Supporting Information: Spatiotemporally programmable metasurfaces via viscoelastic shell snapping

Supplementary Text 1: Material modeling and characterization We used the following incompressible neo-Hookean material model to define the instantaneous constitutive behavior of the shells, = tr − 3, (S1) where W is the strain energy density function, µ is the shear modulus, F is the deformation gradient tensor. To describe the viscoelastic behavior of the shells, Prony series were used and the shear modulus µ can be expressed as = 1 − ∑ 1 − ⁄ , (S2) where µ0 is the instantaneous shear modulus, n is the number of the series terms, is the dimensionless relaxation modulus, t is the time, and τi is the relaxation time constant. Here we characterize the viscoelastic properties of the silicone rubber (Dragon SkinTM30) and urethane rubber (VytaFlexTM 20). We modeled their viscou

Coupling Moving Morphable Voids and Components Based Topology Optimization of Hydrogel Structures Involving Large Deformation

A coupling of moving morphable void and component approach for the topology optimization of hydrogel structures involving recoverable large deformation is proposed in this paper. In this approach, the geometric parameters of moving morphable voids and components are set as design variables to respectively describe the outline and material distribution of hydrogel structures for the first time. To facilitate the numerical simulation of large deformation behavior of hydrogel structures during the optimization process, the design variables are mapped to the density field of the design domain and the density field is then used to interpolate the strain energy density function of the element. Furthermore, the adjoint sensitivity of the optimization formulation is derived and combined with the gradient-based algorithm to solve the topology optimization problem effectively. Finally, two representative numerical examples of the optimization of isotropic hydrogel structures are used to prove the effectiveness of the proposed method, and the optimization design of an anisotropic bionic hydrogel structure is presented to illustrate the applicability of the method. Experimental results are also presented to demonstrate that the explicit topologies obtained from the method can be directly used in the manufacture of hydrogel-based soft devices.

Determination of a Strain Energy Density Function for the Tricuspid Valve Leaflets Using Constant Invariant-Based Mechanical Characterizations

The tricuspid valve (TV) regulates the blood flow within the right side of the heart. Despite recent improvements in understanding TV mechanical and microstructural properties, limited attention has been devoted to developments of TV-specific constitutive models. The objective of this work is to use the first-of-its-kind experimental data from constant invariant-based mechanical characterizations to determine a suitable invariant-based strain energy density function (SEDF). Six specimens for each TV leaflet are characterized using constant invariant mechanical testing. The data is then fit with three candidate SEDF forms: (i) a polynomial model as the transversely isotropic version of the Mooney-Rivlin model, (ii) an exponential model, and (iii) a combined polynomial-exponential model. Similar fitting capabilities were found for the exponential and polynomial forms (R2=0.92-0.99 vs. 0.91-0.97) compared to the combined polynomial-exponential SEDF (R2=0.65-0.95). Furthermore, the polynomial form had larger Pearson's correlation coefficients than the exponential form (0.51 vs. 0.30), indicating a more well-defined search space. Finally, the exponential and combined polynomial-exponential forms had notably smaller but more eccentric model parameter's confidence regions than the polynomial form. Further evaluations of invariant decoupling revealed that the decoupling of the invariant terms within the exponential SEDF leads to a less satisfactory performance. From these results, we conclude that the exponential form is better suited for the TV leaflets due to its superb fitting capabilities and smaller parameter's confidence regions.

Internal Resonance of Hyperelastic Thin-Walled Cylindrical Shells under Harmonic Axial Excitation and Time-Varying Temperature Field

In this paper, the internal resonance characteristics of hyperelastic cylindrical shells under the time-varying temperature field are investigated for the first time, and the evolution of the isolated bubble is carried out. Through the analysis of the influences of temperature on material parameters, the hyperelastic strain energy density function in the unsteady temperature field is presented. The governing equations describing the axisymmetric nonlinear vibration are derived from the nonlinear thin shell theory and the variational principle. With the harmonic balance method and the arc length method, the steady state solutions of shells are obtained, and their stabilities are determined. The influences of the discrete mode number, structural and temperature parameters on the nonlinear behaviors are examined. The role of the parameter variation in evolution behaviors of isolated bubble responses is revealed under the condition of 3:1 internal resonance. The results manifest that both structural and temperature parameters can affect the resonance range of the response curve, and the perturbed temperature has a more significant effect on the stable region of the solution.

A Physics-Informed Neural Network Constitutive Model for Cross-Linked Polymers

The behavior of Cross-linked Polymers in finite deformations is often characterized by nonlinear behaviour. In this paper, we propose to embed an artificial neural network (ANN) within a micro-mechanical platform and thus to enforce certain physical restrictions of an amorphous network such as directional dependency and history-dependency of the constitutive behavior of rubber-like materials during loading and unloading. Accordingly, a strain energy density function is assumed for a set of chains in each direction based on ANN and trained with experimental data set. Summation of the energies provided by ANNs in different directions can determine the strain energy density function of the matrix. Stress-strain relation is derived from strain energy density function. Polyconvexity is enforced to assure minimized potential energy, a global solution for an optimization problem, and thermodynamic consistency that show the model cannot generate energy. The model is validated against multiple sets of experimental data, e.g. uniaxial, shear, and biaxial deformation available in the literature. This model captures not only the loading and unloading paths but also the inelastic response of these materials, such as the Mullins effect and permanent set. The model can be generalized to other materials and other inelastic effects as well.

A Review of Physically Based and Thermodynamically Based Constitutive Models for Soft Materials

In this paper, we review constitutive models for soft materials. We specifically focus on physically based models accounting for hyperelasticity, visco-hyperelasticity, and damage phenomena. For completeness, we include the thermodynamically based viscohyperelastic and damage models as well as the so-called mixed models. The models are put in the frame of statistical mechanics and thermodynamics. Based on the available experimental data, we provide a quantitative comparison of the hyperelastic models. This information can be used as guidance in the selection of suitable constitutive models. Next, we consider visco-hyperelasticity in the frame of the thermodynamic theory and molecular chain dynamics. We provide a concise summary of the viscohyperelastic models including specific strain energy density function, the evolution laws of internal variables, and applicable conditions. Finally, we review the models accounting for damage phenomenon in soft materials. Various proposed damage criteria are summarized and discussed in connection with the physical interpretations that can be drawn from physically based damage models. The discussed mechanisms include the breakage of polymer chains, debonding between polymer chains and fillers, disentanglement, and so on.