Introduction to Elasticity
Introduction to Elasticity
Outline. What is Elasticity?: A Brief History of Elasticity? Tools of the Trade? Engineering Applications of Elasticity? Fundamental Concepts in Elasticity: Assumptions of Elasticity Theory? Geometry of Elastic Solids. Topics That Will Be Covered· Greek Alphabet2
Outline • What is Elasticity? • A Brief History of Elasticity • Tools of the Trade • Engineering Applications of Elasticity • Fundamental Concepts in Elasticity • Assumptions of Elasticity Theory • Geometry of Elastic Solids • Topics That Will Be Covered • Greek Alphabet 2
What is Elasticity?? Concerned with determining stress, strain, anddisplacement distribution in an elastic solid under theinfluence of external forces.FALLinearly ElasticElastic. Using continuum mechanics formulation establishesa mathematical boundary value problem model - aset of governing partial differential field equationswith particular boundary conditions3
• Concerned with determining stress, strain, and displacement distribution in an elastic solid under the influence of external forces. 3 What is Elasticity? Linearly Elastic F ΔL Elastic F ΔL • Using continuum mechanics formulation establishes a mathematical boundary value problem model - a set of governing partial differential field equations with particular boundary conditions
History of Elasticity? Started from early 19th century, primarily by Navier,Cauchy, Saint-Venant, Love, Muskhelisvili.Kirchhoff, Poisson,. Needed for understanding the deformation anddamage mechanisms of bridges, roads, shipsmilitary weapons, and etc: Understanding optical wave propagation requireselastic wave theory.4
• Started from early 19th century, primarily by Navier, Cauchy, Saint-Venant, Love, Muskhelisvili, Kirchhoff, Poisson, . • Needed for understanding the deformation and damage mechanisms of bridges, roads, ships, military weapons, and etc. • Understanding optical wave propagation requires elastic wave theory. 4 History of Elasticity
History of Elasticity: Claude-L0uis Navier (1785-1836): Unified the theory for beam bending (1826).. Founder of Continuum Mechanics: submittedtwo monographs to French Academy ofScience in 1821.: Derived the equations of equilibrium andmotion for elastic and isotropic solidsC(V2u, + 2uj,jt)+ F, = 0: Promoted the method allowable stress forexamining the strength condition.: There is only one elastic constant in hisformulations and no stress and strain concepts5
5 • Claude-Louis Navier (1785−1836) • Unified the theory for beam bending (1826). • Founder of Continuum Mechanics: submitted two monographs to French Academy of Science in 1821. • Derived the equations of equilibrium and motion for elastic and isotropic solids. • Promoted the method allowable stress for examining the strength condition. • There is only one elastic constant in his formulations and no stress and strain concepts. 2 , 2 0 C u u F i j ji i History of Elasticity
History of Elasticity· Augustin-Louis Cauchy (1789-1857): Founder of elasticity theory. (800 papers and7 books). Famous for his mathematical talent: Invented - in limit & continuity analysis: Founder of complex variable theory (CauchyRiemann conditions).: Strain and strain-displacement equations. Stress, principal stress vs.principal strain >two elastic constants.: Generalized Hooke's law: 36 elastic constantsat most.: Equations of equilibrium and BCs in terms ofdisplacements.: Studied the torsion of elastic rods6
6 • Augustin-Louis Cauchy (1789−1857) • Founder of elasticity theory. (800 papers and 7 books) • Famous for his mathematical talent. • Invented ε-δ in limit & continuity analysis. • Founder of complex variable theory (CauchyRiemann conditions). • Strain and strain-displacement equations. • Stress, principal stress vs. principal strain » two elastic constants. • Generalized Hooke’s law: 36 elastic constants at most. • Equations of equilibrium and BCs in terms of displacements. • Studied the torsion of elastic rods. History of Elasticity
History of Elasticity: Simeon Denis Poisson (1781-1840): Solve the Poisson equation.Developed thePoissondistribution inprobability theory.. Longitudinal and transverse waves canpropagate in elastic solids: Theoretically derived the Poisson's ratio(1/4).. First derived the deflection equation ofplates.7
7 • Siméon Denis Poisson (1781−1840) • Solve the Poisson equation: • Developed the Poisson distribution in probability theory. • Longitudinal and transverse waves can propagate in elastic solids. • Theoretically derived the Poisson’s ratio (1/4). • First derived the deflection equation of plates. History of Elasticity
History of Elasticity? Adhémar Jean Claude Barré de Saint-Venant(1797-1886), a student of Navier? Developed the Saint-Venant Principle frompure bending of beams: First verified the precision of the twohypotheses assumed in beam bending: Semi-inverse solution for elasticity (1855): Improved the torsion of (non-circular) elasticrods by Cauchy.: Derived solutions for a large number of elasticproblems and promoted their use inengineering practice8
• Adhémar Jean Claude Barré de Saint-Venant (1797−1886), a student of Navier 8 • Developed the Saint-Venant Principle from pure bending of beams. • First verified the precision of the two hypotheses assumed in beam bending. • Semi-inverse solution for elasticity (1855) • Improved the torsion of (non-circular) elastic rods by Cauchy. • Derived solutions for a large number of elastic problems and promoted their use in engineering practice. History of Elasticity
History of Elasticity: Gustav Robert Kirchhoff (1824-1887)? Contributed to the fundamentalunderstanding of electrical circuitsspectroscopy, and the emission of black-body radiation by heated objects.: Developed Kirchhoff plate theory by theprinciple of virtual displacement in 1850(Straight normal line, two BCs only): Derived the equilibrium equations ofelastic rods under large deflections, inanalogy to the motion of a rigid bodyabout a fixed point.9
• Gustav Robert Kirchhoff (1824−1887) 9 • Contributed to the fundamental understanding of electrical circuits, spectroscopy, and the emission of blackbody radiation by heated objects. • Developed Kirchhoff plate theory by the principle of virtual displacement in 1850 (Straight normal line, two BCs only). • Derived the equilibrium equations of elastic rods under large deflections, in analogy to the motion of a rigid body about a fixed point. History of Elasticity
History of Elasticity: Augustus Edward Hough Love (1863-1940): A treatise on the mathematical theory ofelasticity, 1892-1893, summarized all up-to-date achievements in elasticity: The bending theory of thin-shells in 1888(Kirchhoff-Love Hypothesis): Point-source theory in infinite elastic solids.later found its application in mathematicalmodeling of earthquake source.. Some problems of geodynamics: Love wave(earthquake) and Love number (elasticconstants of earth)10
10 • Augustus Edward Hough Love (1863−1940) • A treatise on the mathematical theory of elasticity, 1892-1893, summarized all up-todate achievements in elasticity • The bending theory of thin-shells in 1888 (Kirchhoff-Love Hypothesis) • Point-source theory in infinite elastic solids, later found its application in mathematical modeling of earthquake source. • Some problems of geodynamics: Love wave (earthquake) and Love number (elastic constants of earth). History of Elasticity