Some Open Problems in Elasticity

By J. Ball. 2002

381 citations recorded by the source corpus.

Subfields: Biomedical Engineering, Computational Theory and Mathematics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/e42579e51c27dac13b1aaa7cc060bd7afa51e05e

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In Biomedical Engineering

In Computational Theory and Mathematics

In Mechanics of Materials

Citations in the corpus

Anisotropic polyconvex energies on the basis of crystallographic motivated structural tensors (2008, background): “A drawback of the quasiconvexity condition is that the growth condition to be satisfied precludes the physically reasonable requirement W (F ) → ∞ for detF → 0, see problem 1 in Ball [3].”
Atomistic to continuum limits for computational materials science (2007, background): “The main difficulty is that quasiconvexity is a nonlocal condition [10], in general difficult to check.”
Injective weak solutions in second-gradient nonlinear elasticity (2009, background): “In our case, the latter is guaranteed only for s ≤ q, directly by finite energy alone, whereas in [6] it is observed that this integrability can be obtained (for arbitrary s > 0), exploiting the fact that a minimizer u∗ is always a weak solution of the Eshelby conservation law [3,4].”
Progress and puzzles in nonlinear elasticity (2010, background): “If Ω is homeomorphic to a ball, ∂Ω = ∂Ω1, W strictly polyconvex, are minimizers (or sufficiently smooth equilibrium solutions) unique? There are well-known counterexamples to uniqueness when ∂Ω1 6= ∂Ω or if Ω has holes (see [7] Section 2.”
Fracture Surfaces and the Regularity of Inverses for BV Deformations (2011, background): “It was shown by Ball [8] (see also [9,24]) that”
Quasiconvex functions incorporating volumetric constraints are rank-one convex (2008, background): “For a review of these concepts see, e.g., [7, 19, 3 , 10].”
Local Minimizers and Quasiconvexity – the Impact of Topology (2005, methodology): “We introduce and exploit the notion of a topologically non-trivial domain and under this establish the first existence and multiplicity result for strong local minimizers of F[·] that in turn settles a longstanding open problem in the multi-dimensional calculus of variations as described in [6].”
Existence results for a class of rate-independent material models with nonconvex elastic energies (2006, background): “Parts (a) and (b) of the following proposition are proved in [3].”
Geometry of Logarithmic Strain Measures in Solid Mechanics (2015, background): “While every such energy function can be taken as a strain measure, many additional conditions for “proper” energy functions are discussed in the literature, such as constitutive inequalities [202, 106, 107, 11, 44, 127], generalized convexity conditions [10, 13] or monotonicity conditions to ensure ”
Regularity of minima: An invitation to the dark side of the calculus of variations (2006, background): “Quasiconvexity plays an important role in the context of non-linear elasticity, as discussed in the fundamental work of Ball [23].”
Discontinuous Galerkin methods for non‐linear elasticity (2006, background): “For a fairly recent perspective on the status of nonlinear elasticity theory we refer the reader to [4], where a number of open problems are stated and discussed.”