slide cơ học vật chất rắn chapter 5 new formulation solution strategies

.c om ng co an th cu u du o ng Chapter 5: Formulation – Solution Strategies TDT  University  -­‐  2015 CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   5.1 Review of basic field equations c om Institute for computational science ng 5.2 Boundary conditions & fundamental problems co 5.3 Stress formulation th an 5.4 Displacement formulation ng 5.5 Principle of superposition du o 5.6 Saint-Venant’s principle cu u 5.7 General solution strategies CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   Institute for computational science c om 5.1 Review of basic field equations ng 5.2 Boundary conditions & fundamental problems an th 5.4 Displacement formulation co 5.3 Stress formulation ng 5.5 Principle of superposition du o 5.6 Saint-Venant’s principle cu u 5.7 General solution strategies CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   Institute for computational science ui , j + u j ,i ) ( eij ,kl + ekl ,ij − eik , jl − e jl ,ik = c om eij = σ ij = ( λ + µ ) ekkδ ij + 2µ eij Equilibrium Equations co an th +ν ν σ ij − σ kkδ ij E E ng Hooke’s Law cu u du o eij = ng σ ij , j + Fi = Strain-Displacement Relations Compatibility Relations 15 Equations for 15 Unknowns σij , eij, ui CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   Institute for computational science c om 5.1 Review of basic field equations ng 5.2 Boundary conditions & fundamental problems an th 5.4 Displacement formulation co 5.3 Stress formulation ng 5.5 Principle of superposition du o 5.6 Saint-Venant’s principle cu u 5.7 General solution strategies CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   Institute for computational science T(n)   S   R   St   c om S   R   ng R   Su   co u   Traction Conditions Displacement Conditions th an Mixed Conditions u y du o ng Symmetry Line cu u=0 x CuuDuongThanCong.com Ty( n ) = https://fb.com/tailieudientucntt     h"p://incos.tdt.edu.vn   Institute for computational science c om On coordinate surfaces the traction vector reduces to simply particular stress components ng σy   co τxy   τrθ   σx   σθ   cu u x   τrθ   du o y   r   σr   ng τxy   th σy   θ   σθ   an σx   σr   Cartesian Coordinate Boundaries CuuDuongThanCong.com Polar Coordinate Boundaries https://fb.com/tailieudientucntt     h"p://incos.tdt.edu.vn   c om Institute for computational science On general non-coordinate surfaces, traction vector will not reduce to individual stress components and general traction vector form must be used co ng   n   Tx( n ) = σ x nx + τ xy n y = S cos α S   cu u du o ng th an y   Ty( n ) = τ xy nx + σ y n y = S sin α α   x   Two-dimensional example CuuDuongThanCong.com https://fb.com/tailieudientucntt     h"p://incos.tdt.edu.vn   Institute for computational science Example boundary conditions c om   Traction  Free  Condition   Traction  Condition an co Tx( n ) = σ x = S , Ty( n ) = τ xy =   ng Fixed  Condition   y   u  =  v  =  0   Traction  Condition Tx( n) = −τ xy = 0, Ty( n) = −σ y = S   ng th S   b   x   l   du o S   a   cu u Tx( n ) = Traction  Free  Condition Tx( n) = −τ xy = 0, Ty( n) = −σ y =   Coordinate    Surface  Boundaries   CuuDuongThanCong.com x   Ty( n ) = y   Fixed  Condition   u  =  v  =  0   Traction  Free  Condition   Non-­‐Coordinate    Surface   Boundary   https://fb.com/tailieudientucntt     h"p://incos.tdt.edu.vn   Institute for computational science c om Interface conditions n th an s Interface Conditions: Perfectly Bonded, Slip Interface, Etc co σ ij(1) , u i(1) Material (1): ng   σ ij( 2) , u i( 2) u cu   du o ng Material (2): Embedded Fiber or Rod CuuDuongThanCong.com Layered Composite Plate Composite Cylinder or Disk https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   Institute for computational science c om Summary of Reduction of Fundamental Elasticity Field General Field Equation System (15  Equa,ons,  15  Unknowns:)   ng ℑ{ui , eij , σ ij ; λ, µ, Fi } = du o ng th an co (ui , j + u j ,i ) σ ij, j + Fi = σ ij = (λ + µ )ekk δ ij + 2µeij eij,kl + ekl ,ij − eik , jl − e jl ,ik = eij = Stress Formulation cu u (6  Equa,ons,  6  Unknowns:)   ℑ ( t ) {σ ij ; λ, µ, Fi } σ ij,kk σ ij, j + Fi = ν + σ kk ,ij = − δij Fk ,k − Fi , j − F j ,i 1+ ν 1− ν CuuDuongThanCong.com Displacement Formulation (3  Equa,ons,  3  Unknowns:  ui)   ℑ( u ) {ui ; λ, µ, Fi } µui ,kk + (λ + µ)uk ,ki + Fi = https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   Institute for computational science c om 5.1 Review of basic field equations ng 5.2 Boundary conditions & fundamental problems an th 5.4 Displacement formulation co 5.3 Stress formulation ng 5.5 Principle of superposition du o 5.6 Saint-Venant’s principle cu u 5.7 General solution strategies CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   Institute for computational science c om For a given problem domain, if the state {σij(1) , eij(1) , ui(1) } is a solution to the fundamental elasticity equations with prescribed body forces Fi (1) and surface tractions Ti (1) , and the ng state {σij( ) , eij( ) , ui( ) } is a solution to the fundamental equations with prescribed body forces   Fi ( ) and surface tractions Ti ( ) , then the state {σij(1) + σij( ) , eij(1) + eij( ) , ui(1) + ui( ) } co will be a solution to the problem with body forces Fi (1) + Fi ( ) and surface tractions =   (1)   +   (2)   cu u (1)+(2)   du o ng th an Ti (1) + Ti ( ) {σij(1) , eij(1) , ui(1) } {σij(1) + σij( 2) , eij(1) + eij( 2) , ui(1) + ui( 2) } CuuDuongThanCong.com {σij( ) , eij( ) , ui( ) } https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   Institute for computational science c om 5.1 Review of basic field equations ng 5.2 Boundary conditions & fundamental problems an th 5.4 Displacement formulation co 5.3 Stress formulation ng 5.5 Principle of superposition du o 5.6 Saint-Venant’s principle cu u 5.7 General solution strategies CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   Institute for computational science FR   P P P 3 co P P 2 T(n)   S*   (3)   (2)   cu u du o (1)   ng th an P ng c om The stress, strain and displacement fields due to two different statically equivalent force distributions on parts of the body far away from the loading points are approximately the same   Stresses approximately the same CuuDuongThanCong.com Boundary loading T(n) would produce detailed and characteristic effects only in vicinity of S* Away from S* stresses would generally depend more on resultant FR of tractions rather than on exact distribution https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   Institute for computational science c om 5.1 Review of basic field equations ng 5.2 Boundary conditions & fundamental problems an th 5.4 Displacement formulation co 5.3 Stress formulation ng 5.5 Principle of superposition du o 5.6 Saint-Venant’s principle cu u 5.7 General solution strategies CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   Institute for computational science ng c om 5.7.1 Direct Method - Solution of field equations by direct integration Boundary conditions are satisfied exactly Method normally encounters significant mathematical difficulties thus limiting its application to problems with simple geometry du o ng th an co 5.7.2 Inverse Method - Displacements or stresses are selected that satisfy field equations A search is then conducted to identify a specific problem that would be solved by this solution field This amounts to determine appropriate problem geometry, boundary conditions and body forces that would enable the solution to satisfy all conditions on the problem It is sometimes difficult to construct solutions to a specific problem of practical interest cu u 5.7.3 Semi-Inverse Method Part of displacement and/or stress field is specified, while the other remaining portion is determined by the fundamental field equations (normally using direct integration) and the boundary conditions It is often the case that constructing appropriate displacement and/or stress solution fields can be guided by approximate strength of materials theory The usefulness of this approach is greatly enhanced by employing Saint-Venant’s principle, whereby a complicated boundary condition can be replaced by a simpler statically equivalent distribution CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   Institute for computational science c om Example 5-1: Direct Integration Example: cu u du o ng th an co ng Stretching of Prismatic Bar Under Its Own Weight As an example of a simple direct integration problem, consider the case of a uniform prismatic bar stretched by its own weight, as shown in Figure 5-11 The body forces for this problem are Fx = Fy = 0, Fz = -ρg, where ρ is the material mass density and g is the acceleration of gravity CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   Institute for computational science (5.7.1) ng σ x = σ y = τ xy = τ yz = τ zx = c om Assuming that on each cross-section we have uniform tension produced by the weight of the lower portion of the bar, the stress field would take the from co The equilibrium equations reduce to the simple result (5.7.2) th an ∂σ z = − Fz = ρ g ∂z du o ng This equations can be integrated directly, and applying the boundary condition σz = at z = gives the result σz(z) = 𝜌gz gz Next, by using Hooke’s law, the strains are easily calculated as cu u ez = CuuDuongThanCong.com exy ρ gz , ex = e y = − E = e yz = exz = νρ gz E (5.7.3) https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   Institute for computational science ρg ⎡ νρ gyz E an E ,v = − (5.7.4) z +ν ( x2 + y ) − l ⎤ ⎦ 2E ⎣ cu u du o ng w= νρ gxz th u=− co ng c om The displacements follow from integrating the strain-displacement relation and for the case with boundary conditions of zeros displacement and rotation at point A ( x = y =0; z = l ), the final result is CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   Institute for computational science Figure 5.12 pure bending problem   th an co ng c om Example 5-2: Inverse Example - Pure Beam Bending Consider the case of an elasticity problem under zero body forces with following stress field σ x = Ay, σ y = σ z = τ xy = τ yz = τ zx = (5.7.5) cu u du o ng Where A is a constant It is easily shown that this simple linear stress field satisfies the equations of equilibrium and compatibility, and thus the field is a solution to an elasticity problem The equation is, what problem would be solved by such a field? A common scheme to answer this question is to consider some trial domain and investigate the nature of the boundary conditions that would occur using the given stress field Therefore, consider the tow-dimensional rectangular domain shown in Figure 5-12 Using the field (5.7.5), the tractions (stresses) on each boundary face give zero loadings on the top and bottom and a linear distribution of normal stresses on the right and left side shown Clearly, this type of boundary loading is related to a pure bending problem, whereby the loading on the right and left sides produce no net force and only a pure bending moment CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   Institute for computational science c om Example 5-3: Semi-Inverse Example: Torsion of Prismatic Bars ng A simple semi-inverse example may be borrowed from the torsion problem that is discussed in detail in Chapter Skipping for now the developmental details, we propose the following displacement field: (5.7.6) an co u = −α yz , v = α xz , w = w( x, y ) du o ng th Where α is constant The assumed field specifies the x and y components of the displacement, while the z component is left to be determined as a function of the indicated spatial variables By using the strain-displacement relations and Hook’s law, the stress field corresponding to (5.7.6) is given by cu u σ x = σ y = τ xy = ⎛ ∂w ⎞ ⎛ ∂w ⎞ τ xz = µ ⎜ −α y ⎟ ⎝ ∂x ⎠ (5.7.7) τ yz = µ ⎜ +αx⎟ ⎝ ∂y ⎠ CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   Institute for computational science + ∂y =0 (5.7.8) co ∂x ∂2w ng ∂2w c om Using these stresses in the equations of equilibrium gives the following results cu u du o ng th an Which is actually the form of Navies’s equations for this case This result represents a single equation (Laplace’s equation) to determine the unknown part of the assumed solution form It should be apparent that by assuming part of the solution field, the remaining equations to be solve are greatly simplified A special domain in the x, y plane along with appropriate boundary conditions is needed to complete the solution to a particular problem Once this has been accomplished, the assumed solution form (5.7.6) has been shown to satisfy all the field equations of elasticity CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   ng co 5.7.4 Analytical Solution Procedures - Power Series Method - Fourier Method - Integral Transform Method - Complex Variable Method c om Institute for computational science th an 5.7.5 Approximate Solution Procedures - Ritz Method cu u du o ng 5.7.6 Numerical SolutionProcedures - Finite Difference Method (FDM) - Finite Element Method (FEM) - Boundary Element Method (BEM) CuuDuongThanCong.com 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 ... fundamental problems an th 5. 4 Displacement formulation co 5. 3 Stress formulation ng 5. 5 Principle of superposition du o 5. 6 Saint-Venant’s principle cu u 5. 7 General solution strategies CuuDuongThanCong.com... equations ng 5. 2 Boundary conditions & fundamental problems an th 5. 4 Displacement formulation co 5. 3 Stress formulation ng 5. 5 Principle of superposition du o 5. 6 Saint-Venant’s principle cu u 5. 7 General... om 5. 1 Review of basic field equations ng 5. 2 Boundary conditions & fundamental problems an th 5. 4 Displacement formulation co 5. 3 Stress formulation ng 5. 5 Principle of superposition du o 5. 6