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MIT 16 01 - Transformation of Stres

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M14 Transformation of Stress (continued). Introduction to Strain Principal Stresses/Axes There is a set of axes (in 2 or 3-D) into which any state of stress can be resolved such that there are no shear stresses. These are known as the principal axes of stress. sI, sII , sIII (By convention, sI is most tensile, sIII is most compressive) Can also see from Mohr's circle that two of these are the largest (and smallest) or extreme values of stress. Side Note: if we thought about stress as a matrix: Ès11 s12 s13 ˘ Í˙dF =sd As= s21Ís22 s23˙ ÍÎs21 s32 s33˚˙ dF =sd AThe principal stresses are the eigenvalues of s and the principal directions the eigenvectors. Important: We are transforming the axis system, not the stress state. Consider a bar with two grid squares (rectangle) on the surface, one rotated through 45°: the bar now undergoes a tensile loading, and this generates a tensile strainxn rectangle is stretched/elongated (angles remain the same = 90°.) x˜m is sheared (i.e angles changed) But both grid squares (rectangles) are experiencing the same stress state (and strain state) Stress Invariants - actually the basis for Mohr's CircleNote that in 2-D, = cons tan ts11 +s22 =sI +sII In 3-D, s= constan t (In var iant)11 +s22 +s33 i.e., the diameter Of the Mohr's circle (will meet other invariants elsewhere in solid mechanics).M15 Introduction to Strain We have examined stress, the continuum generalization of forces, now let's look at the continuum generalization of deformations: Definition of Strain Strain is the deformation of the continuum at a point. Or, the relative deformation of an infinitessimal element. Two ways that bodies deform: By elongation and shear: Elongation (extension, tensile strain) Tensile strain can be thought of as the change in length relative to the original length Ldeformed - Lundeformedtensile strain = Ldeformed but body can also deform in shear – angles change Shear This produces an angle change in the body (with no rotation for pure shear) Consider infinitesimal element.Undeformed: Deformed : Consider change in angle BPA = f f = undeformed - deformed NOTE: By convention positive strain is a reduction in angle (consistent with positive shear stresses) D = ÈPÊP ˆ˘ -Á -f˜˙=f radiansÍÎ2 Ë2 ¯˚NOTE: Strain is a non-dimensional quantity (Just as for stress, strain varies with position and direction. Use tensor formulation to describe) Formally Define emn = strain tensor For complete definition see: Bisplinghoff, Mar and Pian, "Statistics of Deformable Solids", Chapter 5 This is the general definition of strain. But usually we are concerned with small strains: change in length < 10% (0.1), change in angles < 5% (0.05) Good for range of use of most engineering materials, most structural applications. Allows us to neglect higher order terms and leads to:Strain - Displacement Relations (for small strains) Conceptually, want to separate out rigid body translations from deformations e11 = relative elongation in x1 direction e22 = relative elongation in x2direction. e33 = relative elongation to x3 direction. Consider infinitesimal element, side length dx1 undergoing displacements and deformations in the x1 direction defined by u1 is a field variable = u1( x1,x2, x3 ), i.e. displacements vary with position let u1 be the displacement of the left-hand side of the infinitesimal element. ˆ And ÁÊ u1 +∂u1 dx1˜is the displacement of right-hand sideË∂x1 ¯Recall: l deformed - l undeformed ªe1 =e11 l undeformed È∂u ˘ Ídx1 + u1 + dx1 - u1˙- dx1Î∂x1 ˚ e11 = dx1 l undeformed Rigid body∂u1 translation= ∂x1 Similarly, e22 = elongation in x2 l deformed - l undeformed =e22 l undeformed È Ídx2 + u2 +∂u2 dx2 - u2 ˙˘- dx2Î∂x2 ˚ = dx2 ∂u2=e22 ∂x2 ∂u3and e33 = ∂u3Now for shear strain Need to be careful which angles we choose to define shear strain Ê 1 e12 = 1 angle change Ë= 0/12 ˆ etc. 2 2 ¯=1( undeformed - deformed )2 1 ÊPÊP ˆˆ =Á -Á -f12 ˜˜2 Ë2 Ë2 ¯¯ e.g: Angular charge of the x1 edge in the x2 direction e12 +e21 = total angle change in x1 - x2 plane total angle change in x1 - x3 planee13 +e31 = = total angle change in x2 - x3 planee23 +e32 NOTE: The strain tensor is defined such that there seems to be two parts in each angle change. But the stress tensor is symmetric and we would like the strain tensor also to be symmetric. Thus: 1 e12 =e21 =(angle change in x1 - x2 plane)2 1 e13 =e31 =(angle change in x1 - x3 plane)2 1 e23 =e32 =(angle change in x2 - x3 plane)2Consider diagonal of parallelogram undergoing pure shear: Angles change by same amount. Formal Definitions : =q1 +q2f12 Make small angle approximation: tanq=q12 23 u1 u1 dx2 -x2 ∂∂u1 + 1 =qˆ ¸˘ˆ ˜ ˜ ˜ ˜˜ ÔÔ ÔÔ˛ ˙˙ ˙ ˙˚¯ ˝ ˜¯ - ∂∂u1 dx2 x2 ʈ Á˘¯ + 12fu2 dx1 -x1∂∂u2 dx1 -x1∂∂ˆj˜˜¯ ÊÈÏ Ê = q2 ÁËÍ Í Í ÍÎ - P 2 ˆ Ì u1ˆ∂˜¯x2 ∂ ∂u2 u3∂Ê ∂∂∂∂ÁÁË ui u x xi + ÔÔ ÔÔÓ ÁÁÁÁÁ˜¯∂x2 ˆ∂u1˜¯∂x3 j - P 2 + + + Ê 1 Ë ∂Êu3 u2∂Ê x1 and similarly: ∂∂ÁËx3 ∂2 x1 12=eÁË ÁË 2 =eij = e 31 =eor = e 1 =e13 The strain tensor for small strains Strain displacement relation 6 independent strain components (We have dealt with elongation and shear strains but remember, we were concerned to eliminate translation and rotational displacements) u1 u1 + u2 dx1 dx2 u2 dx2 dx1 u2 + u2 + =e21 =e32 1 2 1 2 1 2 Hence:(- translation accounted for by subtracting u1, u2 etc. from strain equations) Rotation u1 + ∂u1 ∂x2 dx2 q1 =q2 u2 +∂u2 dx1 - u2 ∂x1 ∂u2 q1 = = as before dx1 ∂x1Negative because it acts to increase enclosed angle - consistent with shear strain Ï∂u1 ¸ -Ìu1 + dx2 - u1˝ Ó∂x2 ˛ -∂u1or q2 = = dx2 ∂x2 But may also have shear deformation of the cube q1 ≠q2 Define average rotation (of diagonal) 1 Ï∂u2 ∂u1 ¸ q=Ì -˝ note (-) sign2 Ó∂x1 ∂x2 ˛Compatibility One cannot independently define 6 strains from 3 displacements fields. Ïu1(x1, x2, x3 )¸ ÔÔ Ìu2 ( x1, x2, x3 )˝ ÔÔ Óu3(x1, x2, x3 )˛ The strains must be related (by equations) for them to be compatible. for instance in x1 - x2plane take second partial of each: ∂3u1 ∂3u2∂2 e11 =∂2 e22 = ∂x22 ∂x12 ∂x12∂x1dx22 1 Ê∂3u1 ˆ∂2 e12 =Á∂3u2 ˜˜∂x1∂x22 ËÁ∂x1∂x22 +∂x12∂x2 ¯ Substitute first two into latter to get: ∂2 e11 +∂2 e22 = 2∂2 e12 ∂x22 ∂x12 ∂x1∂x2 In general, this can be written in tensor


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MIT 16 01 - Transformation of Stres

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