TABLE OF CONTENTS SECTION 1 INTRODUCTION... 1 1.1 Introduction... 1 1.2 Objectives... 1 1.3 Report organization... 2 SECTION 2 BACKGROUND AND LITERATURE REVIEW... 3 2.1 Introduction... 3 2.2 Wave propagation in layered structures... 3 2.3 Shape and topology optimization... 6 2.4 Genetic algorithm... 8 2.5 Summary... 11 SECTION 3 WAVE REFLECTION AND TRANSMISSION IN RODS... 13 3.1 Introduction... 13 3.2 Wave propagation in thin rods... 13 3.2.1 Spectral analysis... 15 3.3 Required relations in wave propagation analysis... 16 3.3.1 Phase speed... 18 3.3.2 Group Speed... 18 3.3.3 Mechanical relations... 20 3.4 Reflection and transmission at discontinuities of rod structures... 20 3.4.1 End boundaries in rods... 20 3.4.2 Impedance mismatch... 25 ix
3.5 Summary... 30 SECTION 4 WAVE REFLECTION ANDTRANSMISSION IN BEAMS... 31 4.1 Introduction... 31 4.2 Wave propagation in Timoshenko beams... 31 4.2.1 Governing differential equation of Timoshenko beams... 32 4.3 Reflection and transmission at discontinuities of Timoshenko beam structures. 37 4.3.1 End boundaries in Timoshenko beams... 37 4.3.2 Lumped mass at the end of a Timoshenko beam... 41 4.3.3 Stepped Timoshenko beam... 43 4.3.4 Angled joints in Timoshenko beams... 47 4.4 Summary... 60 SECTION 5 OPTIMIZATION METHODOLOGY FOR DESIGNING STRESS WAVE ATTENUATORS... 61 5.1 Introduction... 61 5.2 Elastic stress wave attenuators... 61 5.3 Genetic algorithms... 64 5.4 Proposed stress wave attenuators... 66 5.5 Design of stress wave attenuators using GA and FE... 68 5.5.1 Calculating fitness function... 69 5.6 Summary... 70 SECTION 6 DESIGN PARAMETERS FOR STRESS WAVE ATTENUATORS... 71 6.1 Introduction... 71 6.2 Design parameters... 71 x
6.3 Effect of relative length of each layer ( )... 73 6.4 Effect of the impedance mismatch ratio ( )... 74 6.5 Effect of the wavelength ratio ( )... 74 6.6 Effect of the rigidity of the host structure ( )... 74 6.7 Effect of the in-plane dimension parameter ( )... 75 6.8 Effect of the out-of-plane dimension (PS & PE)... 78 6.9 Examples for elastic stress wave attenuator design... 78 6.10 Summary... 83 SECTION 7 LAYERED COLLINEAR STRESS WAVE ATTENUATORS... 85 7.1 Introduction... 85 7.2 Optimal design parameters and characteristics of collinear stress wave attenuators... 85 7.3 Material properties... 87 7.4 Optimization procedure... 87 7.5 Results and discussion... 88 7.6 Three dimensional structures with layered collinear stress wave attenuators... 93 7.7 Summary... 96 SECTION 8 NON-COLLINEAR STRESS WAVE ATTENUATORS... 97 8.1 Introduction... 97 8.2 Non-collinear stress wave attenuators and effect of symmetry... 97 8.3 Layered diamond-shape stress wave attenuators... 102 8.3.1 Results and discussion... 103 8.4 Three dimensional structure with layered diamond-shape stress wave attenuators... 109 xi
8.5 Non-collinear single-layered stress wave attenuators... 112 8.5.1 Non-collinear stress wave attenuator with 1... 115 8.5.2 Non-collinear stress wave attenuator with 2... 120 8.5.3 Non-collinear stress wave attenuator with 3... 125 8.5.4 Non-collinear stress wave attenuator with 4... 129 8.5.5 Non-collinear stress wave attenuator with 5... 133 8.5.6 Non-collinear stress wave attenuator with 6... 138 8.5.7 Non-collinear stress wave attenuator with 7... 143 8.5.8 Non-collinear stress wave attenuator with 8... 148 8.5.9 Effect of on the attenuation capacity... 152 8.6 Summary... 152 SECTION 9 STRESS WAVE ATTENUATION IN POROUS PLATES... 155 9.1 Introduction... 155 9.2 Geometry optimization of porous plates for stress wave attenuation... 155 9.3 Design parameters for 2D porous stress wave attenuators... 158 9.4 2D porous stress wave attenuators with 1... 160 9.5 2D porous stress wave attenuators with 2... 166 9.6 2D porous stress wave attenuators with 3... 172 9.7 2D porous stress wave attenuators with 4... 178 9.8 Effect of on the attenuation capacity... 183 9.9 2D porous stress wave attenuators with 4, 1... 184 9.10 Verifying the coupled GA-FE optimization methodology using exhaustive search...... 190 9.11 Summary... 192 xii
SECTION 10 INTERFACE PROFILE OPTIMIZATION FOR STRESS WAVE ATTENUATION IN B -LAYERED PLATES...... 193 10.1 Introduction... 193 10.2 Theory and background... 193 10.3 Concept of interface profile optimization... 197 10.4 Problem Definition... 199 10.5 Optimization method... 205 10.6 FE modeling... 206 10.7 Coupled GA-FE methodology... 208 10.8 Results and discussion... 209 10.8.1 Effect of the length of the optimization zone ( )... 209 10.8.2 Effect of grid dimensions... 211 10.8.3 Effect of wavelength ratio ( )... 213 10.9 Summary... 220 SECTION 11 SUMMARY, CONCLUSION, AND RECOMMENDATIONS FOR FUTURE RESEARCH... 221 11.1 Summary... 221 11.2 Conclusion... 222 11.3 Recommendations for Future Research... 223 SECTION 12 REFRENCES... 225 xiii
LIST OF FIGURES 3-1 Straight prismatic thin rod with elastic properties... 13 3-2 Forces acting on a small element of a thin rod... 14 3-3 Elastic boundary condition (spring)... 24 3-4 Effect of lumped mass on reflection and transmission of waves... 25 3-5 Effect of changing material properties and cross section... 27 3-6 Effect of a finite rod between two long bars on wave propagation... 29 4-1 Kinematics of Timoshenko beam under flexure... 33 4-2 Pin, clamped, and free boundary conditions... 37 4-3 Moments and shear forces at boundaries when is the incident amplitude... 40 4-4 Moments and shear forces at boundaries when is the incident amplitude... 40 4-5 Lumped mass at the end of a Timoshenko beam... 41 4-6 Effect of lumped end mass, mass ratios 0, 0.5,1, and 2... 42 4-7 Effect of lumped end mass, mass ratios 5, 10, 50, and 100... 43 4-8 Stepped Timoshenko beam... 44 4-9 Changing of moment and shear when the wave enters from Aluminum to Steel... 46 4-10 Changing of moment and shear when the wave enters from Steel to Aluminum... 47 4-11 Two beams jointed at an arbitrary L joint... 48 4-12 Geometry of the L joint... 49 4-13 Longitudinal incident ( ),, L joint... 51 4-14 Propagating flexural incident ( ),, L joint... 51 xv
4-15 Longitudinal incident ( ),, L joint... 52 4-16 Propagating flexural incident ( ),, L joint... 52 4-17 Longitudinal incident ( ),, L joint... 53 4-18 Propagating flexural incident ( ),, L joint... 53 4-19 Three beams jointed at an arbitrary T joint... 54 4-20 Geometry of the T joint... 55 4-21 Longitudinal incident ( ),, T joint... 57 4-22 Propagating flexural incident ( ),, T joint... 58 4-23 Longitudinal incident ( ),, T joint... 58 4-24 Propagating flexural incident ( ),, T joint... 59 4-25 Longitudinal incident ( ),,, T joint... 59 4-26 Propagating flexural incident ( ),,, T joint... 60 5-1 Eight layered structure... 62 5-2 Stress history at the boundary of the structure in Figure 5-1 for three different setups... 64 5-3 Layered collinear stress wave attenuator... 66 5-4 Layered diamond-shape stress wave attenuator... 67 5-5 Non-collinear stress wave attenuator... 67 5-6 Two-dimensional porous stress wave attenuator... 68 5-7 Schematic of a layered stress wave attenuator... 69 6-1 Schematic of a stress wave attenuator and the design parameters... 71 6-2 Effect of on the peak force at the boundary of a thin stress wave attenuator with.... 73 xvi
6-3 Effect of,, and on the peak force at the boundary of a thin stress wave attenuator with.... 75 6-4 Effect of,, and on the peak force at the boundary of a thick stress wave attenuator with.... 76 6-5 Effect of,, and on the peak force at the boundary of a thin stress wave attenuator with.... 76 6-6 Effect of,, and on the peak force at the boundary of a thick stress wave attenuator with.... 77 6-7 Effect of,, and on the peak force at the boundary of a thin stress wave attenuator with.... 77 6-8 Effect of,, and on the peak force at the boundary of a thick stress wave attenuator with.... 78 6-9 Incident pulse time history and its frequency content for the stress wave attenuator design in Example 6-1... 79 6-10 Schematic of the stress wave attenuator in Example 6-1... 80 6-11 Incident pulse time history and its frequency content for the stress wave attenuator design in Example 6-2... 82 6-12 Schematic of the stress wave attenuator in Example 6-2... 82 7-1 Schematic of a collinear stress wave attenuator and its design parameters... 86 7-2 Optimal material string for the layered collinear stress wave attenuators... 89 7-3 Optimal design of the collinear layered stress wave attenuators... 90 7-4 Force history at the boundary of the optimized structures... 91 7-5 Attenuation vs.... 93 7-6 3D model with collinear stress wave attenuators... 94 7-7 Boundary condition and loading of the 3D model... 94 xvii
7-8 Two types of layered collinear structures used in the 3D model... 95 7-9 Force histories at the boundary of the 3D models with collinear stress wave attenuators... 95 8-1 Non-symmetric inclined structure... 98 8-2 3D Abaqus model of the non-symmetric inclined structure... 98 8-3 Element positions at the boundary... 99 8-4 Symmetric diamond-shape structure... 100 8-5 3D Abaqus model of the symmetric diamond-shape structure... 100 8-6 Schematic of a layered diamond-shape stress wave attenuator and its design parameters... 102 8-7 Optimal material strings for the layered diamond-shape stress wave attenuators... 104 8-8 Optimal design of the layered diamond-shape stress wave attenuators... 105 8-9 Force history at the boundary of the optimized diamond-shape structures... 106 8-10 Attenuation vs., Diamond-shape structure... 109 8-11 3D model with layered diamond-shape stress wave attenuators, a) dimensions, b) whole model... 109 8-12 Boundary condition and loading of the 3D model with diamond-shape stress wave attenuators... 110 8-13 Force histories at the boundary of the 3D models with diamond-shape stress wave attenuators... 111 8-14 Concept of geometry optimization for non-collinear stress wave attenuators 112 8-15 Example for the geometry optimization of non-collinear stress wave attenuators... 113 8-16 Optimization zone for the non-collinear structure with and.. 115 xviii
8-17 Optimal design of the non-collinear stress wave attenuators,... 116 8-18 Force history at the boundary of the optimized non-collinear structures with... 117 8-19 Attenuation vs., non-collinear structure with... 119 8-20 Optimization zone for the non-collinear structure with and.. 120 8-21 Optimal design of the non-collinear stress wave attenuators,... 121 8-22 Force history at the boundary of the optimized non-collinear structures with... 122 8-23 Attenuation vs., non-collinear structure with... 124 8-24 Optimization zone for the non-collinear structure with and.. 125 8-25 Optimal design of the non-collinear stress wave attenuators,... 126 8-26 Force history at the boundary of the optimized non-collinear structures with... 127 8-27 Attenuation vs., non-collinear structure with... 128 8-28 Optimization zone for the non-collinear structure with and.. 129 8-29 Optimal design of the non-collinear stress wave attenuators,... 130 8-30 Force history at the boundary of the optimized non-collinear structures with... 131 8-31 Attenuation vs., non-collinear structure with... 132 8-32 Optimization zone for the non-collinear structure with and.. 133 8-33 Optimal design of the non-collinear stress wave attenuators,... 134 8-34 Force history at the boundary of the optimized non-collinear structures with... 135 8-35 Attenuation vs., non-collinear structure with... 137 8-36 Optimization zone for the non-collinear structure with and... 138 8-37 Optimal design of the non-collinear stress wave attenuators,... 139 xix
8-38 Force history at the boundary of the optimized non-collinear structures with... 140 8-39 Attenuation vs., non-collinear structure with... 142 8-40 Optimization zone for the non-collinear structure with and... 143 8-41 Optimal design of the non-collinear stress wave attenuators,... 144 8-42 Force history at the boundary of the optimized non-collinear structures with... 145 8-43 Attenuation vs., non-collinear structure with... 147 8-44 Optimization zone for the non-collinear structure with and... 148 8-45 Optimal design of the non-collinear stress wave attenuators,... 149 8-46 Force history at the boundary of the optimized non-collinear structures with... 150 8-47 Attenuation vs., non-collinear structure with... 152 9-1 Concept of geometry optimization for 2D porous stress wave attenuators... 156 9-2 Example for geometry optimization of 2D porous stress wave attenuators... 157 9-3 Schematic of a 2D porous stress wave attenuator and its design parameters... 159 9-4 Optimization zone for the 2D porous structure with,, and... 160 9-5 Optimal design of the 2D porous stress wave attenuators,... 162 9-6 Force history at the boundary of the optimized 2D porous structures with... 163 9-7 Attenuation vs., 2D Porous structure with... 165 9-8 Optimization zone for the porous structure with,, and 166 9-9 Optimal design of the 2D porous stress wave attenuators,... 167 xx
9-10 Force history at the boundary of the optimized 2D porous structures with... 169 9-11 Attenuation vs., 2D Porous structure with... 171 9-12 Optimization zone for the porous structure with,, and... 172 9-13 Optimal design of the 2D porous stress wave attenuators,... 173 9-14 Force history at the boundary of the optimized 2D porous structures with... 174 9-15 Attenuation vs., 2D Porous structure with... 177 9-16 Optimization zone for the porous structure with,, and... 178 9-17 Optimal design of the 2D porous stress wave attenuators,... 179 9-18 Force history at the boundary of the optimized 2D porous structures with... 180 9-19 Attenuation vs., 2D Porous structure with... 183 9-20 Optimization zone for the porous structure with,, and... 184 9-21 Optimal design of the 2D porous stress wave attenuators,,... 186 9-22 Force history at the boundary of the optimized 2D porous structures with,... 187 9-23 Attenuation vs., 2D Porous structure with... 189 9-24 Exhaustive search results for 2D porous stress wave attenuators with and... 191 10-1 Reflection and transmission of waves at the interface of two solids, a) dilatational incident wave, b) shear incident wave... 195 xxi
10-2 Reflection and transmission stress coefficients for incident dilatational wave on AL-HDPE interface for varying incident angle... 197 10-3 Reflection and transmission stress coefficients for incident shear wave on AL- HDPE interface for varying incident angle... 197 10-4 General bi-layered rectangular plate and its optimization zone... 199 10-5 Example for the optimized interface between the two layers... 199 10-6 Bi-layered plates with a)., b)., c)... 203 10-7 Bi-layered plates with. and,, and... 205 10-8 Optimal designs of the bi-layered plates in Figure 10-6 for., a)., b)., c).... 211 10-9 Optimal design of the bi-layered plates in Figure 10-7 for.... 213 10-10 Structure, a) schematic of the optimal design, b) force history at the boundary for.... 216 10-11 Structure, a) schematic of the optimal design, b) force history at the boundary for.... 217 10-12 Structure, a) schematic of the optimal design, b) force history at the boundary for.... 218 10-13 Structure, a) schematic of the optimal design, b) force history at the boundary for.... 219 10-14 Attenuation- curve for the optimized structures... 220 xxii
LIST OF TABLES Table 3-1 Summary of wave relations... 17 Table 3-2 Properties of some materials for wave propagation analysis... 19 Table 3-3 Mechanical relationships for an elastic material with small deformations assumption... 20 Table 3-4 Some boundary condition properties (Doyle (1989))... 21 Table 3-5 Comparison of the fixed and free boundary conditions in rods... 23 Table 4-1 End boundary condition properties for Timoshenko beams... 38 Table 6-1 Peak force at the boundary of the structure in Example 6.1, PS = Plane Stress & PE = Plane Strain... 81 Table 6-2 Peak force at the boundary of the structure in Example 6-2... 83 Table 7-1 Materials used for optimal design of the layered collinear stress wave attenuators... 87 Table 7-2 Wavelength ratios and duration of the applied pulses... 89 Table 7-3 Amount of attenuation of the optimized structures for different values of... 93 Table 8-1 Normalized stress at the boundary of the non-symmetric structure... 99 Table 8-2 Normalized stress at the boundary of the symmetric diamond-shape structure... 101 xxiii
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... 103 Attenuation at the optimized layered diamond-shape structures for different values... 108 Wavelength ratios and duration of the applied pulses on the non-collinear structures... 114 Optimized vertical position string for the non-collinear structure with... 115 Attenuation at the optimized non-collinear structures for different values,... 119 Optimized vertical position string for the non-collinear structure with... 120 Attenuation at the optimized non-collinear structures for different values,... 123 Optimized vertical position string for the non-collinear structure with... 126 Attenuation at the optimized non-collinear structures for different values,... 128 Optimized vertical position string for the non-collinear structure with... 129 Attenuation at the optimized non-collinear structures for different values,... 132 Optimized vertical position string for the non-collinear structure with... 134 Attenuation at the optimized non-collinear structures for different values,... 136 Optimized vertical position string for the non-collinear structure with... 139 xxiv
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,... 141 Optimized vertical position string for the non-collinear structure with... 144 Attenuation at the optimized non-collinear structures for different values,... 146 Optimized vertical position string for the non-collinear structure with... 149 Attenuation at the optimized non-collinear structures for different values,... 151 Table 8-22 Attenuation range for various values... 152 Table 9-1 Wavelength ratios and duration of the applied pulses on the 2D porous stress wave attenuators... 158 Table 9-2 Optimized diameter string for the 2D porous structure with... 161 Table 9-3 of the optimized structures for different values of, 2D porous structure ( )... 164 Table 9-4 Table 9-5 Attenuation at the optimized 2D porous structures for different values, ( )... 164 Optimized position and diameter string for the 2D porous structure with... 167 Table 9-6 F F of the optimized structures for different values of R, 2D porous Table 9-7 structure (n 2)... 170 Attenuation at the optimized 2D porous structures for different values, ( )...... 170 Table 9-8 Optimized position and diameter string for the 2D porous structure with... 173 xxv
Table 9-9 of the optimized structures for different values of, 2D porous structure ( )... 176 Table 9-10 Table 9-11 Attenuation at the optimized 2D porous structures for different values, ( )... 176 Optimized position and diameter string for the 2D porous structure with... 179 Table 9-12 of the optimized structures for different values of, 2D porous structure ( )... 182 Table 9-13 Table 9-14 Table 9-15 Attenuation at the optimized 2D porous structures for different values, ( )... 182 Global attenuation range for different values, 2D porous stress wave attenuators... 183 Optimized position and diameter string for the 2D porous structure with and... 184 Table 9-16 of the optimized structures for different values of, 2D porous structure (, )... 188 Table 9-17 Table 9-18 Amount of attenuation of the optimized structures for different values of, 2D porous structure (, )... 188 Position and diameter string for the 2D porous structure with and from the exhaustive search... 191 Table 10-1 Mechanical properties of the plate materials... 200 Table 10-2 Duration of half-sine loadings ( ) and duration of analysis ( ) for different values of wavelength ratios... 204 Table 10-3 Minimum element size of the mesh for different values of... 208 xxvi
Table 10-4 Table 10-5 Optimization string and attenuation capacity for different values of... 210 Optimization string and attenuation capacity for different values of and... 212 Table 10-6 Optimization string and attenuation capacity of the structure in Figure 10-6a for different values of... 214 Table 10-7 Amount of attenuation (%) in the optimized structures for different values of... 219 xxvii