Author

# John M. Ball

Other affiliations: University of California, Berkeley, Brown University, Heriot-Watt University ...read more

Bio: John M. Ball is an academic researcher from University of Oxford. The author has contributed to research in topics: Martensite & Bounded function. The author has an hindex of 46, co-authored 123 publications receiving 13355 citations. Previous affiliations of John M. Ball include University of California, Berkeley & Brown University.

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2,329 citations

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TL;DR: In this article, the authors explore a theoretical approach to these fine phase mixtures based on the minimization of free energy and show that the α-phase breaks up into triangular domains called Dauphine twins which become finer and finer in the direction of increasing temperature.

Abstract: Solid-solid phase transformations often lead to certain characteristic microstructural features involving fine mixtures of the phases. In martensitic transformations one such feature is a plane interface which separates one homogeneous phase, austenite, from a very fine mixture of twins of the other phase, martensite. In quartz crystals held in a temperature gradient near the α-β transformation temperature, the α-phase breaks up into triangular domains called Dauphine twins which become finer and finer in the direction of increasing temperature. In this paper we explore a theoretical approach to these fine phase mixtures based on the minimization of free energy.

1,488 citations

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TL;DR: In this paper, the authors make predictions based on an analysis of a new nonlinear theory of martensitic transformations introduced by the authors, where the crystal is modelled as a nonlinear elastic material, with a free-energy function that is invariant with respect to both rigid-body rotations and the appropriate crystallographic symmetries.

Abstract: Predictions are made based on an analysis of a new nonlinear theory of martensitic transformations introduced by the authors The crystal is modelled as a nonlinear elastic material, with a free-energy function that is invariant with respect to both rigid-body rotations and the appropriate crystallographic symmetries The predictions concern primarily the two-well problem, that of determining all possible energy-minimizing deformations that can be obtained with two coherent and macroscopically unstressed variants of martensite The set of possible macroscopic deformations obtained is completely determined by the lattice parameters of the material For certain boundary conditions the total free energy does not attain a minimum, and the finer and finer oscillations of minimizing sequences are interpreted as corresponding to microstructure The predictions are amenable to experimental tests The proposed tests involve the comparison of the theoretical predictions with the mechanical response of properly oriented plates subject to simple shear Additional crystallographic background is given for the model, and the theory is compared with the `linearized' model of Khachaturyan, Roitburd and Shatalov There are some similarities in the predictions of the two theories, but also some major discrepancies

694 citations

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TL;DR: In this article, the existence of singular solutions to the nonlinear elastostatics problem with respect to radial motion has been studied for a class of strongly elliptic materials by means of the direct method of the calculus of variations, and it has been shown that the only radial equilibrium solutions without cavities are homogeneous.

Abstract: A study is made of a class of singular solutions to the equations of nonlinear elastostatics in which a spherical cavity forms at the centre of a ball of isotropic material placed in tension by means of given surface tractions or displacements. The existence of such solutions depends on the growth properties of the stored-energy function W for large strains and is consistent with strong ellipticity of W . Under appropriate hypotheses it is shown that a singular solution bifurcates from a trivial (homogeneous) solution at a critical value of the surface traction or displacement, at which the trivial solution becomes unstable. For incompressible materials both the singular solution and the critical surface traction are given explicitly, and the stability of all solutions with respect to radial motion is determined. For compressible materials the existence of singular solutions is proved for a class of strongly elliptic materials by means of the direct method of the calculus of variations, an important step in the analysis being to show that the only radial equilibrium solutions without cavities are homogeneous. Work of Gent & Lindley (1958) shows that the critical surface tractions obtained agree with those observed in the internal rupture of rubber.

660 citations

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01 Jan 1989643 citations

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01 Dec 1992TL;DR: In this paper, the existence and uniqueness of nonlinear equations with additive and multiplicative noise was investigated. But the authors focused on the uniqueness of solutions and not on the properties of solutions.

Abstract: Part I. Foundations: 1. Random variables 2. Probability measures 3. Stochastic processes 4. The stochastic integral Part II. Existence and Uniqueness: 5. Linear equations with additive noise 6. Linear equations with multiplicative noise 7. Existence and uniqueness for nonlinear equations 8. Martingale solutions Part III. Properties of Solutions: 9. Markov properties and Kolmogorov equations 10. Absolute continuity and Girsanov's theorem 11. Large time behaviour of solutions 12. Small noise asymptotic.

4,042 citations

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31 Dec 1988

TL;DR: In this article, the authors consider a continuous dynamical system with a global attractor and describe the properties of the flow on the attractor asymptotically smooth and Morse-Smale maps.

Abstract: Discrete dynamical systems: Limit sets Stability of invariant sets and asymptotically smooth maps Examples of asymptotically smooth maps Dissipativeness and global attractors Dependence on parameters Fixed point theorems Stability relative to the global attractor and Morse-Smale maps Dimension of the global attractor Dissipativeness in two spaces Continuous dynamical systems: Limit sets Asymptotically smooth and $\alpha$-contracting semigroups Stability of invariant sets Dissipativeness and global attractors Dependence on parameters Periodic processes Skew product flows Gradient flows Dissipativeness in two spaces Properties of the flow on the attractor Applications: Retarded functional differential equations Sectorial evolutionary equations A scalar parabolic equation The Navier-Stokes equation Neutral functional differential equations Some abstract evolutionary equations A one-dimensional damped wave equation A three-dimensional damped wave equation Remarks on other applications Dependence on parameters and approximation of the attractor.

2,639 citations

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2,329 citations

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TL;DR: In this paper, a variational model of quasistatic crack evolution is proposed, which frees itself of the usual constraints of that theory : a preexisting crack and a well-defined crack path.

Abstract: A variational model of quasistatic crack evolution is proposed. Although close in spirit to Griffith’s theory of brittle fracture, the proposed model however frees itself of the usual constraints of that theory : a preexisting crack and a well-defined crack path. In contrast, crack initiation as well as crack path can be quantified, as demonstrated on explicitly computable examples. Furthermore the model lends itself to numerical implementation in more complex settings.

2,283 citations

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TL;DR: In this article, the authors define a notion of "two-scale" convergence, which is aimed at a better description of sequences of oscillating functions, and prove that bounded sequences in $L^2 (Omega )$ are relatively compact with respect to this new type of convergence.

Abstract: Following an idea of G. Nguetseng, the author defines a notion of “two-scale” convergence, which is aimed at a better description of sequences of oscillating functions. Bounded sequences in $L^2 (\Omega )$ are proven to be relatively compact with respect to this new type of convergence. A corrector-type theorem (i.e., which permits, in some cases, replacing a sequence by its “two-scale” limit, up to a strongly convergent remainder in $L^2 (\Omega )$) is also established. These results are especially useful for the homogenization of partial differential equations with periodically oscillating coefficients. In particular, a new method for proving the convergence of homogenization processes is proposed, which is an alternative to the so-called energy method of Tartar. The power and simplicity of the two-scale convergence method is demonstrated on several examples, including the homogenization of both linear and nonlinear second-order elliptic equations.

2,279 citations