TABLE OF CONTENTS SECTION 2 BACKGROUND AND LITERATURE REVIEW... 3 SECTION 3 WAVE REFLECTION AND TRANSMISSION IN RODS Introduction...

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1 TABLE OF CONTENTS SECTION 1 INTRODUCTION Introduction Objectives Report organization... 2 SECTION 2 BACKGROUND AND LITERATURE REVIEW Introduction Wave propagation in layered structures Shape and topology optimization Genetic algorithm Summary SECTION 3 WAVE REFLECTION AND TRANSMISSION IN RODS Introduction Wave propagation in thin rods Spectral analysis Required relations in wave propagation analysis Phase speed Group Speed Mechanical relations Reflection and transmission at discontinuities of rod structures End boundaries in rods Impedance mismatch ix

2 3.5 Summary SECTION 4 WAVE REFLECTION ANDTRANSMISSION IN BEAMS Introduction Wave propagation in Timoshenko beams Governing differential equation of Timoshenko beams Reflection and transmission at discontinuities of Timoshenko beam structures End boundaries in Timoshenko beams Lumped mass at the end of a Timoshenko beam Stepped Timoshenko beam Angled joints in Timoshenko beams Summary SECTION 5 OPTIMIZATION METHODOLOGY FOR DESIGNING STRESS WAVE ATTENUATORS Introduction Elastic stress wave attenuators Genetic algorithms Proposed stress wave attenuators Design of stress wave attenuators using GA and FE Calculating fitness function Summary SECTION 6 DESIGN PARAMETERS FOR STRESS WAVE ATTENUATORS Introduction Design parameters x

3 6.3 Effect of relative length of each layer ( ) Effect of the impedance mismatch ratio ( ) Effect of the wavelength ratio ( ) Effect of the rigidity of the host structure ( ) Effect of the in-plane dimension parameter ( ) Effect of the out-of-plane dimension (PS & PE) Examples for elastic stress wave attenuator design Summary SECTION 7 LAYERED COLLINEAR STRESS WAVE ATTENUATORS Introduction Optimal design parameters and characteristics of collinear stress wave attenuators Material properties Optimization procedure Results and discussion Three dimensional structures with layered collinear stress wave attenuators Summary SECTION 8 NON-COLLINEAR STRESS WAVE ATTENUATORS Introduction Non-collinear stress wave attenuators and effect of symmetry Layered diamond-shape stress wave attenuators Results and discussion Three dimensional structure with layered diamond-shape stress wave attenuators xi

4 8.5 Non-collinear single-layered stress wave attenuators Non-collinear stress wave attenuator with Non-collinear stress wave attenuator with Non-collinear stress wave attenuator with Non-collinear stress wave attenuator with Non-collinear stress wave attenuator with Non-collinear stress wave attenuator with Non-collinear stress wave attenuator with Non-collinear stress wave attenuator with Effect of on the attenuation capacity Summary SECTION 9 STRESS WAVE ATTENUATION IN POROUS PLATES Introduction Geometry optimization of porous plates for stress wave attenuation Design parameters for 2D porous stress wave attenuators D porous stress wave attenuators with D porous stress wave attenuators with D porous stress wave attenuators with D porous stress wave attenuators with Effect of on the attenuation capacity D porous stress wave attenuators with 4, Verifying the coupled GA-FE optimization methodology using exhaustive search Summary xii

5 SECTION 10 INTERFACE PROFILE OPTIMIZATION FOR STRESS WAVE ATTENUATION IN B -LAYERED PLATES Introduction Theory and background Concept of interface profile optimization Problem Definition Optimization method FE modeling Coupled GA-FE methodology Results and discussion Effect of the length of the optimization zone ( ) Effect of grid dimensions Effect of wavelength ratio ( ) Summary SECTION 11 SUMMARY, CONCLUSION, AND RECOMMENDATIONS FOR FUTURE RESEARCH Summary Conclusion Recommendations for Future Research SECTION 12 REFRENCES xiii

7 LIST OF FIGURES 3-1 Straight prismatic thin rod with elastic properties Forces acting on a small element of a thin rod Elastic boundary condition (spring) Effect of lumped mass on reflection and transmission of waves Effect of changing material properties and cross section Effect of a finite rod between two long bars on wave propagation Kinematics of Timoshenko beam under flexure Pin, clamped, and free boundary conditions Moments and shear forces at boundaries when is the incident amplitude Moments and shear forces at boundaries when is the incident amplitude Lumped mass at the end of a Timoshenko beam Effect of lumped end mass, mass ratios 0, 0.5,1, and Effect of lumped end mass, mass ratios 5, 10, 50, and Stepped Timoshenko beam Changing of moment and shear when the wave enters from Aluminum to Steel Changing of moment and shear when the wave enters from Steel to Aluminum Two beams jointed at an arbitrary L joint Geometry of the L joint Longitudinal incident ( ),, L joint Propagating flexural incident ( ),, L joint xv

8 4-15 Longitudinal incident ( ),, L joint Propagating flexural incident ( ),, L joint Longitudinal incident ( ),, L joint Propagating flexural incident ( ),, L joint Three beams jointed at an arbitrary T joint Geometry of the T joint Longitudinal incident ( ),, T joint Propagating flexural incident ( ),, T joint Longitudinal incident ( ),, T joint Propagating flexural incident ( ),, T joint Longitudinal incident ( ),,, T joint Propagating flexural incident ( ),,, T joint Eight layered structure Stress history at the boundary of the structure in Figure 5-1 for three different setups Layered collinear stress wave attenuator Layered diamond-shape stress wave attenuator Non-collinear stress wave attenuator Two-dimensional porous stress wave attenuator Schematic of a layered stress wave attenuator Schematic of a stress wave attenuator and the design parameters Effect of on the peak force at the boundary of a thin stress wave attenuator with xvi

9 6-3 Effect of,, and on the peak force at the boundary of a thin stress wave attenuator with Effect of,, and on the peak force at the boundary of a thick stress wave attenuator with Effect of,, and on the peak force at the boundary of a thin stress wave attenuator with Effect of,, and on the peak force at the boundary of a thick stress wave attenuator with Effect of,, and on the peak force at the boundary of a thin stress wave attenuator with Effect of,, and on the peak force at the boundary of a thick stress wave attenuator with Incident pulse time history and its frequency content for the stress wave attenuator design in Example Schematic of the stress wave attenuator in Example Incident pulse time history and its frequency content for the stress wave attenuator design in Example Schematic of the stress wave attenuator in Example Schematic of a collinear stress wave attenuator and its design parameters Optimal material string for the layered collinear stress wave attenuators Optimal design of the collinear layered stress wave attenuators Force history at the boundary of the optimized structures Attenuation vs D model with collinear stress wave attenuators Boundary condition and loading of the 3D model xvii

10 7-8 Two types of layered collinear structures used in the 3D model Force histories at the boundary of the 3D models with collinear stress wave attenuators Non-symmetric inclined structure D Abaqus model of the non-symmetric inclined structure Element positions at the boundary Symmetric diamond-shape structure D Abaqus model of the symmetric diamond-shape structure Schematic of a layered diamond-shape stress wave attenuator and its design parameters Optimal material strings for the layered diamond-shape stress wave attenuators Optimal design of the layered diamond-shape stress wave attenuators Force history at the boundary of the optimized diamond-shape structures Attenuation vs., Diamond-shape structure D model with layered diamond-shape stress wave attenuators, a) dimensions, b) whole model Boundary condition and loading of the 3D model with diamond-shape stress wave attenuators Force histories at the boundary of the 3D models with diamond-shape stress wave attenuators Concept of geometry optimization for non-collinear stress wave attenuators Example for the geometry optimization of non-collinear stress wave attenuators Optimization zone for the non-collinear structure with and xviii

11 8-17 Optimal design of the non-collinear stress wave attenuators, Force history at the boundary of the optimized non-collinear structures with Attenuation vs., non-collinear structure with Optimization zone for the non-collinear structure with and Optimal design of the non-collinear stress wave attenuators, Force history at the boundary of the optimized non-collinear structures with Attenuation vs., non-collinear structure with Optimization zone for the non-collinear structure with and Optimal design of the non-collinear stress wave attenuators, Force history at the boundary of the optimized non-collinear structures with Attenuation vs., non-collinear structure with Optimization zone for the non-collinear structure with and Optimal design of the non-collinear stress wave attenuators, Force history at the boundary of the optimized non-collinear structures with Attenuation vs., non-collinear structure with Optimization zone for the non-collinear structure with and Optimal design of the non-collinear stress wave attenuators, Force history at the boundary of the optimized non-collinear structures with Attenuation vs., non-collinear structure with Optimization zone for the non-collinear structure with and Optimal design of the non-collinear stress wave attenuators, xix

12 8-38 Force history at the boundary of the optimized non-collinear structures with Attenuation vs., non-collinear structure with Optimization zone for the non-collinear structure with and Optimal design of the non-collinear stress wave attenuators, Force history at the boundary of the optimized non-collinear structures with Attenuation vs., non-collinear structure with Optimization zone for the non-collinear structure with and Optimal design of the non-collinear stress wave attenuators, Force history at the boundary of the optimized non-collinear structures with Attenuation vs., non-collinear structure with Concept of geometry optimization for 2D porous stress wave attenuators Example for geometry optimization of 2D porous stress wave attenuators Schematic of a 2D porous stress wave attenuator and its design parameters Optimization zone for the 2D porous structure with,, and Optimal design of the 2D porous stress wave attenuators, Force history at the boundary of the optimized 2D porous structures with Attenuation vs., 2D Porous structure with Optimization zone for the porous structure with,, and Optimal design of the 2D porous stress wave attenuators, xx

13 9-10 Force history at the boundary of the optimized 2D porous structures with Attenuation vs., 2D Porous structure with Optimization zone for the porous structure with,, and Optimal design of the 2D porous stress wave attenuators, Force history at the boundary of the optimized 2D porous structures with Attenuation vs., 2D Porous structure with Optimization zone for the porous structure with,, and Optimal design of the 2D porous stress wave attenuators, Force history at the boundary of the optimized 2D porous structures with Attenuation vs., 2D Porous structure with Optimization zone for the porous structure with,, and Optimal design of the 2D porous stress wave attenuators,, Force history at the boundary of the optimized 2D porous structures with, Attenuation vs., 2D Porous structure with Exhaustive search results for 2D porous stress wave attenuators with and Reflection and transmission of waves at the interface of two solids, a) dilatational incident wave, b) shear incident wave xxi

14 10-2 Reflection and transmission stress coefficients for incident dilatational wave on AL-HDPE interface for varying incident angle Reflection and transmission stress coefficients for incident shear wave on AL- HDPE interface for varying incident angle General bi-layered rectangular plate and its optimization zone Example for the optimized interface between the two layers Bi-layered plates with a)., b)., c) Bi-layered plates with. and,, and Optimal designs of the bi-layered plates in Figure 10-6 for., a)., b)., c) Optimal design of the bi-layered plates in Figure 10-7 for Structure, a) schematic of the optimal design, b) force history at the boundary for Structure, a) schematic of the optimal design, b) force history at the boundary for Structure, a) schematic of the optimal design, b) force history at the boundary for Structure, a) schematic of the optimal design, b) force history at the boundary for Attenuation- curve for the optimized structures xxii

15 LIST OF TABLES Table 3-1 Summary of wave relations Table 3-2 Properties of some materials for wave propagation analysis Table 3-3 Mechanical relationships for an elastic material with small deformations assumption Table 3-4 Some boundary condition properties (Doyle (1989)) Table 3-5 Comparison of the fixed and free boundary conditions in rods Table 4-1 End boundary condition properties for Timoshenko beams Table 6-1 Peak force at the boundary of the structure in Example 6.1, PS = Plane Stress & PE = Plane Strain Table 6-2 Peak force at the boundary of the structure in Example Table 7-1 Materials used for optimal design of the layered collinear stress wave attenuators Table 7-2 Wavelength ratios and duration of the applied pulses Table 7-3 Amount of attenuation of the optimized structures for different values of Table 8-1 Normalized stress at the boundary of the non-symmetric structure Table 8-2 Normalized stress at the boundary of the symmetric diamond-shape structure xxiii

16 Table 8-3 Table 8-4 Table 8-5 Table 8-6 Table 8-7 Table 8-8 Table 8-9 Table 8-10 Table 8-11 Table 8-12 Table 8-13 Table 8-14 Table 8-15 Table 8-16 Optimal material strings for the layered diamond-shape stress wave attenuators Attenuation at the optimized layered diamond-shape structures for different values Wavelength ratios and duration of the applied pulses on the non-collinear structures Optimized vertical position string for the non-collinear structure with Attenuation at the optimized non-collinear structures for different values, Optimized vertical position string for the non-collinear structure with Attenuation at the optimized non-collinear structures for different values, Optimized vertical position string for the non-collinear structure with Attenuation at the optimized non-collinear structures for different values, Optimized vertical position string for the non-collinear structure with Attenuation at the optimized non-collinear structures for different values, Optimized vertical position string for the non-collinear structure with Attenuation at the optimized non-collinear structures for different values, Optimized vertical position string for the non-collinear structure with xxiv

17 Table 8-17 Table 8-18 Table 8-19 Table 8-20 Table 8-21 Attenuation at the optimized non-collinear structures for different values, Optimized vertical position string for the non-collinear structure with Attenuation at the optimized non-collinear structures for different values, Optimized vertical position string for the non-collinear structure with Attenuation at the optimized non-collinear structures for different values, Table 8-22 Attenuation range for various values Table 9-1 Wavelength ratios and duration of the applied pulses on the 2D porous stress wave attenuators Table 9-2 Optimized diameter string for the 2D porous structure with Table 9-3 of the optimized structures for different values of, 2D porous structure ( ) Table 9-4 Table 9-5 Attenuation at the optimized 2D porous structures for different values, ( ) Optimized position and diameter string for the 2D porous structure with Table 9-6 F F of the optimized structures for different values of R, 2D porous Table 9-7 structure (n 2) Attenuation at the optimized 2D porous structures for different values, ( ) Table 9-8 Optimized position and diameter string for the 2D porous structure with xxv

18 Table 9-9 of the optimized structures for different values of, 2D porous structure ( ) Table 9-10 Table 9-11 Attenuation at the optimized 2D porous structures for different values, ( ) Optimized position and diameter string for the 2D porous structure with Table 9-12 of the optimized structures for different values of, 2D porous structure ( ) Table 9-13 Table 9-14 Table 9-15 Attenuation at the optimized 2D porous structures for different values, ( ) Global attenuation range for different values, 2D porous stress wave attenuators Optimized position and diameter string for the 2D porous structure with and Table 9-16 of the optimized structures for different values of, 2D porous structure (, ) Table 9-17 Table 9-18 Amount of attenuation of the optimized structures for different values of, 2D porous structure (, ) Position and diameter string for the 2D porous structure with and from the exhaustive search Table 10-1 Mechanical properties of the plate materials Table 10-2 Duration of half-sine loadings ( ) and duration of analysis ( ) for different values of wavelength ratios Table 10-3 Minimum element size of the mesh for different values of xxvi

19 Table 10-4 Table 10-5 Optimization string and attenuation capacity for different values of Optimization string and attenuation capacity for different values of and Table 10-6 Optimization string and attenuation capacity of the structure in Figure 10-6a for different values of Table 10-7 Amount of attenuation (%) in the optimized structures for different values of xxvii

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