Sol mech course text feb10 solid mechanics at harvard. While the above three equations equilibrium equations, straindisplacement. For two dimensional solid mechanics problems, three strain components are. Following development of the equations, applications will be presented that involve airy stress functions and tire mechanics. Statics equilibrium of a rigid body in two dimensions for all forces and moments acting on a twodimensional structure, fz 0 mx m y 0 mz mo equations of equilibrium become. On the direct computation of critical equilibrium states in solid and structural mechanics alexis fedoroff1, reijo kouhia1, heikki marjamaki2 and jari makinen2 1 department of structural engineering and building technology helsinki university of technology, p.
In general, solving a solid mechanics problem must satisfy equations of equilibrium static or dynamic, conditions of compatibility between strains and. We could therefore proceed by particularizing the equations 2. Our results include evolutions that illustrate the coexistence of both positive and negative dislocations in. It is shown that the boundary conditions and differential equations governing the equilibrium of an anisotropic homogeneous elastic solid, whose 21 elastic constants satisfy 14 specially chosen conditions, can be expressed by means of a linear transformation in a form identical with the corresponding equations for an isotropic solid. Basic themes and pretty problems of nonlinear solid mechanics. What does the onedimensional equation of motion say about the stresses in a bar in the absence of any body force or acceleration. Elasticity studies mechanics variables displacements, strains, stresses, internal forces, and. Hydrostatic and deviatoric stress components, octahedral shear stress 8. They are there, hidden at the interface of your particle with the rest of the world. For verifying the new method, the governing equations in cartesian, polar and spherical coordinates systems are derived.
General problems in solid mechanics and nonlinearity. Static equilibrium force and moment mit opencourseware. X i f i 0 or, equivalently, the components of the vectors in three directions which must be linearly indepen. In the second part dealing with non equilibrium processes, the emphasis is on the quantum foundations of markovian behaviour and irreversibility via the paulivan hove master equation. A first course on variational methods in structural. With only one course in solid mechanics, it was felt that a. The stress components acting on these planes are as shown in fig. Scalar equations conveniently obtained from vector equations of equilibrium, f 0 m o r f 0 engineers mechanics equilibrium of rigid bodies.
Pdf effective stress and equilibrium equation for soil. The equations of equilibrium may also be used as a good approximation in the. The second section shows how to transcribe tensorial relations in the engineering notation. Mech 114 statics of solid mechanics, equilibrium1 force. Writing the equation of equilibrium for the stresses in the y direction produces a similar result. Balance of momentum the equilibrium equations balance of linear momentum are given in index form as. For static equilibrium of the isolated particle, the resultant of the two forces w acting downward and r acting upward must be zero. Equilibrium and nonequilibrium statistical mechanics. We could therefore proceed by particularizing the equations and to the case of systems of bodies subject to concentrated forces. An exception to this was the varying stress field in the loaded beam, but there a simplified set of elasticity equations was used. Pdf mechanics of solids vdb1063 equilibrium of deformable. While in the mechanics of materials course, one was introduced. The equilibrium relations to be discussed in this module have this. Any material, fluid or solid, can support normal forces.
Therefore, the strain equations of equilibrium can be converted to displacement equations of equilibrium. This separation, typical of solid mechanics, is performed using imaginary surfaces used to cut the starting body. Statics mech 106 basic mechanics equilibrium of rigid bodies introduction 2 according to newtons first law of motion, if a system of forces acting on a body has zero resultant, the body will either remain at rest or move in a straight line with constant velocity. First, by describing the concept of equilibrium equations and investigating the. This can be expressed by the equilibrium equations. With this we find from moment equilibrium that fa fc w. Introduction to finite element analysis in solid mechanics. Stress components in rectangular and polar coordinate systems 4. A first course on variational methods in structural mechanics. To appear in mathematics and mechanics of solids can equations of equilibrium predict all physical equilibria. Equilibrium equations 2equilibrium equations 2d the 2d stresses are. David fox, in the finite element method for solid and structural mechanics seventh edition, 2014 1. In this article we will prove the equilibrium equations by calculating the resultant force and moment on each axis. Then the equations will be developed in 3d, and also presented in cylindrical coordinates.
Abstract thermodynamic models for viscoplastic solids are often formulated in the context of continuum thermodynamics and the dissipation principle. The equilibrium equations david roylance department of materials science and engineering massachusetts institute of technology cambridge, ma 029. Naturally, in this case it must calculate the motion of the solid as a function of time. Mechanics lecture notes 1 notes for lectures 2 and 3. The classical theory of solid mechanics is based on the assumption of a continuous distribution of mass within a body. Pitarresi, introduction to solid mechanics,prentice hall of india pvt. Mech 114 statics of solid mechanics, equilibrium 1 free download as powerpoint presentation. Anyone who has tried to solve the equations of motion for more than two interacting particles knows already that this is an extremely di cult, if not impossible task. Thermodynamic model formulation for viscoplastic solids as. Strain energy function and its con nection with hookes law. A primary result is the existence of spatially nonperiodic, extremely slowly evolving quasi equilibrium cellwall dislocation microstructures practically indistinguishable from equilibria, which however cannot be solutions to the equilibrium equations of the model, a feature shared with certain types of gl equations. Formulas in solid mechanics tore dahlberg solid mechanics ikp, linkoping university linkoping, sweden this collection of formulas is intended for use by foreign students in the course tmhl61, damage mechanics and life analysis, as a complement to the textbook dahlberg and.
Justifiable linear response expressions and the quantumboltzmann approach are discussed and applied to various condensed matter problems. Equilibrium equations have already been discussed in the chapter 2 dedicated to continuous bodies. Equations can be solved for at most 6 unknowns which generally are reactions. Equilibrium equation an overview sciencedirect topics. Differential equations of equilibrium stress components vary from point to point. M a 0 where a is any point in the plane of the 4 3 structure. Equilibrium of ideal systems, efficiency of simple machines, stable and unstable equilibriums. Morozov, in advanced mechanics of composite materials third edition, 20 2. Writing the equation of equilibrium for the stresses in the y direction produces a. Determine all the forces that are acting on the rigid body. A general solution of equations of equilibrium in linear elasticity. The equations of statics must be satisfied throughout the body. Barber, elasticity, solid mechanics and its applications 172. Equilibrium and dynamic equations for an isotropic elastic solid.
The basic equations describing a linear elastic deformable body are. A rigid body is said to be in equilibrium if both the resultant force and the resultant moment acting on it. Algebraically, this just means that the vector sum of the forces is zero. These equations are based on a model of internal forces within a body in which material points interact with each other directly over nite distances. The vector equation of equilibrium in linear elasticity with no body forces, also known as. Material properties constitutive relations, for example, hookes law must comply with the known behavior of the material involved.
If we begin with the case when the weight of the body is the only body force, the equations to be satisfied are to these equations the boundary conditions 20 should be added. Equilibrium, compatibility, and constitutive law in general, solving a solid mechanics problem must satisfy equations of equilibrium static or dynamic, conditions of compatibility between strains and displacements, and stressstrain relations or. Box 2100, fi02015 tkk, finland 2 department of mechanics and design, tampere university of technology, p. Instead of using these equations directly, we replace them by the. Pdf direct method for deriving equilibrium equations in solid. They do not have any information on the rheology of the body, so these equations apply to viscous fluids, plastics, and elastic solids. The dierence between the particle of the last lecture and the body in this lecture is that all the forces on the particle act through the same. For more contents please read the reference effective stress and equilibrium equation for soil mechanics by. Rw 0 this leads to the not very earth shaking conclusion that the magnitude of the reaction force, acting up, must equal the weight. Further, in chapter 2, the strain components were related to the displacement components. The analysis of the equations of equilibrium and motion under this condition 7, 16 is technically difficult, presenting many unresolved.
This paper provides a brief overview of the basic concepts and equations that will be. The correct stress distribution for a problem in solid mechanics must satisfy these equations of equilibrium. Mathematics and mechanics of solids can equations of. Basic concepts and equations of solid mechanics milan jirasek department of mechanics faculty of civil engineering czech technical university in prague 166 29 prague, czech republic milan. You will find it takes courage, as well as facility with the language of engineering mechanics, to venture forth and construct reaction forces out of thin air. These are forces directed perpendicular, or normal, to a material plane across which they act. Equilibrium, compatibility, and constitutive law in general, solving a solid mechanics problem must satisfy equations of equilibrium static or dynamic, conditions of compatibility between strains and displacements, and stressstrain relations or material constitutive law see figure 7. Finally, the equilibrium equations are used to develop expressions for the speed of stress waves in steel, aluminum, and rubber. Solid mechanics is concerned with the stressing, deformation and failure of solid materials and structures. The purpose of the current work is to show that models for such material behavior can also be formulated in the form of a general equation for non equilibrium revers. The first two sets of equations are universal independent of the material as they depend on geometry straindisplacement and equilibrium equilibrium. A solid body is in static equilibrium when the resultant force and moment on each axis is equal to zero. Introduction to stress analysis in elastic solids 2.
Pdf basic concepts and equations of solid mechanics. Can equations of equilibrium predict all physical equilibria. Hence, stress equations of equilibrium can be converted to strain equations of equilibrium. The free body diagram is a depiction of an object or a. These equations give 6 independent equations in 3d space for each rigid body. Considering a threedimensional case in which stresses act in the x, y, and z directions gives the following set of equations of equilibrium. Solid mechanics part ii 3 kelly equations of motion in part i, balance of forces and moments acting on any component was enforced in order to ensure that the component was in equilibrium. Thermodynamic model formulation for viscoplastic solids 2 a. For static problems the finite element method solves the equilibrium equations f0. Outline this lecture is based on chapter 1 of hibbelersbook mechanics of materials 1.
The problem of solid mechanics is reduced, as follows from the foregoing derivation, to a set of 15 equations, i. Fundamental equations of structural mechanics wiley online library. Box 2100, fi02015 tkk, finland 2 department of mechanics and design, tampere university of. Engineering solid mechanics direct method for deriving equilibrium. The equilibrium equations david roylance department of materials science and engineering massachusetts institute of technology cambridge, ma 029 september 26, 2000. Mechanics of solids vdb1063 equilibrium of deformable body. This matrix equation, which is sometimes called cauchys relation, can be. In particular, see the figure above, for the two body parts thus obtained the following pairs of equations can be written. In the design of the new single course, most of the topics of the original two courses were retained but the coverage and sophistication of the presentation was reduced. Instead, we prefer to derive the equations of equilibrium autonomously by applying the principle of virtual work already discussed. For dynamic problems the finite element method solves the equations of motion for a continuum essentially a more complicated version of fa m.
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