## Continuum Mechanics for Engineers, Third EditionThe second edition of this popular text continues to provide a solid, fundamental introduction to the mathematics, laws, and applications of continuum mechanics. With the addition of three new chapters and eight new sections to existing chapters, the authors now provide even better coverage of continuum mechanics basics and focus even more attention on its applications. Beginning with the basic mathematical tools needed-including matrix methods and the algebra and calculus of Cartesian tensors-the authors develop the principles of stress, strain, and motion and derive the fundamental physical laws relating to continuity, energy, and momentum. With this basis established, they move to their expanded treatment of applications, including linear and nonlinear elasticity, fluids, and linear viscoelasticity Mastering the contents of Continuum Mechanics: Second Edition provides the reader with the foundation necessary to be a skilled user of today's advanced design tools, such as sophisticated simulation programs that use nonlinear kinematics and a variety of constitutive relationships. With its ample illustrations and exercises, it offers the ideal self-study vehicle for practicing engineers and an excellent introductory text for advanced engineering students. |

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Solid Mechanics

### Contents

Continuum Theory | 1 |

12 Continuum Mechanics | 2 |

Essential Mathematics | 3 |

22 Tensor Algebra in Symbolic Notation Summation Convention | 4 |

23 Indicial Notation | 13 |

24 Matrices and Determinants | 16 |

25 Transformations of Cartesian Tensors | 22 |

26 Principal Values and Principal Directions of Symmetric SecondOrder Tensors | 28 |

56 Moment of Momentum Angular Momentum Principle | 181 |

57 Law of Conservation of Energy The Energy Equation | 182 |

58 Entropy and the ClausiusDuhem Equation | 186 |

59 Restrictions on Elastic Materials by the Second Law of Thermodynamics | 190 |

510 Invariance | 194 |

511 Restrictions on Constitutive Equations from Invariance | 204 |

512 Constitutive Equations | 207 |

References | 210 |

27 Tensor Fields Tensor Calculus | 34 |

28 Integral Theorems of Gauss and Stokes | 36 |

Problems | 37 |

Stress Principles | 47 |

32 Cauchy Stress Principle | 48 |

33 The Stress Tensor | 51 |

34 Force and Moment Equilibrium Stress Tensor Symmetry | 57 |

35 Stress Transformation Laws | 59 |

36 Principal Stresses Principal Stress Directions | 62 |

37 Maximum and Minimum Stress Values | 70 |

38 Mohrs Circles For Stress | 73 |

39 Plane Stress | 80 |

310 Deviator and Spherical Stress States | 85 |

311 Octahedral Shear Stress | 87 |

Problems | 89 |

Kinematics of Deformation and Motion | 103 |

42 Material and Spatial Coordinates | 104 |

43 Lagrangian and Eulerian Descriptions | 109 |

44 The Displacement Field | 111 |

45 The Material Derivative | 113 |

46 Deformation Gradients Finite Strain Tensors | 116 |

47 Infinitesimal Deformation Theory | 122 |

48 Stretch Ratios | 131 |

49 Rotation Tensor Stretch Tensors | 136 |

410 Velocity Gradient Rate of Deformation Vorticity | 140 |

411 Material Derivative of Line Elements Areas Volumes | 146 |

Problems | 149 |

Fundamental Laws and Equations | 169 |

52 Material Derivatives of Line Surface and Volume Integrals | 170 |

53 Conservation of Mass Continuity Equation | 172 |

54 Linear Momentum Principle Equations of Motion | 175 |

55 The PiolaKirchhoff Stress Tensors Lagrangian Equations of Motion | 176 |

Linear Elasticity | 219 |

62 Hookes Law for Isotropic Media Elastic Constants | 224 |

63 Elastic Symmetry Hookes Law for Anisotropic Media | 230 |

64 Isotropic Elastostatics and Elastodynamics Superposition Principle | 235 |

65 Plane Elasticity | 238 |

66 Linear Thermoelasticity | 242 |

67 Airy Stress Function | 244 |

68 Torsion | 256 |

69 ThreeDimensional Elasticity | 264 |

Problems | 273 |

Classical Fluids | 285 |

72 Basic Equations of Viscous Flow NavierStokes Equations | 288 |

73 Specialized Fluids | 290 |

74 Steady Flow Irrotational Flow Potential Flow | 291 |

75 The Bernoulli Equation Kelvins Theorem | 295 |

Problems | 296 |

Nonlinear Elasticity | 301 |

82 A Strain Energy Theory for Nonlinear Elasticity | 309 |

83 Specific Forms of the Strain Energy | 314 |

84 Exact Solution for an Incompressible NeoHookean Material | 316 |

References | 324 |

Linear Viscoelasticity | 329 |

92 Viscoelastic Constitutive Equations in Linear Differential Operator Form | 330 |

93 OneDimensional Theory Mechanical Models | 332 |

94 Creep and Relaxation | 336 |

95 Superposition Principle Hereditary Integrals | 342 |

96 Harmonic Loadings Complex Modulus and Complex Compliance | 344 |

97 ThreeDimensional Problems The Correspondence Principle | 350 |

References | 357 |

Problems | 358 |

373 | |

### Common terms and phrases

angle Answer arbitrary assumed axes lexzx3 axis body forces boundary conditions Cartesian Cartesian tensors Cauchy stress constitutive equation continuity equation continuum mechanics coordinate deﬁned deﬁnition deformation gradient deformation tensor determine differential elastic symmetry element equations of motion equilibrium equations Example expressed ﬁrst ﬂow ﬂuid given Hooke’s law incompressible indices indicial notation infinitesimal integral invariant inverse isotropic Kelvin Kronecker delta linear material derivative matrix form modulus obtain orthogonal particle plane strain plane stress principal axes principal directions principal stress values principal values problems reference conﬁguration respect result rigid body motion rotation rubber satisﬁed scalar second-order tensor shear stress Show shown in Figure simple shear solution spatial speciﬁc strain energy strain tensor stress components stress function stress tensor stress vector stretch substitution superposed rigid body surface symmetric temperature unit normal unit vector velocity ﬁeld viscoelastic viscous volume written zero