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

Kinematics

This chapter develops the necessary mathematics for describing general deformations that a solid body may undergo, a topic known as kinematics. Definitions of motion, displacement, velocity, and acceleration which are vectors, and the deformation gradient and displacement gradient which are tensors are given. The mapping of material vectors by the deformation gradient tensor as a basic concept in describing the large deformation kinematics of a deformable body is presented. The powerful polar decomposition theorem is discussed and applied to the deformation gradient tensor to show that it can be decomposed into a stretch followed by a rotation, or a rotation followed by a stretch. Non-linear measures of strain are defined in terms of the stretch tensors. The important case of small deformations, which results in linear measures of strain, is discussed. For small strains the important decomposition of the state of strain that separates a volumetric strain from a non-volumetric or deviatoric strain is presented.

Multi-Trigger Thermo-Electro-Mechanical Soft Actuators under Large Deformations

Dielectric actuators (DEAs), because of their exceptional properties, are well-suited for soft actuators (or robotics) applications. This article studies a multi-stimuli thermo-dielectric-based soft actuator under large bending conditions. In order to determine the stress components and induced moment (or stretches), a nominal Helmholtz free energy density function with two types of hyperelastic models are employed. Non-linear electro-elasticity theory is adopted to derive the governing equations of the actuator. Total deformation gradient tensor is multiplicatively decomposed into electro-mechanical and thermal parts. The problem is solved using the second-order Runge-Kutta method. Then, the numerical results under thermo-mechanical loadings are validated against the finite element method (FEM) outcomes by developing a user-defined subroutine, UHYPER in a commercial FEM software. The effect of electric field and thermal stimulus are investigated on the mean radius of curvature and stresses distribution of the actuator. Results reveal that in the presence of electric field, the required moment to actuate the actuator is smaller. Finally, due to simplicity and accuracy of the present boundary problem, the proposed thermally-electrically actuator is expected to be used in future studies and 4D printing of artificial thermo-dielectric-based beam muscles.

Constitutive Equations for Hyperelastic Materials Based on the Upper Triangular Decomposition of the Deformation Gradient

The upper triangular decomposition has recently been proposed to multiplicatively decompose the deformation gradient tensor into a product of a rotation tensor and an upper triangular tensor called the distortion tensor, whose six components can be directly related to pure stretch and simple shear deformations, which are physically measurable. In the current paper, constitutive equations for hyperelastic materials are derived using strain energy density functions in terms of the distortion tensor, which satisfy the principle of material frame indifference and the first and second laws of thermodynamics. Being expressed directly as derivatives of the strain energy density function with respect to the components of the distortion tensor, the Cauchy stress components have simpler expressions than those based on the invariants of the right Cauchy-Green deformation tensor. To illustrate the new constitutive equations, strain energy density functions in terms of the distortion tensor are provided for unconstrained and incompressible isotropic materials, incompressible transversely isotropic composite materials, and incompressible orthotropic composite materials with two families of fibers. For each type of material, example problems are solved using the newly proposed constitutive equations and strain energy density functions, both in terms of the distortion tensor. The solutions of these problems are found to be the same as those obtained by applying the polar decomposition-based invariants approach, thereby validating and supporting the newly developed, alternative method based on the upper triangular decomposition of the deformation gradient tensor.

A Novel Small-Specimen Planar Biaxial Testing System With Full In-Plane Deformation Control

Simulations of soft tissues require accurate and robust constitutive models, whose form is derived from carefully designed experimental studies. For such investigations of membranes or thin specimens, planar biaxial systems have been used extensively. Yet, all such systems remain limited in their ability to: (1) fully prescribe in-plane deformation gradient tensor F2D, (2) ensure homogeneity of the applied deformation, and (3) be able to accommodate sufficiently small specimens to ensure a reasonable degree of material homogeneity. To address these issues, we have developed a novel planar biaxial testing device that overcomes these difficulties and is capable of full control of the in-plane deformation gradient tensor F2D and of testing specimens as small as ∼4 mm × ∼4 mm. Individual actuation of the specimen attachment points, combined with a robust real-time feedback control, enabled the device to enforce any arbitrary F2D with a high degree of accuracy and homogeneity. Results from extensive device validation trials and example tissues illustrated the ability of the device to perform as designed and gather data needed for developing and validating constitutive models. Examples included the murine aortic tissues, allowing for investigators to take advantage of the genetic manipulation of murine disease models. These capabilities highlight the potential of the device to serve as a platform for informing and verifying the results of inverse models and for conducting robust, controlled investigation into the biomechanics of very local behaviors of soft tissues and membrane biomaterials.