Walter Noll: The Mathematician Who Rebuilt the Foundations of Continuum Mechanics

Walter Noll (1925 to 2017) was a German-born American mathematician who spent most of his career at Carnegie Mellon University in Pittsburgh. Outside a fairly specialised circle his name is not widely known, yet his ideas sit underneath much of modern solid mechanics, fluid mechanics and the theory of materials. Engineers who model rubber, polymer melts, blood, soils or biological tissue on a computer are using a framework that Noll did more than anyone to make precise: the idea that a material's behaviour can be described by a "constitutive law" that must obey certain general principles, the most famous of which he called the principle of material objectivity, now known as the principle of material frame-indifference.

Noll's life also tells a remarkable twentieth-century story. He grew up in a Social Democratic family in Nazi Berlin, survived the war as a teenage conscript, studied mathematics while starving in the ruins of postwar Berlin, finished an American doctorate in less than a year, and then chose a career in the United States over a full professorship in Germany. This article covers his life, his main scientific ideas in plain terms, his books, his views on mathematics, and the common misunderstandings about his work.

Quick facts

Childhood in Berlin

Most of what is known about Noll's early life comes from his own "Autobiographical Notes," written in 1988 and posted on his Carnegie Mellon website.

He was born on 7 January 1925 in Berlin-Biesdorf, in the north-east of the city. His parents, Franz and Martha Noll, had only an eight-year grade-school education. His mother had worked as a maid; his father trained at a trade school, became a tool-and-die maker, rose to foreman and division manager at a Berlin mica-products company, and originated five patents. His father was active in the Social Democratic Party from his youth. In 1917, during the First World War, he gave a speech at a rally of striking munitions workers and spent the last nine months of the war in prison. When the Nazis took power in 1933 he predicted a second world war, that the United States would again fight Germany, and that Germany would again be defeated. Noll wrote that even as a child he never doubted his father was right.

In 1931 the family moved to Zeuthen, a suburb south-east of Berlin, into a house they had built. Noll later thought the move had protected his parents, since their political views were less well known in the new community. He recorded two incidents that shaped him: the family's former doctor in Biesdorf, who had attended his birth, was kidnapped by Nazi stormtroopers and beaten to death in 1933 or 1934; and a friend of his parents was sent to a concentration camp for several months for a critical remark about the regime, returned "a broken man," and later took his own life. Noll learned never to discuss politics with anyone outside his family.

An unlikely mathematician

Noll's account of his schooling is notable for how unpromising it was in parts. He said his facility with arithmetic "was never very good, and still isn't," that his grades in arithmetic almost never exceeded a C, and that his memory for numbers was extremely poor. What caught him was mathematics as a structure. Geometry arrived in the seventh grade, taught by an authoritarian teacher who made the class memorise Euclid's axioms as dogma.

The turning point was a physics and mathematics teacher, Rudolf Hohensee, who recognised Noll's talent, gave him private instruction on Sundays, and recommended a small book on set theory by Kamke that Noll found fascinating. Noll also read popular physics and philosophy, including Kant's Critique of Pure Reason. He passed his Abitur, the school-leaving examination, on 23 March 1943 with the designation "Gut," at a time when, he noted, fewer than 10 percent of people his age obtained the Abitur at all.

War and survival

"My principal concern from 1943 until the end of the war was survival," Noll wrote. Many classmates were killed, including his best friend. In June 1943 he was drafted into the Reichsarbeitsdienst, then essentially a basic military training unit, and sent to southern France. A minor injury became severely infected and nearly cost him his right leg; a surgeon at the military hospital in Perpignan operated within minutes of his arrival. After months in hospital he was sent home just before Christmas 1943, and, apparently because his papers were misfiled, he was left at home for six months. During that time he attended classes at the University of Berlin in modern algebra and advanced calculus without officially enrolling.

On 5 June 1944 he was drafted into the air force signal corps. He saw little action, partly because he contracted diphtheria in late 1944 and spent six weeks in isolation. On 1 May 1945 he was taken prisoner by a British soldier in Lübeck and released on 7 July. His parents did not know whether he was alive until October 1945. While working on a farm and in a factory in north-west Germany, he taught himself English from a borrowed textbook and radio lessons, because, he wrote, "one cannot get anywhere in this world without knowing English." He never had formal instruction in the language.

Studies in Berlin and Paris

On 12 March 1946 Noll returned to Berlin to enrol at the Technische Universität, the Technical University. As a condition of enrolment students had to spend 100 hours clearing rubble left by the war. The next two years were extremely hard: he described freezing in winter and "starving all the time." He chose the Technical University, an engineering school, only because he could start a semester earlier there than at the Humboldt University. The price was four semesters of engineering mechanics, a requirement he would have avoided elsewhere. It turned out to be one of the most consequential accidents of his career.

At the Technical University he came to the attention of Istvan Szabó, an instructor who later became a senior professor of engineering mechanics, and who employed him as a student assistant. He also attended courses as a guest student at the Humboldt University, including number theory with Helmut Hasse and mathematical logic with Karl Schröter, and later quantum mechanics with Günther Ludwig at the Free University.

In 1949 to 1950 he spent a year at the University of Paris on a French government fellowship. There he encountered the work of Nicolas Bourbaki, the collective pseudonym of a group of French mathematicians who were reorganising mathematics around a small number of fundamental structures. He saved enough of his allowance to buy all the volumes of Bourbaki's Éléments de mathématique then published. He passed three demanding examinations and earned the licence ès sciences. In one of them, "Calcul différentiel et intégral," only 49 of 296 candidates passed; Noll received a "bien."

Back in Berlin, in May 1951 he received the degree of Diplom-Ingenieur with the mark "Ausgezeichnet," summa cum laude. Despite the title it was a mathematics degree. His thesis was on reproducing kernels and their applications to analytic functions. Its introduction, in his own later translation, already stated the outlook that would shape his whole career:

"There has been a realization that only a few fundamental structures are at the basis of all branches of mathematics. The discovery of these structures leads to deeper insight and above all to considerable simplification."

Into mechanics, almost by chance

After his diploma Noll looked for a position as a scientific assistant with a professor of mathematics. None was available, but Szabó, now a senior professor of engineering mechanics, offered him one at his institute from October 1951. "Thus, it was almost by chance that I became involved with mechanics," Noll wrote. Much of his work there involved helping Szabó write books, including a major revision of the mathematical formulas section of the engineering handbook Hütte.

Indiana and Truesdell

In late 1952 Noll saw a pamphlet from the Graduate Institute for Applied Mathematics at Indiana University offering research assistantships. It had been sent to the retired mechanician Georg Hamel, who was on the editorial board of the Journal of Rational Mechanics and Analysis, edited at Indiana by Clifford Truesdell. Hamel passed it to Szabó, who showed it to Noll. Truesdell later wrote that he had asked Hamel to send him an advanced graduate student from Germany. Noll applied, won a Fulbright travel grant, and sailed for New York, arriving in Bloomington, according to Truesdell, on 19 September 1953.

Noll had not realised he was enrolling as a graduate student, a category that did not exist in German universities. He learned that a doctorate required a qualifying examination, language examinations, courses and a thesis, and decided to try to finish in one year. He passed the qualifying examination in October. He took courses including hydrodynamics with Truesdell and partial differential equations with David Gilbarg.

He asked Truesdell to be his adviser because he thought a thesis in mechanics would help him on his return to Berlin. He did not know, he later wrote, that Truesdell was already becoming the leading figure in the field, or that he himself was Truesdell's first doctoral student.

The Indiana circle

Noll landed in an unusual place. Truesdell's 1993 account describes the Graduate Institute for Applied Mathematics as recently founded by T. Y. Thomas, with Eberhard Hopf and David Gilbarg among the professors; students had full scholarships and no duties except to study and do research. Two earlier students, James Serrin and Jerald Ericksen, had just finished there and went on to major careers in fluid mechanics and the theory of materials. Noll's autobiographical notes add Max Zorn, Václav Hlavatý and William Gustin to the faculty of the mathematics department.

Truesdell himself was then in his early thirties. Born in Los Angeles in 1919, he had trained at Caltech and Princeton, worked on mechanics for the U.S. Navy in the 1940s, and taught at Indiana from 1950 to 1961 before moving to Johns Hopkins as professor of rational mechanics until 1989. He founded and edited the Archive for Rational Mechanics and Analysis, where much of Noll's later work appeared, and the Archive for History of Exact Sciences. Noll later described him as a "singularity" among twentieth-century scientists, a man who admired the eighteenth century, wrote letters with a quill pen and regarded Leonhard Euler as the greatest mathematician of all time.

Truesdell's account of Noll's qualifying examination, held on 14 October 1953, is revealing. Noll offered a list of more than two dozen subjects, from complex analysis and Lie groups to Galois theory, algebraic topology, quantum mechanics and game theory. He handled almost every question easily. The exception was a question from Truesdell: derive the laws of continuum mechanics from Newton's laws. Noll quickly saw that the standard arguments were not rigorous. Truesdell's point was that continuum mechanics should be treated as an independent science based on Euler's laws of balance, not as a consequence of particle mechanics, and that point became central to Noll's later axiomatic work.

The thesis

Truesdell handed him a paper by the Polish mathematician Stanisław Zaremba and asked him to make sense of it. Truesdell's later account says Zaremba had published the basic ideas in 1903 and the paper he gave Noll dated from 1937. Noll's thesis generalised ideas of Maxwell and Zaremba to a class of constitutive relations in which the rate of change of stress depends on the velocity gradient, the stress and the density. In working it out, Noll realised that the earlier three-dimensional extensions of Maxwell's idea were not invariant under changes of the observer's frame of reference. He formulated a general requirement to fix this, which he called "the principle of isotropy of space." It was the first version of the idea he would later call the principle of material objectivity.

He submitted 114 typewritten pages on 26 July 1954, defended on 9 August and received the Ph.D. on 7 September 1954, less than a year after arriving. The thesis, On the Continuity of the Solid and Fluid States, was published in the Journal of Rational Mechanics and Analysis volume 4 (1955), pages 3 to 81. Truesdell noted that Noll replaced Zaremba's approximate solutions with exact ones, including detailed treatments of simple shearing flow and Poiseuille flow.

When Noll met the mechanician Ronald Rivlin at Brown University shortly afterwards and mentioned how quickly he had finished, Rivlin reportedly said, "They must be giving away their Ph.D. there now."

Choosing America

Noll returned to Berlin for one more year as an assistant. He married Helga Schönberg on 1 April 1955. In late 1954 the University of Southern California had approached him, and in September 1955 the couple emigrated, crossing to Montreal on a small freighter and taking a bus to Los Angeles. At USC he was an instructor on a salary of $4,500.

In December 1955 he received a letter from Walter Leighton, head of mathematics at the Carnegie Institute of Technology in Pittsburgh, saying he had been nominated by Truesdell and Gilbarg. The offer, made in January 1956, was an associate professorship with a five-year contract at $7,000 for the academic year. He accepted on 4 February 1956.

Truesdell wrote to him on 14 February 1956 with advice that Noll kept for the rest of his life: "maintain the level set by your two papers in the Journal. Nowadays so many people are publishing like rabbits that volumes of papers make little impression." Noll's publication list, modest in length by modern standards and dense with foundational papers, suggests he took it to heart.

At almost the same moment, Szabó wrote from Berlin, first offering him a junior academic post at the Technical University and then indicating that he would very likely be offered a Lehrstuhl, a full professorship, the summit of a German academic career and rarely offered to someone so young. Noll wrote that the situation forced him to choose definitively between an academic life in Germany and one in the United States. "I chose the United States, and I have never regretted the decision." He and Helga arrived in Pittsburgh in early September 1956, after a road trip that included a car accident in Idaho. He remained at Carnegie Tech, which became Carnegie Mellon University, for the rest of his career.

The science: what is continuum mechanics?

To understand why Noll matters, it helps to understand the problem he worked on.

Balance laws and constitutive laws

Continuum mechanics treats materials as continuous, ignoring their atomic structure, and describes them with fields such as density, velocity, stress and temperature that vary smoothly from point to point. Two kinds of equations are involved.

1. Balance laws apply to every material. They express conservation of mass, balance of linear momentum and balance of angular momentum, and in thermomechanics balance of energy. They are the same for steel, honey and air.
2. Constitutive laws describe what makes a particular material special: how stress in it responds to deformation, rate of deformation, temperature and history. Hooke's law for a linear elastic spring and the Navier-Stokes relation for a viscous fluid are classical examples.

The balance laws alone are not enough to predict anything. You need a constitutive law to close the system. For most of the nineteenth and early twentieth centuries, constitutive laws were mainly linear and were proposed case by case. After the Second World War, new materials, especially polymers and rubber, behaved in strongly non-linear ways that the classical theories could not describe. Truesdell later wrote that in the late 1940s a new kind of continuum mechanics began to be explored, based on the "maximal, often difficult use of general principles" rather than special cases.

The question Noll took on was: what general rules must any constitutive law obey, whatever the material? And can one define, in mathematically precise terms, what a "material" even is?

Frame-indifference in plain terms

The best known of Noll's general rules is the principle of material objectivity, now called the principle of material frame-indifference. Wikipedia states it as the idea that the constitutive laws governing a system's internal conditions and interactions "should not depend on whatever external frame of reference is used to describe them."

Here is the intuition. Imagine a rubber band stretched between your fingers. Its stress depends on how much it is stretched. Now imagine describing the same stretched band from a merry-go-round that is spinning. From the spinning observer's point of view, the band is rotating, but nothing about the band itself has changed. A correct law for the band's stress must give the same physical answer for both observers, after accounting for the rotation of the description. If a proposed law predicted different stress depending on whether the observer was spinning, the law would be describing the observer, not the material.

Worked example (hypothetical)

Suppose someone proposed, for a fluid, that the stress depends on the fluid's velocity: faster-moving fluid has more stress. This fails frame-indifference immediately. An observer moving alongside the fluid would see it at rest and predict no stress, while an observer standing still would predict a large stress. Same fluid, different predicted stress. The law is ruled out.

Now suppose instead the stress depends on how the velocity changes from point to point, the velocity gradient. That is better, but still not enough. The velocity gradient can be split into a stretching part, which describes how neighbouring bits of fluid move apart or together, and a spin part, which describes local rotation. A rotating observer sees a different spin but the same stretching, rotated. A frame-indifferent law can therefore depend on the stretching part but not, in that simple way, on the spin. The classical law for a Newtonian viscous fluid has exactly this property, which is a quick check that the principle is consistent with well-established physics.

This kind of reasoning lets theorists eliminate whole families of candidate laws before any experiment, and it tells engineers what form a valid constitutive law can take.

History of the principle

Noll did not work in a vacuum, and he and Truesdell were candid about predecessors. Truesdell wrote that a principle of this kind had been stated by the British rheologist James Oldroyd in 1950, "but we did not perceive it." In The Non-Linear Field Theories of Mechanics, Noll and Truesdell included a brief history of two forms of the principle, which they named the Hooke-Poisson-Cauchy form and the Zaremba-Jaumann form; Noll's thesis used the former. Over time Noll's own name for it changed from "principle of isotropy of space" to "principle of objectivity" to "principle of material frame-indifference." Wikipedia, citing Miroslav Šilhavý's 1997 textbook The Mechanics and Thermodynamics of Continuous Media, notes that "material objectivity" is now considered obsolete terminology.

Simple materials and the axiomatic method

The 1958 paper

Noll's most influential single paper is "A mathematical theory of the mechanical behavior of continuous media," published in the Archive for Rational Mechanics and Analysis, volume 2, pages 197 to 226, in 1958. In it he introduced the concept of a simple material: one in which the stress at a point is determined by the history of the deformation in an arbitrarily small neighbourhood of that point, specifically by the history of the deformation gradient. This one definition includes elastic solids, viscous fluids, viscoelastic materials with memory and many others as special cases.

The power of the definition is that it lets one reason about very large classes of materials at once. Once frame-indifference and material symmetry are imposed on the general form, one can prove results that hold for every simple material, and then specialise. In particular the theory defines, in mathematical terms, what it means for a material to be a solid or a fluid, through the symmetry group of the material. A fluid is a material whose response is unchanged by any volume-preserving rearrangement of a reference configuration; a solid has a much smaller symmetry group. The title of his thesis, On the Continuity of the Solid and Fluid States, hints at how long this question had occupied him.

Axiomatic foundations

Around the same time Noll presented "The Foundations of Classical Mechanics in the Light of Recent Advances in Continuum Mechanics" at an International Symposium on the Axiomatic Method held at the University of California, Berkeley, from 27 December 1957 to 4 January 1958, published in 1959. He also wrote a 68-page Carnegie technical report, On the Foundations of the Mechanics of Continuous Media (1957), and a paper in French on classical mechanics based on an "axiom of objectivity," presented at a Paris colloquium in 1959.

The ambition was to do for mechanics what Bourbaki had done for algebra: to build it from a small number of clearly stated mathematical structures and axioms, with concepts such as body, force, motion and material defined precisely rather than taken from intuition. Much of his later work returned to this programme, including "A new mathematical theory of simple materials" (1972) and "Lectures on the foundations of continuum mechanics and thermodynamics" (1973), both in the Archive.

The Coleman-Noll partnership

From 1959 into the mid-1960s Noll wrote a long series of papers with Bernard D. Coleman. Their joint work, as listed in Noll's curriculum vitae, includes:

Two themes stand out.

Fading memory. Real materials such as polymer melts remember their past deformation, but recent events matter more than distant ones. The 1960 approximation theorem made this idea mathematically precise and showed how classical theories, such as the Newtonian fluid and higher-order fluid models, arise as approximations of a general simple material for slow motions. This gave the familiar engineering models a rigorous place inside the general theory.

Thermodynamics as a restriction. The 1963 paper on elastic materials with heat conduction and viscosity is the source of what is widely called the Coleman-Noll procedure. The idea is to require that every possible process allowed by the balance laws and the constitutive equations must satisfy the second law of thermodynamics, in the form of an entropy inequality. Imposing that requirement forces strong restrictions on the constitutive equations, for example that stress in an elastic material can be derived from a free-energy function. The procedure became a standard tool in the field known as rational thermodynamics, and is still used to check new constitutive models.

With Coleman and Hershel Markovitz, Noll also wrote Viscometric Flows of Non-Newtonian Fluids: Theory and Experiment (1966, 130 pages), number 5 in Springer's Tracts in Natural Philosophy. It connected the general theory to measurements in instruments used to test polymer fluids, showing that a wide class of flows used in viscometers could be analysed exactly for every simple fluid.

The Non-Linear Field Theories of Mechanics

Origin

Noll's best-known book was written with Truesdell for Springer's Encyclopedia of Physics (Handbuch der Physik). According to Noll's 2002 lecture on its genesis, Truesdell agreed in 1953 to write for the encyclopedia an exposition of the foundations of continuum mechanics. It grew so large that it was split in two. The first part, The Classical Field Theories, written by Truesdell with the collaboration of Toupin and Ericksen and published in 1960, ran to more than 600 pages. The second was to be The Non-Linear Field Theories of Mechanics, on constitutive equations.

In December 1956 Truesdell wrote to Noll proposing that he join the project, originally planned with Jerald Ericksen, and do a major part of the work, with the order of authors decided by the amount each contributed. Noll accepted on 2 January 1957 and sketched an outline: general principles of constitutive equations first, then finite elasticity and non-linear fluid theory, then more general theories.

A slow, difficult collaboration

The writing took years. Noll explained the tension in his genesis lecture: Truesdell was very good at reading, digesting and summarising existing literature, while Noll could not help "rethinking and recreating whatever subject I wish to understand." That improved the result but slowed it down. At one low point Truesdell, frustrated by long silences, wrote that Noll should withdraw from the article. They persisted.

The book

The Non-Linear Field Theories of Mechanics was published in 1965 as volume III/3 of the Encyclopedia of Physics, 602 pages long. A second edition followed in 1992, a Chinese translation in 2000 and a third edition in 2004. Noll's own website describes it as the standard reference work in the field. It presented the general theory of constitutive equations, the principle of frame-indifference, simple materials, finite elasticity, fluids with memory and much else in a unified way, and it shaped how a generation of researchers in mechanics thought about materials.

Teaching, pure mathematics and later work

Noll was a mathematician first, and much of his later output is in pure mathematics and teaching.

• Finite-Dimensional Spaces. His textbook Finite-Dimensional Spaces: Algebra, Geometry, and Analysis, Volume I (Martinus Nijhoff, 1987, 393 pages), presents linear algebra, geometry and analysis in the coordinate-free style he favoured. A corrected version and preliminary chapters of an unfinished second volume were posted on his website.
• Special relativity. He taught a course on special relativity from about 1970 to 1993, and with Vincent J. Matsko wrote Mathematical Structures of Special Relativity (1993, 239 pages).
• Honours programme. In 1973, with Juan J. Schäffer, he became involved in an undergraduate honours programme at Carnegie Mellon called "Mathematical Studies."
• Geometry of bodies and contact. With Epifanio Virga he wrote on "fit regions" and functions of bounded variation (1988) and on edge interactions and surface tension (1990), and alone on the geometry of contact, separation and reformation of continuous bodies (1993): the question of what mathematical objects should count as the "bodies" and "parts" of continuum mechanics.
• Frame-free formulations. Late in life he worked on formulating elasticity and thermomechanics without any reference frame at all, including "A Frame-Free Formulation of Elasticity" in the Journal of Elasticity (2006), and with his student Brian Seguin.

Sixteen of his mechanics papers were collected in The Foundations of Mechanics and Thermodynamics: Selected Papers (Springer, 1974). In 2004 he posted Five Contributions to Natural Philosophy, 73 pages, dedicated to Truesdell's memory.

He served as a visiting professor for a month or more at Johns Hopkins University, the University of Karlsruhe, the Technion (Israel Institute of Technology), the École Polytechnique in Nancy, the University of Pisa, the University of Pavia and Oxford University. He retired from teaching in 1993, and Truesdell marked the occasion with a lecture at a meeting of the Society for Natural Philosophy that April, later posted as "The Naturalization and Apotheosis of Walter Noll."

Noll's view of mathematics

Structure over computation

Noll's thinking is consistent from his 1951 diploma onward: mathematics should be organised around a few fundamental structures, and good mathematical language makes hard ideas simple. He preferred coordinate-free, "direct" notation to the index-heavy notation common in physics and engineering. Truesdell at first made him add coordinate versions of every formula to his thesis, but in a letter of 4 August 1958 conceded: "I must also admit that the direct notations you use are better suited to fundamental questions than are indicial notations."

Natural philosophy

Noll and Truesdell both used the old term "natural philosophy" for the endeavour of understanding nature with mathematics as a conceptual tool, as Newton's Philosophiae Naturalis Principia Mathematica had. Noll's 2005 essay "On the Past and Future of Natural Philosophy" quotes Truesdell's explanation that he revived the term after finding his work dismissed by physicists as mathematics and by mathematicians as engineering. The distinctive claim of this school, in Truesdell's words, was that "the physical concepts themselves are made mathematical at the outset, and mathematics is used to formulate theories," rather than mathematics being applied only after the physics is settled.

Essays

His website collects short essays with titles that convey his attitudes: "The Role of the Professor" (1997), "The Conceptual Infrastructure of Mathematics" (1995), "Mathematics should not be boring" (2003), "What is Mathematics all about" (2006) and "The Future of Scientific Publication" (2008).

Recognition

Noll was not a prize-collector in the conventional sense, and his influence is best measured by how standard his concepts became. A book by Y. Ignatieff, The Mathematical World of Walter Noll, was published by Springer in 1996. In 2012 he was named a Fellow of the American Mathematical Society He died in Pittsburgh on 6 June 2017, aged 92.

Common misconceptions

"Noll invented frame-indifference out of nothing." Not quite. Truesdell acknowledged that Oldroyd had stated a related principle in 1950, and Noll's own thesis built on Maxwell and Zaremba. What Noll did was state the principle generally and precisely, and make it a cornerstone of the whole theory of materials.

"Frame-indifference means physics is the same in every frame." That is a statement about the laws of motion, and in classical mechanics the balance of momentum does change form in accelerating or rotating frames, picking up fictitious forces. Frame-indifference is a narrower claim about constitutive laws: the material's response must not depend on the observer.

"Noll was an engineer." His Berlin degree was titled Diplom-Ingenieur, but it was a mathematics degree, and he worked in mathematics departments all his career. He came to mechanics by the accident of his first job.

"The Non-Linear Field Theories of Mechanics is mainly Truesdell's book." The book was a genuine collaboration. Noll's letters and genesis lecture show he drafted much of the general theory of constitutive equations, which was his specialty.

"The theory has been superseded." The terminology has changed, and frame-indifference has been debated, but the framework of balance laws plus constitutive laws restricted by general principles remains standard in continuum mechanics and computational mechanics.

Frequently asked questions

What is a constitutive equation?

A constitutive equation is a rule that describes how a particular material responds, for example how its stress depends on its deformation, deformation rate or temperature. Together with the universal balance laws, it lets you predict how a body of that material will behave.

What is the principle of material frame-indifference?

It is the requirement that a constitutive law must not depend on the frame of reference of the observer describing the material. Noll first formulated it in his 1954 thesis as the principle of isotropy of space and later called it the principle of material objectivity.

What is a simple material?

In Noll's 1958 definition, a material in which the stress at a point is determined by the history of the deformation gradient at that point. The class includes elastic solids, viscous fluids and materials with memory.

Who was Noll's adviser, and was he really the first student?

His adviser was Clifford Truesdell at Indiana University. Noll's own curriculum vitae states that he was the first of Truesdell's doctoral students.

Where can his writings be found?

Many of his papers, his textbook on finite-dimensional spaces and his essays, including the autobiographical notes used for this article, are posted on his Carnegie Mellon web page. Sixteen mechanics papers are collected in the 1974 Springer volume of selected papers.

What is the Society for Natural Philosophy?

It is the scholarly society associated with Truesdell's revival of the term "natural philosophy" for mathematically grounded science of nature. Noll's career is threaded through its meetings: Truesdell marked Noll's retirement with a lecture at its April 1993 meeting, and Noll gave his lecture on the genesis of The Non-Linear Field Theories of Mechanics at a meeting held in Truesdell's memory in Pisa, Italy, in November 2000. That lecture was published in the Journal of Elasticity in 2003.

Why did he not return to Germany?

He wrote that the near-simultaneous offers of an associate professorship at Carnegie Tech and a likely full professorship in Berlin forced him to choose, and that he chose the United States and never regretted it. His wife Helga also favoured staying.

Summary

Walter Noll came to mechanics by chance and changed it permanently. From a thesis written in under a year at Indiana, he extracted a general principle, that material laws must not depend on the observer, which became the principle of material frame-indifference. In 1958 he defined the simple material, a single concept broad enough to include solids, fluids and materials with memory, and with Bernard Coleman he showed how fading memory and the second law of thermodynamics constrain how materials can behave. With Clifford Truesdell he wrote The Non-Linear Field Theories of Mechanics, the standard reference of the field. Behind all of it was a conviction, stated in his 1951 diploma thesis, that a few fundamental structures underlie all of mathematics, and that finding them brings both insight and simplicity.

Sources

• Walter Noll (Wikipedia)
• Walter Noll's home page, Carnegie Mellon University
• Walter Noll, "Autobiographical Notes" (1988)
• Walter Noll, "The Genesis of the Non-Linear Field Theories of Mechanics" (2002)
• Clifford Truesdell, "The Naturalization and Apotheosis of Walter Noll" (1993)
• Walter Noll, "On the Past and Future of Natural Philosophy" (2005)
• Walter Noll, Curriculum vitae and list of publications
• Clifford Truesdell (Wikipedia)
• Objectivity (frame invariance) (Wikipedia))