J Elast (2014) 116:103–114 DOI 10.1007/s10659-014-9485-2 BIOGRAPHICAL REMEMBRANCE

Piero Villaggio: Representative of the Italian Tradition of Honored Elasticians Roger Fosdick · Gianni Royer Carfagni

Received: 3 June 2014 / Published online: 13 June 2014 © Springer Science+Business Media Dordrecht 2014

Abstract We recall the scientific work and professional contributions of Piero Villaggio (b. Genoa, Dec 31st 1932, d. Jan 4th 2014), member of the Editorial Board of the Journal of Elasticity for almost 30 years.

Piero Villaggio was appointed Professor of Structural Mechanics at the University of Pisa in 1966. He also was Professor of Fluid Mechanics at the Scuola Normale Superiore in Pisa, Member of the Italian Accademia dei Lincei, Associate Editor of many international scientific journals (including Meccanica, Journal of Elasticity, Stability and Applied Analysis of Continuous Media, Nonlinear Differential Equations and Applications and Journal for the Mathematics and Mechanics of Solids), past Vice-President of AIMETA (Italian Association for Theoretical and Applied Mechanics), accomplished rock climber and member of the most prestigious CAAI (Italian Academic Alpine Club), Oriental Group. Piero Villaggio served as a member of the Editorial Board of the Journal of Elasticity since 1987 until his passing at the beginning of this year. Since 1959, the year he published his first scientific work, until the end of his life, he authored over 140 articles and scientific treatises, and he participated as an invited speaker in many of the most prestigious Italian and International congresses. He was elected as a member of the Italian Accademia dei Lincei in 1998. He has been a visiting Professor at the Johns Hopkins University in Baltimore, at the Herriot-Watt University in Edinburgh and at the University of Minnesota in Minneapolis. Because of his well-established professional reputation and his breadth of international relationships, a number of Italian scholars, especially his students, were able to participate in the scientific renaissance of continuum mechanics that had Clifford Truesdell (1919–2000)

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R. Fosdick ( ) Department of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis, MN, USA e-mail: [email protected] G. Royer Carfagni Department of Industrial Engineering, Università di Parma, Parma, Italy e-mail: [email protected]

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Fig. 1 Piero Villaggio and climbing partners at the top of the Rosskuppe mountain, Austrian Alps (Summer, 1971)

as its leading architect. Most of his former Ph.D. students are now professors at major universities, in Italy and abroad. Indeed, he has been one of the most honored and recognized personalities in the field of continuum mechanics in Italy, continuing in the long tradition of excellence established by Italian elasticians such as Gabrio Piola (1794–1850), Enrico Betti (1823–1892), Eugenio Beltrami (1835–1900), Carlo Alberto Castigliano (1847–1884), Ernesto Cesaro (1859–1906), Carlo Somigliana (1860–1955), Vito Volterra (1860–1840), Emilio Almansi (1869–1948), Gustavo Colonnetti (1886–1968), Antonio Signorini (1888– 1963), Gaetano Fichera (1922–1996) and Edoardo Benvenuto (1940–1998). Piero Villaggio was an engineer who knew much more mathematics and physics than a traditional engineer. Throughout his professional life, he used the theory of linear elasticity as a chisel to shape his interpretative vision of many interesting phenomena and challenging problems. According to him elasticity was a model, indeed the model, to investigate physical phenomena in the material world, to achieve the deduction of results not directly observable through experiments and, at the same time, to acquire information that could be used to design an experiment aimed at the confirmation of a preliminary conjecture. Piero Villaggio was the son of Ettore Villaggio (1905–1992), a talented structural engineer, originally from Palermo, who was one of the artifices of the Italian post-war period of building and bridge reconstruction. His mother, Maria, originally from Venice, was a (German) language teacher and Piero knew well the German language and had a deep knowledge of the relevant German scientific literature. Rather impressive is the list of problems of practical engineering interest that he tackled where the elastic model was the key tool to suggest a correct and effective design. In the early days, when numerical methods for structural engineering were yet to be developed, Villaggio’s interest was in metallic flange joints [12], stress concentration around holes of particular shape [13], simple bolted joints

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Fig. 2 A contact problem between two elastic corrugated bodies to explain the theory of Coulomb friction (from [45])

[19] and prestressed bolted joints [22]. Other areas of practical research in which he was involved included the statics of dams [11], kinetic stability of mechanical systems [15], design of testing apparatus [18], rheological (viscous) response of concrete [17], bending stiffness and the shape of band saws for wood cutting [63], bonded adhesive joints [67], design of shock absorbers [100, 114], wear of mechanical components [101], hammering of nails and pitons [111], brakes [116] and the plasticization of holes [126]. Very extensive is his work devoted to foundation design [20, 99, 125, 141], dynamics of a landslide [139], structural efficiency of bridge pillars [83], tunnels [140], theory of masonry constructions [53] and arches [71, 128, 130], including a particular interest in historical construction works [86]. Remarkable, also, are his studies on the problem of packaging of brittle goods [82]. Piero Villaggio had a profound classical culture. He had attended the Liceo Classico Andrea Doria in Genoa, a prestigious high school with emphasis on humanities, having as classmate’s students that were to become important personalities, including Paolo Fresco, former CEO of General Motors Company and FIAT. He had a strong belief that a classical education is critical to forming the “taste for problems” of a scientist and to friends who asked him for advice whether their children should take scientific or classical studies at high school, he always suggested the classics. He was also very fond of music and a close friend of the famous musical critic Massimo Mila, his climbing partner as well. Therefore, it is not surprising that Piero Villaggio had a strong interest in fundamental classical issues of mechanics. Indeed, he studied a number of basic problems and attempted to provide simple explanations of fundamental physical laws. Certainly, his treatment of Coulomb friction [45], successive to an earlier work on rolling resistance [5], which he reduced to a contact problem in elasticity in order to explain the proportionality of the tangential force with the applied normal force, is an important contribution. Later, he developed a theory of brittle fracture in compression [78]. In this, he studied concrete prisms under compression and he justified the classical hourglass shape of failure on the basis of an energetic competition between the release of elastic energy due to the detachment of lateral parts in a compressed prism, and the surface energy à la Griffith necessary to produce material separation. Other noteworthy fundamental studies concerned an elementary explanation of internal friction in solids [65], the added mass of solids vibrating in a fluid [84] and Newton’s aerodynamic problem in the presence of friction [104]. In [62] he considered an interesting problem associated with the notion of elastic capacity-a

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Fig. 3 Piero Villaggio in the classroom

notion originally introduced by Kirchhoff for an electrical conductor and having analogies with a problem later studied by Thomson & Tait concerning the rigid motion of a solid in a frictionless liquid—in which one is to determine the distribution of stress and strain in an elastic space that corresponds to the rigid displacement of an internal spheroidal inclusion. In [102], developing an idea of Fichera concerning the paradox of heat propagation with infinite speed in the Fourier model, Villaggio reasoned about the apparent speed of an elastic wave from a source as it may be recorded effectively by a measuring instrument at a given reference point. In addition to his position at the University of Pisa, Piero Villaggio also held the position of Professor at the Scuola Normale Superiore di Pisa, where he offered a classical course in the mechanics of fluids. During the period when the calculus of variations was being developed at that prestigious school, he was in close contact with eminent mathematicians, including Guido Stampacchia (1922–1978). In particular, Stampacchia’s extensive studies on variational inequalities inspired many of Villaggio’s works. Among these is the variational formulation of thermodynamic processes [33, 35], the study of a great number of unilateral elastic problems [39, 42, 49, 50, 69, 74] including the search for numerical methods of solutions [52], buckling under unilateral constraints [44], and a general variational approach to the theory of structures [110]. Major applications of the field of variational inequalities are contained in his studies of contact problems for elastic bodies, ranging from the one-dimensional case of bars and beams [24, 51, 97, 123, 136], to the two-dimensional situation involving membranes [57, 59], and arriving at the complete three dimensional class of Signorini problems [60, 75], including the study of the dynamics of interaction [58, 85, 108, 109]. He also applied the variational approach to the field of structural optimization, the subject of which Villaggio was particularly fond. The general optimization problem was discussed on theoretical grounds in [48, 54] and applied to specific cases, such as the optimal distribution of loads in elastic solids in [55, 66], optimal reinforcements in [76, 115] and shape optimization in [77, 87, 88, 96].

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Villaggio’s scientific activity also was devoted to more traditional problems, such as the equilibrium, stability and optimization of plates and shells [1, 2, 8, 9, 41, 61, 94], the De Saint-Venant principle [3, 91, 124, 131, 133], plane linear elastic problems [4, 89, 92], a priori bounds (extension of the maximum principle) in the half space problem [47], effects of reinforcements [7, 142], the complex variable method in classical elasticity [112, 134, 135], energetic bounds in linear [14] and non-linear [32] elasticity, linear elastic fracture mechanics [113, 120], inclusions in elastic media [129], contact of bonded uniformly prestressed half-spaces [64], incompressible elasticity [93, 98], aeroelasticity and aerodynamic stability [16], stability of finite element methods [27], non-linear hypoelastic materials [31] and shock problems [43]. Piero’s twin brother, Paolo Villaggio, is a well-known and celebrated Italian actor, writer, director and comedian who has played numerous comic as well as dramatic roles on stage and screen. In some sense, Piero also liked being on stage. In his lectures, prepared with meticulous care, as well as in his invited conference talks, he was able to hold the attention of the audience while interspersing the scientific explanations with anecdotes, maxims and, sometimes jokes. This characteristic is also reflected in some of his papers. In his wellknown work on the mathematical theory of the Guillotine [70] (“dedicated” to an eminent colleague of his), he imagined a world of fantasy in the year 1989, two centuries after the French revolution, in which the revolutionaries are grappling with the problem of optimizing the shape of the blade of the Guillotine in order to make it as sharp as possible. To attack this problem, Villaggio turned to the linear elastic contact problem of a sharpened punch pressed against an elastic half space. Since this associated problem is singular and predicts an infinite displacement under the blade, Villaggio proposed to find the shape that is optimal in the sense that it maximizes the relative displacement with respect to fixed reference points. An additional condition that the stress intensity factor associated with the (singular) elastic stress field must remain below a prescribed value is proposed, and the best sharpened blade is determined. “Unfortunately” the theory could not be applied to the Guillotine because every citizen could be proved to be a revolutionary. Nevertheless, it was used to grind optimal knives for slicing bread and salami. . . only the pig protested!!! The presentation of the Guillotine problem is certainly quite unusual, but its scientific value is undoubtedly important. Many problems of linear elasticity involve unbounded solutions in terms of stress and strain, or even displacement. In fact, all of linear elastic fracture mechanics is based upon the evaluation of the stress intensity factor, which represents the coefficient of the singular part of the stress field. When the theory predicts infinite displacement, Villaggio’s idea is to consider only the relative displacement with respect to fixed reference points, which somehow is equivalent to accounting for the singular part of the displacement only. Using this rationale, he also studied the penetration of a wedge-shaped indenter applied to a half space [68], and the optimal penetration of a rigid inclusion in an elastic space [73]. Another article with a rather “Villaggian” title is [95], wherein the author pretends to give his advice to a worm in order to optimize its posture, so as to improve its ability to anchor to the ground and avoid, in this way, being pulled away by a bird in search of food. This paper, and the mathematical approach taken, is of great interest nowadays because of the rapidly increasing technique of bonding fiber-reinforced-polymer (FRP) strips to concrete members in order to increase their mechanical stiffness and strength: The optimal shape of the reinforcing stringer is found by following the same advice that Villaggio gave to the worm. For the specific problem of the debonding of a stiffener from an elastic substrate, in [107] Villaggio proposed a criterion à la Griffith that was later followed by other authors. Another side of the character of Villaggio is embedded in his “Variational plebean problems” [118], which contains a rather humorous review of every-day practical problems, addressed from

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Fig. 4 A drawing by Piero Villaggio (from [118]). In the passing of a cliff, the weight on the arms of a climber gradually increases. The problem can be described mathematically and there is an optimal technique for overcoming the obstacle

the mathematical point of view of variational techniques. He discusses optimization problems (in the sense of minimum effort, or minimum time of execution) associated with primitive movements like walking, running, lifting one’s body with the arms and chewing, as well as the optimal design of primitive tools like the hammer, a can-opener, scissors and knives. Our “hero” was an experienced and skilled mountaineer who had climbed many challenging peaks in Italy and abroad. It is not then surprising that he dedicated two papers to a mathematical theory of climbing [119, 121]. He was very fond of nature and there are several of his works in continuum mechanics that reflect this awareness. He gave justification for the equilibrium shape of earth masses in [72], of the flattening of mountain chains in [81], of the form of the roots of trees in [90], of the shape of glaciers in [138] and he studied the dynamics of a landslide in [139]. Although having been mainly dedicated in his professional life to elasticity, Piero also made significant contributions in the theory of plasticity, but his approach was anything but traditional. In [40] he proposed a novel and intriguing mechanical model for an elasticplastic body whose microstructure was conceived of as an orderly array of snap-through arch units: the onset of the plastic phase was correlated with the orderly instability of such units. Other works are related to the general formulation of elasto-plasticity [10], limit analysis [21, 26, 28], fatigue-sensitive materials [46], visco-plastic molding [105] and a theory of plastic splashing [117]. Moreover, his studies concerning the stability of mechanical systems span a wide spectrum of problems, starting from his early work on the stability of linear elastic bodies [6], passing to non-linear continua [29, 30, 36] and to thermoelastic media [34]. A particular attention was dedicated to the stability of fluid mixtures [37], which was followed with a discussion of conditions for wave propagation [38]. As regards to the theory of solid mixtures, he also proposed an extension of the classical Kelvin state and the corresponding nuclei of strain [56] and proposed a general approach for the study of elastic material with sparse rigid reinforcements [106]. Other studies concerned the propagation of elastic waves in a half space with a corrugated boundary [79]. Piero Villaggio was always very attentive to the changing trends in scientific research, doing a scrupulous work of reviewing [127], sometimes not skimping harsh criticisms of malpractices he discovered. In [80] he presented a guide on how to write a paper on a subject in mechanics, illustrating a set of rules, even ethical, to correctly present the results without scientific fraud. In [132], he gave a critical analysis of the best achievements and

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drawbacks in the last sixty years of solids mechanics, wherein he showed a breadth and depth of knowledge of past major and minor activities. In [103] he warned the scientific community about the potential risks of a careless reading of the history of science, including the quite naive telescopic fallacy. Can we talk of a crisis in the mechanics literature? In [137] he asks both himself and the reader questions about the radical change in the method and mentality of facing problems created by the readily available large amounts of processing power at low cost. In another paper [122], he deeply criticizes the current mechanisms of evaluation of scientific results (number of papers, number of citations, bibliometric indexes, journal impact factors) based upon which academic careers are built. The exasperated search for quantity is to the detriment of the quality and Piero’s warnings are timely as ever nowadays. According to Villaggio, science must make difficult things easy, but instead we are witnessing a germ of alleged scientists that complicate trivial problems until they become incomprehensible to most, in order to achieve recognition. Another bad habit that appears to be surfacing, as Villaggio has noted, is to identify the physical problem that can be solved by a complicated equation, best if complex, but for which one already knows the solution. It is not easy to recap the extreme variety of scientific interests of Piero Villaggio. Perhaps a final synthesis and appreciation of his contributions can be made by examining the three books he wrote. It took him 4 years of hard and dedicated work to finish his first opera, “Qualitative Methods in Elasticity”, published in 1977 [A.1]. In his opinion, a qualitative solution of a physical problem should be an analytical solution that, with appropriate simplifications, would capture and explain the essence of the phenomenon under consideration. This book deals with the characterization and solution of general boundary value problem of linear elastostatics, including mean value properties and inequalities, strong formulations, energetic a priori estimates, pointwise bounds and special other topics classified by Villaggio as “behavioral properties” (including perturbation methods, symmetrization properties, estimates at infinity and singular states). The treatise is full of paradigmatic examples and problems (with solutions) and it represents an invaluable synthesis of varied technique and results widely scattered throughout the literature. His second book “Mathematical Models for Elastic Structures” [A.2] was published 20 years later. Written within the period of one year, it aimed at offering a critical collection of the various mechanical theories for modeling the behavior of structures. It includes models of increasing degree of complexity such as rods, strings, membranes, plates and shells. It certainly represents a natural progression to the theory of structures of the work done in [A.1] for the elastic solid. During his last years, Piero completely devoted himself, with great enthusiasm and commitment to a great work of historical character [A.3]: a critical review of selected works by Johann I Bernoulli (1667–1748), with a listing of all the works and manuscripts of Johann I and his son Nicolaus II Bernoulli (1695–1726). This is the sixth volume of the Opera Omnia by the Bernoulli family, published by Birkhäuser, and it deals with the mechanics of rigid and elastic bodies, not encompassing the area of hydraulics, which is contained in the seventh volume. With the exception of one short text by Nicolaus II, all pieces included in this volume were written by Johann I Bernoulli. Villaggio’s approach is, as is characteristic, original, pointed and stimulating. He manages to summarize in a just few introductory pages the historical importance and the scientific achievements of Johann I. The remaining hundreds of pages of the volume are mainly dedicated to the presentation of early selected papers by Johann I Bernoulli, most written in Latin. These papers were fully revisited by Villaggio in that he solved the same problems using modern analytical techniques, making illuminating comparisons with the procedures adopted by the Swiss scholar. This opera is a masterpiece that Piero Villaggio left to us as an everlasting keepsake for time immemorial. Piero Villaggio also was devoted to literature, philosophy, philology, art and music. However, his interest for the “two cultures”, mainly delivered in seminars and private discussions,

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cannot be fully appreciated in this note, it being dedicated solely to his scientific achievements related to his academic and editorial contributions. But even within this limited scope, our review cannot be exhaustive, because it does not consider other noteworthy essays, “hidden” in national journals, special volumes, and newspapers. Slightly adapting the words of Clifford Truesdell, we could say that “Piero was not an interdisciplinarian, because for him there were no disciplines; in a period of increasing specialization he remained a Natural Philosopher”. Our sincere hope is that this remembrance, albeit partial, can stimulate curiosity and serve as a guide to those, especially the youngest, who would like to get closer to the work of this great scholar, an eminent colleague and a sincere friend whose example has shown to all of us that there can be a true passion for knowledge, and this can be ethically pursued without compromise.

Publications of Piero Villaggio Books A.1. Villaggio, P.: Qualitative Methods in Elasticity. Nordhoff, Groningen (1977) A.2. Villaggio, P.: Mathematical Models for Elastic Structures. Cambridge University Press, Cambridge (1997) A.3. Villaggio, P.: Die Werke Von Johann I Und Nicolaus II Bernoulli. Birkhäuser, Basel (2007)

Scientific Papers 1. Villaggio, P.: Sul problema variazionale della lastra curva a simmetria assiale. Atti Accad. Ligure Sci. Lett. 16, 1–11 (1959) 2. Villaggio, P.: Sulla teoria non lineare delle lastre sottili elastiche a doppia curvatura. Rend. Ist. Lomb., Accad. Sci. Lett. 94, 151–165 (1960) 3. Villaggio, P.: Su un problema non lineare relativo alla torsione di solidi prismatici. Atti Accad. Ligure Sci. Lett. Col. 18, 1–29 (1961) 4. Villaggio, P.: Sui problemi di trasmissione dell’elasticità piana. Rend. Ist. Lomb., Accad. Sci. Lett. 95, 493–524 (1961) 5. Villaggio, P.: Su un problema elastico piano della teoria dell’attrito volvente. Rend. Ist. Lomb., Accad. Sci. Lett. 95, 919–940 (1961) 6. Villaggio, P.: Sui problemi al contorno per sistemi di equazioni differenziali lineari del tipo di stabilità dell’equilibrio elastico. Ann. Sc. Norm. Sup. Pisa III XV, I–II, 25–40 (1961) 7. Villaggio, P.: Su un problema misto relativo ad un semipiano elastico collegato ad una nervatura. Atti Ist. Sci. Costr. Pisa 84, 1–16 (1961) 8. Villaggio, P.: Limiti inferiori del carico critico di lastre elastiche a doppia curvatura. Atti Ist. Sci. Costr. Pisa 86, 1–20 (1961) 9. Villaggio, P.: Limiti inferiori del primo moltiplicatore critico di lastre sottili elastiche di rivoluzione. Atti Ist. Sci. Costr. Pisa 89, 1–19 (1961) 10. Villaggio, P.: Sull’esistenza e l’unicità di soluzioni dei problemi al contorno in elastoplasticità. Rend. Ist. Lomb., Accad. Sci. Lett. 96, 712–727 (1962) 11. Villaggio, P.: Su un problema misto relativo alla statica delle dighe cilindriche spesse. Atti Ist. Sci. Costr. Pisa 93, 1–18 (1962) 12. Villaggio, P.: Tensioni deformazioni in flange sottili circolari elastiche. Atti Ist. Sci. Costr. Pisa 95, 1–34 (1962) 13. Villaggio, P.: Limiti superiori del fattore di concentrazione della tensione intorno a fori ipotrocoidali. L’Aerotecnica 6, 119–128 (1962) 14. Villaggio, P.: Su alcune formule di maggiorazione per l’energia di deformazione di un corpo elastico. Matematiche 17, 121–135 (1962) 15. Villaggio, P.: Su un criterio di stabilità cinetica di sistemi meccanici soggetti ad azioni impulsive. Rend. Ist. Lomb., Accad. Sci. Lett. 97, 96–100 (1963) 16. Villaggio, P.: Condizioni di sicurezza all’instabilità aerodinamica di lastre sottili in regime supersonico. Atti Ist. Sci. Costr. Pisa 96, 1–11 (1963)

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17. Villaggio, P.: Sullo scorrimento viscoso del conglomerato in flessione pura. Atti Ist. Sci. Costr. Pisa 100, 1–14 (1963) 18. Villaggio, P.: Studio di un sistema servomotore per prove di rilassamento. Atti Ist. Sci. Costr. Pisa 101, 1–8 (1963) 19. Villaggio, P.: Limiti di collasso plastico-rigido di unioni chiodate. Costr. Met. 1, 3–11 (1964) 20. Villaggio, P.: Limiti di resistenza di pali in terreno plastico-rigido. Cemento 10, 25–30 (1964) 21. Villaggio, P.: Analisi limite di piastre sottili appoggiate plastico-rigide. G. Genio Civ. 3, 133–141 (1965) 22. Villaggio, P.: La nozione di efficienza teorica nei bulloni pretesi. Costr. Met. 4, 284–288 (1965) 23. Villaggio, P.: Alterazione perturbativa dei carichi critici di lastre piane sottili. G. Genio Civ. 12, 607–616 (1965) 24. Villaggio, P.: Monodimensional solids with constrained solutions. Meccanica 2, 65–68 (1967) 25. Villaggio, P.: Stability conditions for elastic-plastic Prandtl-Reuss solids. Meccanica 3, 46–47 (1968) 26. Villaggio, P.: Stabilità rispetto al convesso dei domini plastici delle travature. G. Genio Civ. 2–3, 101– 109 (1968) 27. Villaggio, P.: Proprietà di stabilità e di monotonia nel metodo degli elementi finiti. L’Aerotecnica 3–6, 1–9 (1969) 28. Villaggio, P.: Theorems of convergence for minimal sequences in limit analysis. Int. J. Solids Struct. 5, 833–841 (1969) 29. Villaggio, P.: A criterion of stability for non-linear continua. IUTAM Symp. Herrenalb., 19–24 (1969) 30. Villaggio, P.: Strong ellipticity conditions for the differential operator of the finite isotropic elasticity. Meccanica 5, 191–196 (1970) 31. Villaggio, P.: A criterion of classification for non-linear hypoelastic materials in simple shear. Meccanica 6, 87–91 (1971) 32. Villaggio, P.: Energetic bounds in finite elasticity. Arch. Ration. Mech. Anal. 45, 282–293 (1972) 33. Villaggio, P.: Formulation of some homegeneous thermodynamic processes as variational inequalities. In: Int. Symposium on Fundations of Plasticity, Warsaw, August 30–Sept. 2 (1972) 34. Villaggio, P.: Condition of stability for thermoelastic continua. Meccanica 7, 19–22 (1972) 35. Villaggio, P.: Formulation of some homogeneous thermodynamic processes as variational inequality. Arch. Mech. Stosow. 25, 293–298 (1973) 36. Villaggio, P.: The stability of continuous systems. Meccanica 10, 313–314 (1975) (with L. Salbadori, M. Como and R. Rionero) 37. Villaggio, P.: On stability in the classical linear theory of fluid mixtures. Ann. Mat. Pura Appl. (IV) 111, 57–67 (1976) (with M. Gurtin) 38. Villaggio, P.: Condition of stability and wave speeds for fluid mixtures. Meccanica 11, 191–195 (1976) (with I. Müller) 39. Villaggio, P.: Two-sided estimates in unilateral elasticity. Int. J. Solids Struct. 13, 279–292 (1977) 40. Villaggio, P.: A model for an elastic plastic body. Arch. Ration. Mech. Anal. 65, 25–46 (1977) (with I. Müller) 41. Villaggio, P.: Hadamard’s theorem applied to thin plates. J. Elast. 7, 425–436 (1977) 42. Villaggio, P.: Comparison properties for solutions of unilateral problems. Meccanica 13, 41–47 (1978) 43. Villaggio, P.: Concavity techniques with application to shock problems. J. Elast. 9, 29–41 (1979) 44. Villaggio, P.: Buckling under unilateral constraints. Int. J. Solids Struct. 15, 193–201 (1979) 45. Villaggio, P.: An elastic theory of Coulomb friction. Arch. Ration. Mech. Anal. 70, 135–143 (1979) 46. Villaggio, P.: The thermodynamics of fatigue-sensitive materials. Meccanica 14, 48–54 (1979) 47. Villaggio, P.: Maximum modulus theorems for the elastic half-space. Riv. Mat. Univ. Parma 4/5, 663– 672 (1979) 48. Villaggio, P.: Inverse boundary value problems in structural optimization. In: Free Boundary Problems (Proc. Sem. Pavia 1979), Ist. Naz. Alta Matem., vol. 2, pp. 587–599 (1979) 49. Villaggio, P.: The exact deflection of an elastic beam over a soft obstacle. Meccanica 14, 219–223 (1979) 50. Villaggio, P.: A unilateral contact problem in linear elasticity. J. Elast. 10, 113–119 (1980) 51. Villaggio, P.: Stress diffusion in non-linear interpenetrating bars. Int. J. Solids Struct. 17, 411–420 (1981) 52. Villaggio, P.: The Ritz method in solving unilateral problems in elasticity. Meccanica 16, 123–127 (1981) 53. Villaggio, P.: Stress diffusion in masonry walls. J. Struct. Mech. 9, 439–451 (1981) 54. Villaggio, P.: Are the optimum problems in structural design well posed? Arch. Ration. Mech. Anal. 78, 199–211 (1982) (with W. Velte) 55. Villaggio, P.: Isoperimetric distributions of loads in elastostatics. Rend. Semin. Mat. Univ. Padova 68, 261–268 (1982)

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56. Villaggio, P.: Kelvin’s solution and nuclei of strain in a solid mixture. Ann. Sc. Norm. Pisa. Ser. IV. 10, 109–124 (1983) 57. Villaggio, P.: A free boundary value problem in plate theory. J. Appl. Mech. 50, 297–302 (1983) 58. Villaggio, P.: The motion of a detaching elastic body. Arch. Ration. Mech. Anal. 85, 161–170 (1984) 59. Villaggio, P.: Detachment instabilities in membranes. Meccanica 19, 201–205 (1984) 60. Villaggio, P.: A signorini problem in elasticity with prescribed contact set. Appl. Math. Optim. 13, 163–174 (1985) 61. Villaggio, P.: A correction to the Föppl-v. Kármán equations. Boll. UMI 4-B(6), 761–771 (1985) 62. Villaggio, P.: The main elastic capacities of a spheroid. Arch. Ration. Mech. Anal. 92, 337–353 (1986) 63. Villaggio, P.: Influence of the bending stiffness on the shape of a belt in steady motion. J. Appl. Mech. 53, 266–270 (1986) (with R. Fosdick) 64. Villaggio, P.: Self straining at the interface of two bonded hyperelastic half-spaces uniformly prestressed. Q. J. Mech. Appl. Math. 41, 347–361 (1988) (with J. Dunwoody) 65. Villaggio, P.: The virtual friction of a sphere vibrating in an elastic medium. Arch. Ration. Mech. Anal. 102, 193–203 (1988) 66. Villaggio, P.: Optimal distributors of loads in plane elastostatics. Meccanica 23, 203–208 (1988) 67. Villaggio, P.: How to model a bonded joint. J. Appl. Mech. 56, 590–594 (1989) 68. Villaggio, P.: The penetration of an elastic wedge. Quad. Sc. Norm. Super. Pisa, 791–797 (1989) 69. Villaggio, P.: A note on the motion of a string on a unilaterally reacting foundation. SIAM J. Appl. Math. 49, 1223–1230 (1989) (with R.E.L. Turner) 70. Villaggio, P.: A mathematical theory of a guillotine. Arch. Ration. Mech. Anal. 110, 93–101 (1990) 71. Villaggio, P.: The minimal thickness of a semicircular arch with a tension-free crown. Mech. Struct. Mach. 18, 515–527 (1990) 72. Villaggio, P.: The equilibrium shapes of earth masses. Stab. Appl. Anal. Cont. Mech. 1, 149–164 (1991) 73. Villaggio, P.: The rigid inclusion with highest penetration. Meccanica 26, 149–153 (1991) 74. Villaggio, P.: The shape of a free surface of a unilaterally supported elastic body. Ann. Sc. Norm. Pisa, S.N. 18, 525–539 (1991) 75. Villaggio, P.: On the detachment of an elastic body bonded to a rigid support. J. Elast. 27, 133–142 (1992) (with W. Velte) 76. Villaggio, P.: The best position of a reinforcement in an elastic sheet. Boll. UMI 7, 103–112 (1992) 77. Villaggio, P.: New results in shape optimization. In: Zolésio, J.P. (ed.) Boundary Control and Boundary Variation. Lecture Notes in Control and Inform. Sci., vol. 178, pp. 356–361. Springer, Berlin (1992) (with J.P. Zolésio) 78. Villaggio, P.: A theory for brittle fracture in compression. Contin. Mech. Thermodyn. 5, 243–254 (1993) (with J. Dunwoody) 79. Villaggio, P.: Elastic waves against a corrugated boundary. Meccanica 28, 153–157 (1993) 80. Villaggio, P.: How to write a paper on a subject in mechanics. Meccanica 28, 163–167 (1993) 81. Villaggio, P.: The flattening of mountain chains. In: Baiocchi, C., Lions, J.L. (eds.) Boundary Value Problems for Partial Differential Equations and Applications, pp. 449–454. Masson, Paris (1993) 82. Villaggio, P.: The problem of packaging. In: Marcellini, Talenti, Vesentini (eds.) Partial Differential Equations and Applications. Lecture Notes in Pure and Appl. Math. Series, vol. 177, pp. 319–327 (1996) 83. Villaggio, P.: The pillar of best efficiency. J. Elast. 42, 79–89 (1996) 84. Villaggio, P.: The added mass of a deformable cylinder moving in a liquid. Contin. Mech. Thermodyn. 8, 115–120 (1996) 85. Villaggio, P.: The rebound of an elastic sphere against a rigid wall. J. Appl. Mech. 63, 259–263 (1996) 86. Villaggio, P.: The thickness of Roman arches. In: Batra, R.C., Beatty, M.F. (eds.) Contemporary Research in the Mechanics and Mathematics of Materials, pp. 472–479. CIMNE, Barcellona (1996) 87. Villaggio, P.: A semi-inverse shape optimization problem in linear anti-plane shear. J. Elast. 45, 53–60 (1996) (with C.O. Horgan) 88. Villaggio, P.: Suboptimal shape of a plate stretched by planar forces. In: Da Prato, G., Zolésio, J.P. (eds.) Partial Differential Equations Methods in Control and Shape Analysis. Lecture Notes in Pure Appl. Math., vol. 188, pp. 321–331. Dekker, New York (1997) (with J.P. Zolésio) 89. Villaggio, P.: Some extensions of Carother’s paradox in plane elasticity. Math. Mech. Solids 3, 17–28 (1998) 90. Villaggio, P.: The roots of trees. Contin. Mech. Thermodyn. 10, 233–240 (1998) 91. Villaggio, P.: Transverse decay of solutions in an elastic cylinder. Meccanica 33, 577–585 (1998) (with R.J. Knops) 92. Villaggio, P.: Recovery of stresses in a beam from those in a cone. J. Elast. 53, 65–75 (1999) (with R.J. Knops)

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93. Villaggio, P.: Spatial behaviour in plane incompressible elasticity on a half-strip. Q. Appl. Math. 58, 355–367 (2000) (with R.J. Knops) 94. Villaggio, P.: The slope of roofs. Meccanica 35, 215–227 (2000) 95. Villaggio, P.: The problem of the stiffener and advice for the worm. Math. Comput. Model. 34, 1423– 1429 (2001) 96. Villaggio, P.: The high cantelivered beam of minimal compliance. Q. J. Mech. Appl. Math. 54, 329–339 (2001) (with R.J. Knops) 97. Villaggio, P.: An elementary theory of the oblique impact of rods. Rend. Mat. Accad. Lincei, s. 9 12, 49–56 (2001) (with R.J. Knops) 98. Villaggio, P.: Spatial behaviour in the incompressible linear elastic free cylinder. Proc. R. Soc. Lond. A 457, 2113–2135 (2001) (with Robin J. Knops) 99. Villaggio, P.: How to design a foundation. Int. J. Solids Struct. 38, 8899–8906 (2001) 100. Villaggio, P.: How to design a shock absorber. Struct. Multidiscip. Optim. 23, 88–93 (2001) 101. Villaggio, P.: Wear of an elastic block. Meccanica 36, 243–249 (2001) 102. Villaggio, P.: The apparent propagation velocity of a wave. Rend. Mat. Accad. Lincei, s. 9 12, 191–197 (2001) 103. Villaggio, P.: Distorsions in the history of mechanics. Meccanica 36, 589–592 (2001) 104. Villaggio, P.: Newton’s aerodynamic problem in the presence of friction. Nonlinear Differ. Equ. Appl. 9, 296–307 (2002) (with D. Horstmann and B. Kawohl) 105. Villaggio, P.: On the termodynamics of repetitive visco-plastic moulding. Z. Angew. Math. Phys. 53, 1139–1149 (2002) (with I. Müller and H.-S. Sahota) 106. Villaggio, P.: Elastic materials with sparse, rigid reinforcements and debonding. Contin. Mech. Thermodyn. 15, 287–294 (2003) (with J. Jenkins) 107. Villaggio, P.: Brittle detachment of a stiffener bonded to an elastic plate. J. Eng. Math. 46, 409–416 (2003) 108. Villaggio, P.: The dynamics of a detaching rigid body. Meccanica 38, 595–609 (2003) (with M.J. Leitman) 109. Villaggio, P.: An approximate treatment of blunt body impact. J. Elast. 72, 213–228 (2003) (with R. Knops) 110. Villaggio, P.: Calcolo delle variazioni e teoria delle strutture. Boll. Unione Mat. Ital., A 7, 49–76 (2004) 111. Villaggio, P.: Hammering of nails and pitons. Math. Mech. Solids 10, 461–468 (2005) 112. Villaggio, P.: An extension of the complex variable method in plane elasticity to domains with corners: a notch problem. J. Elast. 81, 205–215 (2005) (with M.J. Leitman) 113. Villaggio, P.: Frobenius’ method for curved cracks. Int. J. Fract. 139, 59–69 (2006) (with R. Ballarini) 114. Villaggio, P.: The dynamics of a membrane shock-absorber. Mech. Based Des. Struct. Mach. 34, 277– 292 (2006) (with M.J. Leitman) 115. Villaggio, P.: Exact solutions in the reinforcement of a circular plate under concentrated loads. Struct. Multidiscip. Optim. 32, 427–433 (2006) (with R. Knops) 116. Villaggio, P.: An optimum braking strategy. Meccanica 41, 693–696 (2006) (with R. Knops) 117. Villaggio, P.: A theory of plastic splashing. Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., Rend. Lincei, Mat. Appl. 17, 89–93 (2006) 118. Villaggio, P.: Problemi variazionali plebei. Boll. Unione Mat. Ital., A 10, 1–20 (2007) 119. Villaggio, P.: A mathematical theory of climbing. IMA J. Appl. Math. (Inst. Math. Appl.) 72, 570–576 (2007) 120. Villaggio, P.: Comment on the paper “Perturbed cracks in two dimensions: a reprise”, by P.A. Martin. Int. J. Fract. 148, 93–94 (2007) (with R. Ballarini) 121. Villaggio, P.: How to climb the face of a mountain. Note Mat. 27, 249–255 (2007) 122. Villaggio, P.: Small perturbations and large self-quotations. Meccanica 43, 81–83 (2008) 123. Villaggio, P.: Axial impact on a semi-infinite elastic rod. Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., Rend. Lincei, Mat. Appl. 19, 205–210 (2008) 124. Villaggio, P.: On Saint-Venant’s principle for elasto-plastic bodies. Math. Mech. Solids 14, 601–621 (2009) (with R. Knops) 125. Villaggio, P.: Deep foundations. Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., Rend. Lincei, Mat. Appl. 20, 413–420 (2009) 126. Villaggio, P.: Plastic zone around circular holes. J. Eng. Mech. 135, 1467–1471 (2009) (with M.J. Leitman) 127. Villaggio, P.: Reviews of books. Meccanica 45, 897–899 (2010) 128. Villaggio, P.: Optimal parabolic arches. Int. J. Eng. Sci. 48, 1433–1439 (2010) (with M.J. Leitman) 129. Villaggio, P.: Elastic stress diffusion around a thin corrugated inclusion. IMA J. Appl. Math. (Inst. Math. Appl.) 76, 633–641 (2011) (with R. Ballarini)

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130. Villaggio, P.: An approximate theory of pseudo-arches. Int. J. Solids Struct. 48, 2960–2964 (2011) (with M.J. Leitman) 131. Villaggio, P.: Illustrations of Zanaboni’s formulation of Saint-Venant’s principle. Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., Rend. Lincei, Mat. Appl. 22, 347–364 (2011) (with R. Knops) 132. Villaggio, P.: Sixty years of solid mechanics. Meccanica 46, 1171–1189 (2011) 133. Villaggio, P.: Zanaboni’s treatment of Saint-Venant’s principle. Appl. Anal. 91, 345–370 (2012) (with R. Knops) 134. Villaggio, P.: Remarks on Neuber’s complex variable procedure for plane elastic solutions. ZAMM Z. Angew. Math. Mech. 92, 196–203 (2012) (with R. Knops) 135. Villaggio, P.: Some ambiguities in the complex variable method in elasticity. J. Elast. 109, 223–234 (2012) (with M.J. Leitman) 136. Villaggio, P.: On oblique impact of a rigid rod against a Winkler foundation. Contin. Mech. Thermodyn. 24, 559–582 (2012) (with R. Knops) 137. Villaggio, P.: Crisis of mechanics literature? Meccanica 48, 765–767 (2013) 138. Villaggio, P.: The shape of a glacier. Boll. Unione Mat. Ital. 6, 299–317 (2013) (with R. Knops) 139. Villaggio, P.: The dynamics of a landslide. AAPP Atti Accad. Pelorit. Pericol., Cl. Sci. Fis. Mat. Nat. 91(1), A11 (2013) (with M.J. Leitman) 140. Villaggio, P.: Parabolic tunnels in a heavy elastic medium. Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., Rend. Lincei, Mat. Appl. 24, 1–10 (2013) (with M.J. Leitman) 141. Villaggio, P.: The optimal shape of a plinth. Int. J. Eng. Sci. 77, 24–29 (2014) (with R. Knops) 142. Villaggio, P.: Deformable stiffner welded to an elastic plate. J. Eng. Mech. (2014). doi:10.1061/ (ASCE)EM.1943-7889.0000787 (with R. Knops). In press