Lecture meets 2 times per week for 75 minutes per meeting for 15 weeks
Specific course information
Catalog description: Studies of stress and strain, failure theories, and yield criteria; flexure and torsion theories for solid and thin-walled members; and energy methods.
Prerequisites: CCE 27000 or equivalent, or graduate standing
Course status: Elective course
Specific goals for the course
Student learning outcomes - Upon successful completion of this course the student shall be able to:
solve simple boundary problems in linear elasticity
estimate stresses and strains in three-dimensional bodies
set-up weighted residual forms of governing equations for numerical analysis
find numerical solutions for elasticity problems
assess accuracy and convergence of numerical solutions
write Matlab programs for numerical analysis of elasticity problems
communicate analysis methods and discuss results in a technical paper
Relationship of course to program outcomes
Outcome 1: An ability to identify, formulate, and solve complex engineering problems by applying principles of engineering, science, and mathematics.
Outcome 3: An ability to communicate effectively with a range of audiences.
Outcome 5: An ability to function effectively on a team whose members together provide leadership, create a collaborative and inclusive environment, establish goals, plan tasks, and meet objectives.
Topics
VECTOR OPERATIONS
Index notation, dot product, cross product, tensor product
TENSORS, TENSOR OPERATIONS
Tensor operations on vectors and tensors, calculating tensor components, tensor and vector calculus
CHANGE OF BASIS, TENSOR INVARIANTS
Transformation matrix, transforming vector and tensor components, invariants
STRETCH AND STRAIN, CHANGE OF AREA AND VOLUME
Definition of stretch and strain in 3D, different measures of strain, linearized strains.
EIGENVALUE PROBLEMS
Tensor eigenvectors and eigenvalues, characteristic equation, orthogonality of eigenvectors, Cayley-Hamilton theorem
PRINCIPAL STRETCH, SPECTRAL DEC
Principal directions and values of stretch and strain, spectral decomposition theorem
TRACTIONS, STRESS TENSOR
Traction vector, Newton’s law, tractions on arbitrary surfaces, the Cauchy stress tensor
PRINCIPAL STRESS
Normal and shear stresses, maximum normal stress, stress tensor eigenvectors and eigenvalues
EQUILIBRIUM, ALTERNATIVE REPRESENTATIONS OF STRESS
Governing differential equations for forces and moments, first and second Piola-Kirchhoff stress tensors
CONSTITUTIVE THEORY
Hyper-elasticity, isotropy, stress-strain relationships for linear elastic materials, definition of material moduli: Lamé parameters, Young’s modulus and Poisson’s ration, bulk and shear modulus
BOUNDARY VALUE PROBLEMS, CLASSICAL SOLUTION
Boundary-value problems in linear elasticity, natural and essential boundary conditions, well-posed problems and unique solutions, torsion
LITTLE BOUNDARY VALUE PROBLEMS
1-D elasticity equations, classical solutions to 1-D problems
VIRTUAL WORK
Weighted residuals for 1-D and 3-D elasticity, divergence theorem, virtual work statement, essential and natural boundary conditions, the Fundamental Theorem of the Calculus of Variations
DISCRETIZATION VIA THE RITZ METHOD
The Ritz method of approximation, discretized governing equations, error and convergence, Lagrangian shape functions, finite element approximations
BEAM THEORY (IF TIME PERMITS)
Beam theory equations in 3D, 2D beam theory as a subset of 3D elasticity
