ElasticityChapterIntroduction
1 Elasticity
IntroductionChapter1 IntroductionS 1-1 The Contents of the ElasticityS 1-2 The Several Basic Concepts of ElasticityS 1-3 The Basic Assumption of the ElasticityExercises Lesson2
2 Chapter 1 Introduction §1-3 The Basic Assumption of the Elasticity §1-2 The Several Basic Concepts of Elasticity §1-1 The Contents of the Elasticity Exercises Lesson
IntroductionThe elasticity is a branch of the solid mechanics. The task isto research the elasticity object's stress, deformation anddisplacement due to external force or change of temperature.This course shows the mathematics modeling process ofmechanics problems completely, and establishes the basicequation and boundary condition of the elasticity andproceeds to beg the solutions of some problems. Thefoundation of the elasticity basic equation lays a foundationfor further number methodThe elasticity is the foundation of studying plasticityfracture mechanics and finite element method3
3 The elasticity is a branch of the solid mechanics. The task is to research the elasticity object’s stress, deformation and displacement due to external force or change of temperature. The elasticity is the foundation of studying plasticity, fracture mechanics and finite element method. This course shows the mathematics modeling process of mechanics problems completely, and establishes the basic equation and boundary condition of the elasticity and proceeds to beg the solutions of some problems. The foundation of the elasticity basic equation lays a foundation for further number method
Introduction 1-1The Contents of the Elasticityl、InvestigativetaskThe elasticity is a branch of the solid mechanics, the task of itis to research the elasticity object's stress, deformation anddisplacement due to external force or change of temperature.2. Investigative objectThe research object of the elasticity is general andcomplicated shape structure piece, entity structure, plate shelletc.4
4 The elasticity is a branch of the solid mechanics, the task of it is to research the elasticity object’s stress, deformation and displacement due to external force or change of temperature. §1-1 The Contents of the Elasticity 1、Investigative task The research object of the elasticity is general and complicated shape structure piece, entity structure, plate shell etc. 2、Investigative object
Introduction3, The relationship to the other courses:Theoretical mechanics: study statics and dynamics of therigid body (constraint force, velocity, acceleration)Material mechanics: research stress and displacement of thebar structure piece that is pulled, pressed, sheared, bent orturned.andStructural mechanics:researchinternalforcedisplacement of the bar structure.Plasticity: plasticity analysis and design of the structureElasticity: stress and displacement analysis of general planeproblem, plate, shell and entity structure etc.5
5 Plasticity: plasticity analysis and design of the structure. 3、The relationship to the other courses： Material mechanics: research stress and displacement of the bar structure piece that is pulled, pressed, sheared, bent or turned. Theoretical mechanics: study statics and dynamics of the rigid body (constraint force, velocity, acceleration). Structural mechanics: research internal force and displacement of the bar structure. Elasticity: stress and displacement analysis of general plane problem, plate, shell and entity structure etc
IntroductionS 1-2 The Several Basic Concepts of the Elasticity1.External forceIt can be divided into the force of volume and plane accordingto the different distribution of the external force, which are calledbody force and surface force, respectively(1).Body force(a) Definition: The force acting on eachinternal particle of body or structuralmembers is so called body force. ForZ40example, gravity and inertia force. It isAVshown in Fig.1-2.(b) Property: Body force is different0from the different locations; theydistribution of the body force iscontinuous.Fig.1-2x6
6 §1-2 The Several Basic Concepts of the Elasticity It can be divided into the force of volume and plane according to the different distribution of the external force, which are called body force and surface force, respectively. (b) Property: Body force is different from the different locations; the distribution of the body force is continuous. 1.External force (1).Body force (a) Definition: The force acting on each internal particle of body or structural members is so called body force. For example, gravity and inertia force. It is shown in Fig.1-2. z x y △V O X P Y Q F Fig.1-2 Z
Introduction(c） Gather degree:The average gather degree of volumetric force:AVAOThe gather degree of volumetric force at point P: F = limAVAV->0The direction of F is the limited one of △Q(d) The components of the body force:The force of F is resolved along with the threecoordinates, which will get the three components ofF= Xi +Yi+Zkstraight intersection:X,Y,Z are called the components of the body force atpoint P. Plus sign and negative sign are separatelydetermined by the direction of the components, and its unitis [Force][Length]-37
7 （c）Gather degree： （d）The components of the body force： The force of F is resolved along with the three coordinates, which will get the three components of straight intersection: F Xi Yj Zk = + + X,Y,Z are called the components of the body force at point P. Plus sign and negative sign are separately determined by the direction of the components, and its unit is [Force][Length]-3 . The average gather degree of volumetric force: . V Q The gather degree of volumetric force at point P: . V Q F V = → lim 0 The direction of F is the limited one of . Q
Introduction(2). Surface force(a) Definition: Surface force is distributed in the surface of theobject. For instance, liquid pressure and contact forceIt is shown in Fig.1-3.(b) Property: In general, surface force is the function of thelocated coordinates of the point in the surface of the object(c) Gather degree of surface force: The average gather degreeof the surface force on △Sis: 4ASAOThe gather degree of the surfacezforce at P:FASAQF = limAS->0 △SPX(d) The components of surface force)The components of surface forceFig.1-3areX , Y, Z and the dimension is[Force][Length]-2.x8
8 (2). Surface force (c) Gather degree of surface force: The average gather degree of the surface force on is: . S S Q The gather degree of the surface force at P: S Q F S = →0 lim (d) The components of surface force: The components of surface force are , , , and the dimension is [Force][Length]-2 . X Y Z x y z P △S X Y Z F Q Fig.1-3 (b) Property: In general, surface force is the function of the located coordinates of the point in the surface of the object. (a) Definition: Surface force is distributed in the surface of the object. For instance, liquid pressure and contact force . It is shown in Fig.1-3. Q
Introduction2.Stress(l).Definition:Theobjectbearstheexternal force.Additional internalforceisproducedamong every cross sections of the interior of the object. For displaying these internalforces, we use a cross section to cut the object, and then take out a part among them. Theeffect of a part on another part expresses the internal force, which are resultant force ofthe distributed forces that distribute on thecross section. When the area of the crosssectioninclinestothezero,the distributedforcesonthecross section is shown asFig.l-4(2).Property: The same point in the object, whose stress is different in different crosssections.(3).Stress gather degree:The average gather degree of the above internal10Zforce in A is:40AAAQmThe stress at the point P is : s = limM=0 △AnThe stress component at the point P is , T.AO---Normal stressT---Shearing stress0Andthedimensionofthestress isFig.1-4[Force][Length]-2.9x
9 2.Stress (3).Stress gather degree： The average gather degree of the above internal force in is: A Q A The stress at the point P is : A Q s A = →0 lim -Normal stress -Shearing stress And the dimension of the stress is [Force][Length]-2 . The stress component at the point P is , . (2).Property：The same point in the object, whose stress is different in different cross sections. (1).Definition：The object bears the external force. Additional internal force is produced among every cross sections of the interior of the object. For displaying these internal forces, we use a cross section to cut the object, and then take out a part among them. The effect of a part on another part expresses the internal force, which are resultant force of the distributed forces that distribute on the cross section. When the area of the cross section inclines to the zero, the distributed force s on the cross section is shown as Fig.1-4. x y z A B P o △A s Q n m Fig.1-4
Introduction(4).The component of the stressStress is not only relevant with the position of point but alsorelevant with the direction of the cross section. It is not a generalvector but a second order tensor(a) For analyzing the stress state ofca point, one small positive parallelhexahedron is taken out from thepoint. The component of the stresswithofeachsectionalongBthecalledsaxisiscoordinatescomponent of the stress.AZThe component of the stress onthe planes face to face is equal in0ysize but contrary in direction at themeaning of omitting the smallxFig.1-5quantity in high level.10
10 (4).The component of the stress Stress is not only relevant with the position of point but also relevant with the direction of the cross section. It is not a general vector but a second order tensor. The component of the stress on the planes face to face is equal in size but contrary in direction at the meaning of omitting the small quantity in high level. (a) For analyzing the stress state of a point, one small positive parallel hexahedron is taken out from the point. The component of the stress of each section along with coordinates axis is called the component of the stress. x y z o Fig.1-5 A B C P