DIRECT AND INVERSE PROBLEMS FOR THE EVERSION OF A SPHERICAL SHELL

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1 THE PUBLISHING HOUSE POCEEDINGS OF THE OMANIAN ACADEMY, Seies OF THE OMANIAN ACADEMY Volume, Numbe /, pp. 33 DIECT AND INVESE POBLEMS FO THE EVESION OF A SPHEICAL SHELL aluca I. OŞC Coina S. DAPACA Depatment of Engineeing Science and Mechanics, Pennsylvania State Univesity Coesponding autho: Coina S. DAPAC Eath-Engineeing Sciences Building, Univesity Pak, PA 68, US Fax: , csd@psu.edu Motivated by an inteest in mateial paamete detemination fo a stuctue with esidual stesses, we evisit the classical evesion poblem of a spheical shell, as studied by Eicksen, 955 and moe ecently Johnson and Hoge, 993. The spheical shell is made of an isotopic, incompessible, hypeelastic, Mooney-ivlin mateial. In the diect poblem, we examine the dependence of the elasticity tenso coefficients on esidual and initial stesses and epesent it fo thee numeical cases. In the invese poblem, we analyse the effect of measuement uncetainties on the detemined values of mateial paametes and we suggest an expeimental potocol that allows fo a obust ecovey of these paametes. Key wods: Invese poblems; Evesion; Mooney-ivlin; esidual stesses.. INTODUCTION Diagnostic adiology is an exciting and apidly expanding multi-disciplinay field of clinical medicine which links medicine to science and engineeing. It enables noninvasive imaging and investigation of stuctue and function of the human body, and a unique insight into disease pocesses in vivo. In paticula, computed tomogaphy (CT), magnetic esonance (M), o othe biomedical imaging modalities can be used fo inspecting the inne sufaces of hollow stuctues such as the colon and stomach. Howeve, taditional visualization techniques can pesent only a vey small potion of inne sufaces in one view, due to the limitations of the opeato s pespective and of the field of view with vitual endoscopy. Thus, navigation is typically equied in a visualization session, which is time-consuming, tedious, and eo-pone paticulaly when featues of inteest ae hidden [3, 5]. Among the many methods poposed in the liteatue to ovecome the above mentioned difficulties in visualizing anatomical cavities, the digital evesion of a hollow stuctue intoduced vey ecently by Zhao et al. [8] is the most pomising. As the name of this technique suggests, the tuning inside out of a stuctue is done via a compute. The pimay advantages of digital evesion ove conventional vitual endoscopy include the diect visualization of a lage potion of the inne suface and the close coelation to the impotant anatomical featues, without the need fo difficult and time-consuming navigation. In addition, the digital evesion method combined with an appopiate mechanical model of an anatomical stuctue could help in finding the mechanical paametes of the imaged stuctue which contain impotant infomation about its pathology since tumos ae hade than the suounding nomal tissue. Thus, digital evesion with the help of mechanics can help impove sceening, diagnosis, sugical planning and medical education. Given the elevance that the evesion poblem of a spheical shell might have to moden medicine, in the pesent pape we evisit this classic mechanical poblem and fomulate its coesponding invese. If fo a hollow stuctue (such as a sock o the uteus), evesion means simply tuning inside out, fo a complete spheical shell this is possible only peceded by a cut along a diamete and followed by a ejoining that keeps fomely neighbouing points still neighbouing. Fo a spheical shell made of an isotopic, incompessible, hypeelastic mateial, Eicksen [] had poven that the evesion defomation is compatible with the equilibium equations. Moeove, when the mateial of the shell is of Mooney-ivlin type, the eveted shell can be maintained in a spheical shape with zeo o non-zeo unifom nomal pessues p applied to the inne and oute shell sufaces.

2 3 aluca I. oşca, Coina S. Dapaca Even when we stat fom an unstessed configuation, in the eveted state the shell maintains nonvanishing stesses. As shown by Johnson and Hoge [], the elasticity tenso poduced by a small defomation fom the eveted configuation will depend on the stesses in the eveted configuation. In section we will epesent this dependence, extending the calculation fom the case of esidual stesses consideed in [] to the case of initial stesses, and show the impotant ole played by the non-lineaity of the mateial in the elationship between the elasticity tenso and the applied pessue p. We epesent this dependence fo thee values of the applied pessue p, as well as thee choices of the Mooney-ivlin mateial constants. In section 3 we fomulate the invese poblem of the eveted spheical shell, in which C and C, the mateial constants of the Mooney-ivlin mateial, ae computed fom measuements in the eveted configuation. In paticula, we analyse the impact that eos in measuements of the evesion constant A o the pessue p have in detemining C and C. To the best of ou knowledge, this is the fist time that such a study has been pefomed. Ou esults suggest novel expeimental potocols that will allow a obust ecovey of the mechanical paametes which ultimately will help diffeentiate between nomal and pathological tissues. The pape ends with a section of conclusions. The pesent wok is an expansion of the wok pesented by the authos in [6].. FOMULATION OF THE DIECT POBLEM As in Johnson and Hoge [], we conside a spheical shell made of a homogenous, isotopic, incompessible, hypeelastic, Mooney-ivlin mateial. We denote by and the intenal and extenal shell adius, with B the elaxed, stess fee configuation of the shell and with B the shell configuation afte an evesion. The evesion defomation maps the mateial coodinates, Θ, Φ in the spatial coodinates, θ, φ as follows: 3 3 = ( A ), θ = π Θ, φ = Φ, () whee π is the constant 3...,, Φ < π and Θ < π, and the constant A> 3. Though defomation () the sufaces initially at, ae tansfomed in the sufaces at, espectively, with < and >. In physical (spheical) coodinates, the gadient of defomation () [F ] has the fom: =, () [ F ] and the coesponding left Cauchy-Geen defomation tenso [B ] and its invese ae given by: T [ B ]: = [ F F ] = B =. (3), [ ] The stain enegy function fo a Mooney-ivlin mateial is: ˆ σ = C ( I 3) + C ( II 3), () with C, C mateial constants and I, II the fist two invaiants of defomation (): I = t B, ( ) II = t B t B. (5) The constitutive equation fo incompessible, isotopic, hypeelastic mateials is: T = p + α B + α B, (6)

3 3 Diect and invese poblems fo the evesion of a spheical shell 35 whee the alpha coefficients depend on the stain enegy function as follows: σˆ I α = ( I II ), = ( I II ), σˆ α,. (7) II Computing (7) fo the Mooney-ivlin stain enegy () and substituting in (6), we obtain the following fom fo the stess due to evesion: p + C B T = C B. (8) As seen in (3), [B ] and [B - ] have diagonal foms in the physical system of coodinates, and fom (8) this is also tue fo T. In fact, fom (3) and (6) we obtain the following physical components of stess: T θθ = T φφ, T θ = T φ = T θφ =. (9) In the eveted configuation, B, the body is still in equilibium. Using (9), the equilibium equations (in physical coodinates) educe to p = p() and dt = ( T θθ T ). () d Substituting (6) and (7) in equation (), as well as a evesed chain ule fo the stain enegy, we obtain the integal fom: 3 ( A ρ ) σˆ dρ T T = +, () A ρ whee is the adius at a point in the shell wall and ρ is an integation vaiable. Until now we have not used any bounday conditions fo the spheical shell. We assume futhe that the stess on the intenal and the extenal sufaces of the shell ae equal to a given pessue p whee, unlike Johnson and Hoge [], p can be non-zeo. Substituting this bounday condition in () and ewiting this integal ove the undefomed domain, we obtain: whee g( ) was calculated afte a tedious integation as: 3 σ ˆ d = g ( A, ) (, ), g A = () A ( 5 A) 3 3 ( A ) 3 ( A ) 3 ( A ) 3 g( ) = C + C. (3) As shown by Eicksen in [], fo given C, C,,, thee exists a unique value of A that satisfies both () and A> 3. With A known, () can be witten explicitly as: T = + p, () g( ) g( ) whee is tansfomed though evesion in. Finally, compaing () and the adial component of (8), we can obtain the following expession fo the pessue: 3 ( A ) 3 ( A ) 3 C 3 g( ) + g( ) p( ) = C p. (5) With configuation B pefectly detemined, let us tun ou attention to an infinitesimal defomation of B into a new configuation B. As shown in [], the state of stess in the new configuation will depend both on the defomation fom B to B.and the state of stess in B, though

4 36 aluca I. oşca, Coina S. Dapaca H : ( W T TW ) + ζ [ ] T = p + T + c E. (6) Hee E and W ae the symmetic and anti-symmetic pat, espectively, of the displacement gadient T T H = F, E = ( H + H ), W = ( H H ). (7) ζ simplifies fo this case to: It can be shown that the elasticity tenso [ ] c c E [ E ] = ( Λ + ΜT + ΝT ) E + E ( Λ + ΜT + ΝT ) ζ, (8) with Λ, Μ, Ν some expessions of C, C, p, I and II, as given in Johnson and Hoge []. ζ c E depends on initial stesses. We conside = cm, = cm and diffeent Mooney- ivlin mateial constants C, C. The computed values fo the evesion constant the eveted adii and the pessue on boundaies (up to the choice of p ) ae pesented in Table, while the dependence of the elasticity tenso on the initial stess is pesented gaphically in Fig.. Using thee numeical examples, we will show futhe how the elasticity tenso [ ] Coefficients of E <, > (kg f/cm ) Coefficients of E <, θ> (kg f/cm ) Coefficients of E <θ, θ> (kg f/cm ) Fig. Vaiation of elasticity tenso coefficients with espect to eveted adius fo diffeent values of the pessue p = kg f/cm,.9 kg f/cm,.8 kg f/cm. Top ow C =.75 kg f/cm, C =.5 kg f/cm, middle ow C =.7 kg f/cm, C =.5 kg f/cm, bottom ow C =.75 kg f/cm, C =. kg f/cm. The case p = kg f/cm coesponds to the esidual stess case of [].

5 5 Diect and invese poblems fo the evesion of a spheical shell 37 We notice that the case p = kg f/cm, coesponding to the pesence of esidual stesses in the shell, is in ageement with the esults pesented by Johnson and Hoge []. In addition, the elasticity tenso coefficients vay little with p compaed with the vaiation though the adius of the shell. Moeove, compaed with Case (see Table ), the magnitude of the vaiation, both with espect to the shell adius and to p, inceases with inceasing C (Case ) o C (Case 3). Even if Cases and 3 conseve a combination of the mateial constants (C + C =.95 which epesents / of the shea modulus G of the coesponding lineaized elastic constitutive model, see []), a vaiation in C poduces a lage vaiation in magnitude of the elasticity tenso coefficients compaed with a simila vaiation in C. Table Thee numeical cases: computed evesion constant eveted adii and, and bounday pessues fo thee given pais of mateial constants C, C Case # C (kg f/cm ) C (kg f/cm ) A (cm) (cm) (cm) p =p( ) (kg f/cm ) p =p( ) (kg f/cm ) INVESE POBLEM FOMULATION In this section we fomulate the invese poblem fo the evesion of the spheical shell pesented peviously. We pesume known the value of the pessue p() on one of the sufaces of the shell (eithe = o = ) and the value of the evesion constant and we look fo the mateial constants C, C. Fo this, we conside the linea system of two equations with the unknowns C, C fomed by equations () and (5) at the chosen adius. While the mathematical poblem of finding C, C is well detemined, only some of its solutions have physical meaning. Accoding to Tuesdell and Noll [7] and Beatty [], based on empiical evidence we must have C >, C. In what follows, we study how a measuement eo in A o p influences the accuacy of the computed constants C, C. Thoughout this study we keep constant the initial geometic body, that is the undefomed spheical shell of adii = cm, = cm. We fist conside the numeical case A=.589 cm and p = p( ) =.3 kg f/cm and ecove C =.75 kg f/cm, C =.5 kg f/cm, the same values that we used in the diect poblem, Case, above. We epeatedly solve fo C, C when A and p ae within of thei nominal values and pesent in Fig. top, the eos in C, C with espect to thei nominal values. We obseve that C is especially sensitive to eos in and a measuement eo in A poduces almost 95 absolute eos in C and, in fact, a negative value that has no physical sense. We epeat the study in a few othe cases, modifying the point whee the pessue is measued (i.e. p = p( ) o p = p( )), and the values fo A and pessue. When A =.589 cm and p = kg f/cm, we ecove C =.75 kg f/cm, C =.5 kg f/cm, but as shown in Fig. bottom, the eos in C, C ae modeate. Once again we obseve that an eo of in A gives unphysical (negative) values fo C. Compaing the top and bottom of Fig. we conclude that, if possible, we pefe to measue p since the esulting eos fo C and C would be smalle in this case. We will examine whethe this conclusion is tue fo the othe consideed values of the evesion constant and of the pessue. When A =.7 cm and p = -. kg f/cm, we ecove C =.7 kg f/cm, C =.5 kg f/cm as used in the diect poblem, Case. We epeatedly solve fo C, C when A and p ae within of thei nominal values and pesent in Fig. 3 top, the eos in C, C with espect to thei nominal values. As above, we obseve that C is especially sensitive to eos in and a measuement eo in A poduces lage eos in C and, in fact, a negative value that has no physical sense. We epeat the study fo p =.93 kg f/cm obtaining the same constants C, C. Once again we pefe to measue p instead of p since the esulting eos fo C and C would be smalle in this case.

6 38 aluca I. oşca, Coina S. Dapaca 6 Eos in C (fo diffeent A) % - 3% Eos in p - - Eos in C (fo diffeent A) % 6% -% - -8% - -3% - 3% Eos in p Eos in C (fo diffeent A) % - 3% Eos in p - - Eos in C (fo diffeent A) % - 3% - Eos in p - - Fig. Pecent eos fo constants C and C, fo up to eos in A=.589 cm and p =.3 kg f/ cm (Case., top) and p = kg f/ cm (Case., bottom). Eos in C (fo diffeent A) % - 3% Eos in p - - Eos in C (fo diffeent A) 6% % -% -6% - -% - -3% - 3% Eos in p - - Eos in C (fo diffeent A) % - 3% Eos in p - - Eos in C (fo diffeent A) % % - Eos in p - - Fig. 3 Pecent eos fo constants C and C, fo up to eos in A =.7 cm and p =. kg f/ cm (Case., top) and p =.93 kg f/ cm (Case., bottom). Finally, fo A =9.988 cm and p =.783 kg f/cm, we ecove C =.7 kg f/cm, C =.5 kg f/cm as used in the diect poblem, Case 3. We epeatedly solve fo C, C when A and p ae within of thei nominal values and pesent in Fig. top, the eos in C, C with espect to thei nominal values. In contast with the pevious cases, now all values computed fo C, C have physical meaning.

7 7 Diect and invese poblems fo the evesion of a spheical shell 39 We epeat the analysis fo A = cm and p = 9.77 kg f/ cm and plot the esults in Fig., bottom. Once again all computed values have physical meaning, but this time the eos ae lage than in Case 3.. Based on this example, we could not ecommend whethe to measue the pessue at o at. Eos in C (fo diffeent A) % - 3% Eos in p - - Eos in C (fo diffeent A) % 8% % -% -8% -% - -3% - 3% Eos in p - - Eos in C (fo diffeent A) % - 3% Eos in p - - Eos in C (fo diffeent A) % - - 3% Eos in p - - Fig. Pecent eos fo constants C and C, fo up to eos in A =9.988 cm and p =.783 kg f/ cm (Case 3., top) and p = 9.77 kg f/ cm (Case 3., bottom) Case. Case. Case. with eos Case. with 3% eos Case. with eos p (kg f/cm) -5 p (kg f/cm) Case. -35 Case. -.6 Case A (cm) Case. with eos A (cm) p (kg f/cm) Case. Case. Case 3. Case. with eos p (kg f/cm) Case. Case. with eos Case. with 3% eos Case. with eos A (cm) A (cm) Fig. 5 epesentation of the thee cases in the evesion constant-pessue space. When the couple A p is measued as input fo the invese poblem, cases - cannot be distinguished with measuement eos of % o moe; the same elative eo have no confounding effect on the thee cases epesented in the space A p. Figues on ight ae zoom-in of figues on the left.

8 aluca I. oşca, Coina S. Dapaca 8 To elucidate the question, we plot the thee cases in the space evesion constant-pessue (A p). As it can be seen in Table, the values of the pessue p fo the Cases and of the diect poblem ae vey close, even if C most than doubled. In ode to see how close those cases ae, we epesent in Fig. 5, top left, the Cases. 3. as points, as well as the ectangula box containing all measuements with elative eos fom the Case.. In Fig. 5 top ight, we zoom aound the Case. and epesent also the ectangles of Case. with, 3% and measuement eos. We use a simila epesentation fo the coesponding cases in the A p space in Fig. 5, bottom. We obseve that when the couple A p is measued as input fo the invese poblem, having measuement eos of % o moe do not let us distinguish between Cases and (Fig. 5 top), while case 3 is clealy identified (Fig. 5 top left). In contast, the same pecentage eo has no confounding effect on the thee cases epesented in the space A p. Based on this obsevation and the geneal eo behavio in Cases and 3, we ecommend measuing the pessue at the intenal adius in the eveted configuation ( ).. CONCLUSIONS In this pape we have evisited the evesion poblem fo a spheical shell and studied, fo the fist time, the coesponding invese poblem. The spheical shell is made of a homogenous, incompessible, hypeelastic Mooney-ivlin mateial. Fo the diect poblem, we expanded the study fom [] to the case of initial stesses, and noticed that the elastic tenso coefficients vay little with vaiation of the initial stess compaed with the vaiation though the adius of the shell. In the thee numeical cases consideed, the magnitude of the vaiation, inceases with inceasing constant mateials C o C and when a combination of the mateial constants equivalent to constant shea modulus is kept constant, a vaiation in C poduces a lage vaiation in magnitude of the elasticity tenso coefficients compaed with a simila vaiation in C. Fo the invese poblem, we studied how a measuement eo in the evesion constant o in pessue influences the accuacy of the computed mateial constants. We found that, in ode to minimize the eos in the calculated constants, it is pefeable to measue the pessue at the intenal adius in the eveted configuation (p =p( )). Ou esults suggest novel expeimental potocols that will allow a obust ecovey of the mechanical paametes which ultimately will help diffeentiate between nomal and pathological tissues. We believe that ou study, combined with the digital evesion method fo medical images can help impove sceening, diagnosis, sugical planning and medical education. EFEENCES. BEATTY, M.F., A Lectue on Some Topics in Nonlinea Elasticity and Elastic Stability, IMA Pepint Seies # 99, Institute fo Mathematics and Its Application, Minneapolis, 98, p. 3.. EICKSEN, J.L., Invesion of a pefectly elastic shell, Z. Angew. Math. Mech, 35, 9, pp , HONG, W., GU, X., QIU, F., JIN, M., KAUFMAN, A., Confomal vitual colon flattening, Poc. 6 ACM Symposium on Solid and Physical Modeling, 6 pp JOHNSON, B.E., HOGE, A., The dependence of the elasticity tenso on esidual stess, Jounal of Elasticity, 33,, pp. 5 65, PAIK, D., BEAULIEU, C., KAADI, C., NAPEL, S., Visualization modes fo CT colonogaphy using cylindical and plana map pojections, J.Comput. Assist. Tomog.,,, pp.79 88,. 6. OSC., DAPAC C., Diect and Invese Poblems fo the Evesion of a Spheical Shell, Poc. SISOM 9 (unde pess). 7. TUESDELL, C., NOLL, W., The Nonlinea Field Theoies of Mechanics, Flugge s Handbuch de Physik III/3, Belin, Heidelbeg, New Yok, Spinge-Velag, pp. 5 53, ZHAO, J., CAO, L., ZHUANG, T., WANG, G., Digital evesion of a hollow stuctue: an application in vitual colonogaphy, Int. J. Biomed. Imaging, Aticle ID 7638, 8. eceived Septembe, 9