λ + μ)
Longitudinal Velocity as a Function of Uniaxial Stress in 7064 Aluminum Longitudinal Velocity [m/s] 6480 6460 6440 640 6400 6380 parallel ( c d,p ) normal ( c d,n ) 6360-100 -50 0 50 100 Uniaxial Stress [MPa]
Lattice Nonlinearity Potential Energy [a. u.] parabolic potential well typical 0 1 Normalized Lattice Distance E( ξ ) = E ( 1) 3 o + ξ E + ( ξ 1) E3 +... (4.17) ξ = r/a is the normalized lattice distance E '( ξ ) = E ( 1) o + ξ E is the ideal quadratic potential E( ξ ) = ε E 3 + ε E3 +... (4.18) ε=ξ 1 is the elastic strain ( E o is neglected) second-order stiffness constant: C11 = E third-order stiffness constant: C111 = 3E3
restoring stress: E() ε σε () = = ε E + 3 ε E3 +... ε (4.19) dynamic stiffness: σ() ε Cd = = C11 + C111 ε +... ε (4.0) E 3 < 0 (positive thermal expansion coefficient) Cd ε = C111 < 0 For example, in aluminum C 11 11 1.1 10 N/ m and C 11 111 11 10 N/ m. While the longitudinal velocity along the direction of tensile stress decreases ( c dp, / σ < 0 ), because of the Poisson effect, it increases perpendicular to the stress direction ( c dn, / σ > 0 ).
Excess Nonlinearity Due to Material Imperfections ηtotal η int + η exc = η int + η int G + η crack (4.3) Δ c = c na3 A( θ ) (4.4) o Δ c velocity change c o velocity of intact material n denotes the crack density a is the average crack radius A( θ ) is an orientation function depending only on the Poisson's c σext = c 3 o na A( θ) a σext (4.5) na6ω4 α crack = A ( ) c4 α θ (4.6) 0 α crack is the excess attenuation ω denotes the angular frequency A α ( θ ) is another orientation function
Linear and Nonlinear Parameters Versus Fatigue Degradation in ABS Polymer Normalized Parameters 0 18 16 14 1 10 8 6 4 0 ABS Nonlinearity Linear Parameters 0 0. 0.4 0.6 0.8 1 1. 1.4 1.6 1.8 Number of Cycles [thousand] failure Normalized Parameters 1. 1.1 1 0.9 0.8 ABS Attenuation Velocity Static Modulus 0 0. 0.4 0.6 0.8 1 1. 1.4 1.6 1.8 Number of Cycles [thousand]
Crack-closure in a defective SMC plate (1/4"-thick, 3"-long segment of an automobile bumper) unloaded state 0 lbs 50 lbs 100 lbs loaded state 150 lbs Experimental arrangement for flaw identification from crack closure Force Sensor Ultrasonic Transducer Force Computer Ultrasonic Transmitter/Receiver
Ultrasonic signal versus external load characteristics of SMC plates and joints 600 Amplitude [mv] 500 400 300 00 100 0 plate good bond plate plate good bond good bond plate 0 0 40 60 80 100 Compressive Force [lb] 600 500 imperfect bonds Amplitude [mv] 400 300 00 100 0 0 0 40 60 80 100 Compressive Force [lb].