Maryam Tabatabaei, Dy Le, Satya N. Atluri
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- Gilbert Goodwin
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1 Neary Exact and Highy Efficient Eastic-Pastic Homogenization and/or Direct Numerica Simuation of Low-Mass Metaic Systems with Architected Ceuar Microstructures Maryam Tabatabaei, Dy Le, Satya N. Aturi Center for Advanced Research in the Engineering Sciences Institute for Materias, Manufacturing, and Sustainabiity Texas Tech University, Lubbock, TX, USA, 7949 Abstract [Submitted to the Journa of Mechanics of Materias and Structures] Additive manufacturing has enabed the fabrication of ightweight materias with intricate ceuar architectures. These materias become interesting due to their properties which can be optimized upon the choice of the parent materia and the topoogy of the architecture, making them appropriate for a wide range of appications incuding ightweight aerospace structures, energy absorption, therma management, metamaterias, and bioscaffods. In this paper we present, the simpest initia computationa framework for the anaysis, design, and topoogy optimization of ow-mass metaic systems with architected ceuar microstructures. A very efficient easticpastic homogenization of a repetitive Representative Voume Eement (RVE) of the micro-attice is proposed. Each member of the ceuar microstructure undergoing arge eastic-pastic deformations is modeed using ony one noninear three-dimensiona (3D) beam eement with 6 degrees of freedom (DOF) at each of the nodes of the beam. The noninear couping of axia, torsiona, and bidirectiona-bending deformations is considered for each 3D spatia beam eement. The pastic-hinge method, with arbitrary ocations of the hinges aong the beam, is utiized to study the effect of pasticity. We derive an expicit expression for the tangent stiffness matrix of each member of the ceuar microstructure, using a mixed variationa principe in the updated Lagrangian co-rotationa reference frame. To sove the incrementa tangent stiffness equations, a newy proposed Newton Homotopy method is empoyed. In contrast to the Newton s method and the Newton-Raphson iteration method which require the inversion of the Jacobian matrix, our This paper is written in honor of Dr. Pedro Marça, a true pioneer in Computationa Mechanics, on the occasion of the ceebration of his Lifetime Achievements at ICCES7, June 7, in Madeira, Portuga 4 Ju 7 9:9:55 PDT
2 Homotopy methods avoid inverting it. We have deveoped a code caed CELLS/LIDS [CELLuar Structures/Large Ineastic DeformationS] providing the capabiities to study the variation of the mechanica properties of the ow-mass metaic ceuar structures by changing their topoogy. Thus, due to the efficiency of this method we can empoy it for topoogy optimization design, and for impact/energy absorption anayses. Keywords: Architected Ceuar Microstructures; Large Deformations; Pastic Hinge Approach; Noninear Couping of Axia, Torsiona, Bidirectiona-Bending Deformations; Mixed Variationa Principe; Homotopy Methods.. Introduction A ot of natura structures such as hornbi bird beaks and bird wing bones are architected ceuar materias to provide optimum strength and stiffness at ow density. Humankind, over the past few years, has aso fabricated ceuar materias with more compex architectures in comparison with previousy deveoped synthetic materias ike open-ce metaic foams and honeycombs []. Properties of these ceuar structures are determined based on their parent materias and the topoogy of the microarchitecture. Additive manufacturing technoogies and progress in threedimensiona (3D) printing techniques enabe the design of materias and structures with compex ceuar microarchitectures, optimized for specific appications. In fact, one of the most interesting characteristics of ceuar structures with pore network is that they can be designed with desirabe properties, making them appropriate for ightweight structures, metamaterias, energy absorption, therma management, and bioscaffods []. For exampe, efforts are under way to fabricate bioscaffods to repair and repace tissue, cartiage, and bone [3-6]. These architected materias shoud be fabricated in such a way that they can meet biocompatibiity requirements in addition to the mechanica properties of the tissues at the site of impantation. Therefore, presentation of a highy efficient computationa method to predict and optimize the mechanica properties of such structures is of interest. Herein, we present a neary exact and highy efficient computationa method to predict the eastic-pastic homogenized mechanica properties of ow-mass metaic systems with architected ceuar microstructures. The framework of the methods presented in this paper is aso germane to the anaysis under static as we as impact oads, design, and topoogy optimization of ceuar soids. 4 Ju 7 9:9:55 PDT
3 The utraow-density metaic ceuar micro-attices have been recenty fabricated at HRL Laboratories [7, 8], suitabe for therma insuation, battery eectrodes, catayst supports, and acoustic, vibration, or shock energy damping [9-4]. They produced nicke ceuar micro-attices, consisting of hoow tubuar members, by preparing a sacrificia poymeric tempate for eectroess Ni deposition and, then, chemicay etching the sacrificia tempate [7]. Using this process they fabricated nove nicke-based micro-attice materias with structura hierarchy spanning three different ength scaes, nm, μm, and mm. They obtained a 93% Ni 7% P composition by weight for micro-attices, using energy dispersive spectroscopic anaysis. They empoyed quasi-static axia compression experiments to measure macroscopic mechanica properties such as Young s modui of nicke micro-attices. The oad, P was measured by SENSOTEC oad ces, and the dispacement, δ was measured using an externa LVDT for moduus extraction. Strain-stress curves were obtained based on engineering stress and strain defined, respectivey, as σ = P/A and ε = δ/l. A and L are the initia cross-sectiona area and ength of the sampe, respectivey. Moreover, Saari-Sharif and Vadevit [5] extracted the Young s moduus of a series of nicke utraight micro-attices by couping experimenta resuts obtained using Laser Dopper Vibrometry with finite eement (ABAQUS) simuations. They [5] fabricated a sandwich configuration by attaching carbon/epoxy face sheets as the top and bottom ayers of the utraight nicke hoow micro-attice thin fim [7]. They [5] detected the resonant frequencies by scanning aser vibrometry and ABAQUS simuations and extracted the reation between Young s moduus and the natura frequencies. Then, the effective Young s modui of sampes were obtained in the direction norma to the face sheets [5]. It is worth noting that for finite eement (FE) modeing, a Representative Voume Eement (RVE) consisting of ony four members of the ceuar microattice with at east ten thousands of 4-node she FEs was empoyed [5], resuting in at east, nodes and, thus, 6, degrees of freedom (DOF). We shoud emphasize that in our methodoogy each member can be modeed by a singe spatia beam eement. In other words, to perform a 4-member RVE anaysis, we use ony 4 spatia beam eements and 5 nodes, with a tota of 3 DOF and, thus, at east times ess DOF than in Ref. [5]. Since the cost of computation in a FE noninear anaysis varies as the nth power (n between and 3) of the number of DOF, it is cear that we seek to present a far more efficient anaysis procedure than any avaiabe commercia software. This provides the capabiity to simuate the ceuar microstructure using repetitive 3 4 Ju 7 9:9:55 PDT
4 RVEs, consisting of an arbitrary number of members, enabing a very efficient homogenization and/or Direct Numerica Simuation (DNS) of a Ceuar Macrostructure. In addition, Schaeder et a. [7] and Torrents et a. [8] showed experimentay that nickephosphorous ceuar micro-attices undergo arge effective compressive strains through extensive rotations about remnant node igaments. Unfortunatey, there are no computationa studies in the iterature on the arge-deformation eastic-pastic anaysis of such metaic ceuar structures, which is the major concern of the present study, athough, there is a vast variety of studies on the arge deformation anaysis of space-frames [6-9], form the era of arge space structures for use in the outer space. In the ream of the anaysis of space-frames, numerous studies have been devoted to derive an expicit expression for the tangent stiffness matrix of each eement, accounting for arbitrariy arge rigid rotations, moderatey arge non-rigid point-wise rotations, and the stretching-bending couping [-4]. Some researchers empoyed dispacement-based approaches using variants of a agrangian for either geometricay or materiay noninear anayses of frames [-]. Kondoh et a. [3] extended the dispacement approach to evauate expicity the tangent stiffness matrix without empoying either numerica or symboic integration for a beam eement undergoing arge deformations. Later, Kondoh and Aturi [4] presented a formuation on the basis of assumed stress resutants and stress coupes, satisfying the momentum baance conditions in the beam subjected to arbitrariy arge deformations. In order to study the eastic-pastic behavior of ceuar members undergoing arge defections, we empoy the mechanism of pastic hinge deveoped by Hodge [5], Ueda et a. [6], and Ueda and Yao [7]. In this mechanism, pastic hinge can be generated at any point aong the member as we as its end nodes, everywhere the pasticity condition in terms of generaized stress resutants is satisfied. It is worthwhie to mention that contours of the Von Mises stress given in Ref. [5] for the 4-member RVE with PBCs show very high concentration of stress at the junction of four members. The stress contours were obtained based on inear eastic FE simuations [5]. Therefore, it ceary mandates an eastic-pastic anaysis, which is undertaken in the present study. A compementary energy approach in conjunction with pastic-hinge method has been previousy utiized to study easto-pastic arge deformations of space-framed structures [4, 8]. Shi and Aturi [8] derived the inearized tangent stiffness matrix of each finite eement in the co-rotationa reference frame in an expicit form and showed that this approach based on assumed stresses is 4 4 Ju 7 9:9:55 PDT
5 simper in comparison with assumed-dispacement type formuations. In contrast to Ref. [8] which presents the inearized tangent stiffness, the current work derives expicity the tangent stiffness matrix under the noninear couping of axia, torsiona, and bidirectiona-bending deformations. One of the extensivey empoyed approaches in the iterature for the anaysis of noninear probems with arge deformations or rotations is based on variationa principes. For instance, Cai et a. [9, 3] utiized the prima approach as we as mixed variationa principe [3] in the updated Lagrangian co-rotationa reference frame to obtain an expicit expression for the tangent stiffness matrix of the eastic beam eements. They [3] showed that the mixed variationa principe in comparison to the prima approach, which requires C continuous tria functions for dispacements, needs simper tria functions for the transverse bending moments and rotations. In fact, they [3] assumed inear tria functions within each eement and obtained much simper tangent stiffness matrices for each eement than those previousy presented in the iterature [, 3, 3]. Whie Ref. [3] considered ony a few macro members, our anaysis is appicabe to metaic ceuar microattices with an extremey arge number of repetitive RVEs. Since pasticity and bucking occur in many members of the micro-attice, we found that the Newton-type agorithm that has been utiized in Ref. [3] fais. In the present study we discovered that ony our Newton homotopy method provides convergent soutions in the presence of the pasticity and bucking in a arge number of members of the micro-attice. To sove tangent stiffness equations, we use a Newton Homotopy method recenty deveoped to sove a system of fuy couped noninear agebraic equations (NAEs) with as many unknowns as desired [33, 34]. By using these methods, dispacements of the equiibrium state are iterativey soved without the inversion of the Jacobian (tangent stiffness) matrix. Newton Homotopy methods are advantageous, in particuar, when the effect of pasticity is going to be studied. Since, it is we-known that the simpe Newton s method as we as the Newton-Raphson iteration method require the inversion of the Jacobian matrix, which fai to pass the imit oad as the Jacobian matrix becomes singuar, and require arc-ength methodoogy which are commony used in commercia off-the-shef software such as ABAQUS. Furthermore, Homotopy methods are usefu in the foowing cases, when the system of agebraic equations is very arge in size, when the soution is 5 4 Ju 7 9:9:55 PDT
6 sensitive to the initia guess, and when the system of noninear agebraic equations is either overor under-determined [33, 34]. The paper is organized as foows. The theoretica background incuding the noninear couping of axia, torsiona, and bidirectiona-bending deformations for a typica ceuar member under arge deformation, mixed variationa principe in the co-rotationa updated Lagrangian reference frame, pastic hinge method, and equation-soving agorithm accompanying Newton Homotopy methods are summarized in Section. Section 3 is devoted to the vaidation of our methodoogy; a threemember rigid-knee frame, the Wiiams togge probem, and a right-ange bent incuding the effect of pasticity are compared with the corresponding resuts given in the iterature. Section 4 anayzes the mechanica behavior of two different ceuar micro-attices subjected to tensie, compressive, and shear oading. Throughout this section, it is shown that our cacuated resuts (Young s moduus and yied stress) under compressive oading are very comparabe with those measured experimentay by Schaeder et a. [7] and Torrents et a. [8]. Moreover, the progressive deveopment of pastic hinges in ceuar micro-attice as we as its deformed structure are presented. Finay, a summary and concusion are given in Section 5. Appendices A, B, C, and D foow.. Theoretica Background Throughout this section, concepts empoyed to derive neary exact and highy efficient easticpastic homogenization of ow-mass metaic systems with architected ceuar microstructures are given. Considering the noninear couping of axia, torsiona, and bidirectiona-bending deformations, strain-dispacement and stress-strain reations in the updated Lagrangian corotationa frame are described in Section.. Section. is devoted to derive an expicit expression for the tangent stiffness matrix of each member of the ceuar structure, accounting for arge rigid rotations, moderate reative rotations, the bending-twisting-stretching couping and eastic-pastic deformations. Soution agorithm is aso given in Section Ju 7 9:9:55 PDT
7 . The Noninear Couping of Axia, Torsiona, and Bidirectiona-Bending Deformations for a Spatia Beam Eement with a Tubuar Cross-Section A typica 3D member of a ceuar structure is considered, spanning between nodes and as iustrated in Fig.. The eement is initiay straight with arbitrary cross section and is of the ength before deformation. As seen from Fig., three different coordinate systems are introduced: () the goba coordinates (fixed goba reference), x i with the orthonorma basis vectors e i, () the oca coordinates for the member in the undeformed state, x i with the orthonorma basis vectors e i, and (3) the oca coordinates for the member in the deformed state (current configuration), x i with the orthonorma basis vectors of e i, (i =,, 3). Fig.. Nomencature for the reference frames corresponding to the goba, undeformed, and deformed states. Loca dispacements at the centroida axis of the deformed member aong e i -directions are denoted as u i, (i =,, 3). Rotation about x -axis (ange of twist) is denoted by θ, and those about x i - axes, i =, 3, (bend ange) are denoted by θ i, i =, 3, respectivey. It is assumed that nodes and of the member undergo arbitrariy arge dispacements, and rotations between the undeformed state of the member and its deformed state are arbitrariy finite. Moreover, it is supposed that oca dispacements in the current configuration (x i coordinates system) are 7 4 Ju 7 9:9:55 PDT
8 moderate and the axia derivative of the axia defection at the centroid, u x with that of the transverse defections at the centroid, u i x (i =, 3). is sma in comparison We examine arge deformations for a cyindrica member with an unsymmetrica cross section around x - and x 3 -axes and constant cross section aong x -axis subjected to torsion around x - axis, T and bending moments M and M 3 around x - and x 3 -axes, respectivey. It is assumed that the warping dispacement due to the torsion T is independent of x variabe, u T (x, x 3 ), the axia dispacement at the centroid is u (x ), and the transverse bending dispacements at the origin (x = x 3 = ) are u (x ) and u 3 (x ) aong e - and e 3 -directions, respectivey. The reason for the consideration of the noninear axia, torsiona, and bidirectiona-bending couping for each spatia beam eement is the frame-ike behavior of these ceuar metaic micro-attices. The scanning eectron microscopy (SEM) images of micro-attices given by Torrents et a. [8] notify the formation of partia fracture at nodes (for a micro-attice with t = 5 nm), ocaized bucking (for a micro-attice with t =.3 μm), and pastic hinging at nodes (for a micro-attice with t = 6 μm). Therefore, 3D dispacement fied for each spatia beam eement in the current configuration is considered as foows using the normaity assumption of the Bernoui-Euer beam theory, u (x, x, x 3 ) = u T (x, x 3 ) + u (x ) x u (x ) x x 3 u 3 (x ) x, u (x, x, x 3 ) = u (x ) θ x 3, () u 3 (x, x, x 3 ) = u 3 (x ) + θ x. The Green-Lagrange strain components in the updated Lagrangian co-rotationa frame e i, (i =,, 3) are ε ij = (u i,j + u j,i + u k,i u k,j ), () where the index notation,i denotes x i and k is a dummy index. Repacement of Eqs. () into Eq. () resuts in the foowing strain components, ε = u, + (u,) + (u,) + (u 3,) u, + (u,) + (u 3,) x u, x 3 u 3,, ε = u, + (u,) + (u,) + (u 3,) = (u T, u, ) + θ, 8 4 Ju 7 9:9:55 PDT
9 ε 33 = u 3,3 + (u,3) + (u,3) + (u 3,3), (3) ε = (u, + u, ) + u 3, u 3, (u T, θ, x 3 ), ε 3 = (u,3 + u 3, ) + u, u,3 (u T,3 + θ, x ), ε 3 = (u,3 + u 3, ) + u, u,3. By defining the foowing parameters Θ = θ,, ℵ = u,, ℵ 33 = u 3,, (4) ε = u, + (u,) + (u 3,) = ε L + ε NL, and empoying them into the Eqs. (3), strain components can be rewritten as beow, ε = ε + x ℵ + x 3 ℵ 33, ε = (u T, Θ x 3 ), ε 3 = (u T,3 + Θ x ), ε = ε 33 = ε 3 =, (5) and as foows in the matrix notation, ε = ε L + ε N, (6) in which, ε L = [ ε N = [ L ε L ε L ε 3 N ε N ε N ε 3 u, + x ℵ + x 3 ℵ 33 ] = [ T, Θ x 3 ) T,3 + Θ x ) ], (7) (u,) + (u 3,) ] = [ ]. (8) Simiary, the member generaized strains are determined in the matrix form as ε ℵ E = E L + E N = [ ], (9) ℵ 33 Θ where, E L = [u, u, u 3, θ, ] T and E N = [ (u,) + (u 3,) ] T. 9 4 Ju 7 9:9:55 PDT
10 We consider for now that the member materia is inear eastic, thus the tota stress tensor (the second Pioa-Kirchhoff stress tensor), S of which is cacuated as S = S +, () here, is the pre-existing Cauchy stress tensor, and S is the incrementa second Pioa-Kirchhoff stress tensor in the updated Lagrangian co-rotationa frame e i given by S = E ε, S = μ ε, S 3 = μ ε 3, S = S 33 = S 3, () in which μ is the shear moduus, μ = E (+ν), E is the eastic moduus, and ν is the Poisson s ratio. Using equations (5) and (), the generaized noda forces for the member shown in Fig. subjected to the twisting and bending moments are cacuated as N = S da = E(Aε A + I ℵ + I 3 ℵ 33 ), M = S x da = E(I A ε + I ℵ + I 3 ℵ 33 ), M 33 = S x 3 da = E(I A 3 ε + I 3 ℵ + I 33 ℵ 33 ), T = (S 3 x S x 3 ) da = μi rr Θ, () A where A is the area of the cross section, I i and I ij (i, j =,3) are the first moment and the second moment of inertia of the cross section, respectivey, I = A x da, I 3 = x 3 da, I = A A x da, I 33 = x 3 da, I 3 = x x 3 da, and I rr is the poar moment of inertia, I rr = A A (x + x 3 )da. Using the eement generaized strains, E, the eement generaized stresses, σ are A aso determined in the matrix form as σ = DE, (3) in which, N M σ = [ ], (4) M 33 T 4 Ju 7 9:9:55 PDT
11 EA EI D = [ EI 3 EI EI EI 3 EI 3 EI 3 EI 33 μi rr ]. (5). Expicit Derivation of Tangent Stiffness Matrix Undergoing Large Easto-Pastic Deformation Throughout this section, mixed variationa principe in the co-rotationa updated Lagrangian reference frame and a pastic-hinge method are empoyed to obtain expicit expressions for the tangent stiffness matrix of each member shown in Fig.. Stiffness matrix is cacuated for each member by accounting for arge rigid rotations, moderate reative rotations, the noninear couping of axia, torsiona, and bidirectiona-bending deformations, and the effect of pasticity. The functiona of the mixed variationa principe in the co-rotationa updated Lagrangian reference frame and the tria functions for the stress and dispacement fieds within each eement are given in Section... Pastic anaysis using pastic hinge method is described in Section... The expicit expression of the stiffness matrix in the presence of pasticity for each ceuar member is aso presented in Section Mixed Variationa Principe in the Co-rotationa Updated Lagrangian Reference Frame Consideration of S ij and u i, respectivey, as the components of the incrementa second Pioa- Kirchhoff stress tensor and the dispacement fied in the updated Lagrangian co-rotationa frame, the functiona of the mixed variationa principe in the same reference frame with orthonorma basis vectors e i is obtained as H R = V { B[S ij ] + ij u k,i u k,j + S ij(u i,j + u j,i ) ρb i u i } dv T iu i ds, (6) S σ where, V is the voume in the current co-rotationa reference state, S σ is the part of the surface with the prescribed traction, T i = T i + T i (i =,, 3) are the components of the boundary tractions, and b i = b i + b i (i =,, 3) are the components of body forces per unit voume in the current 4 Ju 7 9:9:55 PDT
12 configuration. The dispacement boundary conditions prescribed at the surface S u are aso considered as u i (i =,, 3), assumed to be satisfied a priori. Eq. (6) is a genera variationa principe governing stationary conditions, which with respect to variations δs ij and δu i resuts in the foowing incrementa equations in the co-rotationa updated Lagrangian reference frame, B S ij = (u i,j + u j,i ), (7) [S ij + τ ik u j,k ] + ρb,j i = τ ij,j ρb i, (8) n j [S ij + τ ik u j,k ] T i = n j τ ij + T i on S σ, (9) where n is the outward unit norma on the surface S σ. For a group of members, V m (m =,,, N) with common surfaces ρ m, Eq. (6) can be written as H R = ( { B[S ij ] + ij u k,i u k,j + S ij(u i,j + u j,i ) ρb i u i } dv S σ m m V m T iu i ds) m =,,, N. () If the tria function u i and the test function u i for each member, V m (m =,,, N) are chosen in such a way that the inter-eement dispacement continuity condition is satisfied at ρ m a priori, then stationary conditions of H R for a group of finite eements ead to B S ij = (u i,j + u j,i ) in V m, () [S ij + τ ik u j,k ] + ρb,j i = τ ij,j ρb i in V m, () [n i (S ij + τ ik u j,k )] + + [n i (S ij + τ ik u j,k )] = [n i τ ij ] + [n i τ ij ] at ρ m, (3) n j [S ij + τ ik u j,k ] T i = n j τ ij + T i on S σm, (4) here, + and denote the outward and inward quantities at the interface, respectivey. The continuity of the dispacement at the common interface, ρ m between eements, is determined by u i + = u i on ρ m. (5) Appying Eqs. (5) and (3) into the Eq. () and integrating over the cross section area of each eement, Eq. () is rewritten as N H R = { ( m= σt D σ) d + N (u, + u 3, )d + (N ε L + M ℵ + M 33 ℵ 33 + T Θ)d Q q}, (6) in which σ = [N M M 33 T ] T is the initia member generaized stress in the corotationa reference coordinates e i, σ = σ + σ = [N M M 33 T ] T is the tota member 4 Ju 7 9:9:55 PDT
13 generaized stress in the coordinates e i, Q is the noda externa generaized force vector in the goba reference frame e i, and q is the noda generaized dispacement vector in the coordinates e i. Eq. (6) can be simpified by appying integration by parts to the third integra term on the right-hand side of the equation. More detais as how to compute integration by parts are given in Appendix A. Stationary conditions for H R given in Eq. (6) resut in D σ = E, N, = in V m, T, = in V m, (7) M, + [N u, ], = in V m, M 33, + [N u 3, ], = in V m, and the noda equiibrium equations are obtained from the foowing reation, N m= {N δu + M, δu M δu, + M 33, δu 3 M 33 δu 3, + T δθ + (N u, )δu + (N u 3, )δu 3 Q δq} =. (8) Herein, the tria functions for the stress and dispacement fieds within each member, V m (m =,,, N) are discussed. We assume that the components of the member generaized stress, σ obey the foowing reation σ = Pβ, (9) where, + x x P = x x, (3) [ ] β = [n m 3 m 3 m m m ] T. (3) Simiary, the components of the initia member generaized stress, σ are determined σ = Pβ, (3) where, β = [n m 3 i Note that, m i m 3 m i i m m ] T, (33) ( m ) and m 3 ( m 3 ) are, respectivey, bending moments (initia ones) around the x - and x 3 -axes at the ith node. n (n ) and m (m ) are the (initia) axia force and the (initia) 3 4 Ju 7 9:9:55 PDT
14 twisting moment aong the eement, respectivey. Therefore, the incrementa interna noda force vector B for the eement shown in Fig., with nodes and at the ends, can be expressed as B = [ N m m which can be written as m 3 N m m m 3 ] T, (34) B = R β, (35) with R =. (36) [ ] From Eq. (6), it is seen that ony the squares of u, and u 3, appear within each member. Therefore, we assume the tria functions for the dispacement fied in such a way that u, and u 3, become inear for each member. Moreover, we suppose that the bend anges around x - and i x 3 -axes aong the member shown in Fig. change with respect to the noda rotations θ i θ 3 (i =, ) via the foowing reation u θ = N θ a θ = [ x x x ] x [ θ θ 3 θ θ 3 ] and. (37) Therefore, the noda generaized dispacement vector of the member can be expressed in the updated Lagrangian co-rotationa frame e i as a = [ where i a i a a a] T, (38) i = [ u (i =, ) is the dispacement vector of the ith node, i u i u 3 i θ i θ i θ 3 ] T. (39) The noda generaized dispacement vector of the member, a is reated to the vector a θ by a θ = T θ a, (4) in which, 4 4 Ju 7 9:9:55 PDT
15 T θ = [ ]. (4) Appying the tria functions of the stresses, Eq. (9) into the Eq. (6), the functiona of the mixed variationa principe in the co-rotationa updated Lagrangian reference frame can be rewritten as H R = H R + H R + H R3 H R4, (4) here, N H R = ( m= σt D σ) d = ( m= βt P T CPβ) d, (43) N N m= H R = { N u N u + ( m 3 m 3 )( u u ) + m 3 θ 3 m 3 θ 3 + ( m m )( u 3 u 3 ) + m θ m θ + m θ m θ } = N m= {B T I a} = N m= {β T R T I a}, (44) H R3 = N N [ (u,) + (u 3,) m= ] d = N σ [ (θ ) + (θ 3) ] d m= = N σ m= u θ T u θ d = N σ m= at A nn a d, (45) H R4 = N m= (a T F a T I T Rβ ), (46) where, C = D, (47) I = [, (48) ] A nn = T θ T N θ T N θ T θ. (49) Invoking variationa form for the functiona of the mixed variationa principe resuts in the foowing equation, N m= δβ T ( (P T CPβ)d N m= + R T I a) + δa T (I T R β + σ A nn a d F + I T R β ) =, (5) 5 4 Ju 7 9:9:55 PDT
16 By etting H = P T CP d, G = R T I, K N = σ A nn d, F = G T β, Eq. (5) can be rewritten as N m= δβ T ( Hβ + Ga) + N m= δa T (G T β + K N a F + F ) =. (5).. Pasticity Effects in the Large Deformation Anaysis of Members of a Ceuar Microstructure For an eastic-perfecty pastic materia, the incrementa work done on the materia per unit voume is dw = σ ij (dε p ij + dε e ij ) in which ε e ij and ε p ij are eastic and pastic components of strain, respectivey, and σ ij are the stress components. Using pastic hinge method, the pastic deformation is deveoped aong the member wherever the pasticity condition is satisfied. Therefore, the tota work expended in deforming the materia of the body is W = V σ ij (dε ij p + dε ij e ) dv = U(ε ij e ) dv V + dw i p i, (5) where, U(ε ij e ) is the eastic strain energy density function, and dw i p is the increment of pastic work at the ith pastic hinge. When the theory of pastic potentia is appied, the pasticity condition in terms of the stress components at the ith node is expressed as f i (σ xi, σ yi,, τ xyi,, σ Y ) =, (53) the increment of pastic work at the ith node can be expressed as dw i p = du pt x, (54) in which, du p, the increment of pastic noda dispacement at the ith node, is expained in term of the function f i (x, σ Y ), du p = dλ i φ i, (55) φ i = [ f i (x,σ Y ) ], (56) x x is the noda force, and dλ i is a positive scaar. Therefore, Eq. (5) can be rewritten W = V U(ε e ij ) dv + dλ i φ T i p x, (57) 6 4 Ju 7 9:9:55 PDT
17 x = p is the ocation of the pastic hinge. A variationa form for the pastic work can be written as beow N δ { m= ( dλ i φ T i p ) (Pβ + Pβ) } = N m= δ ( dλ i φ T i p ) (Pβ + Pβ) + ( dλ i φ i T p ) P δβ = δdλ i φ i T p (Pβ + Pβ) + ( dλ i φ i T p ) P δβ N m= = T N m= δdλ i φ T i p (Pβ + Pβ) + δβ T P T ( dλ i φ T i p ). (58)..3 Expicit Derivation of Tangent Stiffness Accompanying Pasticity Effects Using the functiona of the mixed variationa principe given in Section.., Eqs. (4-46), Eq. (57) is expressed as W = N { ( βt P T CPβ) d + (β T R T I a) + σ m= at T T θ N T θ N θ T θ a d (a T F a T I T R β ) + ( dλ i φ i T p ) (Pβ + Pβ)}. (59) Then, invoking δw = and using Eqs. (5) and (58), Eq. (5) can be modified to incude the effect of pasticity by introducing new determined matrices β, H, and G as foows, N m= δβ T ( H β + G a) + N m= δa T (G T β + K N a F + F ) =, (6) in which, β T = [β T dλ], (6) H = [ H A T ], (6) A G T = [G T ], (63) where, A T = [ f N f M 3 ( + p ) f M 3 ( p ) f M ( p ) f M Since δβ T in Eq. (6) are independent and arbitrary in each eement, p f M ]. (64) H β = G a, (65) β = H G a, (66) By etting N m= δa T (G T β + K N a F + F ) = and substituting β from Eq. (66), we obtain 7 4 Ju 7 9:9:55 PDT
18 (G T H G + K N )a F + F =, (67) Therefore, the stiffness matrix, Ƙ, in the presence of pasticity is derived expicity as Ƙ = G T H G + K N = G T H G + K N G T H A C T G = K G T H A C T G = K K P, (68) where, K = G T H G + K N = K L + K N, (69) K P = G T H A C T G, (7) and, C T = (A T H A ) A T H. (7) Since we study the noninear couping of axia, torsiona, and bidirectiona-bending deformations for each eement, pasticity condition is introduced by f i (N, M, M, M 3) = at the ocation of the ith pastic hinge, then φ i = [ f i N f i M f i M f i M 3 ] T, (7) and dλ i φ i T p = [ dλ i f i N p f dλ i i M p f dλ i f i dλ i M i p M 3 p ] = [H P θ P θ P θ P3 ], (73) in which, H P is the pastic eongation and θ Pi, i =,, 3 are the pastic rotations at the ocation of pastic hinges. Components of the eement tangent stiffness matrices, K N, K L, and K P are presented in Appendix B. Moreover, transformation matrices reating coordinate systems corresponding to the deformed and undeformed states to the goba coordinates system (Fig. ) are aso given in Appendix C..3 Soution Agorithm To sove the incrementa tangent stiffness equations, we empoy a Newton Homotopy method [33, 34]. One of the most important reasons that we use newy deveoped scaar Homotopy methods is that this approach does not need to invert the Jacobian matrix (the tangent stiffness matrix) for soving NAEs. In the case of compex probems, such as eastic-pastic anaysis of arge 8 4 Ju 7 9:9:55 PDT
19 deformations and near the imit-oad points in post-bucking anayses of geometricay noninear frames, where the Jacobian matrix may be singuar the iterative Newton s methods become probematic and necessitate the use of Arc-Length methods as in software such as ABAQUS. One of the other reasons behind the advantages of recent deveoped Hemotopy methods is showing much better performance than the Newton Raphson method, when the Jacobian matrix is neary singuar or is severey i-conditioned. For instance, when we considered the probem discussed in Section 3. (Three-Member Rigid-Knee Frame) using the Newton-Raphson agorithm, the provided code coudn t converge to capture the critica oad whie it converged rapidy after switching to the Homotopy agorithm. Moreover, we discovered that whie the Newton-type agorithm fais to converge, the Newton Homotopy method provides convergent soutions in the presence of the pasticity and bucking in a arge number of members of the micro-attice. As another benefits of the empoyed agorithm, our deveoped CELLS/LIDS code is not sensitive to the initia guess of the soution vector, unike the Newton Raphson method. The Homotopy method was firsty introduced by Davidenko [35] to enhance the convergence rate from a oca convergence to a goba one for the soution of the NAEs of F(X) = ; X R n is the soution vector. This methodoogy was based on the empoyment of a vector Homotopy function, H(X, t) to continuousy transform a function G(X) into F(X). The variabe t ( t ) was the Homotopy parameter, treating as a time-ike fictitious variabe, and the Homotopy function was any continuous function in such a way that H(X, ) = G(X) = and H(X, ) = F(X) =. More detais on the vector Homotopy functions are given in Appendix D. To improve the vector Homotopy method, Liu et a. [33] proposed a scaar Homotopy function, h(x, t) in such a way that h(x, ) = G(X) = and h(x, ) = F(X) =. They [33] introduced the foowing scaar Fixed-point Homotopy function h(x, t) = (t F(X) ( t) X X ), t. (74) Later, Dai et a. [34] suggested more convenient scaar Homotopy functions which hod for t [, ) instead of t [, ]. We consider the foowing scaar Newton Homotopy function to sove the system of equations F(X) =, h n (X, t) = F(X) + Q(t) F(X ), t, (75) 9 4 Ju 7 9:9:55 PDT
20 resuting in, Q F X = Q B T F B T F, t, (76) where, B is the Jacobian (tangent stiffness) matrix evauated with B = F, and Q(t) is a positive and monotonicay increasing function to enhance the convergence speed. Various possibe choices of Q(t) can be found in [34]. Finay, the soution vector, X can be obtained by numericay integrating of Eq. (76) or using iterative Newton Homotopy methods discussed in Appendix D. X 3. Representative Approach and its Vaidation This section is devoted to consider the vaidity of our proposed methodoogy. To this end, three different probems are anayzed and compared with resuts from other methods given in the iterature. The critica oad of three-member rigid-knee frame is computed in section 3.. Section 3. examines the cassica Wiiams togge probem. Section 3.3 is devoted to consider the accuracy and efficiency of the cacuated stiffness matrix in the presence of pasticity by soving the probem of right-ange bent. 3. Three-Member Rigid-Knee Frame The geometry of the three-member rigid-knee frame and the cross section of eements are shown in Fig.. Using the CELLS/LIDS [CELLuar Structures/Large Ineastic DeformationS] code, the onger eement is divided into 6 eements and shorter eements are divided into 3 eements. A transverse perturbation oading,.p is aso appied at the midpoint of the onger member. Load versus dispacement at the ocation of point oad is potted in Fig. 3 and compared with the corresponding resuts presented by Shi and Aturi [8]. As it is observed, there is a good agreement between present cacuated resuts and those obtained by Ref. [8]. Pease note that, they [8] have aso mentioned that their computed critica oad is a itte higher than that obtained by Maett and Berke [8]. 4 Ju 7 9:9:55 PDT
21 P (b) Fig.. The geometry of three-member rigid-knee frame and the cross section of eements Present study Ref. [8] δ (in) Fig. 3. Load versus dispacement at the ocation of point oad. 4 Ju 7 9:9:55 PDT
22 3. Cassica Wiiams Togge Probem Wiiams [36] deveoped a theory to study the behavior of the members of a rigid jointed pane framework and appied it to the case of the rigid jointed togge. Cassica togge probem is exhibited in Fig. 4, consisting of two rigidy jointed eements with equa engths L and anchored at their remote ends. The ange between the eement and the horizonta axis, b is reated to the ength of the eements via the reation L Sin(b) =.3. The characteristics of the cross section of eements are aso incuded in Fig. 4. The structure is subjected to an externa oad W aong z direction at the apex, as iustrated in Fig. 4. The defection of the apex versus the appied oad is cacuated and compared with resuts given by Wiiams [36] in Fig. 5. As it is seen, good correspondence is obtained. Fig. 4. Cassica togge probem. 4 Ju 7 9:9:55 PDT
23 W (b) Present study Wiiams' study δ (in) Fig. 5. Dispacement at the apex of togge versus the appied oad. 3.3 Eastic-Pastic Right-Ange Bent Throughout this section, the accuracy and efficiency of our methodoogy to consider the effect of pasticity is investigated. To this end, the probem of right-ange bent is cacuated and compared with the resuts from other works. Two equa members of ength with square cross sections are ocated in the xy-pane and are subjected to an externa oad, F aong the z-direction at the midpoint of one eement, as shown in Fig. 6. Both members are anchored at their remote ends. Therefore, they are under both bending, M and twisting, T. The yieding condition for such a perfecty pastic materia subjected to bending and twisting is ( M M ) + ( T T ) =, in which M and T are, respectivey, fuy pastic bending and twisting moments. Empoying our CELLS/LIDS code, each member is simuated by four eements. The formation of pastic hinges via the increase of externa oad is presented in Fig. 6 and the cacuated amounts of F M at the onset of pastic hinges are compared with the resuts given by Shi and Aturi [8]. The variation of δ EI M with respect to F M is aso potted in Fig. 7. Here, δ is the dispacement of the tip of the rightange bent aong the z-direction, and E is the Young s moduus. The resuts aso show good agreement with those in Hodge [5]. 3 4 Ju 7 9:9:55 PDT
24 FL/M Fig. 6. Progressive deveopment of pastic hinges in the right-ange bent Pastic nodes,, & 8 Pastic nodes & Pastic node δ EI/(M L ) Fig. 7. The variation of normaized oad with respect to the normaized dispacement at the tip of right-ange bent. 4 4 Ju 7 9:9:55 PDT
25 4. Low-Mass Metaic Systems with Architected Ceuar Microstructures This section is devoted to study computationay the arge eastic-pastic deformations of nickebased ceuar micro-attices fabricated at HRL Laboratories [7, 8]. To mimic the fabricated ceuar microstructures, we mode repetitive RVEs constructed by the strut members with the same geometry and dimension as the experiment. Each member of the actua ceuar microstructure undergoing arge eastic-pastic deformations is modeed by a singe spatia beam finite eement with DOF, providing the capabiity to decrease consideraby the number of DOF in comparison with the same simuation using commercia FE software. The strut members are connected in such a way that the topoogy of the fabricated ceuar materia is achieved. In the foowing, more detais on the formation of RVE mimicking the actua microstructura sampes are given. The properties of nicke as the parent materia of the architected materia is introduced within the CELLS/LIDS code by the Young s moduus, E s = Gpa and the yied stress, σ y s = 45 MPa. The considered RVE is a Bravais attice formed by repeating octahedra unit ces without any attice members in the basa pane, as shown in Fig. 8. The attice constant parameter of the unit ce is a, Fig. 8. The RVE is constructed by a node-strut representation and incudes the nodes coordinate and the nodes connectivity, which determines the ength of the members as we as the topoogy of the micro-attice. Furthermore, the present RVE approach accuratey captures the microstructura ength scae by introducing the area, the first and the second moments of inertia, and the poar moment of inertia of the symmetrica/unsymmetrica cross section of the hoow tube member within the formuation. Periodic boundary conditions (PBCs) are considered aong the x- j and y-directions of RVE, which are the directions perpendicuar to the depth of the thin fim microattice. PBCs are invoved by i a x= = a x=na aong x direction and by k a y= = a y=ma aong y direction, in which α a (α = i, j, k, ) is the dispacement vector of the αth node (Eq. 39) on the boundary of RVE, and n or m is determined based on the size of the RVE aong the x- or y-direction, respectivey. For exampe, for the Na Ma Ka RVE, n = N and m = M. The depth of RVE is modeed to be equa to the thickness of the thin fim. Section 4. studies the a a a RVE incuding nodes and 3 strut members, and Section 4. examines both the a a a RVE with 6 nodes and 8 strut members and the a a 4a RVE with 36 nodes and 64 members. We study the mechanica behavior of the thin fim ceuar micro-attice under tension, compression, and shear oadings. To this end, nodes on both top and bottom faces of RVE 5 4 Ju 7 9:9:55 PDT
26 are oaded accordingy. Micro-attice members are cyindrica hoow tubes, the dimensions of which are aso incuded in Fig. 8. Torrents et a. [8] tested sampes with the strut member ength of L = 4 mm, strut member diameter of D = 5μm, the wa thickness of t = 5nm, and the incination ange of θ = 6ᵒ. In Sections 4. and 4., we anayze the mechanica behavior of two different fabricated ceuar micro-attices in which the geometry of their strut members (L, D, t, and θ) are expained respectivey. Since noninear couping of axia, torsiona, and bidirectiona-bending deformations is considered for each member, the pasticity condition is determined by the foowing reation f(n, M, M 33, T) = {M M + M 33 + T } + N N =, (77) where, M and N are fuy pastic bending moment and fuy pastic axia force, respectivey. Fig. 8. The unit ce of RVE consisting of an octahedron as we as the geometry of strut members. 4. Architected Materia with More Fexibiity as Compared to Parent Materia An RVE incuding nodes and 3 members with PBCs aong x and y directions is empoyed to mode a ceuar thin fim with the thickness of a, Fig. 9. This figure shows the appication of the externa compressive oading, which changes according to tensie as we as shear oads. The dimensions of each member in the micro-attice is as foows: L = 5 μm, D = 5 μm, and t = 5 nm. 6 4 Ju 7 9:9:55 PDT
27 a z y x a a Fig. 9. a a a RVE incuding nodes and 3 strut members. The engineering stress as a function of the engineering strain is presented in Fig. for the nicke ceuar micro-attice under compressive, tensie, and shear oads. The stress-strain curves corresponding to the tensie and compressive oads resut in the overa yied stress of the RVE, σ y = 5.7 kpa and the Young s moduus, E =.9 MPa. Torrents et a. [8] measured the respective vaues σ y = 4. ±.5 kpa and E =. ±.5 MPa for their tested micro-attice abeed with G (L = 5 ± 3 μm, D = 6 ± 4 μm, t =.55 ±.6 μm). The resuts cacuated from our computationa methodoogy agree exceenty with those obtained from experiment by Torrents et a. [8]. It is found that this architected materia shows a yied stress much smaer than the parent materia, which offers more fexibiity in taioring the response to impusive oads. In addition, we are abe to cacuate the shear moduus of the ceuar micro-attice from our obtained stress-strain curve corresponding to the shear oad, resuting in G =.773 MPa. 7 4 Ju 7 9:9:55 PDT
28 Engineering Stress (kpa) 3 5 D E F C 5 5 A G B Compressive oading Tensie oading Shear oading Engineering strain (%) Fig.. Stress-strain curve of the ceuar micro-attice subjected to tension, compression, and shear. The progressive deveopment of pastic hinges as the tensie and compressive oads increase is shown in Fig.. The tota deformation of RVE considering the effect of pasticity corresponding to the step B of compressive oading, step F of tensie oading, and step G of shear oading is aso given in Figs. (a) through (c), respectivey. Since pastic deformation can absorb energy, this architected materia wi be appropriate for protection from impacts and shockwaves in appications varying from hemets to vehices and sporting gear []. 8 4 Ju 7 9:9:55 PDT
29 (a) Pastic hinges formed at steps A and C shown in Fig.. (b) Pastic hinges formed at steps B and D shown in Fig. (c) Pastic hinges formed at step E shown in Fig.. Fig.. Progressive deveopment of pastic hinges in ceuar micro-attice under tension and compression. 9 4 Ju 7 9:9:55 PDT
30 (a) At the step B of compressive oading 3 4 Ju 7 9:9:55 PDT
31 (b) At the step F of tensie oading 3 4 Ju 7 9:9:55 PDT
32 (c) At the step G of shear oading Fig. (a-c) Tota eastic-pastic deformation of the ceuar microstructure in red coor at different steps of oading shown in Fig.. Initia unoaded state is aso presented by back dashed ine. 4. Architected Materia with Further Increased Reative Density In this case, the fabricated sampe is computationay modeed using an RVE consisting of 6 nodes and 8 members with PBCs aong the x- and y-directions, Fig. 3. The strut member dimensions are L = μm, D = 75 μm, and t = 6 μm. The wa thickness of the member in this case is 5 times greater than that of the previous case in section 4.. The RVE is subjected to both tensie and compressive oading to study the mechanica properties of the architected materia. The engineering stress-engineering strain curve is potted in Fig. 4. Stress anaysis shows biinear eastic modui for this ceuar micro-attice subjected to both tension and compression. Eastic 3 4 Ju 7 9:9:55 PDT
33 moduus for the first phase is cacuated as.69 GPa under both tensie and compressive oading. For the second phase it is cacuated to be.84 GPa under tension and.364 GPa under compression. The yied stress is obtained as 7. MPa and MPa subjected to tensie and compressive oading, respectivey. Pastic-hinges emanate at the stress eve MPa when the micro-attice is under tension and originate at the stress eve MPa when the micro-attice is subjected to compression. It is found that both Young s moduus and the yied stress of the ceuar micro-attice increase significanty by increasing the strut thickness. It is weknown that the eastic moduus and the yied strength of the ceuar materias increase with the increase of their reative density [9]. Reative density is cacuated as ρ ρ s where, ρ is the mass of the attice divided by the tota bounding voume (v), and ρ s is the mass of the attice divided by ony the voume of the constituent soid materia (v s ). Therefore, ρ ρ s = (m v) (m v s ) = v s v in which v s = # of Members π[(d + t) (D ) ] L and v = 8a 3 or a 3 for a a a RVE or a a a RVE, respectivey. We cacuate the reative densities of the ceuar micro-attices examined through this section (Section 4.) and Section 4. as.35 and.66, respectivey. Torrents et a. [8] aso extracted experimentay the strain-stress curve of this micro-attice (abeed A) under compression. They measured the Young s moduus, E =.58 ±.3 GPa and the yied stress, σ y = 8.5 ±.5 MPa for the tested micro-attice with the strut diameter D = 75 ± 6 μm, strut ength L = ± 36 μm, and wa thickness t = 6. ±.6 μm. As it is seen there is a very good correspondence between our cacuated mechanica properties of the sampe under compressive oading and those measured experimentay by Torrents et a. [8]. To investigate the effect of the size of the RVE on the macroscae response of the ceuar microattice, the depth of the a a a RVE (Fig. 3) is increased by a factor of two. Due to the PBCs aong x- and y-directions, the size of the RVE aong these directions is considered to be a. Therefore, a a a 4a RVE consisting of 36 nodes and 64 strut members is modeed, Fig. 5, and the corresponding stress-strain curve under compression is incuded in Fig. 4. The stress anaysis of this a a 4a RVE aso exhibits biinear eastic behavior with the eastic modui of.784 GPa for the first inear phase and.47 GPa for the second inear phase. The yied stress is cacuated to be MPa which becomes coser to the corresponding experimenta vaue, σ y = 8.5 ±.5 MPa, in comparison with MPa cacuated for a a a 33 4 Ju 7 9:9:55 PDT
34 RVE. Figs. 6(a-b) show the eastic-pastic deformation of the a a a RVE under tension and compression, respectivey. a z y a x a Fig. 3. a a a RVE incuding 6 nodes and 8 strut members Ju 7 9:9:55 PDT
35 Engineering Stress (MPa) B A Tensie oading (XX) Compressive oading (XX) Compressive oading (XX4) 3 4 Engineering Strain (%) Fig. 4. Stress-strain curve of the ceuar micro-attice subjected to tension and compression. 4a z y x a a 35 4 Ju 7 9:9:55 PDT
36 Fig. 5. a a 4a RVE consisting of 36 nodes and 64 strut members. (a) At the step A of tensie oading 36 4 Ju 7 9:9:55 PDT
37 (b) At the step B of compressive oading Fig. 6. Eastic-pastic deformation of the ceuar microstructure in red coor at different steps of oading shown in Fig. 4. Initia unoaded state is aso presented by back dashed ine. 5. Concusion We presented a computationa approach for the arge eastic-pastic deformation anaysis of owmass metaic systems with architected ceuar microstructures. Studies on this cass of materias are of interest since they can be optimized for specific oading conditions by changing the base materia as we as the topoogy of the architecture. The repetitive RVE approach is utiized to mimic the fabricated ceuar micro-attices. The RVE is generated by a node-strut representation 37 4 Ju 7 9:9:55 PDT
38 consisting of the coordinate of nodes and the connectivity of nodes. Therefore, we can easiy study the effect of the change of topoogy on the overa mechanica response of the ceuar materia by changing both coordinate and connectivity of nodes. Moreover, the microstructura ength scae of the ceuar materia is accuratey captured by introducing the area, the first and the second moments of inertia, and the poar moment of inertia of the symmetrica/unsymmetrica cross section of the strut member within the formuation. In the current methodoogy, each member of the actua micro-attice undergoing arge eastic-pastic deformations is modeed by a singe FE with DOF, which enabes to study the static and dynamic behavior of the macrostructure directy and high efficienty by using arbitrariy arge number of members. We study the noninear couping of axia, torsiona, and bidirectiona-bending deformations for each 3D spatia beam eement. The effect of pasticity is incuded empoying pastic-hinge method, and the tangent stiffness matrix is expicity derived for each member, utiizing the mixed variationa principe in the updated Lagrangian co-rotationa reference frame. In order to avoid the inversion of the Jacobian matrix, we empoy Homotopy methods to sove the incrementa tangent stiffness equations. The proposed methodoogy is vaidated by comparing the resuts of the eastic and eastic-pastic arge deformation anayses of some probems with the corresponding resuts given in the iterature. Moreover, two fabricated ceuar micro-attices with different dimensiona parameters incuding the unit ce size and the strut thickness are modeed using different RVEs. We study their mechanica behaviors under a tensie, compressive, and shear oading. The comparison of the cacuated mechanica properties utiizing the present methodoogy with the corresponding experimenta measurements avaiabe in the iterature reveas a very good agreement. Using this deveoped computationa approach, we can homogenize any ceuar structure easiy, and we can design the topoogy of microstructure for any designated properties. Appendix A L N ε d = N u, d = N, u d + N u, (A.) M ℵ d M 33 ℵ 33 d = M u, d = M 33 u 3, d = M, u d = M 33, u 3 d + M, u M u,, (A.) + M 33, u 3 M 33 u 3,, (A.3) 38 4 Ju 7 9:9:55 PDT
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