Dr. D. Dinev, Department of Structural Mechanics, UACEG

Transcription

1 Lecture 6 Energy principles Energy methods and variational principles Print version Lecture on Theory of Elasticity and Plasticity of Dr. D. Dinev, Department of Structural Mechanics, UACEG 6.1 Contents 1 Work and energy Work and energy Work and energy Work A material particle is moved from point A to point B by a force F The infinitesimal distance along the path from A to B is a displacement du The work dw performed by the force F is defined as dw = F du The work done is the product of the displacement and the force in the direction of the displacement The total work is B W = F du A 6.3 Work and energy Work 1

2 Work = Force Displacement Work and energy Energy The energy is the capacity to do work It is a measure of the capacity of all forces on a body to do work Work is performed on a body trough a change in energy The work done by forces on an elastic solid is stored inside the body in the form of a strain energy Consider the uniaxial tension test where the deformation increases slowly from zero to σ 6.6 The strain energy stored is equal to the work done on the differential element σ ( du = σd u + u ) σ 0 x dx dydz σdudydz 0 = σ 2 2E dxdydz 6.7 The strain energy density is the strain energy per unite volume U = du dxdydz = 1 2 σε 6.8 2

3 The strain energy is the shaded area under the stress-strain curve U = 1 2 σε 6.9 General expression for the strain energy U = 1 2 (σ xxε xx + σ yy ε yy + σ zz ε zz + 2σ xy ε xy + 2σ yz ε yz + 2σ zx ε zx ) = 1 2 σ i jε i j = 1 2 σ : ε The total strain energy is U T = 1 2 The potential of the applied forces is [ W P = σ : εd ] f ud + t uds S TPE functional Total potential energy of the body Π T PE = U T +W P = 1 2 σ : εd f ud t uds S 6.11 TPE functional The principle states that: The body is in equilibrium if there is an admissible displacement field u that makes the total potential energy a minimum δπ T PE (u) = 0 where δπ T PE is a variation of the functional Π T PE (u) The admissible displacement field is the one that satisfy the displacement BCs. Note The variational operator δ is much like the differential operator d except that it operates with respect to the dependent variable u rather than the independent variable x

4 TPE functional Local minimum of the TPE functional 6.13 The condition for minimum of a functional I(u) is δi = I u δu = 0 Almost the same as the condition for a minimum of a function u(x) du = u x dx = Consider a functional The minimum condition is δi(u) = b I(u) = F [ x,u(x),u (x) ] dx a b a ( x ) δx + δu + u u δu dx = 0 The integral can be manipulated to get the expression for the variation of u (integration by parts of the 3-rd addend) 6.15 The variation of the functional is δi(u) = b a [ u x ( )] u δudx The non-trivial solution gives u x ( ) u = 0 The above expression is called Euler equation

5 A more general 2D case is given by I(u,v) = F (x,y,u,v,u,x,v,x,...,v,yy ) The Euler equations are u + 2 x u,x y u,y x u,xx x y u + 2 x u,x y u,y x u,xx x y A + 2 u,xy y u,xy y 2 u,yy = 0 u,yy = Example Consider a cantilever beam with length of l and subjected to uniform load q. Using the principle of a minimum of TPE work out the equilibrium equation A lot of problems in elasticity there is no analytical solution of the field equations For such cases approximate solution schemes have been developed based on the variational formulation of the problem (i.e. the principle of minimum potential energy) The is based on the idea of constructing a series of trial approximating functions that satisfy the essential (displacement) BCs but not differential equations exactly Note Since the TPE functional includes the force BCs, it is require the trial solution satisfies only the displacement BCs! 6.19 Walter Ritz ( )

6 The original paper 6.21 The displacement can be expressed as u = u 0 + a 1 u 1 + a 2 u a n u n = u 0 + v = v 0 + b 1 v 1 + b 2 v b n v n = v 0 + n i=1 n i=1 n w = w 0 + c 1 w 1 + c 2 w c n w n = w 0 + a i u i b i v i i=1 c i w i The terms of u 0, v 0 and w 0 are chosen to satisfy any non-homogeneous displacement BCs and u i, v i and w i satisfy the corresponding homogeneous BCs These forms are not required to satisfy the stress BCs 6.22 These trial functions are chosen from the some combinations of elementary functions (polynomials, trigonometric or hyperbolic forms) The unknown coefficients a i, b i and c i are to be determined so as to minimize the TPE functional of the problem Thus we approximately satisfy the variational formulation of the problem Using this approximation the TPE functional will be a function of these unknown coefficients Π T PE = Π T PE (a i,b i,c i ) 6.23 The minimizing condition ca be expressed as a series of Π T PE a i = 0, Π T PE b i = 0, Π T PE c i = 0 This set forms a system of 3n equations which gives a i, b i and c i 6

7 Under suitable conditions on the choice of trial functions (completeness) the approximation will improve as the number of included terms is increased When the approximate displacement solution is obtained the strains and stresses can be calculated from the appropriate field equations Note The method is suitable to apply at problems involving one or two displacements (bars, beams, plates and shells) 6.24 Example Consider a bending of simply supported beam of length l carrying a uniform load q 6.25 The End Any questions, opinions, discussions?