The metric-restricted inverse design problem

We study a class of design problems in solid mechanics, leading to a variation on the classical question of equi-dimensional embeddability of Riemannian manifolds. In this general new context, we derive a necessary and sufficient existence condition, given through a system of total differential equations, and discuss its integrability. In the classical context, the same approach yields conditions of immersibility of a given metric in terms of the Riemann curvature tensor. In the present situation, the equations do not close in a straightforward manner, and successive differentiation of the compatibility conditions leads to a new algebraic description of integrability. We also recast the problem in a variational setting and analyze the infimum of the appropriate incompatibility energy, resembling the non-Euclidean elasticity. We then derive a Γ-convergence result for dimension reduction from 3d to 2d in the Kirchhoff energy scaling regime. 1. The metric-restricted inverse design problem Assume T is a manifold of material types, differentiated by their structure, density, swellingshrinkage rates and other qualities. We let any material type T ∈ T be naturally endowed with a prestrain ḡ(T ), where ḡ : T → R2×2 sym,pos is a given smooth mapping, taking values in the symmetric positive definite tensors. Suppose now that we need to manufacture a 2-dimensional membrane S ⊂ R3, where at any given point p ∈ S a material of type T (p) ∈ T must be used for a given T : S → T . The question is how to print a thin film U ⊂ R2 in a manner that the activation u : U → R3 of the prestrain in the film would result in a deformation leading eventually to the desired surface shape S. The above described problem is natural as a design question in various areas of solid mechanics, even though the involved tensors are not intrinsic geometric objects. For example, it includes the subproblems and extensions to higher dimensions: (a) Given the deformed configuration of an elastic 2-dimensional membrane and the rectangular Cartesian components of the Right Cauchy-Green tensor field of a deformation, mapping a flat undeformed reference of the membrane to it, find the flat reference configuration and the deformation of the membrane. (b) Given the deformed configuration of a 3-dimensional body and the rectangular Cartesian components of the Right Cauchy-Green tensor field of the deformation, mapping a reference configuration to it, find the reference configuration and the deformation. (c) Suppose the current configuration of a 3-dimensional, plastically deformed body is given, and on it is specified the rectangular Cartesian components of a plastic distortion Fp. Find a reference configuration and a deformation ζ, mapping this reference to the given current configuration, such that the latter is stress-free. Assume that the stress response of the material is such that the stress vanishes if and only if (∇ζ(Fp)) (∇ζ(Fp)) = Id3. 1We thank Kaushik Bhattacharya for bringing this problem to our attention. 1 Page 1 of 28 CONFIDENTIAL AUTHOR SUBMITTED MANUSCRIPT NON-101035.R1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 2 AMIT ACHARYA, MARTA LEWICKA AND MOHAMMAD REZA PAKZAD In view of [16, 9, 19], the activation u must be an isometric immersion of the Riemannian manifold (U,G) into R3, where G is the prestrain in the flat (referential) thin film. Our design problem requires hence that we find an unknown reference configuration U ⊂ R2, an unknown material distribution T : U → T and an unknown deformation u : U → R3 such that (i) S = u(U), (ii) For any x ∈ U , the point u(x) carries a material of type T (u(x)), i.e. T (u(x)) = T(x), (iii) ∇u(x)T∇u(x) = G(x) := ḡ(T(x)). If the membrane S ⊂ R3 is a smooth surface, then letting g := ḡ◦T : S → R2×2 sym,pos, the conditions (i)-(iii) simplify to finding a domain U ⊂ R2 and a bijection u : U → S, such that: (1.1) (∇u) (∇u)(x) = g(u(x)) ∀x ∈ U. The smoothness of g is determined by the regularity of T and of the mappings ḡ and T . Essentially, in all of the applications defined above (e.g. membrane, 3-d), we are dealing with a general class of nonlinear elastic constitutive assumptions involving pre-strain, with the requirement that the stored energy density evaluated at the Identity tensor (of appropriate dimensionality) attain the value zero; in other words, we look for stress-free deformations of a prestrained body. Given a prestrain field specified on the target configuration, we explore the question of existence of deformations that allow such a minimum energy state to be attained pointwise, as well as the characterization of the constraints on the pre-strain field that allows such attainment. In the language of mechanics, note that the question (1.1) may be rephrased as looking for deformations u of the reference U such that: