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A boundary element technique for incremental, non-linear elasticity Part I: Formulation M. Brun , D. Capuani , D. Bigoni a, Dipartimento di Ingegneria Meccanica e Strutturale, Universit di Trento, Via Mesiano 77, I-38050 Povo, Trento, Italy Dipartimento di Ingegneria, Universit di Ferrara, Via Saragat 1, 44100 Ferrara, Italy Received $9 &ovember 2((2) accepted $$ February 2((* Abstract Incremental elastic deformations superimposed upon a given homogeneous strain are analyzed with a boundary element technique. -his is based on a recently-developed .reen s function for non-linear incremental elastic defor- mations. Plane strain perturbations are considered of a broad class of incompressible material behaviours /including hyper-, hypoelastic and &avier–1tokes constitutive equations3 within the elliptic range. &umerical treatment of the problem is detailed. A possibility of employingthe method in the fully non-linear range is outlined, which yields a boundary element approach where the use of domain integrals is avoided, atleast in a conventional sense. -hemethods for bifurcation and shear band analyses will be reported in Part II. 2((* 4lsevier 1cience B.5. All rights reserved. 1. Introduction -he analysis of the response to perturbations of a pre-stressed, non-linear elastic solid is important in a broad range of technological circumstances. For instance, pre-stress aﬀects the design of Microelectro- mechanical 1ystems 7$$8, it is a concern in the behaviour of geological formations 7*9,*98, biological tissues 7$(,$:8, and various structural elements, includingseismic insulators and rubber bearings 79,$98. Referringto plane strain deformations of incompressible materials, Biot 728 has shown that the incre- mental elastic response is governed by two incremental moduli, functions of the current stretch. Biot constitutive framework was assumed by Bigoni and Capuani 7$8 to obtain a .reen s function and a boundary integral formulation for incremental deformations superimposed upon a given, homogeneous strain. Both .reen s function and integral formulation provide the basis to build a boundary element Correspondingauthor. -el.: ;*9-(:6$-992-5(7) fax: ;*9-(:6$-992-599. E-mail addresses: michele.brun@ing.unitn.it /M. Brun3, cpd@dns.unife.it /D. Capuani3, bigoni@ing.unitn.it /D. Bigoni3. URL: http:AAwww.ing.unitn.itA~bigoniA. ((:5-7925A(*AC - see front matter 2((* 4lsevier 1cience B.5. All rights reserved. doi:$(.$($6A1((:5-7925/(*3((269-9 Comput. Methods Appl. Mech. 4ngrg. $92 /2((*3 2:6$–2:79 www.elsevier.comAlocateAcma

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technique for solvingincremental problems of non-linear elasticity. -his is the purpose of the article. In particular, we formulate a general numerical scheme to handle generic boundary value problems with prescribed nominal tractions andAor displacements. Dhen restricted to perturbations of homogeneously deformed,incompressiblesolids,ourboundaryelementstechniqueretains allwell-knownadvantagesofthe small strain formulation. -hese are: discretization only of the boundary of the body) automatic satisfaction of the incompressibility constraint) possibility of describingsingularities arisingnear corner points of the boundary) possibility of employing meshes thoroughly varying in size throughout the body. 1everal attempts can be found in the literature to analyze non-linear problems usingboundary elements techniques. In some cases the non-linearities were related to the material 7:,5,2(,*:,:$8, in other cases to large elastic 72:,*$,*58 or elastoplastic 76–9,$2,$7,29–*(8 strains. In all cases, in addition to the usual boundary integrals, a domain integral is introduced, leading to the so-called ﬁeld-boundary element method . -he introduction of this term nulliﬁes a main advantage of B4M and originates from the dis- crepancy between the non-linear character of the equations governing the problem and the employed fundamental solution /usually referringto linear, isotropic elasticity3. In the present paper, the focus is on incrementally-linear problems, so that domain integrals do not appear in the formulation that will be presented. Fowever, we believe that the solution of incrementally linear problems should be regarded as the ﬁrst step toward the analysis of fully non-linear situations and, in particular, we anticipate with a simple example that when employed as a tool to analyze large /thus non-linear3 deformations, our incre- mental method naturally leads to a volume discretization diﬀerent––in essence––from all those already known. -he paper is organized as follows: the incremental constitutive framework––which includes hyper- and hypoelasticity and the equations governing 1tokes ﬂow of ﬂuids––is presented in 1ection 2 for plane strain, whereas the fundamental solution and boundary integral equations are summarized in 1ection *. -he boundary element formulation is presented in 1ection : and the discretization detailed in subsequent 1ection 5. A numerical example is given in 1ection 6, which oﬀers the simplest context to illustrate the capability of the method to computationally follow a non-linear path of deformation. 1ystematic investi- gations of bifurcation and shear band phenomena in two-dimensional elastic materials will be reported in Part II. 2. Incremental constitutive equations -he general expression given by Biot 728 for rate constitutive equations of an incompressible material incrementally deformed in plane strain are obtained here with reference to a broad class of material be- haviours. ()1) Elasticity -he relation between the Cauchy stress and the left Cauchy–.reen strain tensor FF , with denotingthe deformation gradient, for a Cauchy-elastic, incompressible and isotropic solid can be written as 7:(8 where is a parameter connected to the hydrostatic pressure tr * through 2:62 M) Brun et al) / ,omput) Met-ods .ppl) Mec-) Engrg) 1/( 0(0031 (421–(47/

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tr tr and and are generic functions of two invariants of -he two particular cases of Mooney–Rivlin and neo-Fookean materials are recovered when and are taken constant and, in addition, when ( in the latter case. Constitutive equation /$3 corresponds to Cauchy elasticity, so that it describes a class of behaviours broader than hyperelasticity. -he requirement of existence of an elastic potential inﬂuences the dependence of coeﬃcients and on the invariants of . -o develop this point, let us consider that the constitutive equation /$3 implies coaxiality of tensors and , so that these share /at least3 one principal reference system––the 4ulerian principal axes––where diag diag in which (, $, 2, * are the principal stretches, satisfyingthe incompressibility constraint 4xpressing4q. /$3 in the 4ulerian principal reference system and solvingfor and yields two equations that can be alternatively expressed employingevery permutation of$, 2 and * as indices.It is clear from the expression /63 that existence of an elastic potential restricts the functional dependence of coeﬃcients on the stretch. In particular, the standard deﬁnition of elastic potential for incompressible materials is 7278 not summed which can be immediately employed into 4q. /63. -akingthe material derivative of /$3 yields where DB BD WB BW and DB WB $( in which is the 4ulerian strain rate and the spin tensor. Ieepinginto account now 4q. /$3 and the deﬁnition of Jaumann derivative $$ M) Brun et al) / ,omput) Met-ods .ppl) Mec-) Engrg) 1/( 0(0031 (421–(47/ 2:6*

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the constitutive equation /$3 becomes DB BD DB $2 &otingthat tr $* where , with $ and tr $: or $5 where with $ are the coeﬃcients and expressed as functions of the principal stretches, we arrive at DB BD DB $6 which is equivalent to DB BD DB $7 -heincrementalconstitutiveequationintheform /$63or/$73isvalidforthree-dimensional,incompressible Cauchy elasticity. De are interested here in the particularization of /$63 to incremental plane strain de4ormations super- imposed on a generic state of -omogeneous deformation. In the 4ulerian principal reference system diag ! $9 so that the out-of-plane stress rate components can be determined as $9 and ** $$ 22 2( or ** !# $$ 22 2$ 2:6: M) Brun et al) / ,omput) Met-ods .ppl) Mec-) Engrg) 1/( 0(0031 (421–(47/

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Finally, the in-plane rate components can be expressed in the Biot 728 form as $2 $2 $$ 22 $$ 22 $$ 22 22 where and aretwoincrementalmodulicorrespondingrespectivelytoshearingparallelto,andat:5 to, the 4ulerian principal axes. -hese can be expressed as functions of the invariants of 2* or as functions of the principal stretches !# 2: An alternative expression for the two incremental moduli and , related to the existence of a strain- energy function, was given by Biot 728 /see Appendix A3 in the form 25 As it will become clear later, constitutive relations of the form /223 with generic coeﬃcients and embrace a much broader class of material behaviours than Cauchy, isotropic elasticity. ()1)1) Mooney–Rivlin material A simple explicit form of strain-energy functions for isotropic rubber-like elastic media was proposed by Mooney 7228 in the form 26 or 27 M) Brun et al) / ,omput) Met-ods .ppl) Mec-) Engrg) 1/( 0(0031 (421–(47/ 2:65

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where and are material parameters and represents the shear modulus in the original unstressed state. In this case, with reference to the representation /$3, we simply obtain: 29 and ** 29 ()1)() 5gden material -he followingclass of strain-energy functions was proposed by 7258 to ﬁt experimental results on rubber: *( where and are material parameters, subKected to the constraints: with ... *$ in which, is a positive integer determining the number of terms in the strain-energy function, are constant shear moduli and are dimensionless parameters, with ... . In particular, Mooney Rivlin material can be recovered as a particular case taking 2, 2 and 2. An excellent correlation with experimental data relative to simple-tension, equibiaxial tension and pure shear of vulcanised rubber is obtained employingthe values 725,*6,*78: $( ($2 $( $( *2 yielding 225 $( &Am Coeﬃcients in the representations /$63 and /223 can be obtained from 4q. /*(3 in the form: ** and *: -he component ** is given in Appendix B. 2:66 M) Brun et al) / ,omput) Met-ods .ppl) Mec-) Engrg) 1/( 0(0031 (421–(47/

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()() 6ypoelasticity and t-e loading 7ranc- o4 elastoplastic constitutive la8s De consider a general incremental constitutive equation relating the Jaumann derivative of the Cauchy stress tensor to the 4ulerian strain rate through a generic tensor 1ym in the form DT TD DT *5 where coeﬃcients ... 9 are polynomial functions of the invariants of . In the particular case in which is identiﬁed with the Cauchy stress , the constitutive equation /*53 describes an incompressible, -ypoelastic material 7:(8. Fowever, even if does not represent the Cauchy stress and the coeﬃcients ... 9, remain completely unspeciﬁed /but independent of 3, in a principal reference system of and for plane, incremental deformation, we get ** $$ *6 where with * denote the principal values of and $2 2$ and $$ 22 can be expressed in the form /223, with *7 ()()1) -de4ormation t-eory o4 plasticity: -yperelastic and -ypoelastic approac-es -deformation theory of plasticity was proposed by Futchinson and &eale 7$58 /see also 7$6,2*83 in the framework of hyperelasticity, to model metals subKect to proportional loading. Dith reference to the representation /$3, the constitutive-law in the -deformation theory of plasticity can be expressed as ; *9 where log arethelogarithmicstrainsand tr *.Inthesameequation isthesecantmodulusto the curve representingthe eﬀective stress versus eﬀective strain *9 where are the principal components of deviatoric stress. -he curve is assumed to be determined by :( where 2 isanhardeningexponent, isapositiveconstitutiveparameter.-hestrain-energyfunction results therefore to be :$ M) Brun et al) / ,omput) Met-ods .ppl) Mec-) Engrg) 1/( 0(0031 (421–(47/ 2:67

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-he above framework corresponds to the followingchoice of parameters and /see /$3 and /*933: :2 and to the incremental shear moduli: coth :* -he out-of-plane stress increment is given by ** :: Diﬀerently from Futchinson and &eale 7$58, 1t oren and Rice 7**8 present the followinghypoelastic law /their 4q. /2633 to model elastoplastic materials subKect to proportional loading: :5 where tr *, isthedeviatoric stress, isawork-hardeningexponentand isthesecantmodulus on the shear stress–strain curve. 4q. /:53 can be written as :6 where tr *, which is, evidently, a particular case of /*53 and can therefore be cast in the form /223. ()()() T-e loading 7ranc- o4 non-associative, elastoplastic la8 A generic constitutive equation for an incompressible isotropic-elastic, plastic material depending on a generic collection of state variables can be written in the form if if :7 where is the yield function, 1ym is the yield function gradient and 1ym the plastic potential gradient. -he Macaulay brackets hi apply to every scalar in such a way that i j j 2 and in- troduce the incremental non-linearity, typical of plasticity. -he scalar ( is the plastic modulus, related to the hardeningmodulus through with :9 Finally, tensors and are subKect to the incompressibility constraints tr tr tr :9 so that tr *. Restrictingthe constitutive equation /:73 to its loadingbranch is synonymous of assuming and eliminatingthe incremental non-linearity, thus obtaining 2:69 M) Brun et al) / ,omput) Met-ods .ppl) Mec-) Engrg) 1/( 0(0031 (421–(47/

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5( Let us assume now that and are coaxial /and not necessarily coaxial with the Cauchy stress3. In the principal reference system of and and for plane incremental deformations, 4q. /5(3 can be cast in the form /223 with 5$ -he constitutive fourth-order tensor in /5(3 contains the non-symmetric term , where corresponds to the so-called non-associative elastoplasticity. Fowever, due to the incompressibility and plane strain constraints and without alteringthe material response, we can add a term 52 where is a symmetric tensor coaxial to and and is the unit vector singling the out-of-plane di- rection. &ow, the choice diag f 5* taken in the principal reference system of and , symmetrizes the constitutive operator for every so that non-associativity does not yield an unsymmetric tangent constitutive operator for incompressible, isotropic-elastic, plane-strain plasticity with coaxial yield function and plastic potential gradients. ()3) 9e8tonian ﬂuids -he constitutive equation describinga &ewtonian, incompressible ﬂuid takes the form 5: where the Cauchy stress is related to the 4ulerian strain rate through the viscous coeﬃcient and the pressure in the ﬂuid tr *. Clearly 4q. /5:3 has a structure identical to a very particular case of /*53, so that, for plane strain ** $$ 22 $$ 22 $$ 22 $2 $2 55 a form akin to /223. -herefore, keepinginto account that we will address the problem of quasi-static de- formations, our framework can be immediatelyemployed to describe mutatis mutandis ––two-dimensional 1tokes ﬂow. As a consequence, several results presented in the followingreduce to ﬁndings by Ladyz- henskaya 7$98 as particular cases. ()4) T-e general 4orm o4 constitutive equations 4or plane, incompressi7le, incremental de4ormations De have shown in the section above that a very broad class of incremental material behaviours––in- cludingCauchy elasticity, hyperelasticity, hypoelasticity, the loadingbranch of coaxial elastoplasticity, and &ewtonian ﬂuids––can be described by constitutive equation /223. It is expedient now to transform the constitutive equation /223 in terms of material derivative of the nominal stress tensor . In particular, a Lagrangean formulation of ﬁeld equations is employed in the following, with the current state taken as reference. As a consequence, constitutive equations /223 become ij ijkl ij ijkl pd ij 56 M) Brun et al) / ,omput) Met-ods .ppl) Mec-) Engrg) 1/( 0(0031 (421–(47/ 2:69

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where is the velocity, ij the Ironecker delta and $$ 22 57 Both and measure the in plane hydrostatic stress rate /positive in tension3 as related respectively to Cauchy and to nominal stresses. -ensor ijkl represents the instantaneous moduli, possessingthe maKor symmetry ijkl klij and havingthe form 7$*8 $$$$ $$22 $$$2 $$2$ 22$$ 2222 22$2 222$ $2$2 $22$ 2$$2 2$2$ 59 with 59 &ote that in 4q. /563 ijkl diﬀers from ijkl only in the followingcomponents: $$$$ 2222 6( so that ijkl also possesses the maKor symmetry. Mnly the forms /593 and /6(3 of the constitutive equation /223 will be needed in the following. -he formulation in the present paper is restricted to the elliptic range /43, where the characteristic equation associated to the equilibrium equations governing perturbations superimposed upon a given homogeneous strain does not admit real solutions. In particular, the elliptic range corresponds to negative or complex coeﬃcients and , functions of the material properties and state of pre-stress as follows: 6$ where 62 4mploying4q. /253 , parameter appearingin 4qs. /6$3 and /623 can be written as 6* and it represents a normalized measure of the pre-stress or––more precisely––a measure of the current state of stretch. -he elliptic range may be further sub-divided into elliptic-imaginary /4I3 and elliptic-complex /4C3 regimes. -hese are deﬁned as follows: (, so that and are both negative in /4I3, (, so that and are a conKugate pair in /4C3. It may be important to remark that shear bands, correspondingto the appearance of discontinuous strain rates, are formally excluded within the elliptic range, i.e. in the context analyzed here. Fowever, as shown 2:7( M) Brun et al) / ,omput) Met-ods .ppl) Mec-) Engrg) 1/( 0(0031 (421–(47/

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by Bigoni and Capuani 7$8 and Radi et al. 7*28, formation of zones of concentrated strains in response to a perturbation becomes possible when the boundary of the elliptic regime is approached. -his will be also demonstrated with numerical examples referringto van Fove conditions in Part II. -he boundary of the elliptic range is characterized by the following conditions: elliptic imaginaryAparabolic /4IAP3 boundary is attained when $, correspondingto () elliptic complexAhyperbolic /4CAF3 boundary is attained when (, correspondingto 3. The fundamental solution Dith reference to the constitutive equation /563, the .reen s function set for an inﬁnite and uniformly pre-strained medium can be written in the form 7$8 pd ig log log cos cos cos 6: where and are the polar coordinates singling out the generic point with respect to the position of the concentrated force and sin hi sin hi sin cot cot 65 cos 2cos 66 &ote that in expression /6:3 of we have introduced the usual symbol to denote the Cauchy principal value integral. -he gradient of the .reen s velocity may be obtained directly from /6:3 and can be written as cos hi sin Ra log cos sin Ra log cos 67 where index is not summed and sin 2cos aK sin aK 69 M) Brun et al) / ,omput) Met-ods .ppl) Mec-) Engrg) 1/( 0(0031 (421–(47/ 2:7$

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-he four components of the velocity gradient not included in 4q. /673 can be obtained using incom- pressibility and symmetry of .reen s tensor: 69 Accordingto constitutive equations /563–/593, from .reen s velocity gradient and pressure rate we obtain the in-plane hydrostatic stress rate 7( and the associated incremental nominal stress $$ $2 2$ 22 7$ It is important to note that the .reen s velocity can be given the form ig log ig 72 where the dependence on the current state is condensed in the coeﬃcients ig and ig . -he former coeﬃcient satisﬁes ig (if , while the latter contains the directional dependence on . It follows from repre- sentation /723 that the gradient of the .reen s velocity, the in-plane hydrostatic stress rate and thus the incremental stress rate can be expressed as explicit functions of $ 4. Boundary element formulation De restrict the presentation to mixed boundary value problems in which velocities and incremental nominal tractions are prescribed functions deﬁned on separate portions and of the boundary on ij on 7* of a solid , currently in a state of homogeneous, ﬁnite deformation. In this context, two integral repre- sentations exist relatingthe velocity and the pressure rate in interior points of the body to the boundary values of nominal traction rates ij and velocities 7$8. -hese are 7: and ijkg ll lr $$ rl $$ 75 If the point is on the boundary, 4q. /7:3 becomes 7$8 76 2:72 M) Brun et al) / ,omput) Met-ods .ppl) Mec-) Engrg) 1/( 0(0031 (421–(47/

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where lim 77 is the so-called -matrix 7*8, dependingon the material parameters, the state of pre-stress and the geometry oftheboundary/inthecaseofasmoothboundary, gi 3.&otethatsymbol introducedin/773 denotes the intersection between a circle of radius centred at and the domain 5. Boundary discretization -he boundary equation /763 is the startingpoint to derive the collocation 7oundaryelement met-od .-o thispurpose,theboundary isdividedinto elements ... ,withsubsets and belonging respectively to and /clearly 3. For simplicity the same discretization is assumed for velocity and traction rates at the boundary. In particular, inside each boundary element we use ... 79 where are the nodal values of velocities and nominal traction rates, respectively, and are the relevant shape functions, selected as polynomials of degree -he discretized form of 4q. /763, collocating the point at , correspondingto the node of the element ,is 79 where indices and are summed and range between ( and and $ and 2, respectively. An analysis of 4q. /793 reveals that the number of unknowns is unk , being prescribed on and on Collocatingnow 4q. /793 at nodes alongthe two directions and , yields the followingalgebraic system: 9( where and are the vectors collecting and 9$ 1olution of system /9(3 gives the nodal velocities on and the nominal traction rates on Mnce system /9(3 has been solved, ﬁelds and can be evaluated at internal points by applying the discretized forms of 4qs. /7:3 and /753. In particular, 4q. /7:3 reduces to 4q. /793 with gi and the integration is straightforward, as is always diﬀerent from zero. De limit the presentation to discretization of the boundary into rectilinear elements and linear shape functions, so that $ and unk -he singular integrals in /793, computed over the elements adKacent to the node in which the equation is collocated, are evaluated analytically. In particular, the strongly singular integral ig strong in the left hand side of 4q. /793 is equal to ig strong 92 M) Brun et al) / ,omput) Met-ods .ppl) Mec-) Engrg) 1/( 0(0031 (421–(47/ 2:7*

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which, with reference to the geometry sketched in Fig. $, can be transformed into ig strong 00 00 9* so that the change of variable 00 9: yields ig strong 95 As far as the elements $and are concerned, the incremental .reen s tractions can be computed as ig 96 which are independent of , so that ig can be calculated as . Introducing4q. /963 into 4q. /953, we obtain an explicit formula for the singular integrals /923 in the form ig strong ig log 97 In all our examples we have always found numerically that $$ 22 (, a result that can be analytically checked in the special case of &ewtonian ﬂuid, i.e. when ( and , but still requires a proof in the general case. -he weakly singular integrals ig weak in the right hand side of 4q. /793 is equal to ig weak 99 which becomes /Fig. $3 ig weak 99 Fig. $. .eometry at node 2:7: M) Brun et al) / ,omput) Met-ods .ppl) Mec-) Engrg) 1/( 0(0031 (421–(47/

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-aking4q. /723 into account, the integral /993 can be analytically evaluated and the results are listed in -able $. De note, in passing, that the use of higher-order polynomials as shape functions is straightforward, since the singularities in the above integrals arise from the constant part of the interpolating functions. ). A numerical example: non-linear elasticity without domain inte.rals An elastic block subKect to homogeneous, increasing deformation is considered, with the purpose of il- lustratingwith a simple example that our formulation does not involve domain integrals. In particular, the elastic block is constrained to plane deformations startingfrom an unstressed square conﬁguration and is subKected to tension or compression in one direction. De refer to Mooney–Rivlin /263 and Mgden /*(3 and /*$3 non-linear elastic laws. -he trivial, analytical solution to this problem is reported by Mgden 7268 9( sothatforuniaxialstress/ (3andplanestraindeformations/ $3,from/273and/*(3 we get and 9$ for Mooney–Rivlin and Mgden materials, respectively. Dithin our framework, the solution may be ob- tained by integrating the incremental equations, using a forward 4uler scheme. 1ince the basic step of the method is a linear increment superimposed upon a homogeneous conﬁguration domain integrals or volume discretizations are completelyavoided) It is worth mentioningthat in the special case of homogeneous de- formations, the domain integrals can be brought to the boundary even within the usual framework, see for instance 72$8, with reference to elastoplasticity. In that case, however, speciﬁc manipulations of the domain integrals are requested to transform these to boundary integrals, while in our method domain integrals are simply absent. Results are presented in Fig. 2, where for the Mooney–Rivlin material we have taken *5:$2 MPa, whereas for the Mgden material we have referred to the values listed in /*23. -he analytical solution is compared to the results given by the numerical procedure with a uniform mesh of $6 boundary elements. -able $ Analytic expression for the weakly singular integrals /993 at the right hand side of 4q. /793 ig Integral ig weak $$ 2log $$ 2log $$ $2 2$ 22 2log 22 2log 22 $$ lp 22 lp M) Brun et al) / ,omput) Met-ods .ppl) Mec-) Engrg) 1/( 0(0031 (421–(47/ 2:75

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.aussian quadrature formulae have been employed with $2 integration points for .reen s functions /6:3 /7$3 and $9 points for integrals in the discretized equation /793. Deareinapositionnowtospeculateonamoregeneralprocedureallowingforincrementssuperimposed upon a generic, non-homogeneous deformation. -his would necessarily lead to some volume discretization to describe the material inhomogeneity, but this should be regarded as diﬀerent––in essence––from the usual domain discretization /see for instance 7*$83. -he demonstration is provided by the above example, showingthat our method does not need any domain discretization when the material properties are homo- geneous. 7. Conclusions A boundary element technique has been proposed for plane strain and incompressible, incremental deformations of a broad class of materials /Cauchy-elastic, hyper- and hypoelastic, and &ewtonian ﬂuids3. Diﬃcultiesinherenttothetreatmentoftheincompressibilityconstraintaresimplyavoided bythepresented method. Dhen used as the basis to analyze non-linear deformations, our approach yields a formulation avoidingdomain integrals. It also allows for the analysis of diﬀerent bifurcation situations, that will be investigated in Part II of this study. Ac0nowled.ements Financial support of the Nniversity of -rento /D.B. and M.B.3 and of the Nniversity of Ferrara /D.C.3 is gratefully acknowledged. Appendix A. Biot s expression ofincremental moduli 1252 -o obtain the expression /253 of the incremental modulus , let us begin by considering the spectral representations of the left Cauchy–.reen strain tensor , and of the Cauchy stress Fig. 2. Nniaxial deformation of Mooney–Rivlin and Mgden elastic blocks. Cauchy stress versus stretch 2:76 M) Brun et al) / ,omput) Met-ods .ppl) Mec-) Engrg) 1/( 0(0031 (421–(47/

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where denote the 4ulerian principal axes. -he time derivative of /A.$3 is -akinginto account that for incremental plane strain deformations is and that , the in-plane out-of-diagonal components result to be $2 $2 Mn the other hand, 4qs. /93 and /$$3 give $2 $2 $2 $2 $2 $2 Considering4qs. /A.:3 and /A.*3 yields $2 $2 $2 $2 from which, eliminatingthe term $2 ,weget $2 $2 A comparison between 4q. /223 and 4q. /A.63 gives the Biot s expression of , 4q. /253 -he incremental moduli can be obtained by takingthe time derivative of /73 written for the in-plane components and takinginto account that ( for plane strain incremental deformations If instead of the potential the potential is used, the same relation 4q. /A.73 is obtained, except that replaces Considering 4qs. /93, /A.23, and /$$3 for the diagonal components, we get ii 22 $$ $$ 22 which, used in 4q. /A.73, yield 22 $$ $$ 22 A comparison between 4q. /223 and 4q. /A.93 gives the Biot s expression of , 4q. /253 M) Brun et al) / ,omput) Met-ods .ppl) Mec-) Engrg) 1/( 0(0031 (421–(47/ 2:77

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Appendix B. The out-of-plane stress increment for the 4.den hyperelastic material -he only non-zero component of the Jaumann derivative of the Cauchy stress in the out-of-plane di- rection is ** "# where in which, due to incompressibility 5eferences 7$8 D. Bigoni, D. Capuani, .reen s function for incremental nonlinear elasticity: shear bands and boundary integral formulation, J. Mech. Phys. 1olids 5( /2((23 :7$–5((. 728 M.A. Biot, Mechanics of Incremental Deformations, J. Diley and 1ons, &ew Oork, $965. 7*8 M. Bonnet, Boundary Integral 4quation Methods for 1olids and Fluids, Diley P 1ons, &ew Oork, $995. 7:8 M. Bonnet, 1. MukherKee, Implicit B4M formulation for usual and sensitivity problems in elasto-plasticity usingthe consistent tangent operator concept, Int. J. 1olids 1truct. ** /$9963 ::6$–::9(. 758 F.D. Bui, 1ome remarks about the formulation of three-dimensional thermoelastoplastic problems by integral equations, Int. J. 1olids 1truct. $: /$9793 9*5–9*9. 768 A. Chandra, 1. MukherKee, An analysis of large strain viscoplasticity problems including the eﬀects of induced material anisotropy, J. Appl. Mech. 5* /$9963 77–79. 778 A. Chandra, 1. MukherKee, A boundary element formulation for large strain problems of compressible plasticity, 4ngrg. Anal. *2 /$9963 7$–79. 798 Q.R. Chen, S. Ji, A new approach to ﬁnite deformation problems of elastoplasticity boundary element analysis method, Comput. Methods Appl. Mech. 4ngrg. 79 /$99(3 $–$9. 798 P.D Ambrosio,D.De-ommasi,1.Marzano,&onlinearelasticdeformationsandstabilityoflaminatedrubberbearings,A1C4J. 4ngrg. Mech. $2$ /$9953 $(:$–$(:9. 7$(8 F. Demiray, An evolution of pulse speed in arteries, Bull. Math. Biol. 59 /$9963 $29–$:(. 7$$8 M. 4lwenspoek, R. Diegerink, Mechanical Microsensors, 1pringer-5erlag, Berlin, 2(($. 7$28 A.Foerster,.. Iuhn,Aﬁeldboundaryelementformulationformaterialnonlinearproblemsatﬁnitestrains,Int. J.1olids1truct. *$ /$99:3 $777–$792. 7$*8 R. Fill, J.D. Futchinson, Bifurcation phenomena in the plane tension test, J. Mech. Phys. 1olids 2* /$9753 2*9–26:. 7$:8 ..A. Folzapfel, -.C. .asser, R.D. Mgden, A new constitutive framework for arterial wall mechanics and a comparative study of material models, J. 4last. 6$ /2(((3 $–:9. 7$58 J.D. Futchinson, I.D. &eale, Finite strain -deformation theory, in: D.4. Carlson, R.-. 1hield /4ds.3, Proc. IN-AM 1ymp. on Finite 4lasticity, Martinus &iKhoﬀ, -he Fague, Boston, London, $979, pp. 2*7–2:7. 2:79 M) Brun et al) / ,omput) Met-ods .ppl) Mec-) Engrg) 1/( 0(0031 (421–(47/

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7$68 J.D. Futchinson, 5. -vergaard, 1hear band formation in plane strain, Int. J. 1olids 1truct. $7 /$99$3 :5$–:7(. 7$78 F. Jin, I. Runesson, I. Mattiasson, Boundary element formulation in ﬁnite deformation plasticity usingimplicit integration, Comput. 1truct. *$ /$9993 25–*:. 7$98 J.M. Ielly, 4arthquake-Resistant Design with Rubber, 1pringer-5erlag, Berlin, $997. 7$98 M.A. Ladyzhenskaya, -he Mathematical -heory of 5iscous Incompressible Flow, .ordon P Breach, &ew Oork, $96*. 72(8 .. Maier, Mn elastoplastic analysis by boundary elements, Mech. Res. Commun. $( /$99*3 :5–52. 72$8 .. Maier, 1. Miccoli, .. &ovati, 1. 1irtori, A galerkin symmetric boundary-element method in plasticity formulation and implementation, in: J.F. Iane, .. Maier, &. -osaka, 1.&. Atluri /4ds.3, Advances in Boundary 4lement -echniques, 1pringer- 5erlag, $992, pp. 299–*29. 7228 M.A. Mooney, A theory of large elastic deformations, J. Appl. Phys. $$ /$9:(3 592–592. 72*8 I.D. &eale, Phenomenological constitutive laws in ﬁnite plasticity, 1M Arch. 6 /$99$3 79–$29. 72:8 .. &ovati, C. Brebbia, Boundary element formulation for for geometrically nonlinear elastostatic, Appl. Math. Model. 6 /$9923 $*6–$*9. 7258 R.D. Mgden, Large deformation isotropic elasticity: on the correlation of theory and experiment for incompressible rubberlike soilds, Proc. R. 1oc. Lond. *26 /$9723 565–59*. 7268 R.D. Mgden, 4lastic deformations of rubberlike solids, in: F... Fopkins, M.J. 1ewell /4ds.3, Mechanics of 1olids, -he Rodney Fill 6(th Anniversary 5olume, Pergamon Press, $992, pp. :99–5*7. 7278 R.D. Mgden, &on-Linear 4lastic Deformations, 4llis Forwood, Chichester, $99:. 7298 F. Mkada, F. RaKiyah, 1.&. Atluri, 1ome recent developments in ﬁnite-strain elastoplasticity usingthe ﬁeld-boundary element method, Comput. 1truct. *( /$9993 275–299. 7298 F. Mkada, F. RaKiyah, 1.&. Atluri, &on-hyper-singular integral-representations for velocity /displacement3 gradients in elasticA plastic solids /small or ﬁnite deformations3, Comput. Methods Appl. Mech. 4ngrg. : /$9993 $65–$75. 7*(8 F. Mkada, F. RaKiyah, 1.&. Atluri, A full tangent stiﬀness ﬁeld-boundary element formulation for geometric and material nonlinear problems of solid mechanics, Int. J. &umer. Methods 4ngrg. 29 /$99(3 $5–*5. 7*$8 C. Polizzotto, A symmetric .alerkin boundaryAdomain element method for ﬁnite elastic deformations, Comput. Methods Appl. Mech. 4ngrg. $99 /2(((3 :9$–5$:. 7*28 4. Radi, D. Bigoni, D. Capuani, 4ﬀects of pre-stress on crack-tip ﬁelds in elastic, incompressible solids, Int. J. 1olids 1truct. *9 /2((23 *97$–*996. 7**8 1. 1t oren, J.R. Rice, Localized neckingin thin sheets, J. Mech. Phys. 1olids 2* /$9753 :2$–::$. 7*:8 J.C.F. -elles, -he Boundary 4lement Method Applied to Inelastic Problems, 1pringer, Berlin, $99*. 7*58 -. -ran-Cong, R. Qheng, &. Phan--hien, Boundary element method for ﬁnite elasticity, Comput. Mech. 6 /$99(3 2(5–2$9. 7*68 L.R... -reloar, 1tress–strain data for vulcanized rubber under various types of deformation, -rans. Faraday 1oc. :( /$9::3 59 7(. 7*78 L.R... -reloar, -he Physics of Rubber 4lasticity, third ed., Mxford Nniversity Press, $975. 7*98 &.-riantafyllidis,F.I.Lehner,Interfacialinstabilityofdensity-stratiﬁed2-layersystemsunderinitialstress,J.Mech.Phys.1olids :$ /$99*3 $$7–$:2. 7*98 &. -riantafyllidis, O.M. Leroy, 1tability of a frictional material layer restingon a viscous half-space, J. Mech. Phys. 1olids :2 /$99:3 5$–$$(. 7:(8 C. -ruesdell, D. &oll, -he non-linear ﬁeld theories of mechanics, in: 1. Fl ugge /4d.3, 4ncyclopedia of Physics, IIIA*, 1pringer- 5erlag, Berlin, $965. 7:$8 D.1. 5enturini, Boundary 4lement Method in .eomechanics, 1pringer, Berlin, $99*. M) Brun et al) / ,omput) Met-ods .ppl) Mec-) Engrg) 1/( 0(0031 (421–(47/ 2:79

Brun D Capuani D Bigoni a Dipartimento di Ingegneria Meccanica e Strutturale Universit di Trento Via Mesiano 77 I38050 Povo Trento Italy Dipartimento di Ingegneria Universit di Ferrara Via Saragat 1 44100 Ferrara Italy Received 9 ovember 22 accept ID: 23848

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A boundary element technique for incremental, non-linear elasticity Part I: Formulation M. Brun , D. Capuani , D. Bigoni a, Dipartimento di Ingegneria Meccanica e Strutturale, Universit di Trento, Via Mesiano 77, I-38050 Povo, Trento, Italy Dipartimento di Ingegneria, Universit di Ferrara, Via Saragat 1, 44100 Ferrara, Italy Received $9 &ovember 2((2) accepted $$ February 2((* Abstract Incremental elastic deformations superimposed upon a given homogeneous strain are analyzed with a boundary element technique. -his is based on a recently-developed .reen s function for non-linear incremental elastic defor- mations. Plane strain perturbations are considered of a broad class of incompressible material behaviours /including hyper-, hypoelastic and &avier–1tokes constitutive equations3 within the elliptic range. &umerical treatment of the problem is detailed. A possibility of employingthe method in the fully non-linear range is outlined, which yields a boundary element approach where the use of domain integrals is avoided, atleast in a conventional sense. -hemethods for bifurcation and shear band analyses will be reported in Part II. 2((* 4lsevier 1cience B.5. All rights reserved. 1. Introduction -he analysis of the response to perturbations of a pre-stressed, non-linear elastic solid is important in a broad range of technological circumstances. For instance, pre-stress aﬀects the design of Microelectro- mechanical 1ystems 7$$8, it is a concern in the behaviour of geological formations 7*9,*98, biological tissues 7$(,$:8, and various structural elements, includingseismic insulators and rubber bearings 79,$98. Referringto plane strain deformations of incompressible materials, Biot 728 has shown that the incre- mental elastic response is governed by two incremental moduli, functions of the current stretch. Biot constitutive framework was assumed by Bigoni and Capuani 7$8 to obtain a .reen s function and a boundary integral formulation for incremental deformations superimposed upon a given, homogeneous strain. Both .reen s function and integral formulation provide the basis to build a boundary element Correspondingauthor. -el.: ;*9-(:6$-992-5(7) fax: ;*9-(:6$-992-599. E-mail addresses: michele.brun@ing.unitn.it /M. Brun3, cpd@dns.unife.it /D. Capuani3, bigoni@ing.unitn.it /D. Bigoni3. URL: http:AAwww.ing.unitn.itA~bigoniA. ((:5-7925A(*AC - see front matter 2((* 4lsevier 1cience B.5. All rights reserved. doi:$(.$($6A1((:5-7925/(*3((269-9 Comput. Methods Appl. Mech. 4ngrg. $92 /2((*3 2:6$–2:79 www.elsevier.comAlocateAcma

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technique for solvingincremental problems of non-linear elasticity. -his is the purpose of the article. In particular, we formulate a general numerical scheme to handle generic boundary value problems with prescribed nominal tractions andAor displacements. Dhen restricted to perturbations of homogeneously deformed,incompressiblesolids,ourboundaryelementstechniqueretains allwell-knownadvantagesofthe small strain formulation. -hese are: discretization only of the boundary of the body) automatic satisfaction of the incompressibility constraint) possibility of describingsingularities arisingnear corner points of the boundary) possibility of employing meshes thoroughly varying in size throughout the body. 1everal attempts can be found in the literature to analyze non-linear problems usingboundary elements techniques. In some cases the non-linearities were related to the material 7:,5,2(,*:,:$8, in other cases to large elastic 72:,*$,*58 or elastoplastic 76–9,$2,$7,29–*(8 strains. In all cases, in addition to the usual boundary integrals, a domain integral is introduced, leading to the so-called ﬁeld-boundary element method . -he introduction of this term nulliﬁes a main advantage of B4M and originates from the dis- crepancy between the non-linear character of the equations governing the problem and the employed fundamental solution /usually referringto linear, isotropic elasticity3. In the present paper, the focus is on incrementally-linear problems, so that domain integrals do not appear in the formulation that will be presented. Fowever, we believe that the solution of incrementally linear problems should be regarded as the ﬁrst step toward the analysis of fully non-linear situations and, in particular, we anticipate with a simple example that when employed as a tool to analyze large /thus non-linear3 deformations, our incre- mental method naturally leads to a volume discretization diﬀerent––in essence––from all those already known. -he paper is organized as follows: the incremental constitutive framework––which includes hyper- and hypoelasticity and the equations governing 1tokes ﬂow of ﬂuids––is presented in 1ection 2 for plane strain, whereas the fundamental solution and boundary integral equations are summarized in 1ection *. -he boundary element formulation is presented in 1ection : and the discretization detailed in subsequent 1ection 5. A numerical example is given in 1ection 6, which oﬀers the simplest context to illustrate the capability of the method to computationally follow a non-linear path of deformation. 1ystematic investi- gations of bifurcation and shear band phenomena in two-dimensional elastic materials will be reported in Part II. 2. Incremental constitutive equations -he general expression given by Biot 728 for rate constitutive equations of an incompressible material incrementally deformed in plane strain are obtained here with reference to a broad class of material be- haviours. ()1) Elasticity -he relation between the Cauchy stress and the left Cauchy–.reen strain tensor FF , with denotingthe deformation gradient, for a Cauchy-elastic, incompressible and isotropic solid can be written as 7:(8 where is a parameter connected to the hydrostatic pressure tr * through 2:62 M) Brun et al) / ,omput) Met-ods .ppl) Mec-) Engrg) 1/( 0(0031 (421–(47/

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tr tr and and are generic functions of two invariants of -he two particular cases of Mooney–Rivlin and neo-Fookean materials are recovered when and are taken constant and, in addition, when ( in the latter case. Constitutive equation /$3 corresponds to Cauchy elasticity, so that it describes a class of behaviours broader than hyperelasticity. -he requirement of existence of an elastic potential inﬂuences the dependence of coeﬃcients and on the invariants of . -o develop this point, let us consider that the constitutive equation /$3 implies coaxiality of tensors and , so that these share /at least3 one principal reference system––the 4ulerian principal axes––where diag diag in which (, $, 2, * are the principal stretches, satisfyingthe incompressibility constraint 4xpressing4q. /$3 in the 4ulerian principal reference system and solvingfor and yields two equations that can be alternatively expressed employingevery permutation of$, 2 and * as indices.It is clear from the expression /63 that existence of an elastic potential restricts the functional dependence of coeﬃcients on the stretch. In particular, the standard deﬁnition of elastic potential for incompressible materials is 7278 not summed which can be immediately employed into 4q. /63. -akingthe material derivative of /$3 yields where DB BD WB BW and DB WB $( in which is the 4ulerian strain rate and the spin tensor. Ieepinginto account now 4q. /$3 and the deﬁnition of Jaumann derivative $$ M) Brun et al) / ,omput) Met-ods .ppl) Mec-) Engrg) 1/( 0(0031 (421–(47/ 2:6*

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the constitutive equation /$3 becomes DB BD DB $2 &otingthat tr $* where , with $ and tr $: or $5 where with $ are the coeﬃcients and expressed as functions of the principal stretches, we arrive at DB BD DB $6 which is equivalent to DB BD DB $7 -heincrementalconstitutiveequationintheform /$63or/$73isvalidforthree-dimensional,incompressible Cauchy elasticity. De are interested here in the particularization of /$63 to incremental plane strain de4ormations super- imposed on a generic state of -omogeneous deformation. In the 4ulerian principal reference system diag ! $9 so that the out-of-plane stress rate components can be determined as $9 and ** $$ 22 2( or ** !# $$ 22 2$ 2:6: M) Brun et al) / ,omput) Met-ods .ppl) Mec-) Engrg) 1/( 0(0031 (421–(47/

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Finally, the in-plane rate components can be expressed in the Biot 728 form as $2 $2 $$ 22 $$ 22 $$ 22 22 where and aretwoincrementalmodulicorrespondingrespectivelytoshearingparallelto,andat:5 to, the 4ulerian principal axes. -hese can be expressed as functions of the invariants of 2* or as functions of the principal stretches !# 2: An alternative expression for the two incremental moduli and , related to the existence of a strain- energy function, was given by Biot 728 /see Appendix A3 in the form 25 As it will become clear later, constitutive relations of the form /223 with generic coeﬃcients and embrace a much broader class of material behaviours than Cauchy, isotropic elasticity. ()1)1) Mooney–Rivlin material A simple explicit form of strain-energy functions for isotropic rubber-like elastic media was proposed by Mooney 7228 in the form 26 or 27 M) Brun et al) / ,omput) Met-ods .ppl) Mec-) Engrg) 1/( 0(0031 (421–(47/ 2:65

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where and are material parameters and represents the shear modulus in the original unstressed state. In this case, with reference to the representation /$3, we simply obtain: 29 and ** 29 ()1)() 5gden material -he followingclass of strain-energy functions was proposed by 7258 to ﬁt experimental results on rubber: *( where and are material parameters, subKected to the constraints: with ... *$ in which, is a positive integer determining the number of terms in the strain-energy function, are constant shear moduli and are dimensionless parameters, with ... . In particular, Mooney Rivlin material can be recovered as a particular case taking 2, 2 and 2. An excellent correlation with experimental data relative to simple-tension, equibiaxial tension and pure shear of vulcanised rubber is obtained employingthe values 725,*6,*78: $( ($2 $( $( *2 yielding 225 $( &Am Coeﬃcients in the representations /$63 and /223 can be obtained from 4q. /*(3 in the form: ** and *: -he component ** is given in Appendix B. 2:66 M) Brun et al) / ,omput) Met-ods .ppl) Mec-) Engrg) 1/( 0(0031 (421–(47/

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()() 6ypoelasticity and t-e loading 7ranc- o4 elastoplastic constitutive la8s De consider a general incremental constitutive equation relating the Jaumann derivative of the Cauchy stress tensor to the 4ulerian strain rate through a generic tensor 1ym in the form DT TD DT *5 where coeﬃcients ... 9 are polynomial functions of the invariants of . In the particular case in which is identiﬁed with the Cauchy stress , the constitutive equation /*53 describes an incompressible, -ypoelastic material 7:(8. Fowever, even if does not represent the Cauchy stress and the coeﬃcients ... 9, remain completely unspeciﬁed /but independent of 3, in a principal reference system of and for plane, incremental deformation, we get ** $$ *6 where with * denote the principal values of and $2 2$ and $$ 22 can be expressed in the form /223, with *7 ()()1) -de4ormation t-eory o4 plasticity: -yperelastic and -ypoelastic approac-es -deformation theory of plasticity was proposed by Futchinson and &eale 7$58 /see also 7$6,2*83 in the framework of hyperelasticity, to model metals subKect to proportional loading. Dith reference to the representation /$3, the constitutive-law in the -deformation theory of plasticity can be expressed as ; *9 where log arethelogarithmicstrainsand tr *.Inthesameequation isthesecantmodulusto the curve representingthe eﬀective stress versus eﬀective strain *9 where are the principal components of deviatoric stress. -he curve is assumed to be determined by :( where 2 isanhardeningexponent, isapositiveconstitutiveparameter.-hestrain-energyfunction results therefore to be :$ M) Brun et al) / ,omput) Met-ods .ppl) Mec-) Engrg) 1/( 0(0031 (421–(47/ 2:67

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-he above framework corresponds to the followingchoice of parameters and /see /$3 and /*933: :2 and to the incremental shear moduli: coth :* -he out-of-plane stress increment is given by ** :: Diﬀerently from Futchinson and &eale 7$58, 1t oren and Rice 7**8 present the followinghypoelastic law /their 4q. /2633 to model elastoplastic materials subKect to proportional loading: :5 where tr *, isthedeviatoric stress, isawork-hardeningexponentand isthesecantmodulus on the shear stress–strain curve. 4q. /:53 can be written as :6 where tr *, which is, evidently, a particular case of /*53 and can therefore be cast in the form /223. ()()() T-e loading 7ranc- o4 non-associative, elastoplastic la8 A generic constitutive equation for an incompressible isotropic-elastic, plastic material depending on a generic collection of state variables can be written in the form if if :7 where is the yield function, 1ym is the yield function gradient and 1ym the plastic potential gradient. -he Macaulay brackets hi apply to every scalar in such a way that i j j 2 and in- troduce the incremental non-linearity, typical of plasticity. -he scalar ( is the plastic modulus, related to the hardeningmodulus through with :9 Finally, tensors and are subKect to the incompressibility constraints tr tr tr :9 so that tr *. Restrictingthe constitutive equation /:73 to its loadingbranch is synonymous of assuming and eliminatingthe incremental non-linearity, thus obtaining 2:69 M) Brun et al) / ,omput) Met-ods .ppl) Mec-) Engrg) 1/( 0(0031 (421–(47/

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5( Let us assume now that and are coaxial /and not necessarily coaxial with the Cauchy stress3. In the principal reference system of and and for plane incremental deformations, 4q. /5(3 can be cast in the form /223 with 5$ -he constitutive fourth-order tensor in /5(3 contains the non-symmetric term , where corresponds to the so-called non-associative elastoplasticity. Fowever, due to the incompressibility and plane strain constraints and without alteringthe material response, we can add a term 52 where is a symmetric tensor coaxial to and and is the unit vector singling the out-of-plane di- rection. &ow, the choice diag f 5* taken in the principal reference system of and , symmetrizes the constitutive operator for every so that non-associativity does not yield an unsymmetric tangent constitutive operator for incompressible, isotropic-elastic, plane-strain plasticity with coaxial yield function and plastic potential gradients. ()3) 9e8tonian ﬂuids -he constitutive equation describinga &ewtonian, incompressible ﬂuid takes the form 5: where the Cauchy stress is related to the 4ulerian strain rate through the viscous coeﬃcient and the pressure in the ﬂuid tr *. Clearly 4q. /5:3 has a structure identical to a very particular case of /*53, so that, for plane strain ** $$ 22 $$ 22 $$ 22 $2 $2 55 a form akin to /223. -herefore, keepinginto account that we will address the problem of quasi-static de- formations, our framework can be immediatelyemployed to describe mutatis mutandis ––two-dimensional 1tokes ﬂow. As a consequence, several results presented in the followingreduce to ﬁndings by Ladyz- henskaya 7$98 as particular cases. ()4) T-e general 4orm o4 constitutive equations 4or plane, incompressi7le, incremental de4ormations De have shown in the section above that a very broad class of incremental material behaviours––in- cludingCauchy elasticity, hyperelasticity, hypoelasticity, the loadingbranch of coaxial elastoplasticity, and &ewtonian ﬂuids––can be described by constitutive equation /223. It is expedient now to transform the constitutive equation /223 in terms of material derivative of the nominal stress tensor . In particular, a Lagrangean formulation of ﬁeld equations is employed in the following, with the current state taken as reference. As a consequence, constitutive equations /223 become ij ijkl ij ijkl pd ij 56 M) Brun et al) / ,omput) Met-ods .ppl) Mec-) Engrg) 1/( 0(0031 (421–(47/ 2:69

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where is the velocity, ij the Ironecker delta and $$ 22 57 Both and measure the in plane hydrostatic stress rate /positive in tension3 as related respectively to Cauchy and to nominal stresses. -ensor ijkl represents the instantaneous moduli, possessingthe maKor symmetry ijkl klij and havingthe form 7$*8 $$$$ $$22 $$$2 $$2$ 22$$ 2222 22$2 222$ $2$2 $22$ 2$$2 2$2$ 59 with 59 &ote that in 4q. /563 ijkl diﬀers from ijkl only in the followingcomponents: $$$$ 2222 6( so that ijkl also possesses the maKor symmetry. Mnly the forms /593 and /6(3 of the constitutive equation /223 will be needed in the following. -he formulation in the present paper is restricted to the elliptic range /43, where the characteristic equation associated to the equilibrium equations governing perturbations superimposed upon a given homogeneous strain does not admit real solutions. In particular, the elliptic range corresponds to negative or complex coeﬃcients and , functions of the material properties and state of pre-stress as follows: 6$ where 62 4mploying4q. /253 , parameter appearingin 4qs. /6$3 and /623 can be written as 6* and it represents a normalized measure of the pre-stress or––more precisely––a measure of the current state of stretch. -he elliptic range may be further sub-divided into elliptic-imaginary /4I3 and elliptic-complex /4C3 regimes. -hese are deﬁned as follows: (, so that and are both negative in /4I3, (, so that and are a conKugate pair in /4C3. It may be important to remark that shear bands, correspondingto the appearance of discontinuous strain rates, are formally excluded within the elliptic range, i.e. in the context analyzed here. Fowever, as shown 2:7( M) Brun et al) / ,omput) Met-ods .ppl) Mec-) Engrg) 1/( 0(0031 (421–(47/

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by Bigoni and Capuani 7$8 and Radi et al. 7*28, formation of zones of concentrated strains in response to a perturbation becomes possible when the boundary of the elliptic regime is approached. -his will be also demonstrated with numerical examples referringto van Fove conditions in Part II. -he boundary of the elliptic range is characterized by the following conditions: elliptic imaginaryAparabolic /4IAP3 boundary is attained when $, correspondingto () elliptic complexAhyperbolic /4CAF3 boundary is attained when (, correspondingto 3. The fundamental solution Dith reference to the constitutive equation /563, the .reen s function set for an inﬁnite and uniformly pre-strained medium can be written in the form 7$8 pd ig log log cos cos cos 6: where and are the polar coordinates singling out the generic point with respect to the position of the concentrated force and sin hi sin hi sin cot cot 65 cos 2cos 66 &ote that in expression /6:3 of we have introduced the usual symbol to denote the Cauchy principal value integral. -he gradient of the .reen s velocity may be obtained directly from /6:3 and can be written as cos hi sin Ra log cos sin Ra log cos 67 where index is not summed and sin 2cos aK sin aK 69 M) Brun et al) / ,omput) Met-ods .ppl) Mec-) Engrg) 1/( 0(0031 (421–(47/ 2:7$

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-he four components of the velocity gradient not included in 4q. /673 can be obtained using incom- pressibility and symmetry of .reen s tensor: 69 Accordingto constitutive equations /563–/593, from .reen s velocity gradient and pressure rate we obtain the in-plane hydrostatic stress rate 7( and the associated incremental nominal stress $$ $2 2$ 22 7$ It is important to note that the .reen s velocity can be given the form ig log ig 72 where the dependence on the current state is condensed in the coeﬃcients ig and ig . -he former coeﬃcient satisﬁes ig (if , while the latter contains the directional dependence on . It follows from repre- sentation /723 that the gradient of the .reen s velocity, the in-plane hydrostatic stress rate and thus the incremental stress rate can be expressed as explicit functions of $ 4. Boundary element formulation De restrict the presentation to mixed boundary value problems in which velocities and incremental nominal tractions are prescribed functions deﬁned on separate portions and of the boundary on ij on 7* of a solid , currently in a state of homogeneous, ﬁnite deformation. In this context, two integral repre- sentations exist relatingthe velocity and the pressure rate in interior points of the body to the boundary values of nominal traction rates ij and velocities 7$8. -hese are 7: and ijkg ll lr $$ rl $$ 75 If the point is on the boundary, 4q. /7:3 becomes 7$8 76 2:72 M) Brun et al) / ,omput) Met-ods .ppl) Mec-) Engrg) 1/( 0(0031 (421–(47/

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where lim 77 is the so-called -matrix 7*8, dependingon the material parameters, the state of pre-stress and the geometry oftheboundary/inthecaseofasmoothboundary, gi 3.&otethatsymbol introducedin/773 denotes the intersection between a circle of radius centred at and the domain 5. Boundary discretization -he boundary equation /763 is the startingpoint to derive the collocation 7oundaryelement met-od .-o thispurpose,theboundary isdividedinto elements ... ,withsubsets and belonging respectively to and /clearly 3. For simplicity the same discretization is assumed for velocity and traction rates at the boundary. In particular, inside each boundary element we use ... 79 where are the nodal values of velocities and nominal traction rates, respectively, and are the relevant shape functions, selected as polynomials of degree -he discretized form of 4q. /763, collocating the point at , correspondingto the node of the element ,is 79 where indices and are summed and range between ( and and $ and 2, respectively. An analysis of 4q. /793 reveals that the number of unknowns is unk , being prescribed on and on Collocatingnow 4q. /793 at nodes alongthe two directions and , yields the followingalgebraic system: 9( where and are the vectors collecting and 9$ 1olution of system /9(3 gives the nodal velocities on and the nominal traction rates on Mnce system /9(3 has been solved, ﬁelds and can be evaluated at internal points by applying the discretized forms of 4qs. /7:3 and /753. In particular, 4q. /7:3 reduces to 4q. /793 with gi and the integration is straightforward, as is always diﬀerent from zero. De limit the presentation to discretization of the boundary into rectilinear elements and linear shape functions, so that $ and unk -he singular integrals in /793, computed over the elements adKacent to the node in which the equation is collocated, are evaluated analytically. In particular, the strongly singular integral ig strong in the left hand side of 4q. /793 is equal to ig strong 92 M) Brun et al) / ,omput) Met-ods .ppl) Mec-) Engrg) 1/( 0(0031 (421–(47/ 2:7*

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which, with reference to the geometry sketched in Fig. $, can be transformed into ig strong 00 00 9* so that the change of variable 00 9: yields ig strong 95 As far as the elements $and are concerned, the incremental .reen s tractions can be computed as ig 96 which are independent of , so that ig can be calculated as . Introducing4q. /963 into 4q. /953, we obtain an explicit formula for the singular integrals /923 in the form ig strong ig log 97 In all our examples we have always found numerically that $$ 22 (, a result that can be analytically checked in the special case of &ewtonian ﬂuid, i.e. when ( and , but still requires a proof in the general case. -he weakly singular integrals ig weak in the right hand side of 4q. /793 is equal to ig weak 99 which becomes /Fig. $3 ig weak 99 Fig. $. .eometry at node 2:7: M) Brun et al) / ,omput) Met-ods .ppl) Mec-) Engrg) 1/( 0(0031 (421–(47/

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-aking4q. /723 into account, the integral /993 can be analytically evaluated and the results are listed in -able $. De note, in passing, that the use of higher-order polynomials as shape functions is straightforward, since the singularities in the above integrals arise from the constant part of the interpolating functions. ). A numerical example: non-linear elasticity without domain inte.rals An elastic block subKect to homogeneous, increasing deformation is considered, with the purpose of il- lustratingwith a simple example that our formulation does not involve domain integrals. In particular, the elastic block is constrained to plane deformations startingfrom an unstressed square conﬁguration and is subKected to tension or compression in one direction. De refer to Mooney–Rivlin /263 and Mgden /*(3 and /*$3 non-linear elastic laws. -he trivial, analytical solution to this problem is reported by Mgden 7268 9( sothatforuniaxialstress/ (3andplanestraindeformations/ $3,from/273and/*(3 we get and 9$ for Mooney–Rivlin and Mgden materials, respectively. Dithin our framework, the solution may be ob- tained by integrating the incremental equations, using a forward 4uler scheme. 1ince the basic step of the method is a linear increment superimposed upon a homogeneous conﬁguration domain integrals or volume discretizations are completelyavoided) It is worth mentioningthat in the special case of homogeneous de- formations, the domain integrals can be brought to the boundary even within the usual framework, see for instance 72$8, with reference to elastoplasticity. In that case, however, speciﬁc manipulations of the domain integrals are requested to transform these to boundary integrals, while in our method domain integrals are simply absent. Results are presented in Fig. 2, where for the Mooney–Rivlin material we have taken *5:$2 MPa, whereas for the Mgden material we have referred to the values listed in /*23. -he analytical solution is compared to the results given by the numerical procedure with a uniform mesh of $6 boundary elements. -able $ Analytic expression for the weakly singular integrals /993 at the right hand side of 4q. /793 ig Integral ig weak $$ 2log $$ 2log $$ $2 2$ 22 2log 22 2log 22 $$ lp 22 lp M) Brun et al) / ,omput) Met-ods .ppl) Mec-) Engrg) 1/( 0(0031 (421–(47/ 2:75

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.aussian quadrature formulae have been employed with $2 integration points for .reen s functions /6:3 /7$3 and $9 points for integrals in the discretized equation /793. Deareinapositionnowtospeculateonamoregeneralprocedureallowingforincrementssuperimposed upon a generic, non-homogeneous deformation. -his would necessarily lead to some volume discretization to describe the material inhomogeneity, but this should be regarded as diﬀerent––in essence––from the usual domain discretization /see for instance 7*$83. -he demonstration is provided by the above example, showingthat our method does not need any domain discretization when the material properties are homo- geneous. 7. Conclusions A boundary element technique has been proposed for plane strain and incompressible, incremental deformations of a broad class of materials /Cauchy-elastic, hyper- and hypoelastic, and &ewtonian ﬂuids3. Diﬃcultiesinherenttothetreatmentoftheincompressibilityconstraintaresimplyavoided bythepresented method. Dhen used as the basis to analyze non-linear deformations, our approach yields a formulation avoidingdomain integrals. It also allows for the analysis of diﬀerent bifurcation situations, that will be investigated in Part II of this study. Ac0nowled.ements Financial support of the Nniversity of -rento /D.B. and M.B.3 and of the Nniversity of Ferrara /D.C.3 is gratefully acknowledged. Appendix A. Biot s expression ofincremental moduli 1252 -o obtain the expression /253 of the incremental modulus , let us begin by considering the spectral representations of the left Cauchy–.reen strain tensor , and of the Cauchy stress Fig. 2. Nniaxial deformation of Mooney–Rivlin and Mgden elastic blocks. Cauchy stress versus stretch 2:76 M) Brun et al) / ,omput) Met-ods .ppl) Mec-) Engrg) 1/( 0(0031 (421–(47/

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where denote the 4ulerian principal axes. -he time derivative of /A.$3 is -akinginto account that for incremental plane strain deformations is and that , the in-plane out-of-diagonal components result to be $2 $2 Mn the other hand, 4qs. /93 and /$$3 give $2 $2 $2 $2 $2 $2 Considering4qs. /A.:3 and /A.*3 yields $2 $2 $2 $2 from which, eliminatingthe term $2 ,weget $2 $2 A comparison between 4q. /223 and 4q. /A.63 gives the Biot s expression of , 4q. /253 -he incremental moduli can be obtained by takingthe time derivative of /73 written for the in-plane components and takinginto account that ( for plane strain incremental deformations If instead of the potential the potential is used, the same relation 4q. /A.73 is obtained, except that replaces Considering 4qs. /93, /A.23, and /$$3 for the diagonal components, we get ii 22 $$ $$ 22 which, used in 4q. /A.73, yield 22 $$ $$ 22 A comparison between 4q. /223 and 4q. /A.93 gives the Biot s expression of , 4q. /253 M) Brun et al) / ,omput) Met-ods .ppl) Mec-) Engrg) 1/( 0(0031 (421–(47/ 2:77

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Appendix B. The out-of-plane stress increment for the 4.den hyperelastic material -he only non-zero component of the Jaumann derivative of the Cauchy stress in the out-of-plane di- rection is ** "# where in which, due to incompressibility 5eferences 7$8 D. Bigoni, D. Capuani, .reen s function for incremental nonlinear elasticity: shear bands and boundary integral formulation, J. Mech. Phys. 1olids 5( /2((23 :7$–5((. 728 M.A. Biot, Mechanics of Incremental Deformations, J. Diley and 1ons, &ew Oork, $965. 7*8 M. Bonnet, Boundary Integral 4quation Methods for 1olids and Fluids, Diley P 1ons, &ew Oork, $995. 7:8 M. Bonnet, 1. MukherKee, Implicit B4M formulation for usual and sensitivity problems in elasto-plasticity usingthe consistent tangent operator concept, Int. J. 1olids 1truct. ** /$9963 ::6$–::9(. 758 F.D. Bui, 1ome remarks about the formulation of three-dimensional thermoelastoplastic problems by integral equations, Int. J. 1olids 1truct. $: /$9793 9*5–9*9. 768 A. Chandra, 1. MukherKee, An analysis of large strain viscoplasticity problems including the eﬀects of induced material anisotropy, J. Appl. Mech. 5* /$9963 77–79. 778 A. Chandra, 1. MukherKee, A boundary element formulation for large strain problems of compressible plasticity, 4ngrg. Anal. *2 /$9963 7$–79. 798 Q.R. Chen, S. Ji, A new approach to ﬁnite deformation problems of elastoplasticity boundary element analysis method, Comput. Methods Appl. Mech. 4ngrg. 79 /$99(3 $–$9. 798 P.D Ambrosio,D.De-ommasi,1.Marzano,&onlinearelasticdeformationsandstabilityoflaminatedrubberbearings,A1C4J. 4ngrg. Mech. $2$ /$9953 $(:$–$(:9. 7$(8 F. Demiray, An evolution of pulse speed in arteries, Bull. Math. Biol. 59 /$9963 $29–$:(. 7$$8 M. 4lwenspoek, R. Diegerink, Mechanical Microsensors, 1pringer-5erlag, Berlin, 2(($. 7$28 A.Foerster,.. Iuhn,Aﬁeldboundaryelementformulationformaterialnonlinearproblemsatﬁnitestrains,Int. J.1olids1truct. *$ /$99:3 $777–$792. 7$*8 R. Fill, J.D. Futchinson, Bifurcation phenomena in the plane tension test, J. Mech. Phys. 1olids 2* /$9753 2*9–26:. 7$:8 ..A. Folzapfel, -.C. .asser, R.D. Mgden, A new constitutive framework for arterial wall mechanics and a comparative study of material models, J. 4last. 6$ /2(((3 $–:9. 7$58 J.D. Futchinson, I.D. &eale, Finite strain -deformation theory, in: D.4. Carlson, R.-. 1hield /4ds.3, Proc. IN-AM 1ymp. on Finite 4lasticity, Martinus &iKhoﬀ, -he Fague, Boston, London, $979, pp. 2*7–2:7. 2:79 M) Brun et al) / ,omput) Met-ods .ppl) Mec-) Engrg) 1/( 0(0031 (421–(47/

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7$68 J.D. Futchinson, 5. -vergaard, 1hear band formation in plane strain, Int. J. 1olids 1truct. $7 /$99$3 :5$–:7(. 7$78 F. Jin, I. Runesson, I. Mattiasson, Boundary element formulation in ﬁnite deformation plasticity usingimplicit integration, Comput. 1truct. *$ /$9993 25–*:. 7$98 J.M. Ielly, 4arthquake-Resistant Design with Rubber, 1pringer-5erlag, Berlin, $997. 7$98 M.A. Ladyzhenskaya, -he Mathematical -heory of 5iscous Incompressible Flow, .ordon P Breach, &ew Oork, $96*. 72(8 .. Maier, Mn elastoplastic analysis by boundary elements, Mech. Res. Commun. $( /$99*3 :5–52. 72$8 .. Maier, 1. Miccoli, .. &ovati, 1. 1irtori, A galerkin symmetric boundary-element method in plasticity formulation and implementation, in: J.F. Iane, .. Maier, &. -osaka, 1.&. Atluri /4ds.3, Advances in Boundary 4lement -echniques, 1pringer- 5erlag, $992, pp. 299–*29. 7228 M.A. Mooney, A theory of large elastic deformations, J. Appl. Phys. $$ /$9:(3 592–592. 72*8 I.D. &eale, Phenomenological constitutive laws in ﬁnite plasticity, 1M Arch. 6 /$99$3 79–$29. 72:8 .. &ovati, C. Brebbia, Boundary element formulation for for geometrically nonlinear elastostatic, Appl. Math. Model. 6 /$9923 $*6–$*9. 7258 R.D. Mgden, Large deformation isotropic elasticity: on the correlation of theory and experiment for incompressible rubberlike soilds, Proc. R. 1oc. Lond. *26 /$9723 565–59*. 7268 R.D. Mgden, 4lastic deformations of rubberlike solids, in: F... Fopkins, M.J. 1ewell /4ds.3, Mechanics of 1olids, -he Rodney Fill 6(th Anniversary 5olume, Pergamon Press, $992, pp. :99–5*7. 7278 R.D. Mgden, &on-Linear 4lastic Deformations, 4llis Forwood, Chichester, $99:. 7298 F. Mkada, F. RaKiyah, 1.&. Atluri, 1ome recent developments in ﬁnite-strain elastoplasticity usingthe ﬁeld-boundary element method, Comput. 1truct. *( /$9993 275–299. 7298 F. Mkada, F. RaKiyah, 1.&. Atluri, &on-hyper-singular integral-representations for velocity /displacement3 gradients in elasticA plastic solids /small or ﬁnite deformations3, Comput. Methods Appl. Mech. 4ngrg. : /$9993 $65–$75. 7*(8 F. Mkada, F. RaKiyah, 1.&. Atluri, A full tangent stiﬀness ﬁeld-boundary element formulation for geometric and material nonlinear problems of solid mechanics, Int. J. &umer. Methods 4ngrg. 29 /$99(3 $5–*5. 7*$8 C. Polizzotto, A symmetric .alerkin boundaryAdomain element method for ﬁnite elastic deformations, Comput. Methods Appl. Mech. 4ngrg. $99 /2(((3 :9$–5$:. 7*28 4. Radi, D. Bigoni, D. Capuani, 4ﬀects of pre-stress on crack-tip ﬁelds in elastic, incompressible solids, Int. J. 1olids 1truct. *9 /2((23 *97$–*996. 7**8 1. 1t oren, J.R. Rice, Localized neckingin thin sheets, J. Mech. Phys. 1olids 2* /$9753 :2$–::$. 7*:8 J.C.F. -elles, -he Boundary 4lement Method Applied to Inelastic Problems, 1pringer, Berlin, $99*. 7*58 -. -ran-Cong, R. Qheng, &. Phan--hien, Boundary element method for ﬁnite elasticity, Comput. Mech. 6 /$99(3 2(5–2$9. 7*68 L.R... -reloar, 1tress–strain data for vulcanized rubber under various types of deformation, -rans. Faraday 1oc. :( /$9::3 59 7(. 7*78 L.R... -reloar, -he Physics of Rubber 4lasticity, third ed., Mxford Nniversity Press, $975. 7*98 &.-riantafyllidis,F.I.Lehner,Interfacialinstabilityofdensity-stratiﬁed2-layersystemsunderinitialstress,J.Mech.Phys.1olids :$ /$99*3 $$7–$:2. 7*98 &. -riantafyllidis, O.M. Leroy, 1tability of a frictional material layer restingon a viscous half-space, J. Mech. Phys. 1olids :2 /$99:3 5$–$$(. 7:(8 C. -ruesdell, D. &oll, -he non-linear ﬁeld theories of mechanics, in: 1. Fl ugge /4d.3, 4ncyclopedia of Physics, IIIA*, 1pringer- 5erlag, Berlin, $965. 7:$8 D.1. 5enturini, Boundary 4lement Method in .eomechanics, 1pringer, Berlin, $99*. M) Brun et al) / ,omput) Met-ods .ppl) Mec-) Engrg) 1/( 0(0031 (421–(47/ 2:79

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