How to surpass the deep-sea glass sponges mechanically

: The biomimetic design of engineering structures is based on biological structures with excellent mechanical properties, which are the result of billions of years of evolution. However, current biomimetic structures, such as ordered lattice materials, are still inferior to many biological materials in terms of structural complexity and mechanical properties. For example, the structure of Euplectella aspergillum , a type of deep-sea glass sponge, is an eye-catching source of inspiration for biomimetic design; however, guided by scientiﬁc theory, how to engineer structures surpassing the mechanical properties of E. aspergillum remains an unsolved problem. The lattice structure of the skeleton of E. aspergillum consists of vertically, horizontally, and diagonally oriented struts, which provide superior strength and ﬂexural resistance compared with the conventional square lattice structure. Herein, the structure of E. aspergillum was investigated in detail, and by using the theory of elasticity, a lattice structure inspired by the bionic structure was proposed. The mechanical properties of the sponge-inspired lattice structure surpassed the sponge structure under a variety of loading conditions, and the excellent performance of this conﬁguration was veriﬁed experimentally. The proposed lattice structure can greatly improve the mechanical properties of engineering structures, and it improves strength without much redundancy of material. This study achieved the ﬁrst surpassing of the mechanical properties of an existing sponge-mimicking design. This design can be applied to lattice structures, truss systems, and metamaterial cells.

goal in the field of engineering. Compared with ordinary man-made materials, many biomaterials have very high strength, yet remain lightweight and porous [15].
Various geometric arrays of porous grid-like periodic structured biomedical implants [16] have been manufactured. Lightweight lattice structures can also be the core of sandwich structures, which has advantages in multifunctional structural applications in the aerospace, automotive, and civil engineering industries [17]. Truss lattice materials are often used for energy absorption, especially in the design of impact and explosion-resistant structures [18]. Sandwich structures with a lattice or foam core have better performance than the same quality compact structure. A hinge design in a lattice can consume energy through buckling [19][20][21][22]. The excellent mechanical properties of the skeleton of E. aspergillum have been extensively studied [23,24]. However, most studies have focused on the performance of sponge spicules [25][26][27][28].
The shape of the sponge skeleton has been applied to the appearance of buildings [29]. To date, only Matheus et al. [11] have carried out research on the diagonally reinforced square grid unit structure of the sponge skeleton, and the optimized configuration is very close to the spongemimicking design. However, the potential of this square grid form in mechanics has not been fully explored, and the key scientific problem of surpassing the performance of the sponge structure has not been solved. In this study, our objective was to develop a sponge-inspired structure with beter mechanical properties than the skeleton of E. aspergillum.
Biological systems usually face limited resources, and the skeleton is designed to provide structural stability at minimal cost, which is the common theme of various biological evolutions. The evolutionary trends in natural biomaterials share many of the same aspirations as the development of engineered materials, such as weight reduction while ensuring effective strength and rigidity [30], as seen in feathers [31], plant stems [32], and porcupines quills [33]. Numerous biological materials are inspiring to scientists and engineers. The experience and knowledge gained from research into the structure of biological materials can guide the design of new bionic structures.
Lattice materials with complex topologies can be used in a wide variety of engineering applications, such as tower structures, lightweight sandwich structures, and filters [13,34].
Biological structures have been optimized over longterm evolution and have hierarchical structures [35]. Regarding the optimization of biological structures in nature, the optimal shape will be different for different load patterns. Using optimal design to discover new unit geometry is an effective way to further improve engineered structures [36]. The concept of structural optimization design reflects the best use of materials and resources.
The optimization goal of this study for the proposed sponge-inspired structure was to maximize the load-bearing capacity of the structure under different loads while using the least material possible. First, we carried out finite element simulations of different lattice structures (including the sponge-mimicking lattice structure developed in [11]), calculated the responses of the structure under various load conditions, and verified the experimental and finite element results of [11]. Then, based on our findings, we developed a sponge-inspired design surpassing the mechanical properties of the sponge, which can be used to guide the cell design of lattice structures and metamaterials. It can also be applied to the design of large man-made structures, such as space grids, towers, and bridges.

Underlying assumptions
In this study, we were primarily concerned with the in-plane performance of the lattice structure of the skeleton of E. aspergillum. Detailed measurements on digital photos of glass sponges had previously been carried out in [11]. In order to design the sponge-mimicking configuration as simply as possible, Matheus et al. established the following basic framework of assumptions: the in-plane geometry is uniform, and all elements have the same shape in the thickness direction; all diagonal elements have the same in-plane dimension; all non-diagonal elements have the same in-plane dimension; and in the volume calculations, the area where the overlapping beams cross is negligible. We also adopted these assumptions in this study. The cross-section of the sponge-mimicking design from [11] (Design A) is shown in Figure 2(a), with the non-diagonal elements in black and the diagonal elements in green. Figure 2

Finite element results and analysis
In order to verify the excellent mechanical properties of Design A (sponge-mimicking design), we conducted finite element simulations using Ansys Workbench and compared the results with the simulation and test results in [11]. The geometries were constructed using twodimensional solid elements, and we captured the material's response using an incompressible Neo-Hookean material model with a shear modulus of µ = 14.5 MPa.   In the modal analysis, specimens were fixedly supported at the bottom ends, and we found that the natural frequency of Design A was second highest after Design C.
This preliminarily indicated that Design A has excellent thermal stability and dynamic stability. Euler's formula F cr = π 2 EI l 2 , which is part of the theory of elasticity, reveals an important law. To increase the buckling critical force F cr under equal mass, the trusses must be thicker (the moment of inertia I is greater) and shorter (the length l is smaller), as shown in Figure 6(a). Inspired by the sponge skeleton and the design rule, we developed Design E, which is shown in Figure 7(a).
The non-diagonal elements were identical to Design A (the sponge-mimicking design). However, the diagonal elements were twice as thick compared to Design A, but the total volume was kept equal to that of Design A.
We numerically simulated the uniaxial compression and shear performance of the 6 × 6 square grid cell of Design E and compared the results with the results of Designs A-D. As shown in Figure 7(b), Design E had the lowest initial compression stiffness but the highest load-bearing capacity at strain 6%. As shown in Figure 7 concluded that the double-diagonal sponge design has the best compressive performance [11].
Does Design A and its optimization results represent the limit of such a square lattice structure? Has this double-diagonal-enhanced square grid design reached its optimal performance? Developing a design surpassing the performance of Design A has become a world problem. We have the courage to challenge and devote ourselves to solving this problem, and finally try to surpass Design A under multi-load conditions. There are three main problems with the work of Matheus et al. [11].
First, it ignored the volume distribution in the horizontal and vertical directions in non-diagonal elements. Sec-ond, the design with a single diagonal number (Design B) still had room for optimization, and the performance had the potential to surpass Design A. Third, it set a single optimization goal (that is, the highest critical buckling stress), but the resulting configuration may not be able to achieve excellent comprehensive performance under multi-load conditions. The problem that needs to be solved is clear: how can a new configuration be developed that fully exceeds Design A under multi-load conditions? In this part of the study, our goal was to create a lattice structure design surpassing Design A under multiple load conditions.
To this end, we developed an idea of optimizing the design on the basis of Design E (diagonal elements have a smaller length-to-thickness ratio, which means they are thicker and shorter, but the diagonal elements do not have a through connection) and Design B. The schematic diagram of the optimization parameters in a 6 × 6 square grid is shown in Figure 8.
In the process of simulating lattice structure, material nonlinearity and geometric nonlinearity are big challenges for convergence and computing, resulting in unbearable consumption of computational resources and time to find the peak load. Therefore, we assumed that the magnitude of the initial stiffness corresponds to the strength of the load-bearing capacity so that the optimization results could be obtained quickly. In order to obtain an excellent configuration under multi-load conditions at the lowest cost, we selected the optimization objectives to be the compression, shear loads under a given small displacement, and the rotation angle under a given small bending moment. We selected three parameters to be optimized: the volume ratio of non-diagonal elements to diagonal elements λ 1 , where F c is the axial force when the compressive strain is 1%, F s is the shear force when the displacement is 0.5 mm, and θ is the angle under the 100 N · mm bending moment.
We adopted the multi-objective genetic algorithm (MOGA) provided in Ansys for multi-objective optimization, and obtained the following optimization design results: λ 1 = 1.55, λ 2 = 2.963, and ∆L = 0.32081. We denoted the corresponding design as Design F.
Matheus et al. [11] took the highest critical buckling stress as the optimization goal and performed seven separate optimizations on the diagonally reinforced square grid structure. The optimal result, which we herein denote as A-Opti1, was in the form of a double diagonal and only differed from Design A in the values of two parameters: the volume ratio of non-diagonal elements to diagonal elements λ and the node distance S. For A-Opti1, λ = 0.9, S = 0.6788, and for Design A, λ = √ 2, , as shown in Figure 2(a). In order to compare with the optimal results of the sponge configuration and maintain the consistency of the simulations in this paper, we used Ansys to optimize Design A with the optimization goal of high buckling resistance, and we denoted the optimal result as A-Opti2 (λ = 0.74914, S = 0.76853).
Next, we performed finite element simulation of the sponge-mimicking design (Design A) and its optimization results (A-Opti1 and A-Opti2) under various load conditions. The results are shown in Figure 9 (the stress contour diagram under each load situation is shown in supplementary Figures 7-12). We set the response of It is important to note that the optimal design results presented in this paper have preconditions, that is, the comprehensive optimal value of multi-objective optimization under certain basic assumptions and given constraints. The result of the optimal design is not unique.
We can change the constraint conditions according to the structural performance requirements so as to obtain the optimal design under specific conditions, and to realize the programmability of the mechanical properties of the lattice structure. and Design G (common design in engineering) at equal volumes (without increasing mass). The loading diagram is shown in Figure 9. The left end was fixed, and the structure was subjected to self-weight only. The thick-ness of the specimen was taken as 6 m, with a density of 7850 kg/m 3 , the Young's modulus was 200 MPa, and the Poisson's ratio was 0.3. The result of the longitudinal displacement contour image magnified by 100 times is shown in Figure 10 (the stress contour diagram is shown in Supplementary Figure 13).   Figure 11. It is evident that the uniaxial compressive load carrying capacity and stiffness of Design F were stronger than those of Design A. Due to the poor quality between layers of 3D printed specimens, there were some defects visible to the naked eye, and the serious ones are shown in Figure 11(c). After considering the larger initial defect (1 mm), the experiment data were still slightly lower than the finite element simulation results, and the laws and trends obtained by the experiment were consistent with the finite element simulation results. Its engineering applications range from various industrial and civil infrastructures to micro-nano-level structural elements.
The evolution of nature and the depth of its scientific laws are amazing to contemplate. We would hazard a guess that for E. aspergillum, the optimized results we achieved may be one of the paths of its future evolution.