slide cơ học vật chất rắn chapter 6 new strain energy and related principles

.c om ng co an th cu u du o ng Chapter 6: Strain energy and related principles TDT  University  -­‐  2015 CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   Institute for computational science c om 6.1 Review of Strain energy and related principles ng 6.2 Strain energy co 6.3 Uniqueness of the elasticity Boundary-Value Problem th an 6.4 Bounds on Elastic Constants ng 6.5 Related Integral Theorems du o 6.6 Principle of Virtual Work cu u 6.7 Principles of Minimum potential & complementary energy 6.8 Rayleigh-Ritz method CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   Institute for computational science c om 6.1 Review of Strain energy and related principles ng 6.2 Strain energy th an 6.4 Bounds on Elastic Constants co 6.3 Uniqueness of the elasticity Boundary-Value Problem ng 6.5 Related Integral Theorems du o 6.6 Principle of Virtual Work cu u 6.7 Principles of Minimum potential & complementary energy 6.8 Rayleigh-Ritz method CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   Institute for computational science c om 6.1 Review of Strain energy and related principles Work done by surface and body forces on elastic solids is stored inside the body ng in the form of strain energy U   ng th an co T  n cu u du o F CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   Institute for computational science c om 6.1 Review of Strain energy and related principles ng 6.2 Strain energy th an 6.4 Bounds on Elastic Constants co 6.3 Uniqueness of the elasticity Boundary-Value Problem ng 6.5 Related Integral Theorems du o 6.6 Principle of Virtual Work cu u 6.7 Principles of Minimum potential & complementary energy 6.8 Rayleigh-Ritz method CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   Institute for computational science 6.2 Strain energy dy - Consider first the simple uniform uniaxial dz dx deformation case with no body forces σ th from zero to σx o σx du o ng we assume that the stress increases slowly z σx cu u dσ dU = ∫ FdΔx = ∫ σ dydz dx = E E 0 σx ∫ (σ dσ ) dydzdx Δx = ε x dx y x ⎧ F = σ dydz ⎪ ⎨ σ Δ x = ε d x = dx x ⎪⎩ E σ σx = σ x dxdydz = σ xε x dxdydz 2E U= dU = σ xε x Strain Energy Density V CuuDuongThanCong.com F = σ dydz an normal stress σ in the x – direction co dy, dz is under the action of a uniform ng - The cubical element of dimensions dx, σ c om σ U  =  Area   Under  Curve e https://fb.com/tailieudientucntt x e h"p://incos.tdt.edu.vn   Institute for computational science 6.2 Strain energy c om We next investigate the strain energy caused by the action of uniform shear stress Choosing the same cubical element as previously analyzed, consider the case an co ng under uniform τxy and τyx loading ⎛ ∂u ⎞ ⎛ ∂u ∂v ⎞ ⎛ ∂v ⎞ y dU = τ xy dydz ⎜ dx ⎟ + τ yx dxdz ⎜ dy ⎟ = τ xy ⎜ + ⎟ dxdydz ⎝ ∂x ⎠ ⎝ ∂y ⎠ ⎝ ∂y ∂x ⎠ τ µγ U = τ xyγ xy = xy = xy Strain Energy Density 2µ 2 th ∂u dy ∂y du o ng τyx τxy τxy cu u dy τ dx U  =  Area   Under  Curve x Shear Deformation CuuDuongThanCong.com ∂v dx ∂x ! https://fb.com/tailieudientucntt xy ! h"p://incos.tdt.edu.vn   Institute for computational science 6.2 Strain energy General Deformation Case Total strain energy U T = ∫∫∫ Udxdydz ng co U ( e ) = λ e jj ekk + µ eij eij 1 1 = λ (ex + ey + ez ) + µ (ex2 + e y2 + ez2 + γ xy2 + γ yz2 + γ zx2 ) 2 2 th an In Terms of Strain V c om 1 σ e + σ e + σ e + τ γ + τ γ + τ γ = σ ij eij ( x x y y z z xy xy yz yz zx zx ) 2 U= du o ng +ν ν σ ijσ ij − σ jjσ kk 2E 2E +ν ν 2 2 = σ + σ + σ + τ + τ + τ − σ + σ + σ ( x y z xy yz zx ) E ( x y z ) 2E U (σ ) = cu u In Terms of Stress Note Strain Energy Is Positive Definite Quadratic Form U ≥0 Relation U ≥ is valid for all elastic materials, including both isotropic and anisotropic solids CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   Institute for computational science   6.2 Strain energy Example Problem Let us consider stress field c om y   2c   an +ν ν 2 +ν σ x + 2τ xy2 ) − σx = σx + τ xy ( 2E 2E 2E E th U= L   co ng 3P 3P ⎛ y2 ⎞ σ x = − xy ,τ xy = − ⎜1 − ⎟ , σ y = σ z = τ yz = τ zx = 2c 4c ⎝ c ⎠ P   x   −c ∫ L 0 ng −c ⎛ +ν ⎞ σx + τ xy ⎟ dxdydz ⎜ E E ⎝ ⎠ ⎛ +ν ⎞ σx + τ xy ⎟ dxdy ⎜ E E ⎝ ⎠ u c ∫ ∫ L cu =∫ c du o U T = ∫∫∫ UdV = ∫ 1 c L 9P2 2 +ν = x y d x d y + E ∫ − c ∫ 4c E c ∫ ∫ −c L 9P2 ⎛ y2 ⎞ ⎜1 − ⎟ dxdy 16c ⎝ c ⎠ P L2 P L(1 + ν ) = + Ec Ec CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   Institute for computational science 6.2 Strain energy c om Derivative Operations on Strain Energy For the Uniaxial Deformation Case: ∂U (σ ) ∂ ⎛ σ x2 ⎞ σ x = = ex ⎜ ⎟= ∂σ x ∂σ x ⎝ E ⎠ E co ng ∂U (e) ∂ ⎛ Eex2 ⎞ = ⎜ ⎟ = Eex = σ x ∂ex ∂ex ⎝ ⎠ ∂eij ∂σ kl = ∂ekl ∂σ ij th ng ∂σ kl ∂eij du o ∂ekl = Cijkl = Cklij u ∂σ ij ∂U (e) ∂U (σ ) , eij = ∂eij ∂σ ij cu σ ij = an For the General Deformation Case: Therefore Cij = Cji, and thus there are only 21 independent elastic constants for general anisotropic elastic materials CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   Institute for computational science 6.6 Principle of Virtual Work c om The surface integral can be changed to a volume integral and combined with the body force term These steps are summarized as δ W = ∫ Ti nδ ui dS + ∫ Fi δ ui dV S V V co S ng = ∫ σ ij n jδ ui dS + ∫ Fi δ ui dV = ∫ (σ ijδ ui ) dV + ∫ Fi δ ui dV ,j V an V th = ∫ (σ ij , jδ ui + σ ijδ ui , j ) dV + ∫ Fi δ ui dV V V ng = ∫ ( − Fiδ ui + σ ijδ eij ) dV + ∫ Fi δ ui dV V du o V = ∫ σ ijδ eij dV V V ij ij cu u ∫ σ δ e dV = ∫ (σ δ e V x x + σ yδ ey + σ zδ ez + τ xyδγ xy + τ yzδγ yz + τ zxδγ zx ) dV Notice that the virtual strain energy does not contain the factor of ½ It is because the stresses are constant during the virtual displacement CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   Institute for computational science c om 6.6 Principle of Virtual Work ) co V St V an (∫ UdV − ∫ Ti nui dS − ∫ Fu i i dV = δ (U T − W ) = th δ ng The external forces are unchanged during the virtual displacements and the region V is fixed cu u du o ng This is one of the statements of the principle of virtual work for an elastic solid The quantity (UT -W) actually represents the total potential energy of the elastic solid, and thus the change in potential energy during a virtual displacement from equilibrium is zero CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   Institute for computational science 6.6 Principle of Virtual Work St V co V σ ijδ eij dV − ∫ Ti nδ ui dS − ∫ Fiδ ui dV = ng ∫ c om The principle of virtual work provides a convenient method for deriving equilibrium equations and associated boundary conditions for various special theories of elastic bodies th an The integrand of the first term can be reduced as σ ijδ eij = σ ij (δ ui , j + δ u j ,i ) = σ ijδ ui , j = (σ ijδ ui ), j − σ ij , jδ ui ∫ (σ + Fi )δ ui dV + ∫ (Ti n − σ ij n j )δ ui dS = S For arbitrary δui cu u V ij , j du o ng ⎡(σ δ u ) − σ δ u ⎤ dV − T nδ u dS − F δ u dV = ∫V ⎣ ij i , j ij , j i ⎦ ∫St i i ∫V i i σ ij , j + Fi = on V and either δ ui = on Su or Ti n = σ ij n j on St CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   Institute for computational science c om 6.1 Review of Strain energy and related principles ng 6.2 Strain energy th an 6.4 Bounds on Elastic Constants co 6.3 Uniqueness of the elasticity Boundary-Value Problem ng 6.5 Related Integral Theorems du o 6.6 Principle of Virtual Work cu u 6.7 Principles of Minimum potential & complementary energy 6.8 Rayleigh-Ritz method CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   Institute for computational science 6.7 Principles of Minimum potential & complementary energy c om Principle of minimum potential energy: th ng Principle of minimum complementary energy: an co ng Of all displacement satisfying the given boundary conditions of an elastic solid, those that satisfy the equilibrium equations make the potential energy a local minimum An additional minimum principle can be developed by reversing the nature of the variation Thus, consider the variation of stresses while holding the displacements constant cu u du o Of all elastic stress satisfying the given boundary conditions of an elastic solid, those that satisfy the equilibrium equations make the complementary energy a local minimum CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   Institute for computational science 6.7 Principles of Minimum potential & complementary energy c om Example 6.1: Euler-Bernoulli Beam theory an co ng In order to demonstrate the utility of energy principles, consider an application dealing with the bending of an elastic beam, as shown in Figure 6-5 The external distributed Loading q will induce internal bending moments M and shear forces V at each section of the beam According to classical Euler-Bernoulli theory, the bending stress σx and moment-curvature and moment-shear relations are given by ng th My d 2w dM σx = − , M = EI , V = I dx dx cu u du o Where I = ∫∫A y 2dA is the area moment of inertia of the cross-section about the neutral axis, and w is the beam deflection (positive in y direction) CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   Institute for computational science 6.7 Principles of Minimum potential & complementary energy c om Example 6.1: Euler-Bernoulli Beam theory Considering only the strain energy caused by the bending stresses EI 2 ⎛ d 2w ⎞ ⎜⎜ ⎟⎟ y ⎝ dx ⎠ ng = E = co 2E M y2 an U= σ x2 th And thus the total strain energy in a beam of length l is du o ng ⎡ ⎤ ⎞2 ⎞2 ⎛ ⎛ E d w EI d w l l U = ∫0 ⎢ ∫∫ ⎜ ⎟ y 2dA⎥dx = ∫0 ⎜ ⎟ dx ⎢ A ⎜⎝ dx ⎟⎠ ⎥ ⎜⎝ dx ⎟⎠ ⎣ ⎦ cu u Now the work done by the external forces (tractions) includes contributions from the distributed loading q and the loading at the ends x = and l W= CuuDuongThanCong.com ⎡ l ∫0 qwdx − ⎢V0 w − M ⎣ l dw ⎤ dx ⎥⎦ (6.6.11) https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   Institute for computational science 6.7 Principles of Minimum potential & complementary energy c om Example 6.1: Euler-Bernoulli Beam theory cu u du o ng th an co ng This result is simply the differential equilibrium equations for the beam, and thus the stationary value for the potential energy leads directly to the governing equilibrium equation in term of displacement and the associated boundary conditions Of course, this entire formulation is based on the simplifying assumption found in Euler-Bernoulli beam theory, and resulting solutions would not match with the more exact theory of elasticity results CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   Institute for computational science c om 6.1 Review of Strain energy and related principles ng 6.2 Strain energy th an 6.4 Bounds on Elastic Constants co 6.3 Uniqueness of the elasticity Boundary-Value Problem ng 6.5 Related Integral Theorems du o 6.6 Principle of Virtual Work cu u 6.7 Principles of Minimum potential & complementary energy 6.8 Rayleigh-Ritz method CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   Institute for computational science 6.8 Rayleigh-Ritz method c om Rayleigh-Ritz method is a method for finding the approximate solution based on the variational form of the problem N N co ng In this method, we construct a series of trial approximating functions that satisfy the boundary conditions but not the differential equations For the elasticity displacement formulation, this concept would express the displacements in the form N an u = u0 + ∑ a j u j ; v = v0 + ∑ b j v j ; w = w0 + ∑ c j w j j =1 j =1 th j =1 cu u du o ng where the functions u0, v0, and w0 are chosen to satisfy any non-homogeneous boundary conditions and uj, vj, wj satisfy the corresponding homogeneous boundary conditions Note that these forms are not required to satisfy the traction boundary conditions Normally, these trial functions are chosen from some combination of elementary functions such as polynomial, trigonometric, or hyperbolic forms The unknown constant coefficients aj, bj, cj are to be determined so as to minimize the potential energy of the problem, thus approximately satisfying the variational formulation of the problem CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   Institute for computational science 6.8 Rayleigh-Ritz method Π = Π ( a j , b j , c j ) (6.7.2) c om Using this type of approximation, the total potential energy will thus be a function of these unknown coefficients ∂Π =0 ∂c j co ∂Π = 0, ∂b j (6.7.3) an ∂Π = 0, ∂a j ng and the minimizing condition can be expressed as a series of expressions du o ng th This set forms a system of 3N algebraic equations that can be solved to obtain the parameters aj, bj, cj Under suitable conditions on the choice of trial functions (completeness property), the approximation will be improved as the number of included terms is increased cu u A big disadvantage of this method is the selection of the approximating functions There exists no systematic procedure of constructing them The selection process becomes more difficult when the domain is geometrically complex and/or boundary conditions are complicated CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   Institute for computational science 6.8 Rayleigh-Ritz method c om Example 6.2: Rayleigh-Ritz solution of a simply supported Euler-Bernoulli Beam co w = at x = 0, l ng Consider a simply supported Euler-Bernoulli beam od length l carrying a uniform loading q0 This one-dimensional problem has displacement boundary conditions an and tractions or moment conditions (6.7.4) ng th d 2w EI = at x = 0, l dx (6.7.5) cu u du o The Ritz approximation for this problem is of form N w = w0 + ∑ c j w j j =1 (6.7.6) With no nonhomogeneous boundary conditions, w0 = For the example, we choose a polynomial form for the trial solution An appropriate choice that satisfies the homogeneous conditions (6.7.4) is wj = xj(l - x) Note this form dose not satisfy the traction conditions (6.7.5) Using the previously developed relation for the potential energy (6.6.12), we get CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   Institute for computational science 6.8 Rayleigh-Ritz method c om Example 6.2: Rayleigh-Ritz solution of a simply supported Euler-Bernoulli Beam ⎡ EI ⎛ d w ⎞ ⎤ Π = ∫ ⎢ ⎜ ⎟ − q0 w⎥ dx (6.7.7) ⎢ ⎝ dx ⎠ ⎥⎦ ⎣ l ⎡ EI ⎤ N ⎛N j −2 j −1 ⎞ j = ∫ ⎢ ⎜ ∑ c j [ j ( j − 1)lx − j ( j + 1) x ] ⎟ − q0 ∑ c j x (l − x ) ⎥ dx j =1 ⎠ ⎣ ⎝ j =1 ⎦ th an co ng l ng Retaining only a two-term approximation (N = 2), the coefficients are found to be cu u du o q0l c1 = , c2 = 24 EI And this gives the following approximate solution: q0l w= x(l − x) 24 EI CuuDuongThanCong.com (6.7.8) https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   Institute for computational science 6.8 Rayleigh-Ritz method c om Example 6.2: Rayleigh-Ritz solution of a simply supported Euler-Bernoulli Beam ng Note that the approximate solution predicts a parabolic displacement distribution, while the exact solution to this problem is given by the cubic relation (6.7.9) th an co q0l w= (l + x − 2lx ) 24 EI cu u du o ng Actually, for this special case, the exact solution can be obtained from b Ritz scheme by including polynomials of degree three 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 ... science c om 6. 1 Review of Strain energy and related principles ng 6. 2 Strain energy co 6. 3 Uniqueness of the elasticity Boundary-Value Problem th an 6. 4 Bounds on Elastic Constants ng 6. 5 Related. .. science c om 6. 1 Review of Strain energy and related principles ng 6. 2 Strain energy th an 6. 4 Bounds on Elastic Constants co 6. 3 Uniqueness of the elasticity Boundary-Value Problem ng 6. 5 Related. .. science c om 6. 1 Review of Strain energy and related principles ng 6. 2 Strain energy th an 6. 4 Bounds on Elastic Constants co 6. 3 Uniqueness of the elasticity Boundary-Value Problem ng 6. 5 Related