Mechanical modeling was carried out on an ANSYS 19.0 by an FE model based on the constitutive equation. In this case, the 3D problem of the stress-strain relations can be calculated as follows{ε}=[C]{σ}(4)or {σ} = [E]{ε} and [C] = [E]−1.

The equation above is known as the generalized Hooke’s law. The matrix C is called the material compliance matrix, while its inverse E is known as the material stiffness matrix or, more commonly, the constitutive matrix.

The compliance matrix for an isotropic elastic material can be described asε11=1E[σ11ν(σ22+σ33)]ε22=1E[σ22ν(σ11+σ33)]ε33=1E[σ33ν(σ11+σ22)]2ε23=σ23G,2ε13=σ13G,2ε12=σ12G(5)E is the Young’s modulus, ν is the Poisson’s ratio, and G is the shear modulus. They are referred to as the engineering constants. The shear modulus G is related to the Young’s modulus E and Poisson’s ratio ν by the expression G=E2(1+ν).

The equations above can be written in the following matrix form[ε11ε22ε332ε232ε132ε12]=1E[1νν0001ν00010002(1+ν)002(1+ν)0symm2(1+ν)] [σ11σ22σ33σ23σ13σ12]

Invert[σ11σ22σ33σ23σ13σ12]=[λ+2μλλ000λ+2μλ000λ+2μ000μ00μ0symmμ] [ε11ε22ε332ε232ε132ε12]

Here, λ and μ are the Lamé parametersλ=(1+ν)(12ν),μ=G=E2(1+ν)(6)

For simplicity, the generalized Hooke’s law can be described as the linear relation among all the components of the stress and strain tensorσij=Cijklεkl(7)where Cijkl are the components of the fourth-order stiffness tensor of the elastic moduli.

The stiffness tensor has the following minor symmetries, which result from the symmetry of the stress and strain tensorsσij=σjiCjikl=Cijkl(8)Such as the matrix above.

σij = Cijklεkl represents[σ11σ22σ33σ23σ13σ12]=[C1111C1122C1133000C2222C2323000C3333000C232300C13130symmC1212] [ε11ε22ε332ε232ε132ε12]

For thermoelastic effectsεijT=αΔTδij(9)where εijT is the thermal strain, α is the thermal expansion coefficient, and δij is the Kronecker delta.

The total strains are those produced by the mechanical strains and the thermal strainsεij=εijM+εijTσij=Cijkl(εklαΔTδkl)(10)

When the thermal effects are considered, the relationship between stress and strain is given byσij=Cijkl(εklαΔTδkl)(11)

Here, the macro was continued to create nodes and FEs. The model with dimensions of 35 mm by 10 mm by 20 μm was used for the PC membrane part, and the model with dimensions of 10 mm by 10 mm by 5 μm was used for the MXCC part and partly embedded in the center of SOLID45 element. Hexahedral meshes were generated across the whole actuator, and more segmentation hexahedral meshes were generated on the MXCC part according to the specifics of the design (fig. S10). Displacement and symmetrical constrains were also implemented for the boundary conditions in the FE model. Uniformed loading and the torque versus y axis were applied to the actuator.

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