1、Theory of Elasticity,Introduction Elasticity of Solids Field Equations of Elasticity - Differential Formulation Prismatic Rods Plane Problems Theory and Solutions Plane Problems Applications Variational Formulation of Elasticity Three-dimensional Problems,Index,0,Variational Formulation,Basic Concep
2、ts of Variation Weak Solutions Principle of Virtual Work Variational Principles Numerical Methods Based on Energy Principles,1,Chapter 7,Basic Concepts of Variation,Concept of Variation Constraints Basic Variational Operations Euler Equations,2,Chapter 7.1,Weak Solutions,Strong SolutionsThe strong s
3、olutions of elasticity refer to the case where the solutions satisfying the complete set of field equations of elasticity in a point-wise manner.,3,Chapter 7.2,equilibrium equation:the kinematical relation: the constitutive relation:the boundary conditions:,Field Equations,Weak Solutions,The weak so
4、lutions, on the other hand, are based on the energy principles. key characteristics: The problem is formulated from a compatible state in the sense of either a kinematical or statically admissible. Possible inclusion of constraints on the compatible state.,4,Chapter 7.2,key characteristics: A suitab
5、le energy principle is selected as the guiding principle. The continuity (or differentiability) requirement for the solution is relaxed to give the name of weak solution. The problem is approximated in a certain sense of truncation. A variety of approximation methods exist, including FEM, Rayleigh-R
6、itz method, the method of weighted residuals, etc.,Weak Solutions,5,Chapter 7.2,Principle of Virtual Work,A Compatible Field,6,Chapter 7.3,Compatible Fields,statically compatible fields If in V and on , then the static field ( ) is called statically compatible fields. kinematical compatible fields i
7、f in V and on , then the kinematical field ( ) is called kinematical compatible fields.,7,Chapter 7.3,Compatible Fields,complete continuum field. The combination of the static field ( ) and the kinematical field ( ) forms the complete continuum field. totally compatible continuum fieldIf , where,8,C
8、hapter 7.3,Theorem for Totally Compatible Fields,Theorem: If a continuum field is not only kinematical compatible (KC) but also statically compatible (SC), then it must be totally compatible. Proof:,9,Chapter 7.3,Deduction,Theorem 1If a continuum field is totally compatible, in the sense of for all
9、kinematical compatible fields, then its static field must be statically compatible. ( PROOF ) Theorem 2If a continuum field is totally compatible, in the sense of for all statically compatible fields, then its kinematical field must be kinematical compatible.,10,Chapter 7.3,Composition Works,the sta
10、tically compatible field (s) the kinematical compatible field (k) The internal composition work produced by the two fields can be shown as equaling to the external composition work:Proof:,11,Chapter 7.3,Remarks:,The static compatible field and the kinematical compatible field can be selected indepen
11、dently;The derivation does not involve any constitutive relations; it is the general principle suitable for any material constitutions;It is another manifestation for the theorem proved before.,12,Chapter 7.3,Principle of Virtual Displacement,A kinematical compatible filed (KC):a virtual kinematics
12、field: that satisfies,13,Chapter 7.3,equality between the internal and external virtual works :called the principle of virtual displacements, or the principle of virtual work. From this principle, can be derived as the Euler equations.,Principle of Virtual Displacement,14,Chapter 7.3,Principle of Vi
13、rtual Stresses,A virtual statically compatible field that satisfies andPrinciple of Virtual Stressesfor,15,Chapter 7.3,Principle of Admissible Work,Two compatible states,16,Chapter 7.3,The external composition work due to the interaction of two compatible fields isThe internal composition work due t
14、o the interaction of two compatible fields is The principle of compatible work is,Principle of Admissible Work,17,Chapter 7.3,Reciprocal Theorem,For the same geometry and material composition, one can assign the statically compatible solution as the actual solution under a prescribed loading (1), an
15、d the kinematical compatible solution as the actual solution under the prescribed loading (2). By the principle of compatible work and the linear elastic constitution, one can derive: Bettis reciprocal theorem or the reciprocal theorem,18,Chapter 7.3,Beam Theory,ConfigurationFor a general shape of t
16、he cross-section, the and axes should be selected as the principal axes of the cross-section with origin at the centroid. Accordingly, one has,19,Chapter 7.3,Bending in the x1-x3 plane, a kinematical compatible field can be constructedThe strain field:,Beam Theory,20,Chapter 7.3,The internal work an
17、d the external work,Beam Theory,21,Chapter 7.3,Some definitions:The corresponding resultant forces and moments at the ends areThe loading densities over the cross-section are defined as,Beam Theory,22,Chapter 7.3,the total compatibility can be phrased as Euler equations:the boundary conditions,Beam
18、Theory,or,or,or,23,Chapter 7.3,For a linear elastic and isotropic beam theoryAnd substituting them into the Eulers equations:,Beam Theory,24,Chapter 7.3,Timoshenko Beam,Configuration and Eulers equations:,25,Chapter 7.3,Bernoulli-Euler Beams,The governing equation:The constitutive relations:,26,Chap
19、ter 7.3,Examples in Textbook,Prob. 2 in page 409 Solving the bar contraction by an auxiliary problem.where,27,Chapter 7.3,Example 1 in page 412 (Details) Deflection of a beam under a central load and an end moment:We choose the following expression for the compatible displacement:,Examples in Textbo
20、ok,28,Chapter 7.3,Example 2 in page 412 (Details) Deflection of a beam under a central load and the end clamp The former problem can be utilized to solve this one by setting that slope as zero, i.e.,Examples in Textbook,29,Chapter 7.3,Example 3 of page 416 (Details) Deflection of a beam under unifor
21、m pressure and a center hinge support:We now start from a statically compatible solution listed as follows:,Examples in Textbook,30,Chapter 7.3,Variational Principles,Reference Washizu, Variational Method in Elasticity and Plasticity, 2nd ed., 1968 Variation Revisited Take a functional g that depend
22、s on a field variable labeled by a tensor f . By the variation of the field, the functional is changed to with . The variation of the functional can be written as,31,Chapter 7.4,Variation Revisited A similar argument can be extended to functionals of multiple fields. Define a functional , where , an
23、d . Again the tensors , and denote the directions of variations in displacement vector u, strain tensor e and stress s , respectively. Thus, the variation of the functional g is,Variational Principles,32,Chapter 7.4,Boundary Value Problems of Elasticity,Recall the basic equations and boundary condit
24、ions of linear elasticity. If they are all imposed a priori, one has the task of searching a strong solution. If parts of the equations are imposed a priori, and parts of them are derived via certain energy principles as the Euler equations and/or natural boundary conditions, one encounters weak sol
25、utions of various forms. Different variational principles come from different classifications of equations to the above mentioned two categories: imposed a priori or derived naturally by the employed variational principles.,equilibrium equation:the kinematical relation: the constitutive relation:the
26、 boundary conditions:,33,Chapter 7.4,HuWashizu Variational Principle,None of the equations needs to be imposed a priori. The Hu-Washizu energy functional can be defined asThe Hu-Washizu variational principle can be stated as follows: basic equations can be satisfied if and only if for any continuous
27、ly differentiable a, b and g.(PROOF),34,Chapter 7.4,Hellinger-Reissner Variation Principle,Adopting a Legendre transformation and imposing the following as a priorithe Hellinger-Reissner functional can be written as,35,Chapter 7.4,If the elastic constitutive relation (7.3) is imposed a priori, then
28、the Hellinger-Reissner variational principle can be stated as follows: Equations (7.1), (7.2), (7.4) and (7.5) can be satisfied if and only if The proof can be constructed in a way similar to that for the Hu-Washizu variational principle.,Hellinger-Reissner Variation Principle,36,Chapter 7.4,Princip
29、le of Complementary Energy,The principle of complementary energy imposes a priori the equilibrium (7.1), the elasticity relation (7.3) and the traction boundary condition (7.4). The complementary energy can be constructed asThe principle of complementary energy can be stated as follows: equations (7
30、.2) and (7.5) can be satisfied if and only if (PROOF),37,Chapter 7.4,Variational Principle of Potential Energy,Let us start from the Hu-Washizu functional and assume the pre-requirement of the kinematical relation (7.2), the elastic law (7.3) and the kinematical boundary condition (7.5). The Hu-Wash
31、izu functional is reduced to exactly the form of potential energyThe variational principle for potential energy states that the equilibrium equation (7.1) and the traction boundary condition (7.4) can be satisfies if and only if,38,Chapter 7.4,Summary,39,Chapter 7.4,Principle of Minimum Potential En
32、ergy,the principle of minimum potential energyIn all kinematical compatible fields, the actual kinematical fields correspond to the smallest potential energy, i.e.where,For any kinematical compatible fields,40,Chapter 7.4,Principle of Minimum Complementary Energy,the principle of minimum Complementa
33、ry energyIn all static compatible fields, the actual static fields correspond to the smallest complementary energy, i.e.where,For any static compatible fields,41,Chapter 7.4,Numerical Methods Based on Energy Principles,42,Chapter 7.5,Rayleigh-Ritz Method Gallerkin Method Method of Weighted Residuals
34、,Rayleigh-Ritz Method,We delineate the Rayleigh-Ritz method through the approach of principle of minimum potential energy. The displacements are chosen assatisfying the inhomogeneous displacement boundary condition on Su;leading only homogeneous displacements on Su.,43,Chapter 7.4,The principle of m
35、inimum potential energy requires P appears to be the quadratic form of , leading to above a linear algebraic equation system. RemarksThe Rayleigh-Ritz method is the first numerical scheme in the history based on the energetic approach. Its drawback lies in the global nature of its trial functions. T
36、hat nature hinders rapid converge under local disturbance.,Rayleigh-Ritz Method,44,Chapter 7.4,Gallerkin Method,The stationary condition of the principle of minimum potential energy, i.e. , necessarily requires where . If are required to satisfy not only the displacement boundary conditions on Su bu
37、t also the stress boundary conditions on St , then,45,Chapter 7.4,Method of Weighted Residuals,A generic formulation for a strong solution for unknowns collectively labeled as u can be proposed in the following The key is to relax the strong requirement of point-wise satisfaction of the above equati
38、ons so that a weak solution can be obtained. Trial function representation :,46,Chapter 7.4,Denote residuals asThe method of weighted residuals requires the weighted values of the residuals to vanish, namelythe interior weighted residual method the boundary weighted residual method. the mixed weight
39、ed residual method.,Method of Weighted Residuals,47,Chapter 7.4,The most influential methods Collocation Method Sub-domain Method Gallerkin Method Least Square Method Moment Method,Method of Weighted Residuals,The weighted functions are chosen as the Diracs delta functions centered at scattered poin
40、ts. Accordingly, the weighted residuals vanish at selected locations.,The weighted functions are selected as patch functions that are non-zero consecutively at N sub-domains. In contrast to the Rayleigh-Ritz method, the trial functions in the sub-domain method only have local support. It is this met
41、hod that forms the basis of the finite element method that overwhelms all computations for solids and structures.,The method of Gallerkin corresponds to the selection of weighted residuals of in the interior.,The method of the least square takes the weighted functions exactly as the residuals. The s
42、quared norm of the residual is minimized by imposing One can proceed a similar formulation for the boundary residuals.,The moment method selects the weighting functions as a power series, namely By requiring all weighted residuals vanished, one achieves the neutralization for moments of residuals of the different orders. The method can be extended to the generalized moments by selecting the weight functions as any complete function series.,48,Chapter 7.4,
