slide cơ học vật chất rắn chapter 7 new two dimensional formulation

.c om ng co an th cu u du o ng Chapter 7: Two-dimensional Formulation TDT  University  -­‐  2015 CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   Institute for computational science c om 7.1 Review of Two-dimensional formulation 7.3 Plane stress ng 7.5 Anti-plane strain th an 7.4 Generalized plane stress co ng 7.2 Plane strain du o 7.6 Airy stress function cu u 7.7 Polar coordinate formulation CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   Institute for computational science c om 7.1 Review of Two-dimensional formulation ng 7.2 Plane strain co 7.3 Plane stress th du o ng 7.5 Anti-plane strain 7.6 Airy stress function an 7.4 Generalized plane stress cu u 7.7 Polar coordinate formulation CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   Institute for computational science c om 7.1 Review of two-dimensional formulation du o ng th an co ng - Three-dimensional elasticity problems are very difficult to solve Thus, most solutions are developed for reduced problems that typically include axisymmetric or two-dimensionality We will first develop governing equations for two-dimensional problems, and will explore four different theories: -  Plane Strain -  Plane Stress -  Generalized Plane Stress -  Anti-Plane Strain cu u - Since all real elastic structures are three-dimensional, theories set forth here will be approximate models The nature and accuracy of the approximation will depend on problem and loading geometry - The basic theories of plane strain and plane stress represent the fundamental plane problem in elasticity While these two theories apply to significantly different types of twodimensional bodies, their formulations yield very similar field equations CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   Institute for computational science 7.1 Review of two-dimensional formulation Three-Dimensional c om Two-Dimensional ng x co y y z du o ng th an z x cu u z Spherical Cavity y x CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   7.1 Review of Two-dimensional formulation ng 7.2 Plane strain c om Institute for computational science co 7.3 Plane stress th du o ng 7.5 Anti-plane strain 7.6 Airy stress function an 7.4 Generalized plane stress cu u 7.7 Polar coordinate formulation CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   Institute for computational science 7.2 Plane strain y   u du o ng th an co u = u ( x, y ) , v = v ( x, y ) , w = cu   ng c om - Consider an infinitely long cylindrical (prismatic) body as shown in Figure If the body forces and tractions on lateral boundaries are independent of the z-coordinate and have no zcomponent, then the deformation field can be taken in the reduced form x   R   z   CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   Institute for computational science c om 7.2 Plane strain Plane Strain Field Equations Note that: Although ez = 0, the normal stress σz will not in general vanish ∂u ∂v ⎛ ∂u ∂v ⎞ , ey = , exy = ⎜ + ⎟ , ez = exz = eyz = ∂x ∂y ⎝ ∂y ∂x ⎠ ex = Stresses ⎧σ x = λ (ex + ey ) + 2µ ex , σ y = λ (ex + ey ) + 2µ ey ⎪ ⎨σ z = λ (ex + ey ) = ν (σ x + σ y ) ⎪ ⎩τ xy = 2µ exy , τ xz = τ yz = th an co ng Strains Navier Equations du o Strain Compatibility ∂ exy ∂ ex ∂ ey + =2 ∂y ∂x ∂x∂y u cu ∂σ x ∂τ xy + + Fx = ∂x ∂y ∂τ xy ∂σ y + + Fy = ∂x ∂y ng Equilibrium Equations µ∇ 2u + (λ + µ ) ∂ ⎛ ∂u ∂v ⎞ ⎜ + ⎟ + Fx = ∂x ⎝ ∂x ∂y ⎠ µ∇ v + (λ + µ ) ∂ ⎛ ∂u ∂v ⎞ ⎜ + ⎟ + Fy = ∂y ⎝ ∂x ∂y ⎠ CuuDuongThanCong.com Beltrami-Michell Equation ∇ (σ x + σ y ) = − ⎛ ∂Fx ∂Fy ⎞ + ⎜ ⎟ −ν ⎝ ∂x ∂y ⎠ https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   Institute for computational science 7.2 Plane strain c om Examples of Plane Strain Problems ng y x x y cu z z u du o ng th an co P Long Cylinders Under Uniform Loading CuuDuongThanCong.com Semi-Infinite Regions Under Uniform Loadings https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   7.1 Review of Two-dimensional formulation ng 7.2 Plane strain co 7.3 Plane stress th an 7.4 Generalized plane stress du o ng 7.5 Anti-plane strain 7.6 Airy stress function c om Institute for computational science cu u 7.7 Polar coordinate formulation CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   Institute for computational science 7.3 Plane stress c om Correspondence Between Plane Formulations Plane Strain ∂ ⎛ ∂u ∂v ⎞ ⎜ + ⎟ + Fy = ∂y ⎝ ∂x ∂y ⎠ co µ∇ v + (λ + µ ) µ∇ 2u + an ∂ ⎛ ∂u ∂v ⎞ ⎜ + ⎟ + Fx = ∂x ⎝ ∂x ∂y ⎠ du o ng th µ∇ 2u + (λ + µ ) ng Plane Stress cu u ∂σ x ∂τ xy + + Fx = ∂x ∂y ∂τ xy ∂σ y + + Fy = ∂x ∂y ⎛ ∂Fx ∂Fy ⎞ ∇ (σ x + σ y ) = − + ⎜ ⎟ −ν ⎝ ∂x ∂y ⎠ CuuDuongThanCong.com µ∇ v + E ∂ ⎛ ∂u ∂v ⎞ ⎜ + ⎟ + Fx = 2(1 −ν ) ∂x ⎝ ∂x ∂y ⎠ E ∂ ⎛ ∂u ∂v ⎞ ⎜ + ⎟ + Fy = 2(1 −ν ) ∂y ⎝ ∂x ∂y ⎠ ∂σ x ∂τ xy + + Fx = ∂x ∂y ∂τ xy ∂σ y + + Fy = ∂x ∂y ⎛ ∂F ∂Fy ⎞ ∇ (σ x + σ y ) = −(1 + ν ) ⎜ x + ⎟ ∂y ⎠ ⎝ ∂x https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   Institute for computational science 7.3 Plane stress c om Transformation Between Plane Strain and Plane Stress du o ng th an co ng Plane strain and plane stress field equations had identical equilibrium equations and boundary conditions Navier’s equations and compatibility relations were similar but not identical with differences occurring only in particular coefficients involving just elastic constants So perhaps a simple change in elastic moduli would bring one set of relations into an exact match with the corresponding result from the other plane theory This in fact can be done using results in the following table cu u Plane Stress to Plane Strain Plane Strain to Plane Stress E ν E − ν2 E (1 + 2ν) (1 + ν) ν 1− ν ν 1+ ν Therefore the solution to one plane problem also yields the solution to the other plane problem through this simple transformation scheme CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   7.1 Review of Two-dimensional formulation ng 7.2 Plane strain c om Institute for computational science co 7.3 Plane stress th an 7.4 Generalized plane stress du o 7.6 Airy stress function ng 7.5 Anti-plane strain cu u 7.7 Polar coordinate formulation CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   Institute for computational science 7.4 Generalized plane stress h φ ( x, y ) = ∫ φ ( x, y, z )dz 2h − h ng th Using the averaging operator defined by an co ng c om The plane stress formulation produced some inconsistencies in particular out-of-plane behavior and resulted in some three-dimensional effects where in-plane displacements were functions of z We avoided these issues by simply neglecting some of the troublesome equations thereby producing an approximate elasticity formulation In order to avoid this unpleasant situation, an alternate approach called Generalized Plane Stress can be constructed based on averaging the field quantities through the thickness of the domain cu u du o all plane stress equations are satisfied exactly by the averaged stress, strain and displacements variables; thereby eliminating the inconsistencies found in the original plane stress formulation However, this gain in rigor does not generally contribute much to applications CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   7.1 Review of Two-dimensional formulation ng 7.2 Plane strain c om Institute for computational science co 7.3 Plane stress th du o 7.6 Airy stress function ng 7.5 Anti-plane strain an 7.4 Generalized plane stress cu u 7.7 Polar coordinate formulation CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   Institute for computational science 7.5 Anti-plane strain ng c om An additional plane theory of elasticity called Anti-Plane Strain involves a formulation based on the existence of only out-of-plane deformation starting with an assumed displacement field u = v = , w = w( x, y ) th an ∂w ∂w , eyz = ∂x ∂y ng exz = co Strains ex = ey = ez = exy = du o Equilibrium Equations σ x = σ y = σ z = τ xy = τ xz = 2µ exz , τ yz = 2µ eyz Navier’s Equation µ∇ w + Fz = cu u ∂τ xz ∂τ yz + + Fz = ∂x ∂y Fx = Fy = Stresses This theory is sometimes used in geomechanic applications to model deformations of portions of the earth’s interior CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   7.1 Review of Two-dimensional formulation ng 7.2 Plane strain c om Institute for computational science co 7.3 Plane stress th an 7.4 Generalized plane stress ng 7.5 Anti-plane strain du o 7.6 Airy stress function cu u 7.7 Polar coordinate formulation CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   Institute for computational science ng c om 7.6 Airy stress function Numerous solutions to plane strain and plane stress problems can be determined using an Airy Stress Function technique The method will reduce the general formulation to a single governing equation in terms of a single unknown The resulting equation is then solvable by several methods of applied mathematics, and thus many analytical solutions to problems of interest can be found du o ng th an co The method is started by reviewing the equilibrium equations for the plane problems We retain the body forces but assume that they are derivable from a potential function V such that ∂V ∂V Fx = − , Fy = − ∂x ∂y This assumption is not very restrictive because many body forces found in applications (e.g gravity loading) fall into this category Under this form, the plane equilibrium equations can be written as cu u ∂τ xy ∂ (σ y − V ) ∂ (σ x − V ) ∂τ xy + =0 ; + =0 ∂x ∂y ∂x ∂y CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   Institute for computational science 7.6 Airy stress function c om These equations will be identically satisfied by choosing a representation ∂ 2φ ∂ 2φ ∂ 2φ σ x = + V ; σ y = + V ; τ xy = − ∂y ∂x ∂x∂y co ng where ϕ=ϕ(x,y) is an arbitrary form called the Airy stress function In the case of zero body forces, then we have ng th an ∂ 2ϕ ∂ 2ϕ ∂ 2ϕ σ x = , σ y = , τ xy = − ∂y ∂x ∂x∂y du o It is easily shown that this form satisfies equilibrium (zero body force case) and substituting it into the compatibility equations gives cu u ∂ 4ϕ ∂ 4ϕ ∂ 4ϕ + + = ∇ ϕ =0 2 ∂x ∂x ∂y ∂y This relation is called the biharmonic equation and its solutions are known as biharmonic functions CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   Institute for computational science 7.6 Airy stress function c om Airy Stress Function Formulation th an co ng The plane problem of elasticity can be reduced to a single equation in terms of the Airy stress function This function is to be determined in the two-dimensional region R bounded by the boundary S as shown in the figure Appropriate boundary conditions over S are   necessary to complete a solution Traction boundary conditions would involve the specification of second derivatives of the stress function; however, this condition can be reduced to specification of first order derivatives ng ∂ 4ϕ ∂ 4ϕ ∂ 4ϕ + 2 + = ∇4ϕ = ∂x ∂x ∂y ∂y So   du o Ty( n ) ∂ϕ ∂ϕ = τ xy nx + σ y ny = − nx + n y ∂x∂y ∂x cu u T ∂ 2ϕ ∂ 2ϕ = σ x nx + τ xy ny = nx − ny ∂y ∂x∂y (n) x Si   2 y   S  =  Si  +  So   R   τxy σx x   CuuDuongThanCong.com σy https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   7.1 Review of Two-dimensional formulation ng 7.2 Plane strain c om Institute for computational science co 7.3 Plane stress th du o ng 7.5 Anti-plane strain 7.6 Airy stress function an 7.4 Generalized plane stress cu u 7.7 Polar coordinate formulation CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   Institute for computational science 7.7 Polar coordinate formulation c om Plane Elasticity Problem ng ⎧ ∂ur ⎪er = ∂r ⎪ ⎪ ∂uθ ⎞ 1⎛ ⎨eθ = ⎜ ur + ⎟ r⎝ ∂θ ⎠ ⎪ ⎪ ⎛ ∂ur ∂uθ uθ ⎞ + − ⎟ ⎪erθ = ⎜ ⎝ r ∂θ ∂r r ⎠ ⎩ th an co Strain-Displacement du o Plane strain ng Hooke’s Law cu u σ r = λ (er + eθ ) + 2µ er σ θ = λ (er + eθ ) + 2µ eθ σ z = λ (er + eθ ) = ν (σ r + σ θ ) τ rθ = 2µ erθ , τ θ z = τ rz = CuuDuongThanCong.com Plane stress (σ r −νσ θ ) E eθ = (σ θ −νσ r ) E er = ez = − ν (σ r + σ θ ) = − ν E −ν +ν erθ = τ rθ , eθ z = erz = E (er + eθ ) https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   Institute for computational science Navier’s  Equa-ons   th an co ng ⎧ ∂ ⎛ ∂ur ur ∂uθ ⎞ µ ∇ u + ( λ + µ ) + + r ⎜ ⎟ + Fr = ⎪ ∂ r ∂ r r r ∂ θ ⎪ ⎝ ⎠ Plane strain ⎨ ⎪ µ∇ 2u + (λ + µ ) ∂ ⎛ ∂ur + ur + ∂uθ ⎞ + F = θ ⎜ ⎟ θ ⎪⎩ r ∂θ ⎝ ∂r r r ∂θ ⎠ c om 7.7 Polar coordinate formulation cu u du o ng ⎧ E ∂ ⎛ ∂ur ur ∂uθ ⎞ ⎪ µ∇ ur + 2(1 −ν ) ∂r ⎜ ∂r + r + r ∂θ ⎟ + Fr = ⎝ ⎠ Plane stress ⎪⎨ ⎪ µ∇ 2u + E ∂ ⎛ ∂ur + ur + ∂uθ ⎞ + F = θ ⎜ ⎟ θ ⎪⎩ 2(1 −ν ) r ∂θ ⎝ ∂r r r ∂θ ⎠ Equilibrium  Equa-ons   ⎧ ∂σ r ∂τ rθ (σ r − σ θ ) + Fr = ⎪⎪ ∂r + r ∂θ + r ⎨ ⎪ ∂τ rθ + ∂σ θ + 2τ rθ + F = θ ⎪⎩ ∂r r ∂θ r CuuDuongThanCong.com Compa-bility  Equa-ons   ∇ (σ r + σ θ ) = − ⎛ ∂Fr Fr ∂Fθ ⎞ + + ⎜ ⎟ −ν ⎝ ∂r r r ∂θ ⎠ https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   Institute for computational science 7.7 Polar coordinate formulation Airy Representation Biharmonic Governing Equation co ng ⎧ ∂ϕ ∂ 2ϕ ⎪σ r = r ∂r + r ∂θ ⎪ ∂ 2ϕ ⎪ ⎨σ θ = ∂r ⎪ ⎪ ∂ ⎛ ∂ϕ ⎞ τ = − r θ ⎜ ⎟ ⎪ ∂r ⎝ r ∂θ ⎠ ⎩ ⎛ ∂ ∂ ∂ ⎞⎛ ∂ ∂ ∂ ⎞ ∇ ϕ =⎜ + + + + ϕ =0 ⎟⎜ 2 ⎟ ∂ r r ∂ r r ∂ θ ∂ r r ∂ r r ∂ θ ⎝ ⎠⎝ ⎠ th ng du o R Traction Boundary Conditions u y S Tr = f r (r , θ ) , Tθ = fθ (r , θ ) cu τrθ   an   σr   c om Airy Stress Function Approach φ = φ(r,θ) σθ   ! r θ CuuDuongThanCong.com x https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   cu u du o ng th an co ng c om Institute for computational science CuuDuongThanCong.com https://fb.com/tailieudientucntt ... science c om 7. 1 Review of Two- dimensional formulation 7. 3 Plane stress ng 7. 5 Anti-plane strain th an 7. 4 Generalized plane stress co ng 7. 2 Plane strain du o 7. 6 Airy stress function cu u 7. 7 Polar... h"p://incos.tdt.edu.vn   7. 1 Review of Two- dimensional formulation ng 7. 2 Plane strain c om Institute for computational science co 7. 3 Plane stress th du o ng 7. 5 Anti-plane strain 7. 6 Airy stress function an 7. 4... h"p://incos.tdt.edu.vn   7. 1 Review of Two- dimensional formulation ng 7. 2 Plane strain co 7. 3 Plane stress th an 7. 4 Generalized plane stress du o ng 7. 5 Anti-plane strain 7. 6 Airy stress function