slide cơ học vật chất rắn chapter 2 new deformation displacements and strain

.c om ng co an th cu u du o ng Chapter 2: Deformation: Displacements and strain TDT  University  -­‐  2015 CuuDuongThanCong.com https://fb.com/tailieudientucntt .c om h"p://incos.tdt.edu.vn   2.1 General deformations   co ng 2.2 Geometric construction of small deformation theory an 2.3 Strain transformation th 2.4 Principal strains du o ng 2.5 Spherical & Deviatoric strains 2.6 Strain compatibility cu u 2.7 Curvilinear strain-displacement relations cylindrical coordinates   CuuDuongThanCong.com https://fb.com/tailieudientucntt .c om h"p://incos.tdt.edu.vn   2.1 General deformations   co ng 2.2 Geometric construction of small deformation theory an 2.3 Strain transformation th 2.4 Principal strains du o ng 2.5 Spherical & Deviatoric strains 2.6 Strain compatibility cu u 2.7 Curvilinear strain-displacement relations cylindrical coordinates   CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   co ng This is in contrast to rigid-body motion where the distance between points remains the same cu u du o ng th an An elastic solid is said to be deformed or strained when the relative displacements between points in the body are changed .c om Deformations: non-homogeneous Fig Rigid-body motion Fig Deformed or strained   CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   c om Small Deformation Theory ng - Consider two neighboring material points P0 and P connected with the relative position vector r as shown in Fig - Through a general deformation, these points are mapped to locations P’0 and P’ in the deformed configuration -  In linear elasticity, only small deformation theory is necessary P r an co P0 r' u0 P0′ Fig General deformation between two neighboring points ng th P′ u du o Taylor series expansion around point P0 to express the components of u as cu u f ( ui + Δui ) = f ( ui ) + f ,ui ( ui ) Δui ∂u ∂u ∂u rx + ry + rz ∂x ∂y ∂z ∂v ∂v ∂v v = v o + rx + ry + rz ∂x ∂y ∂z ∂w ∂w ∂w w = wo + rx + ry + rz ∂x ∂y ∂z u = uo + CuuDuongThanCong.com The change in the relative position vector r prove Δr = r′ − r Δri = ui , j rj   https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   c om Small Deformation Theory co ∂u ∂u ∂u u = u + rx + ry + rz ∂x ∂y ∂z ∂v ∂v ∂v v = v o + rx + ry + rz ∂x ∂y ∂z ∂w ∂w ∂w w = wo + rx + ry + rz ∂x ∂y ∂z ng f ( ui + Δui ) = f ( ui ) + f ,ui ( ui ) Δui an o P r P0 r' u0 du o u ∂u ∂u ∂u rx + ry + rz ∂x ∂y ∂z ∂v ∂v ∂v Δry = r ' y − ry = rx + ry + rz ∂x ∂y ∂z ∂w ∂w ∂w Δrz = r 'z − rz = rx + ry + rz ∂x ∂y ∂z cu Δr = r ′ − r = u − u0 Δrx = r 'x − rx = CuuDuongThanCong.com P0′ Fig General deformation between two neighboring points ng th ui = ui0 + ui, j rj P′ u Δri = ui , j rj https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   c om Small Deformation Theory - Tensor ui,j is called the displacement gradient tensor th an co ng ∂u ⎤ ∂z ⎥⎥ ∂v ⎥ 1 = ( u + u ) + (ui , j − u j ,i ) = eij + ωij i, j j ,i ∂z ⎥ 2 ⎥ ∂w ⎥ ∂z ⎥⎦ ⎧ e = ⎪ ij (ui , j + u j ,i ) , strain tensor ⎨ ⎪ωij = (ui , j − u j ,i ) , rotation tensor ⎩ ng ∂u ∂y ∂v ∂y ∂w ∂y du o ⎡ ∂u ⎢ ∂x ⎢ ⎢ ∂v ui , j = ⎢ ∂x ⎢ ⎢ ∂w ⎢ ∂x ⎣ u - Choose ri = dxi, we can write the general result in the form cu ui = ui0 + ui , j rj = ui0 + eij dx j + ωij dx j - Using a dual vector ωi = -1/2 εijkωjk, we have ⎛ ∂u ∂u ⎞ ⎛ ∂u ∂u ⎞ ⎛ ∂u ∂u ⎞ ω1 = ω32 = ⎜ − ⎟ ; ω2 = ω13 = ⎜ − ⎟ ; ω3 = ω21 = ⎜ − ⎟ ⎝ ∂x2 ∂x3 ⎠ ⎝ ∂x3 ∂x1 ⎠ ⎝ ∂x1 ∂x2 ⎠ ω= ( ∇ × u)   CuuDuongThanCong.com https://fb.com/tailieudientucntt .c om h"p://incos.tdt.edu.vn   2.1 General deformations   co ng 2.2 Geometric construction of small deformation theory an 2.3 Strain transformation th 2.4 Principal strains 2.6 Strain compatibility du o ng 2.5 Spherical & Deviatoric strains cu u 2.7 Curvilinear strain-displacement relations cylindrical coordinates   CuuDuongThanCong.com https://fb.com/tailieudientucntt .c om h"p://incos.tdt.edu.vn   co an (Undeformed Element) (Rigid Body Rotation) (Horizontal Extension) (Vertical Extension) (Shearing Deformation) cu u du o ng th Consider the common deformational behavior of a rectangular element Rigid-body motion does not contribute to the strain field, and hence does not affect the stresses We therefore focus our study on the extensional and shearing deformation ng Examples of Continuum Motion & Deformation Fig Typical deformations of a rectangular element   CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   c om Consider a 2D deformation of a rectangular element with original dimensions dx by dy Point A(x,y) with displacement components u(x,y) and v(x,y) Point B has displacement u(x+dx,y) and v(x+dx,y) co ng In small deformation theory, (Taylor series expansion) u(x + dx, y) ≈ u(x, y) + (∂u / ∂x)dx ng du o u cu ∂u ⎞ ⎛ ∂v ⎞ ⎛ A' B' = ⎜ dx + dx ⎟ + ⎜ dx ⎟ ∂x ⎠ ⎝ ∂x ⎠ ⎝ ⎛ ∂u ⎞ ⎛ ∂v ⎞ = dx ⎜1 + ⎟ + ⎜ ⎟ ⎝ ∂x ⎠ ⎝ ∂x ⎠ ⎛ ∂u ⎞ ≈ ⎜ + ⎟ dx ⎝ ∂x ⎠ β v(x,y+dy) From the geometry D' C' th The normal strain in x-direction A ' B '− AB εx = AB ∂u dy ∂y an   y D C B' α A' dy ∂v dx ∂x v(x,y) A dx u(x+dx,y) B u(x,y) x 10   Fig Two-dimensional geometric strain deformation CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   c om cosθ 0⎤ 0⎥ ⎥ ⎥⎦ ng sin θ ⎧e′x = ex cos θ + e y sin θ + 2exy sin θ cosθ ⎪ 2 ⎨e′y = ex sin θ + ey cos θ − 2exy sin θ cos θ ⎪e′xy = −ex sin θ cosθ + e y sin θ cosθ + exy (cos θ − sin θ ) ⎩ co ⎡ cosθ Qij = ⎢ − sin θ ⎢ ⎢⎣ ng th an eij′ = QipQ jq e pq cu u du o ex + e y e x − e y ⎧ ′ e = + cos 2θ + exy sin 2θ ⎪ x 2 ⎪⎪ ex + e y e x − e y − cos 2θ − exy sin 2θ ⎨e′y = 2 ⎪ e −e ⎪e′xy = y x sin 2θ + exy cos 2θ ⎪⎩ 19   CuuDuongThanCong.com https://fb.com/tailieudientucntt .c om h"p://incos.tdt.edu.vn   2.1 General deformations   co ng 2.2 Geometric construction of small deformation theory an 2.3 Strain transformation th 2.4 Principal strains 2.6 Strain compatibility du o ng 2.5 Spherical & Deviatoric strains cu u 2.7 Curvilinear strain-displacement relations cylindrical coordinates 20   CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   c om eij n j = eni co ⇒ −e3 + ϑ1e − ϑ2e + ϑ3 = du o ng th an ϑ1 = e1 + e2 + e3 ϑ2 = e1e2 + e2e3 + e3e1 ϑ3 = e1e2e3 u ⎡e1 eij = ⎢ e2 ⎢ ⎢⎣ 0 cu => Principle strains n1 ng det ⎡⎣eij − eδ ij ⎤⎦ = n2 0⎤ 0⎥ ⎥ e3 ⎥⎦ 21   CuuDuongThanCong.com https://fb.com/tailieudientucntt .c om h"p://incos.tdt.edu.vn   2.1 General deformations   co ng 2.2 Geometric construction of small deformation theory an 2.3 Strain transformation th 2.4 Principal strains du o ng 2.5 Spherical & Deviatoric strains 2.6 Strain compatibility cu u 2.7 Curvilinear strain-displacement relations cylindrical coordinates 22   CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   co ng ⎛ ⎞ 1 eij = ⎜ eij − ekkδ ij ⎟ + ekkδ ij ⎝ ⎠ $ !# #"## $ !" e% c om In particular applications it is convenient to decompose the strain tensor into two parts called spherical and deviatoric strain tensors eˆij an th represents only volumetric deformation ng e!ij = ekkδ ij du o The spherical strain ij cu u The deviatoric strain eˆij = eij − ekkδ ij accounts for changes in shape of material elements Note: principal directions of the deviatoric strain are the same as those of the strain tensor 23   CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   c om Example 2-2: Determine the principle, spherical, deviatoric strains of the following state of strain cu u du o ng th an co ng ⎡ −2 ⎤ eij = ⎢ −2 −4 ⎥ ⎢ ⎥ ⎢⎣ ⎥⎦ 24   CuuDuongThanCong.com https://fb.com/tailieudientucntt .c om h"p://incos.tdt.edu.vn   2.1 General deformations   co ng 2.2 Geometric construction of small deformation theory an 2.3 Strain transformation th 2.4 Principal strains du o ng 2.5 Spherical & Deviatoric strains 2.6 Strain compatibility cu u 2.7 Curvilinear strain-displacement relations cylindrical coordinates 25   CuuDuongThanCong.com https://fb.com/tailieudientucntt .c om h"p://incos.tdt.edu.vn   cu u du o ng th an co ng Normally we want continuous single-valued displacements; i.e a mesh that fits perfectly together after deformation   Undeformed State Deformed State 26   CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   c om Mathematical Concepts Related to Deformation Compatibility Strain-Displacement Relations co ng ∂u ∂v ∂w ⎛ ∂u ∂v ⎞ ⎛ ∂v ∂w ⎞ ⎛ ∂w ∂u ⎞ , e y = , ez = , exy = ⎜ + ⎟ , e yz = ⎜ + , e = + ⎟ zx ⎜ ⎟ ∂x ∂y ∂z ⎝ ∂y ∂x ⎠ ⎝ ∂z ∂y ⎠ ⎝ ∂x ∂z ⎠ an ex = du o ng th Given the Three Displacements: We have six equations to easily determine the six strains cu u Given the Six Strains: We have six equations to determine three displacement components This is an over-determined system and in general will not yield continuous single-valued displacements unless the strain components satisfy some additional relations The strains must satisfy additional relations called integrability or compatibility equations CuuDuongThanCong.com https://fb.com/tailieudientucntt 27   .c om h"p://incos.tdt.edu.vn         an co ng Physical Interpretation of Strain Compatibility   (b)  Undeformed  Configuration                 cu u du o ng th (a)  Discretized  Elastic  Solid   (c)  Deformed  Configuration             Continuous  Displacements   CuuDuongThanCong.com     (d)  Deformed  Configuration           Discontinuous  Displacements   https://fb.com/tailieudientucntt 28   ng (ui, j + u j ,i ) eij ,kk + ekk ,ij − eik , jk − e jk ,ik = Saint Venant Compatibility Equations an co eij = ⎧ e = ⎪ ij ,kl ( ui , jkl + u j ,ikl ) ⎪ e = uk ,lij + ul ,kij ) ( ⎪⎪ kl ,ij ⎨ ⎪e jl ,ik = ( u j ,lik + ul , jik ) ⎪ ⎪eik , jl = ( ui ,kjl + uk ,ijl ) ⎪⎩ c om h"p://incos.tdt.edu.vn   u du o ng th 81 individual equations, most are either simple identities or repetitions, and only are meaningful cu These six relations may be determined by letting k = l, and in scalar notation, they become ∂ 2exy ∂ ex ∂ ⎛ ∂eyz ∂ezx ∂exy ⎞ ∂ ex ∂ e y = ⎜− + + + =2 ; ⎟ ∂ y ∂ z ∂ x ∂ x ∂ y ∂z ⎠ ∂y ∂x ∂x∂y ⎝ ∂ e y ∂ ez ∂ 2eyz ∂ 2ey ∂ ⎛ ∂ezx ∂exy ∂e yz ⎞ + = ; = + + ⎜− ⎟ 2 ∂z ∂y ∂y∂z ∂z∂x ∂y ⎝ ∂y ∂z ∂x ⎠ ∂ 2ezx ∂ ez ∂ ex ∂ ez ∂ ⎛ ∂exy ∂e yz ∂ezx ⎞ + = ; = + + ⎜− ⎟ ∂x ∂z ∂z∂x ∂x∂y ∂z ⎝ ∂z ∂x ∂y ⎠ 29   CuuDuongThanCong.com https://fb.com/tailieudientucntt .c om h"p://incos.tdt.edu.vn   2.1 General deformations   co ng 2.2 Geometric construction of small deformation theory an 2.3 Strain transformation th 2.4 Principal strains du o ng 2.5 Spherical & Deviatoric strains 2.6 Strain compatibility cu u 2.7 Curvilinear strain-displacement relations cylindrical coordinates 30   CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   u = ur e r + uθ eθ + u z e z x3   z   th ng du o r   u x1   θ   x2   e2 an eˆr cu e1 ⎡ er e = ⎢erθ ⎢ ⎢⎣ erz erθ eθ eθ z erz ⎤ eθ z ⎥ ⎥ ez ⎥⎦ co eˆθ e3 where ng eˆz c om The cylindrical coordinate system ∂ur ∂r ∂u ⎞ 1⎛ eθ = ⎜ ur + θ ⎟ r⎝ ∂θ ⎠ ∂u ez = z ∂z ⎛ ∂ur ∂uθ uθ ⎞ erθ = ⎜ + − ⎟ ⎝ r ∂θ ∂r r ⎠ ⎛ ∂u ∂u z ⎞ eθ z = ⎜ θ + ⎟ ⎝ ∂z r ∂θ ⎠ ⎛ ∂u ∂u ⎞ ezr = ⎜ r + z ⎟ ⎝ ∂z ∂r ⎠ er = 31   CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   The spherical coordinate system c om where eRφ eφ eφθ eRθ ⎤ ⎥ eφθ ⎥ eθ ⎥⎦ co ng u = u Re R + uφ eφ + uθ eθ ⎡ eR ⎢ e = ⎢ eRφ ⎢⎣eRθ ∂u R ∂R ∂u ⎞ 1⎛ eφ = ⎜ u R + φ ⎟ R⎝ ∂φ ⎠ ⎛ ∂uθ ⎞ eθ = + sin φ u + cos φ u R φ ⎟ ⎜ R sin φ ⎝ ∂θ ⎠ ⎛ ∂u R ∂uφ uφ ⎞ eRφ = ⎜ + − ⎟ ⎝ R ∂θ ∂R R ⎠ ⎞ ⎛ ∂uφ ∂uθ eφθ = + − cos φ u θ ⎟ ⎜ R ⎝ sin φ ∂θ ∂φ ⎠ ⎛ ∂u R ∂uθ uθ ⎞ eθ R = ⎜ + − ⎟ ⎝ R sin φ ∂θ ∂R R ⎠ cu u du o ng th an eR = 32   CuuDuongThanCong.com https://fb.com/tailieudientucntt .c om ng co an th ng du o u cu CuuDuongThanCong.com https://fb.com/tailieudientucntt ...   2. 1 General deformations   co ng 2. 2 Geometric construction of small deformation theory an 2. 3 Strain transformation th 2. 4 Principal strains du o ng 2. 5 Spherical & Deviatoric strains 2. 6 Strain. ..   2. 1 General deformations   co ng 2. 2 Geometric construction of small deformation theory an 2. 3 Strain transformation th 2. 4 Principal strains du o ng 2. 5 Spherical & Deviatoric strains 2. 6 Strain. ..   2. 1 General deformations   co ng 2. 2 Geometric construction of small deformation theory an 2. 3 Strain transformation th 2. 4 Principal strains du o ng 2. 5 Spherical & Deviatoric strains 2. 6 Strain