MIT - 16.20 Fall, 2002 Unit 3 (Review of) Language of Stress/Strain Analysis Readings : B, M, P A.2, A.3, A.6 Rivello 2.1, 2.2 T & G Ch. 1 (especially 1.7) Paul A. Lagace, Ph.D. Professor of Aeronautics & Astronautics and Engineering Systems Paul A. Lagace ? 2001 MIT - 16.20 Fall, 2002 Recall the definition of stress : σ = stress = “intensity of internal force at a point ” Figure 3.1 Representation of cross-section of a general body F n F s ? ? F ? Stress = lim ? ? A ? ? A → 0 There are two types of stress: ? σ n ( F n ) 1. Normal (or extensional): act normal to the plane of the element ? σ s (F s ) 2. Shear: act in-plane of element Sometimes delineated as τ Paul A. Lagace ? 2001 Unit 3 - p . 2 MIT - 16.20 Fall, 2002 And recall the definition of strain : ε = strain = “ percentage deformation of an infinitesimal element ” Figure 3.2 Representation of 1-Dimensional Extension of a body ? ? L ? ε = lim ? L ? L → 0 Again, there are two types of strain: ε n 1. Normal (or extensional): elongation of element ε s 2. Shear: angular change of element Sometimes delineated as γ Figure 3.3 Illustration of Shear Deformation shear deformation! Paul A. Lagace ? 2001 Unit 3 - p . 3 MIT - 16.20 Fall, 2002 Since stress and strain have components in several directions, we need a notation to represent these (as you learnt initially in Unified) Several possible ? Tensor (indicial) notation ? Contracted notation will review here ? Engineering notation and give examples ? Matrix notation in recitation IMPORTANT : Regardless of the notation, the equations and concepts have the same meaning ? learn, be comfortable with, be able to use all notations Tensor (or Summation) Notation ? “Easy” to write complicated formulae ? “Easy” to mathematically manipulate ? “Elegant ”, rigorous ? Use for derivations or to succinctly express a set of equations or a long equation Paul A. Lagace ? 2001 Unit 3 - p . 4 MIT - 16.20 Fall, 2002 Example : x i = f i j y j ? Rules for subscripts NOTE: index ≡ subscript ? Latin subscripts (m, n, p, q, …) take on the values 1, 2, 3 (3-D) ? Greek subscripts ( α , β , γ … ) take on the values 1, 2 (2-D) ? When subscripts are repeated on one side of the equation within one term , they are called dummy indices and are to be summed on Thus: 3 f ij y j = ∑ f ij y j j= 1 But f ij y j + g i ... do not sum on i ! ? Subscripts which appear only once on the left side of the equation within one term are called free indices and represent a separate equation Paul A. Lagace ? 2001 Unit 3 - p . 5 MIT - 16.20 Fall, 2002 Thus: x i = ….. ? x 1 = ….. x 2 = ….. x 3 = ….. Key Concept : The letters used for indices have no inherent meaning in and of themselves Thus: x i = f ij y j is the same as: x r = f r s y s or x j = f y i ji Now apply these concepts for stress/strain analysis: 1. Coordinate System Generally deal with right-handed rectangular Cartesian: y m Paul A. Lagace ? 2001 Unit 3 - p . 6 MIT - 16.20 Fall, 2002 Figure 3.4 Right-handed rectangular Cartesian coordinate system y 3 , z y 2 , y Compare notations y 1 , x z y 3 y y 2 x y 1 Engineering Tensor Note : Normally this is so, but always check definitions in any article, book, report, etc. Key issue is self-consistency, not consistency with a worldwide standard (an official one does not exist!) Paul A. Lagace ? 2001 Unit 3 - p . 7 MIT - 16.20 Fall, 2002 2. Deformations/Displacements (3) Figure 3.5 p(y 1 , y 2 , y 3 ), small p (deformed position) P(Y 1 , Y 2 , Y 3 ) Capital P (original position) u m = p(y m ) - P( y m ) --> Compare notations Tensor Engineering Direction in Engineering u 1 u x u 2 v y u 3 w z Paul A. Lagace ? 2001 Unit 3 - p . 8 ??? σσσ MIT - 16.20 Fall, 2002 3. Components of Stress (6) σ mn “ Stress Tensor” 2 subscripts ? 2nd order tensor 6 independent components Extensional σ 11 σ 22 σ 33 Note : stress tensor is symmetric mn = σ nm σ Shear σ 12 = σ 21 σ 23 = σ 32 σ 13 = σ 31 due to equilibrium (moment) considerations Meaning of subscripts: σ mn stress acts in n-direction stress acts on face with normal vector in the m-direction Paul A. Lagace ? 2001 Unit 3 - p . 9 MIT - 16.20 Fall, 2002 Figure 3.6 Differential element in rectangular system NOTE : If face has a “negative normal ”, positive stress is in negative direction --> Compare notations = τ yz sometimes = τ xz used for = τ xy shear stresses Tensor Engineering σ 11 σ x σ 22 σ y σ 33 σ z σ 23 σ yz σ 13 σ xz σ 12 σ xy Paul A. Lagace ? 2001 Unit 3 - p . 1 0 ??? MIT - 16.20 Fall, 2002 4. Components of Strain (6) ε mn “ Strain Tensor ” 2 subscripts ? 2nd order tensor 6 independent components Extensional Shear ε 11 ε 22 ε 33 ε 12 = ε 21 ε 23 = ε 32 ε 13 = ε 31 NOTE (again): strain tensor is symmetric ε mn = ε nm due to geometrical considerations (from Unified) Paul A. Lagace ? 2001 Unit 3 - p . 1 1 MIT - 16.20 Fall, 2002 Meaning of subscripts not like stress ε mn m = n ? extension along m m ≠ n ? rotation in m-n plane BIG DIFFERENCE for strain tensor: There is a difference in the shear components of strain between tensor and engineering (unlike for stress). Figure 3.7 Representation of shearing of a 2-D element angular change Paul A. Lagace ? 2001 Unit 3 - p . 1 2 MIT - 16.20 Fall, 2002 --> total angular change = φ 12 = ε 12 + ε 21 = 2 ε 12 (recall that ε 12 and ε 21 are the same due to geometrical considerations) But , engineering shear strain is the total angle: φ 12 = ε xy = γ xy --> Compare notations = γ yz sometimes = γ xz used for = γ xy shear strains Thus, factor of 2 will pop up When we consider the equations of elasticity, the 2 comes out naturally. (But, remember this “physical ” explanation) Tensor Engineering ε 11 ε x ε 22 ε y ε 33 ε z 2 ε 23 = ε yz 2 ε 13 = ε xz 2 ε 12 = ε xy Paul A. Lagace ? 2001 Unit 3 - p . 1 3 MIT - 16.20 Fall, 2002 CAUTION When dealing with shear strains, must know if they are tensorial or engineering … DO NOT ASSUME ! 5. Body Forces (3) f i internal forces act along axes (resolve them in this manner -- can always do that) --> Compare notations f z f 3 f y f 2 f x f 1 Engineering Tensor Paul A. Lagace ? 2001 Unit 3 - p . 1 4 MIT - 16.20 Fall, 2002 6. Elasticity Tensor (? … will go over later) E mnpq relates stress and strain (we will go over in detail, … recall introduction in Unified) Other Notations Engineering Notation ? One of two most commonly used ? Requires writing out all equations (no “ shorthand ” ) ? Easier to see all components when written out fully Contracted Notation ? Other of two most commonly used ? Requires less writing ? Often used with composites ( “reduces” four subscripts on elasticity term to two) ? Meaning of subscripts not as “physical ” ? Requires writing out all equations generally (there is contracted “ shorthand ” ) Paul A. Lagace ? 2001 Unit 3 - p . 1 5 14 3 256 MIT - 16.20 Fall, 2002 --> subscript changes Tensor Engineering Contracted 11 x 22 y 33 z 23, 32 yz 13, 31 xz 12, 21 xy --> Meaning of “4, 5, 6 ” in contracted notation ? Shear component ? Represents axis ( x n ) “about which ” shear rotation takes place via: m = 3 + n γ m or x n τ m Figure 3.8 Example: Rotation about y 3 Paul A. Lagace ? 2001 Unit 3 - p . 1 6 ? ? MIT - 16.20 Fall, 2002 Matrix notation ? “Super ” shorthand ? Easy way to represent system of equations ? Especially adaptable with indicial notation ? Very useful in manipulating equations (derivations, etc.) Example: x i = A i j y j x = A y ~ ~ ~ ~ ? matrix (as underscore) tilde ? x 1 ? ? A 11 A 12 A 13 ? ? y 1 ? ? ? ? ? ? ? ? x 2 ? = ? A 21 A 22 A 23 ? ? y 2 ? ? ? ? ? ? x 3 ? ? A 31 A 32 A 33 ? ? y 3 ? (will see a little of this … mainly in 16.21) KEY : Must be able to use various notations. Don ’ t rely on notation, understand concept that is represented. Paul A. Lagace ? 2001 Unit 3 - p . 1 7