A new example for the Lavrentiev phenomenon in nonlinear elasticity

Z. Angew. Math. Phys. (2024) 75:2 c 2023 The Author(s)  https://doi.org/10.1007/s00033-023-02132-4 Zeitschrift für angewandte Mathematik und Physik ZAMP A new example for the Lavrentiev phenomenon in nonlinear elasticity Stefano Almi, Stefan Krömer and Anastasia Molchanova Abstract. We present a new example for the Lavrentiev phenomenon in context of nonlinear elasticity, caused by an interplay of the elastic energy’s resistance to infinite compression and the Ciarlet–Nečas condition, a constraint preventing global interpenetration of matter on sets of full measure. Mathematics Subject Classification. 74B20, 46E35. Keywords. Nonlinear elasticity, Local injectivity, Global injectivity, Ciarlet–Nečas condition, Lavrentiev phenomenon, Approximation. 1. Introduction Following the by-now classical theory of nonlinear elasticity [1,2,12,36], we consider an elastic body occupying in its reference configuration an open bounded set Ω ⊆ Rd with Lipschitz boundary ∂Ω, subject to a prescribed boundary condition on a part Γ ⊂ ∂Ω with positive surface measure, i.e., Hd−1 (Γ) > 0. A possible deformation of the body is described by a mapping y : Ω → Rd such that y = y0 on Γ, where y0 is the imposed boundary data. Its associated internally stored elastic energy is given by the functional  (1.1) E(y) := W (∇y(x)) dx, Ω with a function W representing material properties: the local energy density, which is here assumed to be only a function of the deformation gradient and not of the position x. A crucial aspect of this mathematical model [6] is to define a suitable class of admissible deformations that capture relevant features, such as non-interpenetration of matter, which mathematically translates into injectivity of y. However, considering different admissible classes can lead to a Lavrentiev phenomenon, i.e., the functional infima differ when restricting the minimization of (1.1) to more regular deformations, such as W 1,∞ in place of W 1,p . Functionals demonstrating this behavior were first discovered in the early twentieth century [29,31]. There the minimum value over W 1,1 is strictly less than the infimum over W 1,∞ . For an extensive survey on the Lavrentiev phenomenon in a broader context, we refer the interested reader to [11]. In the context of nonlinear elasticity, the Lavrentiev phenomenon was first observed with admissible deformations that allow cavitations, i.e., the formation of voids in the material [4]. For the study of cavitations, we refer to [9,23] and references therein. A natural question raised in [7] and [5] is: Can the Lavrentiev phenomenon occur for elastostatics under growth conditions on the stored-energy function, ensuring that all finite-energy deformations are continuous? This is indeed the case, and the first example of this kind has been given in two dimensions [17– 19]. It features an energy density with desirable properties: W is smooth, polyconvex, frame-indifferent, isotropic, W (F )  |F |p with p > 2, and W (F ) → ∞ as det F → 0+. Moreover, admissible deformations 0123456789().: V,-vol 2 Page 2 of 21 ZAMP S. Almi, S. Krömer and A. Molchanova are almost everywhere (a.e.) injective. In these examples, the reference configuration is represented by a disk sector Ωα := {r(cos θ, sin θ) : 0 < r < 1, 0 < θ < α}. A crucial aspect for the emergence of the Lavrentiev phenomenon in that example is the local behavior of (almost) minimizers near the tip at r = 0, interacting with a particular choice of boundary conditions. The latter fix the origin y(0, 0) = (0, 0), β θ) and y(Ωα ) ⊂ Ωβ , where 0 < β < 34 α. y(1, θ) = (1, α In the current paper, we provide examples of the Lavrentiev phenomenon in elasticity both in two and three dimensions. The elastic energy is of a simple neo-Hookean form with physically reasonable properties as described above, and admissible deformations are continuous and a.e.-injective. Differently from [17–19], the Lavrentiev phenomenon in our example is not related to the local behavior of almost minimizers near prescribed boundary data, but to a possible global self-intersection of the material that still maintains a.e. injectivity by compressing two different material cross sections to a single point (or line in 3D) of self-contact in deformed configuration. It turns out to be energetically favorable due to our particular choice of boundary conditions but is no longer possible if we restrict to a sufficiently smooth class of admissible deformations. This then leads to a higher energy infimum. Throughout the paper, we consider locally orientation preserving deformations with p-Sobolev regularity W+1,p (Ω; Rd ) := {y ∈ W 1,p (Ω; Rd ) | det ∇y > 0 a.e. in Ω} ⊂ W 1,p (Ω; Rd ). (1.2) 1,p If p > d, the Sobolev embedding theorems ensure the continuity of W -mappings. The question of injectivity of deformations, i.e., non-interpenetration of matter, is more delicate, and it has been extensively studied. Let us mention just a few references. For local invertibility conditions, see [8,16,25]. As for global injectivity, one may ask some coercivity with respect to specific ratios of powers of a matrix F , its cofactor matrix cof F , and its determinant det F combined with global topological information from boundary values [3,24,27,28,33,38] or second gradient [21], as well as other regularity [13,38,39] and topological restrictions such as (INV)-condition [8,14,22,34,37] and considering limits of homeomorphisms [10,15,27,33]. In this paper, we adopt the approach from [13], where the authors investigate a class of mappings y ∈ W+1,p (Ω; Rd ) satisfying the Ciarlet–Nečas condition:  det(∇y(x)) dx ≤ |y(Ω)| , (CN) Ω and prove that the mappings of this class are a.e.-injective. In the examples, we consider W (F )  |F |p + (det F )−q , the reference configuration Ω in dimension d = 2, 3. The boundary data y0 are chosen in such a way that the energy E favors deformations that have non-empty sets of non-injectivity. In particular, we construct in Sect. 3 (resp. Sect. 4) a competitor y ∈ W+1,p (Ω; R2 ) (resp. y ∈ W+1,p (Ω; R3 )) satisfying the Ciarlet–Nečas condition (CN) and having a line (resp. a plane) of non-injectivity. The energy of such deformation is shown to be strictly less than that of Lipschitz deformations, for which injectivity is ensured everywhere. The global injectivity in this case follows from the Reshetnyak theorem for mappings of finite distortion [30]. Specifically, a mapping 1,d (Ω; Rd ) with det ∇y ≥ 0 a.e. has finite distortion if |∇y(x)| = 0 whenever det ∇y(x) = 0. If, y ∈ Wloc |∇y| κ in addition, the distortion Ky := det ∇y ∈ L with κ > d − 1, then y is either constant or open and d 1,d (Ω; Rd ) discrete. Furthermore, it is not difficult to see that an a.e.-injective and open mapping y ∈ Wloc is necessarily injective everywhere, as pointed out in [20, Lemma 3.3]. For a general theory of mappings of finite distortion, the reader is referred to [26]. Our example also shows that, depending on the precise properties of the energy de (...truncated)