### Final Exam Questions in Solid Mechanics Module

#### Analytical Mechanics

- Dynamical effects in strength of materials; rigid bodies, elastic bodies, and the application of D'Alembert's Principle.
- Lagrangian equation, its linearization at equilibria, stability in conservative systems.
- Vibration modes and natural frequencies of multi-degree-of-freedom systems. Rayleigh's Ratio, Stodola Iteration, and Dunkerley's method for estimation of natural frequencies
- Bending vibration of continuum beams. Partial differential equations derived by D'Alembert's Principle. Typical boundary conditions.
- Bending vibration modes and natural frequencies of continuum beams. Fourier solution. The Sturm-Liouville problem and its solution.
- Estimation of the first natural frequency of continuum beams. Rayleigh's Principle.
- Longitudinal vibration of continuum beams. Partial differential equations derived by D'Alembert's Principle. Typical boundary conditions. Vibration modes and natural frequencies.
- Longitudinal vibration of continuum beams with rigid body at boundary condition. Impact of rigid body and elastic beam. D'Alembert solution. Travelling waves.
- Bending vibration of rotating shafts. Matrix differential equation of small oscillations of rotors. Gyroscopic effects. Critical speeds. Forward and backward whirl.

#### Beam Structures

- Axial warping of an open rectangular section. Distribution of sector area function and shear center calculation, 3D plot of warping.
- Constrained torsion of thin-walled open sections, concept of Vlasov. Normal stress from constrained torsion. Definition of the bimoment, explanation based on an I-section beam.
- The governing DE of constrained torsion, the possible kinematic and dynamic boundary conditions. Effect of free and built-in cross sections.
- Statical moment, shear flow and stress distribution in a U-section beam. Calculation of the shear center by moment equilibrium. Shear center of the L-section beam, demonstration through real models.
- Shear stresses in a closed rectangular section with nonuniform thickness, shear flow equation. Calculation of the statical moment and shear center.
- The basic concept of the nonlinear bending analysis of slender beams, the nonlinear DE of flexure, force parameter. Practical examples.
- Nonlinear bending of a cantilever beam, solution steps, initial parameters, load-displacement curves and bending moment diagrams.
- The basic concept of Timoshenko beam theory. Transverse shear effect and shear correction factor.

#### Continuum Mechanics

- Descriptions of motion. The referential (Lagrangian) and spatial (Eulerian) descriptions. Deformation gradient.
- Stretch ratio. Deformation and strain tensors.
- Area, volume and angle changes.
- Polar decomposition of deformation gradient.
- Principal stretches and principal directions, spectral representation of the stretch tensors.
- Displacement vector. Displacement gradient. Geometrical linearization.
- Velocity and acceleration fields. Material time derivative.
- Velocity gradient tensor. Rate of deformation and spin (vorticity) tensors.
- Material time derivatives of the line, area, and volume elements.
- Material derivatives of deformation and strain tensors.
- Force, traction stress. Cauchy, first and second Piola-Kirchhoff stresses.
- Objectivity. Objective stress rates.
- Conservation of mass. Continuity equation. Reynolds' transport theorem.
- Balance of linear and angular momentum. Cauchy's equations of motion.
- Balance of mechanical energy.

#### Coupled Problems in Mechanics

- Diffusion-type problems. Governing equations.
- Constitutive equations of coupled piezo-thermo-mechanical problems. Helmholtz free-energy. Linearization.
- Field equations in coupled piezo-thermo-mechanical problems. Initial and boundary conditions. Finite element discretization.
- Microelectro-Mechanical Systems 1. Beam- and plate-type structures.
- Microelectro-Mechanical Systems 2. Physical principles of sensing. Finite element estimation of sensitivity.
- Contact problems. Contact between circular and flat/circular surfaces. Finite element modelling.
- Fluid-structure interaction. Acoustic problems. Governing equations and derivation of finite element discretization.
- Shape memory alloys. Smart structures.

#### Elasticity and Plasticity

- Governing equations of elasticity.
- Rotating disc.
- Airy stress function.
- Torsion of prismatic bars.
- Elastic-plastic constitutive relation. Prandtl-Reuss model.
- Yield function. Tresca and von Mises yield criteria. Isortopic and kinematic hardening.
- Cyclic uniaxial tension-compression with linear isotropic or kinematic hardening.
- Elastic-plastic thick-walled tube with internal pressure

#### Experimental methods in solid mechanics

- Compliance and energy release rate calculation in the end-notched flexure test by the simple beam theory by means of linear elastic fracture mechanics concept.
- Application of the path independent J-integral to fracture tests, example: double-cantilever beam test.
- The area method to calculate the energy release rate in crack propagation tests using a typical load-displacement curve.
- The theoretical concept of virtual crack closure, definition of the energy release rates, numerical scheme based on a fractal-type mesh.
- Control process of fracture tests, load and displacement controlled tests. The necessary and sufficient conditions of stable crack propagation, relation to the R-curve.
- Stability of crack propagation tests under force and displacement control, examples. Calculation of the limit crack lengths, stability diagrams of fracture specimens.
- The free and constrained mode models in the vibration analysis of delaminated composite beams, the system of exact kinematic conditions. Normal force, parametric excitation and dynamic buckling.

#### Finite element analysis

- Linear stability (bifurcation) analysis of elastic systems, Eulers method. Initial stress and perturbed states, field increments. The role of Green-Lagrange strain tensor.
- FE formulation of linear stability problems, the geometric stiffness matrix and its physical meaning. Derivation of the eigenvalue-eigenvector problem, application examples.
- Lateral-torsional buckling of an I-section beam subjected to pure bending. Application of linear stability analysis, displacement field of the perturbed state, critical load.
- Dynamic stability of linear elastic systems, the role of load stiffness matrix. Stability diagrams of a massless column and Becks column, divergence and flutter.
- Parametric excitation in linear elastic systems. Mathieu type equation of motion for multi-DOF systems. 2T and T periodic solutions, stability diagrams and time response of a periodically compressed beam.
- Nonlinear dynamics by FEM, direct time integration, equation of motion. The central difference method, basic concept of Newmark scheme. Application examples.
- The diverse nature of linear and nonlinear structural problems. Classification of nonlinear structural problems, material and geometric nonlinearities. The basic concept of step-by-step (incremental) solution.
- Solution algorithms of nonlinear structural problems. Tangent stiffness matrix, Newton-Raphson and modified Newton-Raphson iteration schemes, graphical solution through a well-behaved 1DOF system.
- Degenerated elastic beam element for moderately large displacements and rotations using von Kármán-type nonlinearity. Discretization steps using the virtual work principle, Bubnov-Galerkin approach, calculation of the tangent stiffness matrix.
- Nonlinear (large amplitude) vibration of elastic beams using von Kármán type nonlinearity. FE equation of motion and Duffing equation. Amplitude dependence of frequencies, solution methods.
- Geometrically nonlinear beam element for large displacements and rotations. Application of the Green-Lagrange strain tensor. Displacement and strain fields, deformation gradient, FE discretization. Nonlinear load-displacement curves of a cantilever beam.

#### Nonlinear Vibrations

- Typical nonlinearities in mechanical oscillatory systems, their mathematical description. Structure of differential equations of nonlinear one degree of freedom mechanical systems.
- Phase plane method for nonlinear one degree of freedom systems. Equilibria, local linearization, classification of local phase portraits, topological methods in the phase plane. Basic types of loss of stability, saddle-node and Hopf-bifurcation.
- Equilibria in one degree of freedom conservative systems. Pitchfork bifurcation, the basic idea of chatastrophy theory. Construction of phase portraits based on the potential function.
- Estimation of the time period of large amplitude oscillations in conservative systems. Linearization methods, Poincaré's asymptotic method. Change of the time period of oscillation in cases of progressive and degressive spring characteristics.
- Construction of the phase portrait in nonlinear dissipative systems. Liénard's phase-plane method. Damped oscillations in the phase plane for different damping characteristics. Phase portrait and non-uniqueness of the solutions for Coulomb friction.
- Nonlinear forced oscillations. Resonance diagram (forced vibration amplitude against excitation frequency) for hardening and softening spring characteristics and harmonic excitation. Application of Poincaré's small parameter method.
- Self-excited vibration and the idea of the limit cycle. Linénard's criterion for the sufficient condition of the existence of limit cycles, Bendixson's criterion for the necessary condition of the existence of limit cycles. The phenomenon of stick-slip.
- Hopf bifurcation theorem for the necessary and sufficient condition of the existence of limit cycles, approximation of the limit cycle. Supercritical and subcritical bifurcations. Application for stick-slip, the phenomenon of unstable self-excited oscillation and its relevance in engineering practice.