6.7 Construction of the Charts of Vibration Characters Verusus
Amplitude and Frequency of the Exciting Load and Their
Analysis (First-, Second- and Third-Order Kinematic
Hypotheses) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
6.7.1 Analysis of the Charts of the Vibration Regimes for the
Euler–Bernoulli, Timoshenko and Sheremetev–Pelekh
Modes Versus k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
6.7.2 Comparison of the Charts of the Vibration Regimes
for One Chosen Model Versus the Relative Length k
with Account of the Size-Dependent Parameter l=h . . . . . . 190
6.8 Influence of a Type of Kinematic Models of the Zero-,
First- and Third-Order Approximations on the Scenario
of Transition from Periodic to Chaotic Vibrations . . . . . . . . . . . . . 192
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
7 Mathematical Models of Functionally Graded Beams
in Temperature Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
7.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
7.3 Laws of Properties Change of FGM . . . . . . . . . . . . . . . . . . . . . . 202
7.3.1 Properties of Material P-FGM . . . . . . . . . . . . . . . . . . . . . 202
7.3.2 Properties of Material E-FGM . . . . . . . . . . . . . . . . . . . . . 204
7.3.3 Properties of S-FGM Material . . . . . . . . . . . . . . . . . . . . . 205
7.3.4 Properties of Porous Materials . . . . . . . . . . . . . . . . . . . . . 206
7.3.5 Homogenization of Properties of Graded Material
Based on Mori-Tanaka and Self-consistent Methods . . . . . 207
7.3.6 Dependence of Material Properties on Temperature . . . . . . 208
7.4 Chaotic Dynamics of Size-Dependent Timoshenko Beams
with Functionally Graded Properties Along Their Thickness . . . . . 210
7.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
7.4.2 Mathematical Background . . . . . . . . . . . . . . . . . . . . . . . . 212
7.4.3 Derivation of the Equations of Motion . . . . . . . . . . . . . . . 215
7.4.4 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . 217
7.4.5 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . 219
7.5 Stability of the Size-Dependent and Functionally Graded
Curvilinear Timoshenko Beams . . . . . . . . . . . . . . . . . . . . . . . . . . 231
7.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
7.5.2 Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . 233
7.5.3 Derivation of Equations of Motion . . . . . . . . . . . . . . . . . . 235
7.5.4 The Methods of Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 238
7.5.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
Contents
xix
Amplitude and Frequency of the Exciting Load and Their
Analysis (First-, Second- and Third-Order Kinematic
Hypotheses) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
6.7.1 Analysis of the Charts of the Vibration Regimes for the
Euler–Bernoulli, Timoshenko and Sheremetev–Pelekh
Modes Versus k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
6.7.2 Comparison of the Charts of the Vibration Regimes
for One Chosen Model Versus the Relative Length k
with Account of the Size-Dependent Parameter l=h . . . . . . 190
6.8 Influence of a Type of Kinematic Models of the Zero-,
First- and Third-Order Approximations on the Scenario
of Transition from Periodic to Chaotic Vibrations . . . . . . . . . . . . . 192
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
7 Mathematical Models of Functionally Graded Beams
in Temperature Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
7.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
7.3 Laws of Properties Change of FGM . . . . . . . . . . . . . . . . . . . . . . 202
7.3.1 Properties of Material P-FGM . . . . . . . . . . . . . . . . . . . . . 202
7.3.2 Properties of Material E-FGM . . . . . . . . . . . . . . . . . . . . . 204
7.3.3 Properties of S-FGM Material . . . . . . . . . . . . . . . . . . . . . 205
7.3.4 Properties of Porous Materials . . . . . . . . . . . . . . . . . . . . . 206
7.3.5 Homogenization of Properties of Graded Material
Based on Mori-Tanaka and Self-consistent Methods . . . . . 207
7.3.6 Dependence of Material Properties on Temperature . . . . . . 208
7.4 Chaotic Dynamics of Size-Dependent Timoshenko Beams
with Functionally Graded Properties Along Their Thickness . . . . . 210
7.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
7.4.2 Mathematical Background . . . . . . . . . . . . . . . . . . . . . . . . 212
7.4.3 Derivation of the Equations of Motion . . . . . . . . . . . . . . . 215
7.4.4 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . 217
7.4.5 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . 219
7.5 Stability of the Size-Dependent and Functionally Graded
Curvilinear Timoshenko Beams . . . . . . . . . . . . . . . . . . . . . . . . . . 231
7.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
7.5.2 Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . 233
7.5.3 Derivation of Equations of Motion . . . . . . . . . . . . . . . . . . 235
7.5.4 The Methods of Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 238
7.5.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
Contents
xix