Characterization of Acoustic Band Structure in Layered Composites Subjected to DynamicLoading

Jimmy Pan, Ruize Hu, Dr. Caglar OskayCivil and Environmental Engineering Department, Vanderbilt University, Nashville, TN

Motivation and Objectives

I Motivation. Continuation of a nonlocal homogenization model for bimaterial composite

structures with the purpose of blast and impact mitigation. Examination of the effect of material microstructure and properties on wave

dispersion and attenuation responses. Numerous military and structural applications including cloaking, impact and

blast resistance, and health monitoring

I Objectives. Model the bandgap structure arising from the difference in material properties

(e.g. density and modulus of elasticity) of bilayered materials. Define a function of material properties (e.g. impedance and wave velocity)

that approximates a parameter ν used to model bandgap structure

Problem Statement and Approach

I Problem Statement. Analyze the dispersion and attenuation characteristics of a bilayer composite

structure experiencing one-dimensional wave propagation

Figure: One dimensional periodic composite structure (Hui 2013)

. For each case in a range of material combinations, determine the value of theparameter ν between 0 and 1 that gives the experimental model the closest fitto the Floquet-Bloch reference model

. ν is a parameter which determines the respective contribution of two nonlocalequilibrium equations that compose a sixth order dispersion equation (i.e. aweight factor)

Figure: Effect of ν on Bandgap Model Figure: Dispersion Relation for Aluminum -Polymer Combination

I Research Approach

1. Compose MATLAB script that calculates the ν value resulting in the best fitfor all desired material combinations

2. Plot the best ν values against parameters defined as material properties inorder to give ν physical meaning

3. Curve fit the data to obtain a function that approximates ν

Terminology

Bandgap: the frequency band within which the dynamic response is significantly attenuated.Arises from interaction between incoming waves and scattered waves due to reflection andrefraction at constituent material interfaces

Attenuation: reduction in the strength of the dynamic response

Nonlocal: in mathematical homogenization theory, refers to defining a mean displacement basedon the macroscale displacements and solving for the mean displacement rather than solvingeach equilibrium equation sequentially at each order; only one nonlocal equilibrium equation issolved

Generating Best Fit ν Data

I Minimizing the Objective Function. Objective Function:

Obj = C |y1exp − y1FB | + |y2exp − y2FB |y1exp: The experimental model’s bandgap initiation pointy2exp: The experimental model’s bandgap endpointy1FB: The Floquet-Bloch reference model’s bandgap initiation pointy2FB: The Floquet-Bloch reference model’s bandgap endpoint

C: Weight factor. Set to 1 because an equally weighted objective function yielded thelowest errors

. Used MATLAB’s fminbnd function to minimize Obj and return the corresponding ν valueI Range of Material Combinations. Held one layer constant as a specific material and varied the second layer’s material

properties within the desired material class according to the Ashby Materials Selection plot. For example: Aluminum - PolymerE1 = 68 GPaρ1 = 2700 kg/m3

E2 = 1.568 GPaρ2 = 1225 kg/m3

α (volume fraction) = 0.5l (unit cell length) = 0.01 mThe ν value resulting in the best fit for thisscenario was calculated to be 0.3154. Togenerate the full set of Aluminum - Polymerdata, set E2 as an array of equally spacedintervals between 0.08 and 10 GPa, and set ρ2 asan array of equally spaced intervals between 800and 2500 kg/m3 Figure: Young’s Modulus (E ) vs. Density (ρ) Ashby

Materials Selection Plot (University of CambridgeDepartment of Engineering 2002)

I Best ν Data. For each material combination, the best ν value was recorded and plotted against

impedance z and wave velocity c

Figure: Best Fit ν Data as a Function of z and c

. Impedancez =√E × ρ

. Wave velocityc =

√E/ρ

. z and c are parameters that measurethe contrast between the compositestructure materials’ impedance andwave velocity, respectively

z = max(z1

z2,z2

z1)

c = max(c1

c2,c2

c1)

Curve Fitting

I Fitted Function. Used MATLAB’s curve fitting toolbox

and nonlinear least squares method toestablish a function that canapproximate ν from z and c

ν(z, c) =1

−0.6181z + 2.559c+ 0.1161

. ν(z, c) is capable of approximating allbest fitting ν data except for Al-Metals,which proved difficult to model due tosmall bandgap sizes

Figure: Curve Fitted Function ν(z, c) Overlaying DataI Calculating Error. Error data was calculated by using ν(z, c) to project each best fitting ν value and

applying the error equation:

Ψ =

√(y1exp − y1FB)2 + (y2exp − y2FB)2

y1FB2 + y2FB2

Figure: Overall Error as a Function of z and c

Figure: Mean and Standard Deviation of ErrorData

Predicting Bandgap Size

I Impedance Contrast Effect on Bandgap Size

. Greater contrast in the materials’impedances implies larger bandgap size

. For this figure, z = z1/z2 and c =c1/c2 inorder to avoid graphical confusion (i.e.the surface plot ”folding back” overitself). Values closer to unity indicatelower contrast

Figure: z and c Effect on Bandgap Size

Conclusions, Future Work, and References

I Conclusions. Established a function ν(z, c) that returns a ν value giving the experimental model the

closest fit to the reference Floquet-Bloch model with error below 10% and typicallybetween 2-5%

. Demonstrated relationship between impedance contrast and bandgap sizeI Future Work. Implement three or more material layers into the sixth order model and progress into core

shell structureI References. Tong Hui, Caglar Oskay. ”A nonlocal homogenization model for wave dispersion in

dissipative composite materials.” International Journal of Solids and Structures, 2013:50(1):38-48.

Multiscale Computational Mechanics Laboratory / Vanderbilt Multiscale Modeling and Simulation (MUMS) Center