The Basic Theoryof the Plane ProblemS2-5Physical EquationIn the isotropic body of the complete elasticity, the relationbetween the deformation components and the stress components isestablished according to the Hooke's law as followsx-u(o,+o.)]0SE1-u(o,+o))E1-μ(α +o,))CE1Y-G1VYG1xyTxyG16
16 §2-5 Physical Equation In the isotropic body of the complete elasticity, the relation between the deformation components and the stress components is established according to the Hooke’s law as follows: = = = = − + = − + = − + xy xy z x z x yz yz z z x y y y z x x x y z G G G E E E 1 1 1 [ ( )] 1 [ ( )] 1 [ ( )] 1
the Basic Theory of the Plane ProblemIn the formula, the E is a elastic modulus: the G is a stiffnessmodulus; the μ is a poisson ratio. The relation of the above threeEquantities is :G=2(1 + μ)1.The physical equations of the plane stress problem1CE1E2(1 + μ)YLEuAnd :CE17
17 In the formula, the E is a elastic modulus; the G is a stiffness modulus; the is a poisson ratio. The relation of the above three quantities is : 2(1+ ) = E G 1.The physical equations of the plane stress problem + = = − = − xy xy y y x x x y E E E 2(1 ) ( ) 1 ( ) 1 ( ) z x y E And : = − +
The Basic Theory of the Plane Problem2.The physical equations of the plane strain problem1-μu8ELuE2(1 + μ)x1E3.Thetransformationrelation of therelation between theplane stress and theplane strainThe relation of the plane stress:EE1+μYxyxy2E18
18 2.The physical equations of the plane strain problem + = − − − = − − − = xy xy y y x x x y E E E 2(1 ) ) 1 ( 1 ) 1 ( 1 2 2 3.The transformation relation of the relation between the plane stress and the plane strain. The relation of the plane stress: + = = − = − xy xy y y x x x y E E E 2 1 ( ) 1 ( ) 1
The Basic Theory of the Plane ProblemEFor changeE-1-u?Because of the similarity, whileuu-solving plane strain problem, the1-μcorresponding equation of the planeThe relation in the planeproblem and the elastic constant instrain can be obtained:the answer can be exchanged asabove. and the solution of thehomologous plane strain problem1-μucan be obtained.E1L1- μ?LE1-μ2(1 + μ)E19
19 For change − → − → 1 1 2 E E The relation in the plane strain can be obtained: + = − − − = − − − = xy xy y y x x x y E E E 2(1 ) 1 1 1 1 2 2 Because of the similarity, while solving plane strain problem, the corresponding equation of the plane problem and the elastic constant in the answer can be exchanged as above, and the solution of the homologous plane strain problem can be obtained