Material planning is an important and developing part
of applied mechanics. The problem is usually solved by FEM, which
leads to highly non-linear equations  causing  several numerical
problems. This paper presents a method for the determination
of the coefficients of a Green-type material model using mathematical
programming based on the theorem of virtual force. The material
model is included in the compatibility equations. The material
parameters appear as unknowns in the non-linear objective function,
while the conditions remain linear. The uniqueness and stability
of the material model (e.g. Drucker postulates) are  assured by
inequality conditions. The boundary conditions are considered.
Sample problems were run by a program for non-linear mathematical
programming problems written in Fortran 77

Fejér, Tamás

Vásárhelyi, Anna

English

Periodica Polytechnica (Budapest University of Technology and Economics)

PERIODICA POLYTECHNICA SER. CIV. ENG. VOL. 44, NO. 1, PP. 3–12 (2000)DETERMINATION OF THE COEFFICIENTS OF THEGREEN-TYPE MATERIAL MODEL BY NON-LINEARPROGRAMMINGTamás FEJÉRand Anna VÁSÁRHELYILaboratory of InformaticsFaculty of Civil EngineeringTechnical University of BudapestH–1521 Budapest, HungaryPhone: (+36-1) 463–1933Fax: (+36-1) 463–3697E-mail: fejert@epito.bme.huReceived: December 20, 1998AbstractMaterial planning is an important and developing part of applied mechanics. The problem is usuallysolved by FEM, which leads to highly non-linear equations causing several numerical problems. Thispaper presents a method for the determination of the coefficients of a Green-type material model usingmathematical programming based on the theorem of virtual force. The material model is included inthe compatibility equations. The material parameters appear as unknowns in the non-linear objectivefunction, while the conditions remain linear. The uniqueness and stability of the material model (e.g.Drucker postulates) are assured by inequality conditions. The boundary conditions are considered.Sample problems were run by a program for non-linear mathematical programming problems writtenin Fortran 77.Keywords: mathematical programming, material design, non-linear material models, Green model.1. IntroductionMaterial design is now an important and developing part of applied mechanics.There are several ways to achieve our design goals, but the problem is usuallysolved with FEM. The greatest disadvantages of this method are that they lead tohighly non-linear equations causing several numerical problems [7]. On the otherhand, the problem can be formulated using the concept of virtual energy. Thisformulation leads to a mathematical programming problem [6].The state of equilibrium of a structure is usually determined by solving thesystem of the equilibrium and compatibility equations. Another possibility is to usethe extremum principles of virtual energy with the strain energy and complementarystrain energy function [9].At first we assume that the principle of small displacement is valid and theconstitutive law is linear. The primal problem expresses the virtual force theoremby the help of a quadratic programming problem [2], where the equality conditionsare the equilibrium equations concerning the structure and the objective function is4 T. FEJÉR and A. VÁSÁRHELYIthe complementary strain energy. In this problem, we are looking for the minimumof the objective function.Formulating the dual problem, one can find that it contains the compatibilityequations as equality condition with the strain energy function as objective func-tion. Again, the problem is to find the minimum of the objective function. Thecomplementary problem gives the usual formulation.This formulation allows wide opportunities in• delimiting stresses and displacements,• satisfying the boundary conditions,• defining yield domain,• choosing either elastic or elasto-plastic constitutive laws.They can be built into the mathematical programming problem as equality orinequality conditions, iff the conditions are consistent.The main advantage of the mathematical programming formulation can beseen if we introduce different material parameters (e.g. elastic modulus, cross-section parameters, etc.) as unknowns. Using the primal formulation, the objectivefunction becomes of higher order even in the case of linear material law, but theequality conditions remain linear.In the rest of the paper this formulation is shown, and applied to a simple,statically indeterminate beam structure with a Green-type hyperelastic materialmodel [1]εi j = ∂∂σi j.The integration of the stresses (σ ) concerning the volume was calculated by theGauss–Legendre scheme, and thus the coefficients can be represented by the diag-onal matrix〈ρ〉. These coefficients are independent of the stresses and thereforethe order of the derivation and the numerical integration can be changed.2. The Mathematical Programming ProblemThe elastic state of a structure can be determined through the equilibrium andcompatibility equations in the following form [5,8]:T∗ · B∗ · 〈ρ〉 · σ + P = 0, (equilibrium equations)B · T · u + 〈ρ〉 · ∇(σ) = 0, (compatibility equations)P∣∣∣∣SP= 0, (stress boundary conditions)u∣∣∣∣Su= 0. (displacement bound. conditions)GREEN-TYPE MATERIAL MODEL 5Hereσ is the stress vector,P is the vector of external forces,u is the displacementvector, B is the geometric matrix,ε is the strain vector,T is the transformationmatrix between the local and global coordinate systems. The unknowns are vectorsu andσ .The structure of this system of equations can be visualized as inFig. 1. MatrixF is hyperdiagonal, and if the constitutive law is linear, the elements are constants,otherwise they are functions.Fig. 1. The structure of the equilibrium and compatibility equations. The boundary condi-tions can be considered by clearing the appropriate rows and columns.The static and kinematic boundary conditions can be considered at the ade-quate equations. The simplest method in the case of prescribed zero displacementor zero stress is to clear the appropriate row and column as shown inFig. 1. This isimpossible unless the constitutive law is linear and thus matrixF can be explicitlystated. In this case, the appropriate stress in the material law must be substituted withzero, but in the geometric matrixB the appropriate row can be deleted. Otherwise,the boundary conditions can be included as additional equality constraints.Based on the principle of virtual force theorem, it is possible to give a pair ofmathematical programming problems, either of them defines the equilibrium stateof a structure, and which are in primal–dual relationship. The primal problem ex-presses the minimum of virtual complementaryenergy6 T. FEJÉR and A. VÁSÁRHELYImin∫V(σ) dVT∗ · B∗ · 〈ρ〉 · σ + P = 0, (equilibrium equations)P∣∣∣∣SP= 0, (stress boundary conditions)u∣∣∣∣Su= 0. (displacement bound. conditions)The Wolf-dual of this problem is:min∫VW (ε) dV − P∗uB · T · u + 〈ρ〉 · ∇(σ) = 0, (compatibility equations)P∣∣∣∣SP= 0, (stress boundary conditions)u∣∣∣∣Su= 0. (displacement bound. conditions)which expresses the minimum of the virtual energy. The complementary problem,which includes the equality conditions of both the primal and dual formulation,recalls the usual system of equilibrium and compatibility equations.The theorem of virtual forces and virtual displacements gives a necessarycondition for the existence of the optimum for the primal and dual problem, respec-tively. On the other hand, the convexity of the feasible domain defines the sufficientcondition. Moreover, if the energy function is convex in the domain of materialstability, the optimum is global optimum and it is the same for the primal and dualproblem.3. The Material ModelThe most simple and convenient material model is based on Hook’s law, and ex-presses a linear function between strains and stresses:ε = ∇(σ) = F · σ.In this case, the equality constraints are linear either in the primal or in the dualproblem, and the objective function is quadratic.GREEN-TYPE MATERIAL MODEL 7min{ ∑elements12σ ∗ · F · σ}(primal problem)T∗ · B∗ · 〈ρ〉 · σ + P = 0, (equilibrium equations)min{ ∑elements12σ ∗ · F · σ − P∗ · u}(dual problem)B · T · u + 〈ρ〉 · ∇(σ) = 0, (compatibility equations)The boundary conditions for both problems are:P∣∣∣∣SP= 0, (stress boundary conditions)u∣∣∣∣Su= 0. (displacement bound. conditions)In the linear case, the stability of the material is automatically satisfied. From thecomputational point of view, one of the most simple non-linear material behaviors isa Green-type hyperelastic model. In a three-parameter quadratic model, the relationbetween stresses and strains can be stated asεi j = ∇(σi j ) = 1δi j + 2σi j + 3σikσ j k,where1, 2 and3 are material parameters. The complementary virtual energyis:(σ) =σ∫0∇(σi j ) dσi j .The integral was calculated numerically with a three-point Gauss–Legendre scheme:(σ) = σ23∑i=1ci∇[σ2(ξi + 1)].Here,ci are the Gauss weights at the integration pointsξi , andσ is the unknownupper limit of the integration.As it can be seen inFig. 2, where1 = 0 and3 = constant, the effect ofnormal stress is influenced by the value of its coefficient at a given value of theenergy function very strongly.If the constitutive law is non-linear, the stability of the material is not automat-ically satisfied. The stability can be ensured by the Drucker postulate (Fig. 3) [3],which states that the work of any external effect rate (Ṫi , Ḟi ) on the displacementrate (̇ui ) is positive, ∫AṪi u̇i dA +∫VḞi u̇i dV > 0.8 T. FEJÉR and A. VÁSÁRHELYIFig. 2. The linear coefficient (2) as function of stresses at constant energy levelUsing the principle of virtual work, one can rewrite this in the following form:σ̇i j ε̇i j > 0.The rate of stresses and strains can be understood as the change of the effect in eachiteration step (r) of the non-linear programming solver and thereforeσi j = σ r+1i j − σ ri j .Using the definition of the Green model one can writeH′i j klσi j σkl > 0,whereH′i j kl is the Hessian of the complementary energy function. This non-linearinequality condition ensures the stability and convexity of the material, however, itis stricter than it would follow form the laws of thermodynamics.4. A Simple Beam StructureThe above described method was applied at the calculation of the a very simple,1D, statically indeterminate structure shown inF g. 4. The structure consists of twoelements, each with three internal Gauss–Legendre integration points for the inte-gration of the energy function. Each node has three degrees of freedom (εx , εy, γxy)GREEN-TYPE MATERIAL MODEL 9Fig. 3. The inner part of the tube is determined by the Drucker postulate.Fig. 4. The beam structureand the stress state of each node is described by the three-element vectorσ =[σxσyτxy].The modified geometric matrix (B̃), concerning the boundary conditions (shadedrows and columns) in the global coordinate system is as follows:10 T. FEJÉR and A. VÁSÁRHELYIThe vector of external forces (P) can be formed by concentrating the dis-tributed load to the nodes.The material law described in chapter 3 is now simply:εx = 1 + 2σx + 3(σ 2x + τ 2xy),γxy = 2τxy + 3σxτxy.The complementary virtual energy can be written as the function of the stresses:(σ, τ) =σ∫0τ∫0εx dτxy dσx +σ∫0τ∫0γxy dτxy dσx,where the upper limits of the integrals are unknowns. Using three-point Gauss–Legendre integration scheme:(σ, τ) = σ2τ23∑i=13∑j=1ci c jεx(σ2(ξi + 1), τ2(ξi + 1))++σ2τ23∑i=13∑j=1cic jγxy(σ2(ξi + 1), τ2(ξi + 1)).GREEN-TYPE MATERIAL MODEL 11The total complementary strain energy of the structure can be obtained by thevolume integration along the two elements:t =2∑n=1ln23∑i=1ci(σi , τi ).Again, a three-point Gauss–Legendre integration scheme was used for the volumeintegration.σi andτi denote here the stress state in thei-th Gauss point.The non-linear mathematical programming problem now can be stated asmin{t}, (objective function)B̃∗ · 〈ρ〉σ + P = 0, (equality conditions)σ ∗ · H′ · σ > 0. (inequality conditions)The unknowns are the stress vectors in the Gauss–Legendre integration points andthe three material parameters. The structure has one degree of statical indetermina-tion and therefore, eliminating the equality conditions, one variable remains free.Altogether, there are 4 independent variables, and two inequality constraints foreach integration point.The problem was run on a non-linear programming solver [4] written in For-tran 77 in a Silicon Graphics workstation environment. Since there was not anydefined yield function, the Drucker postulate was the active criterion at the stop ofthe iteration.The example data were the following:Distributed load p = 5 kN/mSpan L = 2 mCross-sectional area F = 125 cm2Inertia / cross-section extremum distanceK = 12500 cm3The solutions were 2 = 5, 3 = 0.01Moment at the middle support: −8.98 kNmEnergy of the structure: 88460 kNmThe efficient (convex) domain of2 and3 is−2 < 2 < 8.8, 3.34 · 10−3 < 3 < 9.9.The numerical computation has shown clearly that this formulation gives the desiredresults and thus it can be the basis of future extensions.12 T. FEJÉR and A. VÁSÁRHELYI5. ConclusionA mathematical programming formulation has been shown for the calculation ofstatically indeterminate structures in the framework of virtual force and virtualdisplacement theorems. A Green-type hyperelastic material model has been usedwith three unknown parameters to be optimized. This simple model has shown theapplicability of the mathematical programming formulation.This may serve as a basis to a further extension which includes the use ofmore sophisticated material models, the yield condition and plasticity features, andthe consideration of time dependency.AcknowledgementThis research was supported by the Hungarian Scientific Research Foundation (OTKA)under project number T029638.References[1] BOJTÁR, I.: Material Laws of Mechanics, Tankönyvkiadó, Budapest, 1988 (in Hungarian).[2] COHN, M. Z. – MAIER, G.: Engineering Plasticity by Mathematical Programming, PergamonPress Inc., Waterloo, 1979.[3] DRUCKER, D. C.: Basic Concepts in Plasticity, Handbook of Eng. Mech., W. Flügge, (Ed.)McGraw-Hill, 1962.[4] FIACCO, A. V. – MCCORMICK, G. P.: Non-Linear Programming, Sequential UnconstrainedMinimization Techniques, Wiley and Sons, New York, 1968.[5] GÁSPÁR, ZS.: Statics of Beam Structures, M̋uegyetemi Kiadó, Budapest, 1995 (in Hungarian).[6] K ARUSH, W.: Minima of Functions of Several Variables with Inequalities as Side Condition,M. S. Thesis, Dept. of Mathematics, University of Chicago, 1939.[7] POPPER, GY. – CSIZMÁS, F.: Numerical Methods for Engineers, Akadémiai Kiadó, Budapest,1993 (in Hungarian).[8] SZABÓ, J. – ROLLER, B.: Anwendung der Matrizen-Rechnung für Stahlwerke, AkadémiaiKiadó, Budapest, 1978.[9] VÁSÁRHELYI, A. – LÓGÓ, J.: Analysis of Elastic Structures by Mathematical Programming,Periodica Polytechnica Civil Eng. Vol. 33, No. 3–4: pp. 149–156, 1989.

Determination of the coefficients of the green-type material model by non-linear programming

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Determination of the coefficients of the green-type material model by non-linear programming

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