2024 Timoshenko Medal Acceptance Lecture: Mechanics, the French Way Free

I grew up in Grenoble, at the foot of the Alps. My mother was a kindergarten teacher, and my father was a marine engineer. He designed dams and hydraulic structures. He had Timoshenko's books on “strength of materials” on his shelves. I wish I could tell you a wonderful story about how I read them when I was young, but it is not true. I was more attracted by his collection of Jules Verne novels, and my favorite was “The Mysterious Island,” which tells the story of five men who escape by balloon during the American Civil War. Among them is the engineer Cyrus Smith, who solves all the problems—and there are many—that come across their way through a clever combination of observation, knowledge, and thought. Captain Nemo, at the cutting edge of technology and science, plays a central role in this novel and is also a fascinating character. Part of my attraction to science and to our field can be traced back to these novels. That is the magic of reading.

My parents, a middle-class couple, were living happily when illness struck: in 1958 (I was then 4 years old), my father contracted polio, which left him completely paralyzed in all four limbs and his lungs. My mother devoted her life to her husband, and with her help he continued to live for another 32 years, whereas life expectancy for such serious illnesses is only 3 or 4 years. However, life for and with a severely disabled person is difficult, both financially and socially. We benefited from a lot of solidarity, especially from my mother's sister and her husband, Anne-Marie and Jean Essayan, thanks to whom my brother and I spent several wonderful holidays at the seaside. We learned important lessons for the rest of our lives.

After two years of “classes préparatoires” and fierce competition, I was admitted to the Ecole Polytechnique, the top engineering school in France, and to the Ecole Normale Supérieure, the alma mater of many French scientists, which I chose to attend because of my attraction to Mathematics. There we had a lot of freedom. I tried to do both math and physics, but the “noble” fields in the two subjects (algebraic geometry in math, quantum physics in physics) were too far apart from my interests. Through Applied Mathematics, with brilliant courses by H. Brézis or L. Tartar, I discovered, or rediscovered, Mechanics. My case is not isolated, as it is a French tradition to come to Mechanics after a background in Mathematics.

It is probably the appropriate time to say a word about French Mechanics. It is known abroad through the great names of the 18th and 19th centuries. A few names will sound familiar: Joseph-Louis Lagrange (1736–1813), to whom we owe analytical mechanics and a variational vision of mechanics, Augustin Cauchy (1789–1857), who introduced the stress tensor (before tensors even existed), Sadi Carnot (1796–1832), who laid the foundations of thermodynamics, Claude Navier (1785–1836), Gabriel Lamé (1795–1870), Adhémar Barré de Saint Venant (1797–1886), Joseph Boussinesq (1842–1929), the Cosserat brothers (1852–1914 and 1866–1931), who contributed among others to the foundations of the theory of elasticity and continuum mechanics. The 19th century in France was the golden age of the “ingénieurs-savants” (“engineers-scientists”). They received a strong training in mathematics, most of them at the Ecole Polytechnique, a military engineering school since they were supposed to be artillery officers. They began their careers as engineers and civil servants (Cauchy, for example, began his career as a junior engineer in Cherbourg, where Napoleon was building a naval base to face England). But many of them kept science in mind and came back to it after several years of service to the Republic. The typical French curriculum is therefore to favor mathematics at the beginning, then to be confronted with engineering problems, and finally to find one's own way in between the two.

The 19th century was the time of reversible behavior (elasticity or perfect fluids). The 20th century was marked by the recognition of irreversibility (viscous fluids, plasticity and fracture), and French mechanics during the 20th century made its contribution in its own mathematical style; Jean Leray (1906–1998) in fluids or Jean Mandel (1907–1982) in solids are two well-known examples, but I will give you other names in a moment, which are less well-known, although they are most deserving.

I studied continuum mechanics at the University of Paris-Sorbonne. Georges Duvaut (born 1934), a remarkable teacher, offered me a thesis on plasticity. Georges was a student of Paul Germain (1920–2009), an important figure in French Mechanics who had spent in the 50s a sabbatical year at Brown in the Department of Applied Mathematics that William Prager had just founded. There, and in the neighboring Engineering Department, he enjoyed interacting with the best specialists in our field, D. Drucker, W. Prager, P. Hodge, E. Lee, and R. Rivlin, to name just a few. This year reinforced Paul Germain's conviction of the benefits of close interactions between Mechanics and Applied Mathematics. A few years later, he recommended to his student Georges Duvaut to work with Jacques-Louis Lions, a famous applied mathematician. The result was the landmark book “Variational Inequalities in Mechanics and Physics.” I belong to this line of thought.

As we all know, the partnership between mechanics and applied mathematics has had a considerable impact on our discipline, with both eminent scientific achievements and industrial successes. As an example of the latter, again at Brown, the development of abaqus, the first commercial code based on the nonlinear finite element method (FEM) which is used world-wide in Industry and Academia (and now belongs to the French company Dassault Systèmes), was created by PhD students in Solid Mechanics at Brown, and has had an international impact on our field and well beyond.

With my background in Mathematics, my first research was to study the equations of perfect plasticity from the perspective of functional analysis. This may look as far from Engineering, so let me explain. Why should we study the existence and uniqueness of solutions that nature proves to us to exist? Is not the essential question to compute them? Computing a solution requires knowledge of its regularity, and its possible nonuniqueness should lead us to reflect on the value of the “natural” solution. Perfect plasticity concentrates all these difficulties: classical functional spaces are useless, and there may be an infinite number of solutions or no solution at all. To address these issues, I was led to propose a functional space for plasticity with discontinuous fields. It remained unnoticed for at least 20 years in the mathematical community but now, with sophisticated extensions, it plays an important role in the modern “calculus of variations,” crucial in particular in the modern modeling of fracture. This work on plasticity received an immediate and warm welcome from Mechanics, especially from Paul Germain, Nguyen Quoc Son (1944–2021) and Jean-Jacques Moreau (1923–2014).

Nguyen Quoc Son (everybody called him Son) taught me his modern vision of constitutive relations and nonlinear mechanics through thermodynamics, not the axiomatic version developed in the 1950s and 1960s, but a pragmatic version of it, with a particular emphasis on internal variables. Son, originally from Vietnam, studied at the Ecole Polytechnique and at the Ecole des Ponts et Chaussées (as Cauchy). He began his career as a civil engineer, designing bridges and motorway junctions. Soon he realized that his calling was elsewhere, in the new and exciting research in theoretical mechanics, applied mathematics, and computer science. His doctoral thesis under the supervision of Jean Mandel contains two brilliant results: the theory of generalized standard materials (with B. Halphen), where he pursued ideas of Biot, Kestin, Moreau, and Ziegler on the formulation of constitutive relations from internal variables and two potentials, and a general algorithm for computational plasticity with hardening, which he called the double projection algorithm, now known as the closest point algorithm. He subsequently went on to make important contributions to fracture, stability, and friction. Son was a beautiful person, a selfless being of great kindness and discretion.

Another prominent figure in French Mechanics, especially in Plasticity, was Jean-Jacques Moreau (1923–2014). The title of one of Batchelor's texts devoted to G. I. Taylor, “Research as a lifestyle,” fits Jean-Jacques perfectly. He began his career in fluid mechanics with a real breakthrough: the discovery of a new invariant of Euler's equations (perfect fluids). This invariant was later given the name of helicity. His interest in cavitation problems led him to unilateral problems and then to convex analysis, of which he was one of the founders, together with Terry Rockafellar. The mathematical theory of convex analysis found immediate applications in plasticity and friction problems, where Jean-Jacques Moreau made major contributions. Then he devoted the last 20 years of his life to granular media and particle impact problems with again major contributions, mostly oriented toward Computational Mechanics, a field that he discovered and embraced in his 70s! In 1983, I was recruited by the University of Montpellier where Jean-Jacques worked and was lucky to spend a few years (1983–1988) with him.

At the beginning of the 1980s, two subjects were attracting a lot of attention in France: homogenization, which had been pioneered by Evariste (also called Enrique or Henri) Sanchez-Palencia (born 1941) and Ivo Babuska in the USA, and mechanics of materials, which was rapidly taking off in the USA. With regard to homogenization, Evariste was the first to apply asymptotic expansions, well known for multiscale problems in time, to multiscale problems in space. The subject attracted many applied mathematicians and mechanicians who benefited greatly from his work. After pioneering homogenization, he moved on to several other different problems involving a small parameter, in particular couplings in vibrating systems or shells with the same creativity.

During the same period (beginning of the 1980s), I had two important encounters, not only because of a certain vision of mechanics that we share but also because of the friendly ties that we forged. Jean-Jacques Marigo was working on damage using micromechanical methods. Together, we had our first PhD student, Jean-Claude Michel, with whom we did our first work combining damage and plasticity. The second person was Gilles Francfort, who was working on homogenization and had some very original results on high-frequency oscillations in thermoelastic composites. Gilles and Jean-Jacques met a few years later, and their joint work is at the origin of an absolutely beautiful variational theory of fracture.

One of my contributions in the mid-1980s has been to show the convergences between mathematical homogenization and micromechanics in the style of R. Hill or Z. Hashin. I was deeply inspired by the work of John Willis in Cambridge (UK) and André Zaoui in France. In 1982, I defended my habilitation thesis “Plasticity and Homogenization,” a good summary of my ambitions at the time, which were to understand in more depth how macroscopic internal variables reflect all sorts of dissipative phenomena at the microscopic scale.

In 1991, I had other encounters with colleagues, now friends, working in homogenization (all of us were in our 30s). Nick Triantafyllidis was the first to address the relations between instabilities and homogenization. I learned from him that a single unit cell is not sufficient when the strain energy loses convexity. The same year, I met Pedro Ponte Castañeda at a conference. He had just finished a post doc with J. Willis. In two papers, he introduced the concept that would later prove instrumental: the linear comparison composite. I could obtain the same results as him, but with a less general method. My consolation was to show a few years later that his method could be interpreted simply in terms of field fluctuations. Pedro came to Marseille for a few months. We worked together on low-contrast expansions with some interesting ideas, which were the seeds for future work. Later on, several visits of Pedro and Nick at Laboratoire de Mécanique et d'Acoustique (LMA) started a fruitful scientific collaboration with Jean-Claude Michel on homogenization and stability.

In 1988, I joined the Laboratoire de Mécanique et d'Acoustique in Marseille, which I headed from 1993 to 2000 and where I still work (as an emeritus researcher). Together with Jean-Claude Michel, my former student who had become an expert in finite elements, with Hervé Moulinec, a research engineer who had a PhD in image processing for astronomy, and with a lab technician, Frédéric Mazerolle, who was the first in France to investigate materials in 3D with computed tomography, we setup a research group with the ambitious program of achieving all the steps of a true micromechanical approach: starting from the observation of microstructures using X-ray tomography, numerically simulating the response of these microstructures and finally comparing these simulations on the one hand with in situ tests and on the other hand with nonlinear homogenization models. The numerical simulations were initially performed with the FEM. But we soon realized that the mesh needed to interface the computations with images, whether from tomography or microscopy, a 3D mesh in the case of tomography, was a formidable obstacle. We skipped it using fast Fourier transforms and devised a method for simulating the response of heterogeneous materials directly from images of their microstructures. The method has proved successful and is now incorporated in several codes around the world, mostly in the USA, Germany, and France.

This numerical work was also instrumental in the more theoretical project of assessing the suitability of different nonlinear homogenization methods, a subject that was born at the end of the 1980s and was undergoing major developments in particular on the initiative of John Willis and Pedro Ponte Castañeda, and in which I was naturally interested.

In 2000, freed from the directorship of my department, I was able to take a sabbatical at Caltech at the invitation of Kaushik Bhattacharya, whom I had met during a program at the Isaac Newton Institute in Cambridge, where John Willis was the principal organizer. Caltech is a place blessed by God, both for the beauty of the grounds and the quality of the people who frequent it, students and professors alike. Kaushik taught me a lot about phase transformations, and our discussions revived my old curiosity about internal variables. How dissipative phenomena occurring at a small scale can be embedded into macroscopic internal variables? I had more time to think about methods to do so that matured in the following years, incremental variational principles and effective potentials on the one hand, model reduction on the other hand.

On my return to France, preoccupied with other matters, it took me several years to sort out these ideas on internal variables. I knew G. Dvorak's work on transformation field analysis, but it had its limitations, which were the phase uniformity of the fields, whereas we had shown with Pedro that one of the keys to proper nonlinear homogenization was the fluctuations of these fields. Together with Jean-Claude Michel, we proposed a method that we called “nonuniform transformation field analysis.” Not only did we have a clear interpretation of our internal variables, but the computational cost of the reduced-order model compared to full-field simulations was cut by several orders of magnitude (typically a factor of 10⁴). The method was extended and improved by several other groups. At the same time, Noël Lahellec and I used incremental variational principles to introduce the notion of effective internal variables.

Now, some general thoughts about our field. People sometimes ask me why I do research in Mechanics. The answer is because Mechanics is useful and because it is beautiful. Let me elaborate briefly.

Mechanics spans three related fields: a science, a set of technologies, and an industry. Mechanics as a science is therefore naturally close to applications. The term “applications” may have different meanings. It may be through collaborations with industrial companies or through the creation of start-ups, but it may as well be through the development of concepts or methods used in other fields of science. No climate change modeling without mechanics, no seismology without deep knowledge of rheology, waves, and tribology, no insight into the development of an organism without understanding the role of mechanical forces at the cellular level, and no electronics industry without proper modeling of the thermal, electronic, and electromagnetic properties of thin film semiconductors. These applications often involve a fruitful collaboration with other disciplines and even though one can regret that mechanics is not always the most visible, and therefore the most rewarded, partner, these partnerships are vital to mechanics.

Mechanics is not only useful, it is also beautiful, in that it combines the rigor of mathematics with curiosity and observation of nature. For a long time, experiments were scarce and extensive data were missing, so that people's intelligence and imagination were the essential tools. Now, computational mechanics and artificial intelligence are at a stage of development such that accurate predictions can be made about various phenomena. But as René Thom said “Prediction is not explanation” echoing what Henri Poincaré, a precursor of the theory of relativity, wrote “Science is no more an accumulation of facts than a pile of stones is a house.” Machine learning is a fantastic tool and, like many people, I am amazed by what it can do for us. However, the ultimate goal is “understanding” and the Mechanics I find beautiful is that which enables us to understand.

Before wrapping up, I would like to add a word about the interactions that I had along my career. When I started out in research, it was mainly to experience a stimulating intellectual adventure, which I did. I discovered along the way that it was also an equally exciting human adventure. I have already mentioned work relationships that have become lasting friendships dear to my heart. But there have also been shorter encounters with brilliant students. A journalist once asked Paul Germain what his greatest discovery had been, expecting to hear about some theory or other. Paul replied “Jean-Pierre Guiraud”! It was the name of one of his very talented collaborators. One of the great rewards of education and research is meeting and exchanging ideas with brilliant young people. I have been lucky enough to do so, especially at Ecole Polytechnique where I taught for about 20 years, and although I cannot name them all, I would like to thank them all. It is up to us, as educators, to present young people with challenging problems in mechanics, both at the interfaces and at the core of our domain, and thus form the next generation of our field.

I cannot end this talk without thanking my wonderful wife Jeanne and our children for letting me do what I like. They have always been supportive and understanding. Our children are now beautiful adults, and they are certainly the achievements (as a team with my wife) I am the most proud of.

It is now time for me to let you go. Let me thank all of you for the Timoshenko medal and for your time and patience during this speech.