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Lecture Note 2 Problem Setting First Semester, Academic Year 2012 Department of Mechanical Engineering Chulalongkorn University Objectives
#1 Set well-posed elasticity problems (e.g. for finite element analyses) Concept Well-posed problems can be formulated by substituting appropriate constitutive relationships, loads, initial conditions andboundaryconditionsintothegoverningequations 2
and boundary conditions into the governing equations.
Well-posed Problems
#1 In mathematics, the definition of well-posed problems was defined by Hadamard (1865-1963) such that A solution exists. The solution is unique. ThesolutiondependscontinuouslyontheproblemdataThe mathematical models of physical phenomena should have the following characteristics. 3
The solution depends continuously on the problem data.
Well-posed Problems
#2 The continuum mechanics problems are usually formulated using the partial differential equations (PDEs). AsolutiontoaPDEcanbedescribedassimplyafunctionthatA solution to a PDE can be described as simply a function that reduces that PDE to an identity on some region of the independent variables. A PDE, without any auxiliary boundary or initial conditions, will usually have infinite numbers or no solutions. Thus, the formulation of a PDE problem requires three components. 4The PDE from the governing equations and constitutive relationships, The spatial and temporal domains or the region of space-time on which the PDE is required to be satisfied, The auxiliary boundary and initial conditions to be met.
copoets
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Well-posed Problems
#3 If one of these conditions is not satisfied, the PDE problem is said to be ill-posed. InpracticethequestionofwhetheraPDEproblemiswell If there are too many imposed conditions, the solution may not exist. IftherearetoofewimposedconditionsthesolutionmaynotIn practice, the question of whether a PDE problem is well posed can be difficult to decide. Usually, they can be determined from the auxiliary conditions. 5
If there are too few imposed conditions, the solution may not unique. The conditions must be correctly matched to the type of the PDE, otherwise the solution will not depend continuously on the problem data.
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Domains StildiThe problem framework in space and time Spatial domain Temporal domain 6
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Mathematical Model In physics, mathematical models are mathematically expressions of physical theories and obtained from an idealization of experimental observations. deaatooepeetaobseatos A mathematical model that can simulate the behavior of a body is usually composed of 3 parts, the governing equations, the constitutive relationships of the medium and the problem- specific loads and conditions. Emphasis 7The governing equations are the basic conservation laws that are valid regardless of the physical medium of the body. The constitutive relationships are materially specific. The domain, loads and conditions are problem specific.
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Governing Equations #1 The governing equations, which are applicable to any media, is the basis of mathematically modeling of the system behavior. Even though they are based on the law of beaoetougteyaebasedoteao conservation, they can be expressed in many forms. The governing equations that are commonly employed in mechanics are the conservation of mass, linear momentum, moment of momentum and energy. 8The governing equations can be expressed in the global or integral form. The governing equations are usually derived in this form and they present good physical meanings of the problem, witnessing the use of control volume in fluid dynamics. However, it can be difficult to solve, particularly by the analytical methods.
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Governing Equations #2 Laws of conservations can be categorized into the exact laws, which have never been shown to be inexact, an the approximated laws, which are true in certain conditions. appoatedas,caetuecetacodtos There are 4 laws in physics that are commonly used as the governing equations in mechanical simulations. Conservation of Mass Conservation of Momentum ConservationofMomentofMomentum 9Conservation of Moment of Momentum Conservation of Energy
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Constitutive Relationships #1 Upon specify the mechanical and thermal properties of media based upon their constitution, idealized materials, which serve as models for the behaviours of real materials, can beasodesotebeaousoeaateas,cabe defined. That is, the constitutive relationships describe relationships among the kinematic, mechanical and thermal equations and permit the formulation of well-posed problems 10Components
Constitutive Relationships #2 Physical Admissibility: All constitutive equations must be consistent with physical laws. AxiomofCausality:TherecannotbeeffectswithoutcausesAxiom of Causality: There cannot be effects without causes. Determinism: Values of constitutive variables are determined by histories of motion, force and temperature experienced by the body. Principle of Equipresence: An independent variable assumed to be present in one constitutive equation of a material should be assumed to be present in all constitutive equations of the 11same material, unless its presence contradicts an assumed symmetry of the material, the principle of material frame indifference or some other fundamental principle. Still a controversial topic, but should be adopted in the early stages of formulation to aim for generality and reveal possible coupling effects between different kinds of phenomena.
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Constitutive Relationships #3 Principle of Neighbourhood (Local Action): The principle imposes certain restrictions on the smoothness of the constitutive functions in the neighbourhood of a material point.costtuteuctosteegbouoodoaateapot St. Venant Principle: The dependent constitutive variables at position x are not appreciably affected by the values of the independent variables at material points far away from x. Work Conjugation: Appropriate selection of stress and strain measures. Material Frame Indifference: A quantity is invariant if it 12remains unchanged under some transformation.
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Linear Elasticity Domain –Space & Time Governing Equation –Conservations of Moment & Energy ConstitutiveRelationsConstitutive Relations Stress-Strain Relationships (Hooke’s Law) Strain-Displacement Relationships Heat Conduction Equation (Fourier’s Law) Initial Conditions Boundary Conditions 13Displacement-based Stress/force based Symmetry Plane
