Bi-Material Negative Thermal Expansion Inverted Trapezoid Lattice based on A Composite Rod

Negative thermal expansion (NTE) lattices are widely used in aerospace engineering where the structures experience large temperature variation. However, the available range of NTE of the current lattices is quite narrow, which severely limits their engineering application. In this paper, we report an inverted trapezoid lattice (ITL) with large NTE. The NTE of the ITL is 2.6 times that of a typical triangular lattice with the same height and hypotenuse angle. Theoretically, with a pin-jointed assumption, the ITL can improve the NTE by order of magnitude if the length ratio of the composite rod is changed. In the presented ITL, a composite rod is utilized as the base of the ITL. The composite rod has large inner NTE. The inverted trapezoid structure converts the inner NTE to the vertical direction contraction and obtains an extra NTE. Finite element simulations and experimental verification by interferometric measurement were conducted to verify the large thermal expansion of the ITL.


S1. Thermal Expansion of The Composite Rod
The virtual rod's coefficient of thermal expansion (CTE) is decided by the thermal deformation of the long rod and the two short rods. For homogeneous temperature increment, the elongation of the long rod is smaller than that of the short rod, which generates shrinkage of the inner ends of the two short rods. This make the virtual rod with negative thermal expansion (NTE). The virtual rod has thermal expansion α defined such that an increment in temperature ΔT causes a length change. The formula for calculating the equivalent CTE of the virtual rod is given as follows.
For an increment in temperature ΔT, the elongation of the long rod is L D , the elongation of the short rod is S D , and the elongation of the virtual rod is V D , thus and in general: Therefore, combining the equations:

Thermal Expansion of The Bi-Materials Triangular Lattice
The parameters of the inverted trapezoid lattice (ITL) are as follows. The ratio of L/r is 5, the virtual rod length is 24 mm, the upper base length is 120 mm, the height is 30 mm, the rod width is 5 mm, and all rods have a thickness of 5 mm. The parameters of the triangular lattice are the same as with the ITL, as shown in Figure S1. Thus, the angle of the hypotenuse and the height of the lattice are the same. The two constituents are aluminum alloy and titanium. According to the CTE equation of a triangular unit, which was derived by Miller [1], the thermal expansion of the y-direction, α , is (6) When α = 8.6 × 10 −6 K −1 , α = 23.1 × 10 −6 K −1 , a = 4 + ℎ ⁄ = 56.6 mm, according to Equation (6), the thermal expansion of the triangular lattice is α = ( α − 4 * ⁄ α ) ℎ = −28.5 ⁄ × 10 −6 K −1 .

Thermal Expansion of The Bi-Materials Inverted Trapezoid Lattice with The Varying Length Ratio
It is difficult to theoretically figure out the CTE of the ITL with a rigid connection assumption. However, it can be approximated with a pin-jointed assumption. According to the CTE calculation method of Miller [1], considering that the rods of the ITL can freely rotate at the nodes, the thermal expansivities along the y-axis are a function of the geometry of ITL. The thermal expansivity of the ITL can be calculated by considering half of the isosceles trapezoid to form a Pythagorean triangle, thus: and in general: Therefore, combining Equations (7)-(9), the thermal expansion of the bi-material inverted trapezoid lattices along the y-axis direction is given as: The following calculation provides the results of CTE with the length ratio changes. The CTEs of ITL are calculated according to Equation (10). The CTEs used in the calculation are the property of the two constituents: aluminum alloy and titanium. The other structural parameters except the length of long rod are fixed: the virtual rod length is 24 mm, the upper base length is 120 mm, the bi-material ITL height is 30 mm, and all rods are 5 mm wide and 5 mm thick. The length ratio is changed by varying the length of the long rod. The results are presented in Table S1. With length ratio increasing, the NTE of the bi-material inverted trapezoid lattices increases continuously. Considering that the CTE of the triangular lattice is -28.5 × 10 -6 K -1 (Section S2.1), when the length ratio is 28, the NTE of the bi-material inverted trapezoid lattices is -286.3 × 10 -6 K -1 . Thus, the NTE of the bi-material inverted trapezoid lattices will reach to 10 that of the traditional triangular lattice.
It is easy to achieve a large length ratio by changing the length of the long rod. However, there are still some restricting factors in practical applications. 1. Buckling.
With a large length ratio, the length of the short rod will become long. Then, the short rod becomes slender and unstable under compressive stress. This will make the short rod buckle and lead to structural failure. The pin-jointed lattice allows free rotation at the nodes. It makes no stress in the lattice. This connection is a possible solution to the buckling. However, the joint gaps at the nodes will generate remarkable CTE error. 2. Geometrical limits.
First, when the length ratio is large enough, the inner end of the short rod will meet together as temperature increases. This will restrict the negative thermal expansion of the virtual rod and generate large stress.
Second, large length ratio makes the structure's ratio of length to width seriously out of balance.