TAM 551: Solid Mechanics I

Class Description:

Mechanics of elastic deformable bodies, based on the fundamental concepts of modern continuum mechanics: kinematics, balance laws, constitutive equations; classical small-deformation theory; formulation of initial boundary-value problems of linear elastodynamics and boundary-value problems of linear elastostatics; variational formulations, minimum principles; applications of theory to engineering problems. Prerequisite: TAM 251; MATH 380; MATH 385, MATH 386, or MATH 441. 4 graduate hours.

Textbook:

Recommended: Robert J. Asaro and Vlado A. Lubarda, Mechanics of Solids and Materials, Cambridge

P. Chadwick, Continuum Mechanics, Dover Publications

Topics:

Mathematical preliminaries (8 hr)
Indicial notation, vector and Cartesian tensors

Kinematics (10 hr)
Motions of a continuum, finite deformation, rigid-body motion and deformation, polar decomposition, homogenous and nonhomogeneous deformation, infinitesimal deformation, small-strain tensor, compatibility, transport and localization theorems

Balance laws (8 hr)
Conservation of mass, Cauchy's stress principle and Euler's laws, stress measures, Piola-Kirchhoff stress tensors, Cauchy's stress equations of motion, principle of virtual work, average theorem, applications, energy balance

Constitutive relationships (10 hr)
Invariance under superposed rigid-body motion, material symmetry, hyperelasticity, hypoelasticity, general anisotropic linear elasticity, incompressible elasticity, Cauchy relation, linearization of the finite elasticity equations, residual stresses, isotropic linear elasticity

Linear elastostatics (16 hr)
Fundamental field equations, elastic state, work and energy, the reciprocal theorem, boundary-value problems, uniqueness, the mixed problem in terms of displacements, the traction problem in terms of stress, variational formulation (minimum potential energy, minimum complementary energy); methods of Ritz, Galerkin, Kantorovich; the finite-element method

Linear elastodynamics (6 hr)
Fundamental field equations, elastic processes, boundary-initial-value problem, uniqueness

Midterm exam (2 hr)
TOTAL HOURS: 60

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