Introduction to Theory of. Elasticity. Kengo Nakajima Summer

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1 Introduction to Theor of lasticit Summer Kengo Nakajima Technical & Scientific Computing I (48-7) Seminar on Computer Science (48-4)

2 elast Theor of lasticit Target Stress Governing quations

3 elast 3 Theor of lasticit Continuum Mechanics, Solid Mechanics lastic Material Theor of lasticit lastomechanics Theor of lasticit, lastomechanics

4 elast 4 What is lastic Material? Deformation is proportional to load Load Hooke s law ample Spring k -mg Metal, Fiber, Resin If load is removed, deformation goes to. Original shape Deformation

5 elast 5 If load (deformation) increases, material is not elastic an more Yield Yield point Load lastic limit Yield Point Inelastic Plastic Deformation

6 elast 6 Deformation does not go to with removed load, after elastic limitation. Initial shape is not recovered an more. Load Permanent deformation Yield Point Permanent Deformation Deformation

7 elast 7 Theor of lasticit covers Up to Yield Point, lastic Limitation Load Small deformation Infinitesimal theor Shape does not change Linear Plastic/Inelastic Nonlinear More interesting part of research Deformation lasticit it is more important t in practical engineering i To control load/deformation below elastic limitation is important t Plastic/Inelastic: Accident condition

8 elast 8 Theor of lasticit Target Stress Governing quations

9 elast 9 Stress (/6) If eternal force is elastic bod, the bod deforms, and resists against eternal force b internal force generated b intermolecular forces. Deformation of the bod reach stead state, when eternal force and internal force are balanced. ternal Force Surface force Bod force ternal/internal forces are vectors.

10 elast Stress (/6) An elastic bod in under balanced condition with eternal forces at n points. P n- P P n P

11 elast Stress (3/6) If we assume an arbitrar surface S, internal force between part-a and part-b acts on through surface S. P n- P n S P P n A B

12 elast Stress (4/6) Consider small surface ΔS on surface S of part-a, A and resultant force vector ΔF If p is considered d as averaged force per area ΔF/ΔS with infinitesimal ΔS, p is called stress vector Δ F Δ P n- p lim ΔF S Δ S n S P n A ΔS

13 elast 3 Stress (5/6) Stress: Force Vector per Unit Surface Positive for etension, negative for compression On a surface Normal: Normal stress) Parallel: Shear stress) Yield Stress is an important design parameter. Δ F Δ P n- p lim ΔF S Δ S n S P n A ΔS

14 elast 4 Stress (6/6) Stress components in orthogonal coordinate sstem 9 t i 3D 9 components in 3D normal stress shear stress {}

15 elast 5 Theor of lasticit Target Stress Governing quations

16 elast 6 Governing quations in Theor of lasticit quilibrium quations Compatibilit Conditions Displacement-Strain Constitutive quations Stress-Strain D eample

17 elast 7 quilibrium d d d quations in X-ais Infinitesimal d G d d d lement d d d d d d X d d Bod Force in X-direction X

18 elast 8 quilibrium d d quations in Y-ais G Infinitesimal d d d d lement d d d d d d Y d d Bod Force in Y-direction Y

19 elast 9 d d Moment around Z-ais G at point-g d d d d d d d d d d d d d d

20 elast quilibrium quations in D X Y

21 elast quilibrium quations in 3D 6 Independent Stress Components {} X Y Z

22 elast What is Strain? Solid Mechanics Load Deformation Stress Load/Force per unit surface Strain Rate of Deformation/Displacement

23 elast 3 Strain: Rate of Displacement Normal strain L ΔL ΔL L Shear strain Δ L Δ L

24 elast 4 Strain-Displacement Displacement in 3D: (u, v, w) f D I fi it i l l t for D Infinitesimal lement Before Deformation: P, Q, R, After Deformation: P, Q, R R R P : (, ) d u / P Q P d Q v / Q : R : P': ( ( (, d u,, d ) ) v ) u v Q': ( d u d, v d u v R' : ( u d, d v d ) )

25 elast 5 Normal Strain - Displacement PQ P Q u d u d ( u ) d d u d R u / P R v / Q P d Q u u v w

26 elast 6 Shear Strain - Displacement d R u / P R v / Q u v w v w u P d Q

27 elast 7 Compatibilit Conditions p D 3D,,

28 elast 8 Constitutive qn s: Stress-Strain (/3) Young s Modulus Stress Strain: Proportional Stress-Strain: Proportional Proportionalit: (depends on material), Poisson s Ratio Bod deforms in Y- and Z- directions, even if eternal force is in X-direction. Poisson s s ratio is proportionalit for this lateral strain. depends on material Metal:.3 Rubber, Water:.5 (incompressible)

29 elast 9 Constitutive qn s: Stress-Strain (/3) q ( ) ffect of normal stress components in 3 directions ( ) (,, ) accumulation of each strain component ( ) { } ( ) { } ( ) { }

30 elast 3 Constitutive qn s: Stress-Strain (3/3) Shear strain components do not depend on normal stress components. The are proportional to shear stress. Lateral lastic Modulus: G,, G G G ( ) G

31 elast 3 Stress-Strain Relationship ( ) ( ) ( ) ( ) ( )

32 elast 3 Strain-Stress Relationship ( )( ) ( ) ( ) ( ) ( ) [ ] D { } [ ]{} D Incompressible Material (~.5): Special Treatment Needed

33 elast 33 Some Assumptions in this Class Isotropic Material Uniform, and (~.3) CFRP (Carbon Fiber Reinforced Plastics) Orthotropic

34 elast 34 Finite-lement Method Displacement-based FM Dependent Variable: Displacement Generall used approach This class adopts this approach Stress-based FM Dependent Variable: Stress

35 elast 35 D Problem tension of D truss element onl deforms in X-dir. Uniform sectional area A Young s Modulus u@x, ternal Force F@XL u X F Displacement-based FM u X

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Dr. D. Dinev, Department of Structural Mechanics, UACEG

Lecture 4 Material behavior: Constitutive equations Field of the game Print version Lecture on Theory of lasticity and Plasticity of Dr. D. Dinev, Department of Structural Mechanics, UACG 4.1 Contents