Two-subsystem thermodynamics for the mechanics of aging amorphous solids

The effect of physical aging on the mechanics of amorphous solids as well as mechanical rejuvenation is modeled with nonequilibrium thermodynamics, using the concept of two thermal subsystems, namely a kinetic one and a configurational one. Earlier work (Semkiv and Hütter in J Non-Equilib Thermodyn 41(2):79–88, 2016) is extended to account for a fully general coupling of the two thermal subsystems. This coupling gives rise to hypoelastic-type contributions in the expression for the Cauchy stress tensor, that reduces to the more common hyperelastic case for sufficiently long aging. The general model, particularly the reversible and irreversible couplings between the thermal subsystems, is compared in detail with models in the literature (Boyce et al. in Mech Mater 7:15–33, 1988; Buckley et al. in J Mech Phys Solids 52:2355–2377, 2004; Klompen et al. in Macromolecules 38:6997–7008, 2005; Kamrin and Bouchbinder in J Mech Phys Solids 73:269–288 2014; Xiao and Nguyen in J Mech Phys Solids 82:62–81, 2015). It is found that only for the case of Kamrin and Bouchbinder (J Mech Phys Solids 73:269–288, 2014) there is a nontrivial coupling between the thermal subsystems in the reversible dynamics, for which the Jacobi identity is automatically satisfied. Moreover, in their work as well as in Boyce et al. (Mech Mater 7:15–33, 1988), viscoplastic deformation is driven by the deviatoric part of the Cauchy stress tensor, while for Buckley et al. (J Mech Phys Solids 52:2355–2377, 2004) and Xiao and Nguyen (J Mech Phys Solids 82:62–81, 2015) this is not the case.     

Strichartz Estimates and Moment Bounds for the Relativistic Vlasov–Maxwell System

We consider the relativistic Vlasov–Maxwell system with data of unrestricted size and without compact support in momentum space. In the two-dimensional and the two-and-a-half-dimensional cases, Glassey–Schaeffer proved (Commun Math Phys 185:257–284, 1997; Arch Ration Mech Anal 141:331–354, 1998; Arch Ration Mech Anal. 141:355–374, 1998) that for regular initial data with compact momentum support this system has unique global in time classical solutions. In this work we do not assume compact momentum support for the initial data and instead require only that the data have polynomial decay in momentum space. In the two-dimensional and the two-and-a-half-dimensional cases, we prove the global existence, uniqueness and regularity for solutions arising from this class of initial data. To this end we use Strichartz estimates and prove that suitable moments of the solution remain bounded. Moreover, we obtain a slight improvement of the temporal growth of the \({L^\infty_x}\) norms of the electromagnetic fields compared to Glassey and Schaeffer (Commun Math Phys 185:257–284, 1997; Arch Ration Mech Anal 141:355–374, 1998). In the three-dimensional case, we apply Strichartz estimates and moment bounds to show that a regular solution can be extended as long as \({{\|p_0^{\theta} f \|_{L^{q}_{x}L^1_{p}}}}\) remains bounded for \({\theta > \frac{2}{q}}\), \({2 < q \leqq \infty}\). This improves previous results of Pallard (Indiana Univ Math J 54(5):1395–1409, 2005; Commun Math Sci 13(2):347–354, 2015).     

Thermodynamic model formulation for viscoplastic solids as general equations for non-equilibrium reversible–irreversible coupling

Thermodynamic models for viscoplastic solids are often formulated in the context of continuum thermodynamics and the dissipation principle. The purpose of the current work is to show that models for such material behavior can also be formulated in the form of a General Equation for Non-Equilibrium Reversible–Irreversible Coupling (GENERIC), see, e.g., Grmela and Öttinger (Phys Rev E, 56:6620–6632, 1997), Öttinger and Grmela (Phys Rev E, 56:6633–6655, 1997), Grmela (J Non-Newtonian Fluid Mech, 165:980–986, 2010). A GENERIC combines Hamiltonian-dynamics-based modeling of time-reversible processes with Onsager–Casimir-based modeling of time-irreversible processes. The result is a model for the approach of non-equilibrium systems to thermodynamic equilibrium. Originally developed to model complex fluids, it has recently been applied to anisotropic inelastic solids in Eulerian (Hütter and Tervoort, in J Non-Newtonian Fluid Mech, 152:45–52, 2008; Hütter and Tervoort, in J Non-Newtonian Fluid Mech, 152:53–65, 2008; Hütter and Tervoort, in Adv Appl Mech, 42:254–317, 2008) and Lagrangian (Hütter and Svendsen, in J Elast 104:357–368, 2011) settings, as well as to damage mechanics. For simplicity, attention is focused in the current work on the case of thermoelastic viscoplasticity. Central to this formulation is a GENERIC-based form for the viscoplastic flow rule. A detailed comparison with the formulation based on continuum thermodynamics and the dissipation principle is given.     

Topological and Measure-Theoretical Entropies of Nonautonomous Dynamical Systems

We consider the dynamics of a nonautonomous dynamical system determined by a sequence of continuous self-maps \(f_n:X \rightarrow X,\) where \( n \in {\mathbb {N}},\) defined on a compact metric space X. Applying the theory of the Carathéodory structures, elaborated by Pesin (Dimension Theory in Dynamical Systems. Chicago Lectures in Mathematics. The University of Chicago Press, Chicago, 1997), we construct a Carathéodory structure whose capacity coincides with the topological entropy of the considered system. Generalizing the notion of local measure entropy, introduced by Brin and Katok (in: Palis (ed) Geometric Dynamics, Lecture Notes in Mathematics. Springer, Berlin 1983) for a single map, to a nonautonomous dynamical system we provide a lower and upper estimations of the topological entropy by local measure entropies. The theorems of the paper generalize results of Kawan (Nonautonomous Stoch Dyn Syst 1:26–52, 2013) and of Feng and Huang (J Funct Anal 263:2228–2254, 2012). Also, we construct a new entropy-like invariant such the entropy of a sequence \(\{f_n:X \rightarrow X\}_{n=1}^{\infty }\) of Lipschitz continuous maps with the same Lipschitz constant \(L >1,\) restricted to a subset \(Y\subset X,\) is upper bounded by Hausdorff dimension of Y multiplied by the logarithm of the Lipschitz constant L. This gives a generalizations of results of Dai et al. (Sci China Ser A 41:1068–1075, 1998) and Misiurewicz (Discret Contin Dyn Syst 10:827–833, 2004).     

On a Nonlinear Model for Tumor Growth in a Cellular Medium

We investigate the dynamics of a nonlinear model for tumor growth within a cellular medium. In this setting the “tumor” is viewed as a multiphase flow consisting of cancerous cells in either proliferating phase or quiescent phase and a collection of cells accounting for the “waste” and/or dead cells in the presence of a nutrient. Here, the tumor is thought of as a growing continuum \(\Omega \) with boundary \(\partial \Omega \) both of which evolve in time. In particular, the evolution of the boundary \(\partial \Omega \) is prescibed by a given velocity \({{{\varvec{V}}}.}\) The key characteristic of the present model is that the total density of cancerous cells is allowed to vary, which is often the case within cellular media. We refer the reader to the articles (Enault in Mathematical study of models of tumor growth, 2010; Li and Lowengrub in J Theor Biol, 343:79–91, 2014) where compressible type tumor growth models are investigated. Global-in-time weak solutions are obtained using an approach based on penalization of the boundary behavior, diffusion, viscosity and pressure in the weak formulation, as well as convergence and compactness arguments in the spirit of Lions (Mathematical topics in fluid dynamics. Compressible models, 1998) [see also Donatelli and Trivisa (J Math Fluid Mech 16: 787–803, 2004), Feireisl (Dynamics of viscous compressible fluids, 2014)].     

On the Generalization of Reissner Plate Theory to Laminated Plates,Part I: Theory

This is the first part of a two-part paper presenting the generalization of Reissner thick plate theory (Reissner in J. Math. Phys. 23:184–191, 1944) to laminated plates and its relation with the Bending-Gradient theory (Lebée and Sab in Int. J. Solids Struct. 48(20):2878–2888, 2011 and in Int. J. Solids Struct. 48(20):2889–2901, 2011). The original thick and homogeneous plate theory derived by Reissner (J. Math. Phys. 23:184–191, 1944) is based on the derivation of a statically compatible stress field and the application of the principle of minimum of complementary energy. The static variables of this model are the bending moment and the shear force. In the present paper, the rigorous extension of this theory to laminated plates is presented and leads to a new plate theory called Generalized-Reissner theory which involves the bending moment, its first and second gradients as static variables. When the plate is homogeneous or functionally graded, the original theory from Reissner is retrieved. In the second paper (Lebée and Sab, 2015), the Bending-Gradient theory is obtained from the Generalized-Reissner theory and comparison with an exact solution for the cylindrical bending of laminated plates is presented.     

On the Generalization of Reissner Plate Theory to Laminated Plates,Part II: Comparison with the Bending-Gradient Theory

In the first part of this two-part paper (Lebée and Sab in On the generalization of Reissner plate theory to laminated plates, Part I: theory, doi: 10.1007/s10659-016-9581-6, 2015), the original thick plate theory derived by Reissner (J. Math. Phys. 23:184–191, 1944) was rigorously extended to the case of laminated plates. This led to a new plate theory called Generalized-Reissner theory which involves the bending moment, its first and second gradients as static variables. In this second paper, the Bending-Gradient theory (Lebée and Sab in Int. J. Solids Struct. 48(20):2878–2888, 2011 and 2889–2901, 2011) is obtained from the Generalized-Reissner theory and several projections as a Reissner–Mindlin theory are introduced. A comparison with an exact solution for the cylindrical bending of laminated plates is presented. It is observed that the Generalized-Reissner theory converges faster than the Kirchhoff theory for thin plates in terms of deflection. The Bending-Gradient theory does not converge faster but improves considerably the error estimate.     

The peculiar elongational viscosity of concentrated solutions of monodisperse PMMA in oligomeric MMA

Concentrated solutions of nearly monodisperse poly(methyl methacrylate), PMMA-270k and PMMA-86k, in oligo(methyl methacrylate), MMA o-4k and MMA o-2k, investigated by Wingstrand et al. (Phys Rev Lett 115:078302, 2015) and Wingstrand (2015) do not follow the linear-viscoelastic scaling relations of monodisperse polystyrenes (PS) dissolved in oligomeric styrene (Wagner in Rheol Acta 53:765–777, 2014a, in Non-Newtonian Fluid Mech 222:121–131, 2014b; Wagner et al. in J Rheol 59:1113–1130, 2015). Rather, PMMA-270k shows an attractive interaction with MMA, in contrast to the interaction of PMMA-86k and MMA. This different behavior can be traced back to different tacticities of the two polymers. The attractive interaction of PMMA-270k with o-4k creates pseudo entanglements, which increase the interchain tube pressure, and therefore, the solution PMMA-270k/o-4k shows, as reported by Wingstrand et al. (Phys Rev Lett 115:078302, 2015), qualitatively a similar scaling of the elongational viscosity with \( {\left(\dot{\varepsilon}{\tau}_R\right)}^{-1/2} \) as observed for polystyrene melts. For the solution PMMA-270/o-2k, this effect is only seen at the highest elongation rates investigated. The elongational viscosity of PMMA-86k dissolved in oligomeric MMA is determined by the Rouse time of the melt, as in the case of polystyrene solutions.

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Graphical abstract ?

     

Statistical and Numerical Considerations of Backus-Average Product Approximation

In this paper, we examine the applicability of the approximation, \(\overline{f g}\approx \overline{f}\,\overline{g}\), within Backus (J. Geophys. Res. 67(11):4427–4440, 1962) averaging. This approximation is a crucial step in the method proposed by Backus (J. Geophys. Res. 67(11):4427–4440, 1962), which is widely used in studying wave propagation in layered Hookean solids. According to this approximation, the average of the product of a rapidly varying function and a slowly varying function is approximately equal to the product of the averages of those two functions.Considering that the rapidly varying function represents the mechanical properties of layers, we express it as a step function. The slowly varying function is continuous, since it represents the components of the stress or strain tensors. In this paper, beyond the upper bound of the error for that approximation, which is formulated by Bos et al. (J. Elast. 127:179–196, 2017), we provide a statistical analysis of the approximation by allowing the function values to be sampled from general distributions.Even though, according to the upper bound, Backus (J. Geophys. Res. 67(11):4427–4440, 1962) averaging might not appear as a viable approach, we show that—for cases representative of physical scenarios modelled by such an averaging—the approximation is typically quite good. We identify the cases for which there can be a deterioration in its efficacy.In particular, we examine a special case for which the approximation results in spurious values. However, such a case—though physically realizable—is not likely to appear in seismology, where Backus (J. Geophys. Res. 67(11):4427–4440, 1962) averaging is commonly used. Yet, such values might occur in material sciences, in general, for which Backus (J. Geophys. Res. 67(11):4427–4440, 1962) averaging is also considered.     

Complete Global and Bifurcation Analysis of a Stoichiometric Predator–Prey Model

Loladze et al. (Bull Math Biol 62:1137–1162, 2000) proposed a highly cited stoichiometric predator–prey system, which is nonsmooth, and thus it is extremely difficult to analyze its global dynamics. The main challenge comes from the phase plane fragmentation and parameter space partitioning in order to perform a detailed and complete global stability and bifurcation analysis. Li et al. (J Math Biol 63:901–932, 2011) firstly discussed its global dynamical behavior with Holling type I functional response and found that the system has no limit cycles, and the internal equilibrium is globally asymptotically stable if it exists. Secondly, for the system with Holling type II functional response, Li et al. (2011) fixed all parameters (with realistic values) except K to perform the bifurcation analysis and obtained some interesting phenomena, for instance, the appearance of bistability and many bifurcation types. The aim of this paper is to provide a complete global analysis for the system with Holling type II functional response without fixing any parameter. Our analysis shows that the model has far richer dynamics than those found in the previous paper (Li et al. 2011), for example, four types of bistability appear: besides the bistability between an internal equilibrium and a limit cycle as shown in Li et al. (2011), the other three bistabilities occur between an internal equilibrium and a boundary equilibrium, between two internal equilibria, or between a boundary equilibrium and a limit cycle. In addition, this paper rigorously provides all possible bifurcation passways of this stoichiometric model with Holling type II functional response.     

Second Order Corrections to Mean Field Evolution for Weakly Interacting Bosons in The Case of Three-body Interactions

In this paper, we consider the Hamiltonian evolution of N weakly interacting bosons. Assuming triple collisions, its mean field approximation is given by a quintic Hartree equation. We construct a second order correction to the mean field approximation using a kernel k(t, x, y) and derive an evolution equation for k. We show global existence for the resulting evolution equation for the correction and establish an a priori estimate comparing the approximation to the exact Hamiltonian evolution. Our error estimate is global and uniform in time. Comparing with the work of Rodnianski and Schlein (Commun Math Phys 291:31–61, 2009), and Grillakis, Machedon and Margetis (Commun Math Phys 294:273–301, 2010; Adv Math 288:1788–1815, 2011), where the error estimate grows in time, our approximation tracks the exact dynamics for all time with an error of the order \({O(1/\sqrt{N}).}\)     

Estimating void fraction in a hydraulic jump by measurements of pixel intensity

A hydraulic jump is a sudden transition from supercritical to subcritical flow. It is characterized by a highly turbulent roller region with a bubbly two-phase flow structure. The present study aims to estimate the void fraction in a hydraulic jump using a flow visualization technique. The assumption that the void fraction in a hydraulic jump could be estimated based on images’ pixel intensity was first proposed by Mossa and Tolve (J Fluids Eng 120:160–165, 1998). While Mossa and Tolve (J Fluids Eng 120:160–165, 1998) obtained vertically averaged air concentration values along the hydraulic jump, herein we propose a new visualization technique that provides air concentration values in a vertical 2-D matrix covering the whole area of the jump roller. The results obtained are found to be consistent with new measurements using a dual-tip conductivity probe and show that the image processing procedure (IPP) can be a powerful tool to complement intrusive probe measurements. Advantages of the new IPP include the ability to determine instantaneous and average void fractions simultaneously at different locations along the hydraulic jump without perturbing the flow, although it is acknowledged that the results are likely to be more representative in the vicinity of sidewall than at the center of the flume.     

On Lie symmetries and soliton solutions of $$$$-dimensional Bogoyavlenskii equations

The present article is devoted to find some invariant solutions of the \((2+1)\)-dimensional Bogoyavlenskii equations using similarity transformations method. The system describes \((2+1)\)-dimensional interaction of a Riemann wave propagating along y-axis with a long wave along x-axis. All possible vector fields, commutative relations and symmetry reductions are obtained by using invariance property of Lie group. Meanwhile, the method reduces the number of independent variables by one, which leads to the reduction of Bogoyavlenskii equations into a system of ordinary differential equations. The system so obtained is solved under some parametric restrictions and provides invariant solutions. The derived solutions are much efficient to explain the several physical properties depending upon various existing arbitrary constants and functions. Moreover, some of them are more general than previously established results (Peng and Shen in Pramana 67:449–456, 2006; Malik et al. in Comput Math Appl 64:2850–2859, 2012; Zahran and Khater in Appl Math Model 40:1769–1775, 2016; Zayed and Al-Nowehy in Opt Quant Electron 49(359):1–23, 2017). In order to provide rich physical structures, the solutions are supplemented by numerical simulation, which yield some positons, negatons, kinks, wavefront, multisoliton and asymptotic nature.     

Linear Pantographic Sheets: Existence and Uniqueness of Weak Solutions

The well-posedness of the boundary value problems for second gradient elasticity has been studied under the assumption of strong ellipticity of the dependence on the second placement gradients (see, e.g., Chambon and Moullet in Comput. Methods Appl. Mech. Eng. 193:2771–2796, 2004 and Mareno and Healey in SIAM J. Math. Anal. 38:103–115, 2006).The study of the equilibrium of planar pantographic lattices has been approached in two different ways: in dell’Isola et al. (Proc. R. Soc. Lond. Ser. A 472:20150, 2016) a discrete model was introduced involving extensional and rotational springs which is also valid in large deformations regimes while in Boutin et al. (Math. Mech. Complex Syst. 5:127–162, 2017) the lattice has been modelled as a set of beam elements interconnected by internal pivots, but the analysis was restricted to the linear case. In both papers a homogenized second gradient deformation energy, quadratic in the neighbourhood of non deformed configuration, is obtained via perturbative methods and the predictions obtained with the obtained continuum model are successfully compared with experiments. This energy is not strongly elliptic in its dependence on second gradients. We consider in this paper also the important particular case of pantographic lattices whose first gradient energy does not depend on shear deformation: this could be considered either a pathological case or an important exceptional case (see Stillwell et al. in Am. Math. Mon. 105:850–858, 1998 and Turro in Angew. Chem., Int. Ed. Engl. 39:2255–2259, 2000). In both cases we believe that such a particular case deserves some attention because of what we can understand by studying it (see Dyson in Science 200:677–678, 1978). This circumstance motivates the present paper, where we address the well-posedness of the planar linearized equilibrium problem for homogenized pantographic lattices. To do so: (i) we introduce a class of subsets of anisotropic Sobolev’s space as the most suitable energy space \(E\) relative to assigned boundary conditions; (ii) we prove that the considered strain energy density is coercive and positive definite in \(E\); (iii) we prove that the set of placements for which the strain energy is vanishing (the so-called floppy modes) must strictly include rigid motions; (iv) we determine the restrictions on displacement boundary conditions which assure existence and uniqueness of linear static problems. The presented results represent one of the first mechanical applications of the concept of Anisotropic Sobolev space, initially introduced only on the basis of purely abstract mathematical considerations.     

On Reducibility of 1d Wave Equation with Quasiperiodic in Time Potentials

In this paper we use a KAM theorem of Grébert and Thomann (Commun Math Phys 307:383–427, 2011) to prove the reducibility of the 1d wave equation with Dirichlet boundery conditions on \([0,\pi ]\) with a quasi-periodic in time potential under some symmetry assumptions. From Mathieu–Hill operator’s known results (Eastham in The spectral theory of periodic differential operators, Hafner, New York, 1974; Magnus and Winkler in Hill’s equation, Wiley-Interscience, London, 1969) and Bourgain’s techniques (Commun Math Phys 204:207–247, 1999), we prove that for any \(\epsilon \) small enough, there exist a \(0<m_{\epsilon }\le 1\) and one solution \(u_{\epsilon }(t,x)\) with

$$\begin{aligned} \Vert u_{\epsilon }(t_n,x)\Vert _{H^1({\mathbb {T}})}\rightarrow \infty , \qquad |t_n|\rightarrow \infty , \end{aligned}$$

where \(u_{\epsilon }(t,x)\) satisfies 1d wave equation

$$\begin{aligned} u_{tt}-u_{xx}+m_{\epsilon }u-\epsilon \cos 2t u=0, \end{aligned}$$

with Dirichlet boundery conditions on \([0,\pi ]\).

     

Far-Field Boundary Conditions for Calculation of Hole-Drilling Residual Stress Calibration Coefficients

The Hole-Drilling method for residual stress measurement, both in its standard version based on strain gauge rosettes (ASTM E837-08e1 2008) and its derivative using optical methods for estimating the displacement field around the hole (Baldi (2005) J Eng Mater Technol 127(2):165–169; Schajer and Steinzig (2005) Exp Mech 45(6):526–532; Schajer (2010) Exp Mech 50(2):159–168), relies on numerical calibrated coefficients (A and B) to correlate the experimentally acquired strains (displacements) with residual stress components. To estimate the A and B coefficients, two FEM (Finite Element Method) computations are required, the former related to a hydrostatic stress state, the latter to a pure shear case. Both can be implemented using either a semi-analytical approach (i.e. an axis-symmetric mesh expanded in the tangential direction using a Fourier series) or a tri-dimensional mesh, usually exploiting the double symmetry of the problem. Whatever the approach selected, the definition of constraints to be applied to the outer boundary is critical because the hole-drilling method assumes an infinite plate, thus both the usual solutions—fully constrained or free boundaries—are unable to correctly describe the theoretical situation. In the following, the problem of correct simulation of the infinite domain will be discussed and two simple and effective solutions will be proposed.     

A Rigidity Result for a Reduced Model of a Cubic-to-Orthorhombic Phase Transition in the Geometrically Linear Theory of Elasticity

In this article we study a simplified two-dimensional model for a cubic-to-orthorhombic phase transition occurring in certain shape-memory-alloys. In the low temperature regime the linear theory of elasticity predicts various possible patterns of martensite arrangements: Apart from the well known laminates this phase transition displays additional structures involving four martensitic variants—so called crossing twins.Introducing a variational model including surface energy, we show that these structures are rigid under small energy perturbations. Combined with an upper bound construction this gives the optimal scaling behavior of incompatible microstructures. These results are related to papers by Capella and Otto (Commun. Pure Appl. Math. 62(12):1632–1669, 2009; Proc. R. Soc. Edinb., Sect. A, Math. 142:273–327, 2012) as well as to a paper by Dolzmann and Müller (Meccanica 30:527–539, 1995).     

Anisotropic Regularity Conditions for the Suitable Weak Solutions to the 3D Navier–Stokes Equations

We are concerned with the problem, originated from Seregin (159–200, 2007), Seregin (J. Math. Sci. 143: 2961–2968, 2007), Seregin (Russ. Math. Surv. 62:149–168, 2007), what are minimal sufficiently conditions for the regularity of suitable weak solutions to the 3D Navier–Stokes equations. We prove some interior regularity criteria, in terms of either one component of the velocity with sufficiently small local scaled norm and the rest part with bounded local scaled norm, or horizontal part of the vorticity with sufficiently small local scaled norm and the vertical part with bounded local scaled norm. It is also shown that only the smallness on the local scaled L ² norm of horizontal gradient without any other condition on the vertical gradient can still ensure the regularity of suitable weak solutions. All these conclusions improve pervious results on the local scaled norm type regularity conditions.     

Comment on “New method to obtain periodic solutions of period two and three of a rational difference equation” [Nonlinear Dyn 79:241–250]

In this paper, we discuss the results of Elsayed (Nonlinear Dyn 79:241–250, 2015) for periodic solution of period two and three of rational difference equation. Also, we study the existence of periodic solutions of some difference equation by using old method and new method and comparison of results. The results obtained here correct and improve some known results in Elsayed (Nonlinear Dyn 79:241–250, 2015). Some numerical examples will be given to illustrate our results.     

Spontaneous bending of pre-stretched bilayers

We discuss spontaneously bent configurations of pre-stretched bilayer sheets that can be obtained by tuning the pre-stretches in the two layers. The two-dimensional nonlinear plate model we use for this purpose is an adaptation of the one recently obtained for thin sheets of nematic elastomers, by means of a rigorous dimensional reduction argument based on the theory of Gamma-convergence (Agostiniani and DeSimone in Meccanica. doi: 10.1007/s11012-017-0630-4, 2017, Math Mech Solids. doi: 10.1177/1081286517699991, arXiv:1509.07003, 2017). We argue that pre-stretched bilayer sheets provide us with an interesting model system to study shape programming and morphing of surfaces in other, more complex systems, where spontaneous deformations are induced by swelling due to the absorption of a liquid, phase transformations, thermal or electro-magnetic stimuli. These include bio-mimetic structures inspired by biological systems from both the plant and the animal kingdoms.