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

A FORTRAN Program to Model Magnetic Gradient Tensor at High Susceptibility Using Contraction Integral Equation Method

The magnetic gradient tensor provides a powerful tool for detecting magnetic bodies because of its ability to emphasize detailed features of the magnetic anomalies. To interpret field measurements obtained by magnetic gradiometry, the forward calculation of magnetic gradient fields is always necessary. In this paper, we present a contraction integral equation method to simulate the gradient fields produced by 3-D magnetic bodies of arbitrary shapes and high susceptibilities. The method employs rectangular prisms to approximate the source region with the assumption that the magnetization in each element is homogeneous. The gradient fields are first solved in the Fourier domain and then transformed into the spatial domain by 2-D Gauss-FFT. This calculation is performed iteratively until the required accuracy is reached. The convergence of the iterative procedure is ensured by a contraction operator. To facilitate application, we introduce a FORTRAN program to implement the algorithm. This program is intended for users who show interests in 3D magnetic modeling at high susceptibility. The performance of the program, including its computational accuracy, efficiency and convergence behavior, is tested by several models. Numerical results show that the code is computationally accurate and efficient, and performs well at a wide range of magnetic susceptibilities from 0 SI to 1000 SI. This work, therefore, provides a significant tool for 3D forward modeling of magnetic gradient fields at high susceptibility.

Technology of determining the Earth’s gravitational field parameters using gradiometric measurements Part 6. Calculation the components of the gravitational potential tensor in the earth’s spatial rectangular coordinate system

This is the sixth one in a series of articles describing the technology of determining the Earth’s gravitational field parameters through gradiometric measurements performed with an onboard satellite electrostatic gradiometer. It provides formulas for calculating the components of the gravitational potential tensor in a geocentric spatial rectangular earth coordinate system in order to convert them into a gradiometric one and obtain a free term for the equations of correcting gradiometric measurements when determining the parameters of the Earth’s gravitational field. The components of the gravitational gradient tensor are functions of test masses accelerations measured by accelerometers and relate to the gradiometer coordinate system, while the desired parameters of the Earth’s gravitational field model relate to the Earth’s coordinate one. The components of the gravitational gradient tensor are the second derivatives of the gravitational potential in rectangular coordinates. The calculated values of the gravitational potential tensor components in the earth’s spatial rectangular coordinate system are obtained through double differentiation of the gravitational potential formula. Basing on the obtained formulas, an algorithm and a program in the Fortran algorithmic language were developed. Using this program, experimental calculations were performed, the results of which were compared with the data of the EGG_TRF_2 product.