Rastislav Ďuriš, Justín Murín. A Solution for 1D Non-linear Problems for Finite Elements With Full Stiffness Matrices

Transcription

1 Rastislav Ďuriš, Justín Murín A Solution for D Non-linear Problems for Finite lements With Full Stiffness Matrices

2 Scientific Monographs in Automation an Computer Science ite by Prof. Dr. Peter Husar Ilmenau University of Technology an Dr. Kvetoslava Resetova Slova University of Technology in Bratislava Vol. 9

3 A SOUTION FOR D NON-INAR PROBMS FOR FINIT MNTS WITH FU STIFFNSS MATRICS Rastislav Ďuriš Justín Murín Universitätsverlag Ilmenau

4 Impressum Bibliographic information of the German National ibrary The German National ibrary lists this publication in the German national bibliography, with etaile bibliographic information on the Internet at Author s acnowlegement to Juraj Miština for translation. This scientific monograph originate from the author's issertation thesis efene at the Slova University of Technology in Bratislava, Faculty of Materials Science an Technology in Trnava. Reviewers: Prof. Ing. Pavel Élesztös, CSc. Prof. Dr. Ing. Vlaimír Kompiš, CSc. Prof. Ing. Milan Žminá, CSc. Author s contact aress: Ing. Rastislav Ďuriš, PhD. Slova University of Technology in Bratislava Faculty of Materials Science an Technology in Trnava Prof. Ing. Justín Murín, DrSc. Slova University of Technology in Bratislava Faculty of lectrical ngineering an Information Technology Ilmenau Technical University / University ibrary Universitätsverlag Ilmenau Postfach Ilmenau Prouction an elivery Verlagshaus Monsenstein un Vannerat OHG Am Haweramp 4855 Münster ISSN ISBN URN 9-49 Print Print urn:nbn:e:gbv:ilm-5 Titelfoto: photocase.com

5 Abstract The reuirements for the avance esign an prouction of structures with high performance, material an economic efficiency lea to the nee for use of the structural elements with stiffness variation. Varying stiffness can be cause by continuous change of a cross-section an/or by using materials with varying material properties. Structural analysis of truss an frame structures consisting of structural parts with stiffness variation can be ifficult. The stiffness variation of structural parts can be moelle by applying the fine beam F mesh or D soli finite elements with average values of cross-sectional an material parameters. If classical finite elements are use, F analyses of these structures reuire application of moels with extremely fine mesh; thus the preparation of a computational moel is very time-consumption an computational time can be too large, particularly in non-linear analyses. For elimination of the mentione isavantages of classical finite element applications, a two-noe non-linear bar element with varying stiffness is evelope in the first part of the monograph. Concerning the bar element, the continuous longituinal variation of cross-sectional area an material properties is consiere. The stiffness matrices of the evelope finite element were erive using full geometric non-linear, non-incremental formulations of euilibrium euations without any linearization for a linear elastic loaing state. New shape functions erive from the moifie concept of transfer functions an constants allow the accurate escription of 5

6 polynomial variation of the cross-sectional area an material properties in the bar element. Further, this approach was extene to the solution of physically non-linear problems. The matrices of non-linear bar elements were moifie for the application of materials with a uni-axial, bilinear stress-strain relationship an isotropic or inematic harening. A non-incremental solution algorithm was formulate for geometric an physically non-linear analysis using the erive bar element matrices. For the implementation of varying properties of the composite sanwich an functionally grae materials FGM into erive stiffness matrices the etermination of effective homogenize material properties is necessary. Macro-mechanical moelling of composite material properties is base on the ifferent homogenization techniues. In this thesis, a two-component composite with longituinal variation of elasticity moulus an volume fractions was consiere. The effective homogenize properties of the chosen composite were calculate using the new extene mixture rule formulate at the Department of Mechanics of the Faculty of lectrical ngineering an Information Technology, Slova University of Technology in Bratislava, Slovaia. The homogenization of thermo-mechanical material properties was carrie out for multilayers sanwich bars with polynomial variation of the effective Young s moulus an volume fractions of fibre an matrix in the layers. The proceure for incluing the varying temperature fiel by means of thermal noal forces was evelope as well. The normal stress istribution in composite layers of the original non-

7 homogenize sanwich bar was calculate by effective computational metho. In the secon part of the thesis, the stiffness matrices of a geometrically non-linear beam finite element were erive using a full non-linear, nonincremental formulation without any linearization. The matrices of the twonoe plane beam element with a ouble-symmetric cross-section an constant stiffness were formulate. The suitability of the concept of transfer constants implementation, the accuracy an efficiency of a geometrically an physically non-linear bar element, a geometrically non-linear beam finite element, an applicability of the extene mixture rule were compare an assesse by several numerical experiments against ANSYS analyses with classical finite elements. A goo agreement between results obtaine by newly evelope elements an the reference solutions in the commercial FM coe ANSYS was achieve. Moreover, the high efficiency of the evelope proceures was prove. Key wors geometric non-linear problems, finite elements, stiffness matrix, plasticity bar element, beam element 7

8 ist of abbreviations an symbols TF total agrange formulation UF upate agrange formulation GT Green-agrange strain tensor CS local coorinate system GCS global coorinate system II.PKT secon Piola-Kirchhoff stress tensor ij e ij η ij S ij u i φ ij C Green-agrange strain tensor linear part of Green-agrange strain tensor non-linear part of Green-agrange strain tensor secon Piola-Kirchhoff stress tensor local noal isplacement rotation global noal isplacement rotation shape function tensor of elastic material properties F, F ext vector of local external noal forces F int F Gext F Gint K K K T vector of local internal noal forces vector of global external noal forces vector of global internal noal forces vector of local noal isplacements local stiffness matrix in the invariant form local stiffness matrix as function of noal isplacement local tangent stiffness matrix in the invariant form 8

9 K T N local tangent stiffness matrix with isplacement epenent elements shape function matrix T; T T transformation matrix; transpose transformation matrix ε A Ax f m x T f x m x T x strain vector inuce temperature fiel cross-sectional area in initial configuration uneforme continuously variable cross-sectional area uneforme elasticity moulus Young s moulus elasticity moulus of fibres elasticity moulus of matrix elasticity moulus continuously varie along the longituinal axis of the element tangent moulus elasticity moulus of fibres continuously varie along the longituinal axis of the element elasticity moulus of matrix continuously varie along the longituinal axis of the element tangent moulus continuously variable along the longituinal axis of the element H x effective homogenize elasticity moulus continuously variable along the longituinal axis of the element x effective longituinal variable elasticity moulus of composite material x effective longituinal variable elasticity moulus of th composite layer I Z I Z secon moment of area secon moment of inertia thir moment of area 9

10 I Z t v f x v m x V α α Τ x bi-uaratic moment of area element length in initial uneforme configuration element length in current eforme configuration length of the line connecting element en noes in the eforme state polynomial escribing the change of fibre volume ratio polynomial escribing the change of matrix volume ratio boy volume in initial configuration initial angle between r axis of local coorinate system an x axis of global coorinate system thermal expansion coefficient continuously varie along the longituinal axis of the element H α T x effective homogenize thermal expansion coefficient continuously varie along the longituinal axis of the element α Τ x effective longituinal variable thermal expansion coefficient of composite material α Tx effective longituinal variable thermal expansion coefficient of th composite layer β total rotation angle connector of element en points stretching σ y stretching on yiel stress ρ σ y σ y x material ensity yiel stress, "mean" yiel stress yiel stress continuously varie along the longituinal axis of the element

11 T T i right superscript matrix/vector T right superscript T enotes tangent matrix right superscript i enotes i th iteration enotes matrix transpose operation of

12 INTRODUCTION If the relationship between the isplacement of the boy points an the external loa is not linear, we are taling about the non-linear behaviour of the boy. The euations necessary to escribe the process of the material behaviour must inclue the boy movement inematics escription of the eformation process by the vector fiel of isplacements, eformation tensors, inetics continuum inetic euations, stress tensor an its increment, thermoynamics an constitutive euations generalize Hooe s law, the relationships between stress increment an eformation in elasticplastic loaing area. Base on the inematics of the eformation process an the use constitutive relations, non-linear behaviour of the boy can be cause by physical material non-linearity, geometric non-linearity an mutual contact of the boies. Physical non-linearity occurs when the correlation between the loa an eformation is not linear, e.g. if the stress in the boy excees the limit of proportionality, which is associate with the emergence of plastic eformation or ue to creep of the material. Geometric non-linearities lea to changes in the boy configuration ue to the loa, an the isplacements of u i points of the boy an their graient u i,j are no longer infinitely small but finite. The term finite eformations means a complex of the boy s rigi motion an strain. Accoring to the extent of isplacements of boy points an the components of strain, we istinguish:

13 - Theory of the st ran comprising the area of infinitely small isplacements an strains. The total strain can be aitively ivie into elastic an plastic components. - Theory of the n ran comprising the area of finite isplacements an small strains. The aitive ivision of the strain into elastic an plastic elements is possible. - Theory of the r ran, comprising a small isplacements an large strains, where the elastic element of the strain is infinitely small an the plastic element is finite. The aitive ivision of the eformation tensor into elastic an plastic elements is acceptable after the moification of the constitutive relations, b both large isplacements an large strains. The aitive ivision of the eformation tensor into elastic an plastic elements is not possible. For this case, multiplicative ivision is use 7. The first part of monograph focuses on the erivation of the stiffness matrices of geometrically non-linear bar elements with variable stiffness. The secon part is evote to the erivation of the stiffness matrices of geometrically non-linear beam element with constant stiffness.

14 . NON-INCRMNTA FORMUATION FOR SOVING GOMTRICAY NON-INAR PROBMS. An overview of the current state of nowlege Numerous papers using ifferent approaches have been publishe on solutions for non-linear tass when isplacements affect the structural stiffness. Among the most important stuies are 5, 8, 4, 78. The solution of geometrically non-linear problems publishe in 5 is base on an incremental formulation. In the erivation of the stiffness relations, the increment variation GT is linearize its non-linear component is neglecte an at the same time, the increment II.PKT is linearize, too. The solution proceure an erivation of the element stiffness matrices is possible by the Total agrange Formulation TF when all static an inematic variables are relate to initial, uneforme configuration of the boy, or using the Upate agrange Formulation UF when all static an inematic variables are relate to the last nown boy configuration. It is nown that the accuracy of the results using these elements in nonlinear problems epens on the mesh ensity. However, coarse mesh can give convergent results. The iscrepancy between the calculate values an the reality is usually significant. In aition to the choice of the shape function representation, the linearization of non-linear expressions of the finite element metho is the main reason for achieving inaccurate 4

15 solutions an increasing the number of iterations to achieve a balance of internal an external forces. Another approach publishe in 4, 5 is base on neglecte terms of higher orer in variations of the increment of the Green-agrange eformation tensor GT. To eliminate the inaccuracy cause by the linearize incremental nonlinear FM euations, a non-incremental formulation of non-linear euilibrium euations without linearization was establishe 4. The obtaine euations contain the full non-linear stiffness matrices.. Non-incremental formulation of solving geometrically nonlinear problems without linearization for the elastic zone of loaing an a bar of constant cross section In orer to minimize the negative impact of linearization in the incremental formulation of euations to erive the stiffness matrices, the non-linearize non-incremental formulation was erive 4, 47. t K δu t u x K K initial uneforme configuration t K configuration in eformation state t x i, t x i Fig. Deformation state of the boy non-incremental metho 5

16 Fig. shows the eformation state of the boy. Unlie the incremental formulation, in the erivation there is omitte the inter-configuration of the boy in eformation step t t. Upon the principle of virtual wor, the static balance of the boy in its immeiate position t K see Fig. can be erive from euality of virtual wor of internal an external forces the Generalize agrange Formulation - GF Sij δij V = V A r F δu A F δ i i, [.] where the integral on the left sie of the euation represents the virtual wor of internal forces an the expression on the right sie the virtual wor of external forces only surface an concentrate to the noe. The integration is performe over the initial volume V surface A of the boy. If for the escription of the eformation process we use GF, the eformation state of the iniviual points of the boy will be escribe by the GT of finite eformations ij = e ij η ij = u i, j u j, i u, i u, j Its variation is expresse fully without linearization [.] δ = δ e δη. [.] ij ij ij Using the ecomposition [.], non-linear euilibrium euations in the current eforme configuration in non-incremental form can be expresse in the moifie relation, 4, 49

17 V = C A ijl i e l δe i ij V V C r F δu A F δ ijl η l δe ij e l δη η ij l δη V = ij [.4] where C ijl is the tensor of material properties efining the constitutive relation between II. Piola-Kirchhoff stress tensor S ij = C ijl rs in configuration t K an the Green-agrange strain tensor ij [.]. Furthermore, u i,j is the current eformation graient, δu i is a variation of isplacement, F i are the surface tractions an δ are virtual isplacements of points of the boy in which the concentrate forces F r operate. The integration is performe through the initial uneforme volume V an the initial area A of the finite element. By iscretization of the boy into finite elements, the isplacement of an arbitrary point of u i element can be expresse by interpolation of the noal point isplacements of the element using shape functions φ i u i = φ i [.5] After substituting the shape functions an their erivation into the euation [.4] an after necessary ajustments, substitution of inices an exclusion of variation δu i to get a steay relationship escribing the nonlinear epenence between the isplacements of points of the boy an the external loa, from which it is possible to erive the shape of the 7

18 stiffness matrices, vali for an arbitrary type of element, the isplacements of which can be escribe by the expression [.5] 4 4 V V V V C C C C ijl ijl ijl ijl φ φ φ φ m, l pm, pr, i pm, φ φ φ pr, l pn, j φ lm, pv, l φ φ φ φ in, j m, l r, i in, j φ φ φ rn, j φ jn, i lm, jn, i m m m v m V r V r V V = A r F φ A F i in n. [.] uation [.] can be rewritten for a single element into component K nm m = F n, or in matrix form K = F, [.7] The local non-linear stiffness matrix consists of one linear an three nonlinear components K = K K N K N K N = K K N [.8] Terms of the nonlinear stiffness matrix K N epen on the local noal isplacement vector in the form of linear an uaratic functions. The system of non-linear euations [.7] has to be solve by one of the iterative methos. In case of Newton's or Newton-Rapson iterative scheme, to achieve a better rate of convergence of solution, it is necessary 8

19 to compile a tangential stiffness matrix K T of the element accoring to the following proceure: T F K NT K = = K = K K, [.9] where K NT is non-linear part of the tangential stiffness matrix K T. 9

20 . BAR MNT WITH VARIAB STIFFNSS FOR SOVING GOMTRICAY AND PHYSICAY NONINAR PROBMS At present, we encounter in technical practice the use of mechanical parts with variable stiffness, either ue to economic an technological reasons, or because of the use of new avance materials such as composites an sanwich structures or Functionally Grae Materials FGM. When moelling the bar an frame structures, the problem of stiffness variability can be reuce by using "average" values of the section an material properties, by choice of the greater ensity of networ elements with ifferentiation of section an material properties, or by moelling the parts of structures with variable stiffness by planar or soli finite elements. However, apart from the increase of time reuire to prepare the moel an the solution itself, it as the problem with inter-element compatibility. Therefore, base on the results obtaine in numerical experiments, which confirm the accuracy an efficiency of the new, geometrically non-linear bar element with constant stiffness loae in the area of elastic eformations, this solution was extene to allow for continuous variation of the cross-sectional area an material properties. Further, the solution was also extene to solve the physical non-linear problems an the possibility to apply the temperature fiel with a prescribe temperature istribution along the longituinal axis of the bar as the loa to the element was applie. This proceure aims to regar the continuous variation of the bar stiffness using a single bar element without having to create a relatively ense networ of finite elements.

21 . Geometrically non-linear elastic bar element with variable stiffness.. Defining the variability of input parameters In orer to meet an escribe the continuous variation of the cross-sectional area an material properties along the longituinal axis of the oneimensional element in the erivation stiffness relationship, the concept of transfer functions an constants publishe by H. Rubin in 9 an extene by J. Murín an V. Kutiš, 4, 47, 5 was use. This leas to the establishment of new shape functions interpolating isplacement of an arbitrary point of the element from the isplacements of the bar element noal points. New shape functions inclue the so-calle transfer functions, their erivatives an transfer constants values of shape functions in the terminal noe of element. Using this approach is conitione by the escription variability of cross-sectional characteristics an material properties in the polynomial shape. We can assume that the continuous variation of the cross-sectional area Ax or elastic moulus x in the axis of the element can be escribe by the polynomial in the form n = = P x P i η P x Pi η P x [.] = where η P are the coefficients of polynomial members η P x escribing the variability of mechanical or geometric parameter, an P i is the size of the variable is the initial noe i of the element.

22 j j = Nj = N Aj x i = ux Ax Nx j i x Ni = N Ai i Fig. Bar with variable stiffness for solving of elastic problems Then the variability of elastic axial stiffness can be expresse in the relationship A x x = A η x η x = A η x [.] i i A where η A x = Ax x/a i i is the polynomial escribing stiffness variation along the longituinal axis of the bar. i i A.. New shape functions of the two-noe bar element The inematic relationship between the axial isplacement at the location x of an arbitrary point of element ux an axial force Nx in place x is expresse by the ifferential euation

23 u x N x N x u x = = = e x [.] x A x x A We can efine the secon erivative of the transfer function e x for the tension/compression loaing in the linear elastic loaing area subscript e as i i x e e x = = [.4] x η x Then the solution of ifferential euations [.], provie that all the loas of the element are transforme into noes of the element an the axial force in the bar is constant Nx = N i = N j, the function escribing the axial isplacement of any point of the element is A Ni u x = i e x [.5] A where is the first erivation of transfer function, for which is vali e x e x = x i i x. η x By substitution of x = in euation [.5] the isplacement is u = j, an e = e A inicates the value of the first erivative of the transfer function in the terminal noe of the element an will be further calle the transfer constant for tension/compression. Calculation of the transfer constant e can be one numerically using a simple algorithm, publishe in, 5. By eriving an axial force N i z from this way moifie euation, an by bacwar substitution into euation [.5] we obtain the epenence

24 between the axial isplacement of an arbitrary point of the bar in the irection of the x-axis an the axial isplacement of noal points i an j, expresse by the new shape functions φx e x e x u x = i j = φ ui x i φuj x j [.] e e The first erivatives of the shape functions for two-noe bar element with varying stiffness are then eual to e x e x φ, x =, φ, x =. [.7] e e.. ocal non-linear stiffness matrix of bar element with variable stiffness By substituting the erivatives of shape functions [.7] for the two-noe bar element an by the substitution for the one-imensional element V = Ax x = A i η A xx an for the tensor of elastic behaviour of the material C ijl x x = i η x into the first integral in euation [.], we obtain an expression for the calculation of the linear members of the stiffness matrix bar element with variable stiffness: K nm = A i i η x φ x φ x x. A m, n, [.8] Using the same proceure we get members of the non-linear stiffness matrices epenent on the unnown isplacements of terminal noes of the 4

25 bar element by the substitution an moification of other members in the euation [.]: K K K N nm N nm N nm = = = 4 φ x φ x Ai i η A x φm, x,, φ n, x x [.9] φ x φ x Ai i η A x φn, x,, φ m, x x [.] φ x φ x Ai i η A x,, φm, x φn, x x [.] Substituting erivatives of shape functions [.7] into euations [.9] an [.] results in the nee for calculation of the integrals e x x an e x x. By substitution of integrants, where in the enominator of the relation [.4] we replace the polynomial η A x for η x = η x or η x = η x entails the nee for the A A A calculation of integrals x = e an η x = e. A x η x This way, we get new transfer constants A A e an e of the bar element for variable stiffness an the elastic loaing area. To calculate these transfer constants, the algorithm mentione in,, 49, 5, 5, can be use, since η A x an η A x are also polynomials. The final shape of the local 5

26 non-linear stiffness matrix of the bar element with variable stiffness can be expresse as = e e e e e i i A K. [.] The vector of local noal isplacement of the bar is = [ ] T an the vector of external local noal forces F = [N N ] T...4 ocal non-linear tangential stiffness matrix The system of non-linear algebraic euations is usually solve using the iterative Newton's or Newton-Raphson metho. Using this way of solution, the local non-linear tangential stiffness matrix is reuire. This tangential stiffness matrix can be calculate using the relation [.9], by erivation of matrix K [.]. After the relevant erivations accoring to noal isplacements i we get the expression for the local non-linear tangential stiffness matrix of the two-noe bar element with variable stiffness = e e e e e i i T A K [.]..5 Global non-linear tangential stiffness matrix For the calculation of the global isplacements of noal points of the bar element from the system of global non-linear euilibrium euations we use

27 Newton's iteration metho. This proceure uses the global non-linear tangential stiffness matrix of the boy to calculate the noal isplacements. Since the non-linear component of tangential stiffness [.] is not invariant to rigi motion of the boy, for transformation between the local an the global stiffness matrix, it is not possible to use stanar transformation, nown from the linear theory. Therefore, we apply replacement of noal isplacements by the value invariant to the rigi motion of the boy - by stretching = = =. [.4] By substituting the relation [.4] into the relationship [.] we obtain invariant component of non-linear tangential stiffness in the form K T Ai i = e e e e e. [.5] From such expresse local non-linear tangential stiffness matrix K T, the global non-linear tangential stiffness matrix classical transformation using the transformation matrix T. T K G can be expresse by.. Internal forces In the iteration process of the solution it is necessary to compute the vector of global internal forces. The global internal forces are formulate from the local euilibrium stiffness relation [.7], where local non-linear stiffness 7

28 8 matrix K is efine by the formula [.]. Internal forces in a bar cannot be calculate from the global stiffness relationship, in relation to the impact of large rotations. Due to the elimination of the problems associate with the transformation of internal forces from the global to the local coorinate system, we rewrite the local non-linear stiffness matrix K using stretching [.4] into the form of local stiffness matrix K, invariant to the rigi motion rotation of the bar = e e e e e i i A K. [.] The local internal axial force can be calculate by multiplying the total nonlinear stiffness of the bar an its eformation A N e e e e e i i i =. [.7] The stress in a bar can be etermine from the relationship i = σ [.8] where,, are transfer constants for variation of elastic moulus x. They are esigne by the same proceure as the transfer constants erive in Sections.. an.., however, base on the secon erivative of the transfer function / x x η = efine only by a polynomial of

29 the variability of moulus of elasticity x. The polynomial escribing the variability of moulus of elasticity η x is in accorance with euation [.]. The local internal force vector is in the form int [ N N ] T = [ N N ] T F =. i j i i. Geometrically non-linear bar element stresse in the elastoplastic area.. Defining the variability of input parameters et us consier the irect two-noe bar element with variable stiffness, loae in the elasto-plastic area Fig.. Variability of material properties is extene in the continuous variation of elastic-plastic moulus T x an yiel strength σ y x along the longituinal axis of the bar. Variability of cross-section Ax an the moulus of elasticity x is efine in accorance with the polynomial [.]. Tangent moulus T x is efine by a similar polynomial. Then, the elasto-plastic stiffness variation along the longituinal axis of the bar in the loaing area above the yiel strength can be expresse by expression similar to the euation [.] x = A η x η x = A η x A x [.9] T i Ti A i Ti T A T In further solution, we will eal only with material moels escribe by bilinear stress-strain iagram with isotropic or inematic harening. 9

30 j j = N j = N ux Ax A j j i x i = Nx x σ yj Tj σyx N i = N A i Tx i σ yi Ti Fig. Bar with variable stiffness for solving of elasto-plastic problems In the case of variability of the moulus of elasticity x an tangent moulus of elasticity T x, the plastic moulus T H = woul not have polynomial shape reuire for application of the concept of transfer functions an constants 9. Therefore it woul be necessary to approximate the change in the plastic moulus H using the Taylor series to the polynomial shape. T.. Moification of the stiffness matrix for non-incremental solution If the normal stress in a bar excees the yiel stress of material σ y, it is necessary to set up a new epenence between stress increment an relative eformation in the bar.

31 With preserving non-incremental solution also in the area of elasto-plastic stress, after reaching the state of plasticity in the bar, the elastic stiffness of the bar e will be change into the elasto-plastic stiffness. The local stiffness matrix of the elasto-plastic stiffness of the bar will have the form K ep Ai = Ti σ ep y σ y ep ep ep ep [.] where expression represents the size of elasto-plastic relative σ y eformation of the bar ε = ε ε above the yiel stress, expresse with ep σ y the help of stretching [.4]. Parameter σ y inicates stretching in the bar at reaching the yiel stress an is erive from the solution of cubic euation e e σ y = [.] 4 σ y σ y σ y e e e.g. by factorisation, where σ y is the mean value of the yiel stress of material of the bar e σ yi σ y = x x y η σ [.] where σ yi is the value of yiel stress in the initial noe i of the element an η σ y x is the polynomial escribing the variability of yiel stress along the i length of the element. The transfer constants, are constants ep ep, ep for elasto-plastic stress conition. These transfer constants can be etermine in a similar way as the transfer constants for the case of elastic

32 loaing see subsections.. an... The ifference is only in the efinition of the suare erivative of the transfer function, which in this case has the form x = / efine by the polynomial of ep η AT x variability of elasto-plastic bar stiffness Ax T x. The polynomial escribing variability of elasto-plastic bar stiffness x stress is efine by the euation [.9]. η above the yiel A T.. Internal force in the bar, stress in the bar Internal force in the bar element uner the stress in elastic area is calculate from euation [.7]. The force in the ro when reaching yiel stress is eual to N A i i e e iσ = y σ y σ [.] y σ y e e e an elasto-plastic component of axial force is expresse by the relation N A i Ti ep ep iep = σ y σ y σ [.4] y ep ep ep The total force in the bar in elasto-plastic state is given by the sum of N i = N σ N. The total normal stress in the bar in elasto-plastic state i y iep can be calculate by the sum of elasto-plastic stress increment σ ep above the yiel stress an the "mean" value of yiel stress σ y [.], which can be expresse by the relationship

33 y Ti y T T y T T y T σ σ σ σ σ = [.5] where T T T,, are the transfer constants for variability elastoplastic moulus T x. They are etermine by the same proceure as the transfer constants erive in subsections.. an.., but base on the suare erivative of the transfer function / x x T T η = efine only by the polynomial of variability of tangential moulus T x...4 ocal tangential elasto-plastic stiffness matrix The stiffness matrix K T of the elastic area, expresse by euation [.5], in the elasto-plastic state will change to = ep ep ep ep ep Ti i T ep A y σ y σ K [.]..5 Proceure of non-incremental solution of elasto-plastic problems The proceure for the non-incremental solution of elasto-plastic problems is shown in the chart in Fig. 4.

34 . Calculation of starting solution isplacements Q from global linear stiffness euation transformation of element local matrices/vectors into global matrices/vectors T int T KG = T K T FG = T F preparation of elastic stiffness euation of the boy calculation of starting isplacements from global linear euations K Q = F Gcel ext G. Preparation of global non-linear stiffness euations calculation of actual lenght t from global noal coorinates of each element = t / preparation of local non-linear stiffness matrices an vectors K [.], σ [.8], K T [.5] preparation of global nonlinear stiffness matrices an vectors preparation of elastic stiffness euation of the boy calculation of starting isplac ements from global euations F stress test in every element yes σ i- > σ y no element was yiele in i- iteration element wasn t yiele in i- iteration 4

35 . valuation of the stress status of each element element was yiele in i- iteration element wasn t yiele in i- iteration calculation of actual length ti from global noal coorina tes of each element i = ti / preparation of local non-linear stiffness matrices an vectors K i [.], σ i [.5], K Ti [.] calculation of actual length ti from global noal coorinates of each element i = ti / prepa ration of loc al non-linear stiffness matrices an vectors K i [.], σ i [.8], K Ti [.5] no σ i > σ i- yes no σ i > σ y yes A B C D 4. Calculation of axial stresses in elements an moification of stiffness matrices A C element loa is ecrease elastic loaing of the element i- = ma x, σ i- = σmax, c alculation of actual lenght ti from global noal coorina tes of each element i = ti / replace elastoplastic stiffness matrix of element to elastic preparation of local non-linear stiffness matrices an vectors K i [.], σ i [.8], K Ti [.5] calculation of actual lenght ti from global noal coorinates of each element i = ti / element stiffness matrix not change prepa ration of loc al non-linear stiffness matrices an vectors K i [.], σ i [.8], K Ti [.5] σ i = σ i σmax 5

36 B D element is loae in elasto-plastic fiel element was currently yiele calculation of actual length ti from global noal coorina tes of each element i = ti / element stiffness matrix not change preparation of local non-linear stiffness matrices an vectors K i [.], σ i [.5], K Ti [.] calculation of actual length ti from global noal coorinates of each element i = ti / replace elastic stiffness matrix of ele ment to elasto-plastic prepa ration of loc al non-linear stiffness matrices an vectors K i [.], σ i [.5], K Ti [.] 5. Preparation of new global non-linear stiffness euations, calculation of new global isplacements transformation of element local matrices/vectors into global matrices/vectors T T T int T KG = T K T FG = T F preparation of global non-linear stiffness matrices an vectors preparation of stiffness euations of whole boy calculation of isplacement increments from non-linear global euations T res K Gcel Q = FG calculation of vector of resiual forces res ext int F = F F G G calculation of global isplacements Q i = Q i- Q chec the conitions for terminating the solution process G F Fig. 4 Flowchart of elasto-plastic problem solution with our bar element with variable stiffness

37 .. Numerical experiments with the bar of variable stiffness loae in elastic an elasto-plastic area As a typical problem for stuying geometrically non-linear behaviour of a structure, there was chosen a planar structure of triple joint connection of two bars, referre to in the literature as von Mises structure Fig. 5. In the solution of the problem the small angle α an symmetry of the structure were consiere. Any imperfections causing a change of the straight shape of bar wasn t consiere. F x, Tx,σ yx uy α h Ax x x Fig. 5 Von Mises structure with bars of variable stiffness Depenence between axial force in the bar N or force F an isplacement of the joint noe u y, for solution in elastic area of loaing an bar of constant stiffness is nown for the construction accoring to the Fig. a. In the literature e.g. 5, 4, a number of approaches to analytical or numerical solutions of appointe problem can be foun. The result of these solutions is the euilibrium epenence between the force in the bar of the strut frame or global reaction in the common joint, an vertical isplacement of the joint, which course, when consiering material with linear elastic behaviour, correspons to the course isplaye by the blac line in Fig. c,. 7

38 When consiering loaing in the elasto-plastic area, this epenence varies consierably, as is presente in Fig. c, at the replacement of the iagram of tension test by the bilinear iagram with isotropic an inematic harening Fig. b. However, in available literature sources, any analytical solutions for the specific geometrically non-linear problem for the bar of variable stiffness an the loaing in elasto-plastic area have not been foun. Therefore, it was possible to assess the accuracy of the new element only by comparison with results obtaine by numerical solution of the commercial finite element program ANSYS. F a b c a u y b c Fig. uilibrium relationship between axial an/or global reaction an isplacement of common joint for elastic an elasto-plastic material with isotropic an inematic harening 8

39 ... Moels use in the numerical experiments To assess how accurately the stiffness matrix of the new geometrically nonlinear bar element, erive in Sections. an. escribe the variability of the real bar stiffness, in MATHMATICA software are create separate programs. The programs allow solving geometrically non-linear problems uner elastic an elasto-plastic loaing of the new finite bar element. In solution of all numerical experiments only one our bar element was use by proceure: one line = one element. The reference solutions for obtaining comparative results were create in ANSYS by simulation moels: Moel BAM - one-imensional moel, the element stiffness matrix is erive using Hermite shape functions. The moel allows approximation of the longituinal stiffness variation of the bar in variants with ifferent numbers of elements. The moel was esigne to solve problems in elastoplastic loaing state. Moel BAM88 - one-imensional moel using the iso-parametric beam elements BAM88 base on the Timosheno theory an with ivision into selecte number of elements. The element can be use for solution of problems with large isplacements rotations an elasto-plastic tass. Moel SOID45 - D moel consisting of 4 volume elements SOID45. The geometric moel accurately approximates the variability of the bar cross-section. Material properties were ivie into 5 iscrete groups, replacing the continuous variation of elasticity moulus an tangential moulus, while values of mouli in each element were 9

40 consiere to be eual to the value of the moulus in the coorinate corresponing to the centre of the element. All the moels were moelle as the irect ones, with no imperfections that woul cause istortion of the straight shape of the bar moel. The isplacement of structure common noe was prescribe within the u y in range sin α ; sin α so that to obtain epenence of axial force N, or global reaction F on the isplacement u y an so the ivergence solution problem near the bifurcation point was remove.... Bar with variable stiffness loae in elasto-plastic area We consier following initial geometric parameters of Mises structure in Fig. 5: α = 7, = m. Polynomials escribing the variation section an material properties are liste in Table. The material of the bar was uner consieration in the bilinear moel with isotropic an inematic harening Fig. b. VARIATION OF CROSS-SCTIONA ARA AND MATRIA PROPRTIS Table variation of geometric parameters an material properties [m, Pa] Ax =.8.988x.4x x =.54 x. x T x =.54 x. x σ y x = x x 4

41 The maximum an minimum ratio of the bar stiffness [Ax x] max /[ Ax x] min = [Ax T x] max /[ Ax T x] min =. MAXIMA VAUS OF FORCS OBTAIND USING OUR MNT AND ANSYS MODS WITH VARIAB STIFFNSS Table axial force N [N] global reaction F [N] No. of elem. BAM BAM88 BAM BAM our element SOID Global reaction values F correspon to areas of extreme values in the first half of the course, where, as can be seen from the presente graphs, ifferences in the results obtaine by the ifferent moels are greatest. Axial forces N in the tables correspon to the position of the bar α t =, i.e. to the loaing sub-step when the local x-axis of the bar is the same as the x-axis of the global coorinate system of the whole structure. At the problem solution, the results were obtaine using only one new nonlinear bar element. As an increment of the isplacement of upper noe of the structure u y = mm was chosen, an the calculation was carrie out steaily " substeps = iteration." 4

42 Fig. 7 Global reaction F common hinge isplacement u y response for the bar with isotropic harening Fig. 8 Axial force N common hinge isplacement u y response for the bar with isotropic harening 4

43 Fig. 9 Global reaction F common hinge isplacement u y response for the bar with inematic harening Fig. Axial force N common hinge isplacement u y response for the bar with inematic harening 4

44 valuation The results of numerical experiments of the von Mises structure stress in the elasto-plastic area are presente graphically in the form of the process of axial forces N an reactions F in epenence on the vertical isplacement u y of common noe. From the epenence of axial forces N, as well as from the results presente in it can be conclue that the stiffness of our bar element is higher by approximately.5% than the stiffness of the spatial moel consisting of SOID45 elements. Higher stiffness than our bar element an thus higher levels of axial force N an reaction F is shown by moels with one beam element BAM or BAM88, for which the eviation from the results obtaine by spatial moel is above %. The resulting percentage ifferences between the maximum values of axial forces N obtaine by one-imensional moels an spatial moels are summarize in Table. PRCNT DIFFRNC IN MAXIMUM VAUS OF AXIA FORCS N DTRMIND BY NW BAR MNT AND BY ANSYS MODS Table percentage ifference of axial forces BAM / e SOID45 SOID45 BAM / e SOID45 SOID45 BAM88 / e SOID45 SOID45 BAM88 /e SOID45 SOID45 our bar elem. SOID45 SOID45.%.4%.9%.84%.4% It can be conclue from the results that the courses of axial forces an reactions etermine by our element are in better compliance with the solution obtaine by spatial moel SOID45 than the courses obtaine by 44

45 moels with beam elements BAM, 88, especially for moels with isotropic harening material. Significant ifferences in the course of forces between the moels with one element an the moels with fine mesh occur in the area of transition from elastic to elasto-plastic state. This ifference is cause by: - in the case of moels with one element new bar element, beam moels is necessary to provie "mean" value of the yiel stress σ y to etermine limits of transition into elasto-plastic state for the entire element, when after exceeing the stress limit, the elastic stiffness matrix [.] of new element turns into elasto-plastic matrix [.], - in contrast to moels with more elements that result in a graual transition of the moel into elasto-plastic state by plasticization of particular elements in epenence from stiffness an yiel stress of the element. This eviation also occurs in other moels consisting of one beam elements BAM, 88. A more significant increase of the ifference in the course of axial force can be observe in case of consiering the material with inematic harening comparing to the isotropic harening material. With increasing axial force i.e. an increase in the loaing of a bar in elastoplastic area the ifference between the results obtaine by one new bar element an results from moels ecreases. In the case of consiering material with isotropic harening, the process of axial force obtaine from solution by one new non-linear bar element is in goo agreement with the results obtaine from the multi-element moels also in area of elastic 45

46 unloaing of the bar an tensile stress after exceeing yiel stress in the tensile area. This compliance is not significantly affecte by the egree of the polynomial of the stiffness variability or by the ratio of maximum an minimum stiffness in the bar. The ifference in maximum values of axial forces etermine by a new bar element an SOID45 spatial moel is less than.% only in cases where the ratio of the maximum an minimum stiffness in the bar is less than.. The results of numerical experiments lea to the following conclusions: - eviation between the value of the axial force maximum in a nonlinear bar of variable stiffness an the value obtaine by a spatial moel SOID45 compare for all variations was at the level of.% or less at a stress of the bar in flexible area, - with increasing egree of the polynomial of variability stiffness, this ifference ecreases slightly, - numerical experiments show a goo agreement with the results of solving a new bar in the elasto-plastic eformation areas with spatial moel in the ANSYS programme, especially when consiering the material with isotropic harening, - significant ifference in the course of axial forces or reactions etermine by a new element compare to the spatial moel an the moel with higher ensity of ivision by BAM elements can be reporte when consiering the material with inematic harening an stiffness ratio in the bar greater than., - significant ifferences between the results obtaine by the new bar element an the moels with ivision into a larger number of elements 4

47 occur after exceeing the yiel stress, an are cause by the transition of the entire bar from elastic to elasto-plastic state. This phenomenon occurs even in moels with one classic BAM element. During further increase of loaing, this ifference in the course of forces ecreases more significantly, namely in the bars from material with isotropic harening.. Sanwich bar element with variable stiffness Implementation of the new avance materials such as composites, sanwich structures or functionally grae materials into the esign an prouction calls for esigning of an appropriate moel of the material. Due to a number of variables that control the esign of a functionally grae microstructure, the full potential of FGM reuires the evelopment of appropriate strategies for moelling of their mechanical or thermomechanical properties. Functionally grae materials FGM are a new generation of structural materials, in which the microstructure is purposefully spatially change ue to the uneven istribution of harening components. Achieving such a state is possible by using harening components of ifferent properties, size an shape of the particles or by continuous graual substitution of harening components fibres an matrices. This is most often achieve by creating FGM by plasma spraying or power metallurgy. The result is a microstructure that is forme by a continuous or iscrete change in macroscopic, electrical, thermal an mechanical properties. 47

48 FGM are suitable materials e.g. for the so-calle thermal barrier in applications involving large temperature graients, from the aircraft an rocet engines up to the use in microelectronic circuits or MMS. Without homogenizing the material properties of such materials, the nonlinear component analysis woul reuire creation of very fine finite element mesh, an even the time-consuming preparation of the moel an the solution process itself woul be significant. Macro-mechanical moelling of effective material properties of such composites is base on ifferent homogenization proceures. By mixing two or more components we can achieve a synergic effect when the properties of newly prouce material are better than the properties of iniviual components. These new materials are characterize by continuous or iscontinuous variation of material properties. Along with the evelopment of these materials to improve the calculation precision an escription of material properties in the numerical simulations, new methos of homogenization are being evelope 7,, 4, or the alreay existing proceures are being improve 54, 55. Recently, also multi-scale methos are elaborate an are starting to be applie, 5. One way of macroscopic moelling of mechanical properties of materials with heterogeneous microstructure is their homogenization an etermination of effective material properties of the composite using mixing rules. The simplest mixing rule by which we can etermine the average effective material properties is base on the assumption that the material properties of the composite are the sum of material properties of each component multiplie by its volume fraction. The resulting effective 48

49 property p e of the bi-component composite consisting of the matrix with property p m an the harening phase p f is etermine by p e = v f p f v m p m [.7] where v f, v m are the volume fractions of the matrix an the harening phase, for which in each point of the material is vali v f v m =. In this part of the stuy, the extene mixing rule publishe in 54, 55 is use. In the first step is consiere a bi-component composite material with a variable change in the moulus of elasticity an the volume ratio of both components along the longituinal axis of a bar composite single-irection composite. For homogenization of material properties of such composite, in section.. is escribe the proceure allowing to erive the relation escribing variability of the effective longituinal elastic moulus x of the composite. In the secon part, the process of homogenization of the material properties of the sanwich element for the extene multi-layer composite is esigne. For such a bar of constant ouble symmetric cross-section with variation of material properties of iniviual composite layers, the homogenization process of material properties of multilayer material is then erive. The proceure of material property homogenization is in accorance with the laminate theory, 54, 55. Delamination of the sanwich material is not consiere. This proceure allows homogenising properties of composite an sanwich materials with spatially varying volume ratio of the components an their material properties. 49