Materials Constitutive Relationship Using Micromechanics And Its Structural Topology Optimization

Structural topology optimization plays an important role in the field of engineering design with the wide application of computational mechanics.In general,more cost can be saved with the application of optimization algorithm through removing materials with lower efficiency.This paper established the constitutive relationship of continuum material using granular micromechanics,and the mathematical formulation for structural topology optimization is given consequently.The finite element method related topology optimization has zig-zag effect in the optimal topology.This paper gives B-spline post-processing method for structure topology optimization,and the topologies have smoother boundary.Main research topics are listed as following.1.The constitutive relationship of granular material based on micromechanics is established.The diversity of biological structures in nature leads to complex constitutive relationships.Using granular material models can predict the mechanical behavior of biological structures.The Taylor expansion is used to link the displacements of two adjacent particles,and a local coordinate system is established at the particle contact position.The displacement of particle is obtained by using the Cauchy stress and the Taylor series.Then,the constitutive relationship of granular material model is derived.2.The mathematical formulation for topology optimization is established with minimization of compliance and volume constraint.According to the constitutive relation of the derived material constitute,the mathematical formulation of topology optimization is established,and the optimization algorithm of this model is proposed.Some benchmark problems in topology optimization are computed according to the different parameters in the material constitute.According to the final results,the influence of the tension-compression ratio on the final topology is analyzed.The asymmetry of final topologies demonstrates the effectiveness of proposed algorithm and comply with some bio-structure.The results of benchmark problems show that the mathematical formulation is feasible for the proposed constitutive relationship of granular material.3.B-spline post-processing method is developed.The bivariate B-spline basis function is used to represent the density within the element,and the B-spline curve is used to depict the material boundary.The post-processing method is applied to the topology optimization of the granular material.Compared with the original results of topology optimization,B-spline post-processing method make the boundary smoother and more convenient for subsequent analysis and design.In conclusion,the micro-mechanics behavior between particles is considered to establish the granular material constitutive and integrated with the topology optimization.The model considers the microstructure and the interaction of the particles,and it can accurately predict the constitutive relationship of the continuum material.Applying this constitutive relationship,the mathematical formulation of topology optimization is established,and the B-spline post-processing method is applied to represent the boundary of material boundary.The computational results of benchmark problems in topology optimization show that the validity and effectiveness of constitutive relationship and the mathematical formulation of topology optimization proposed in this paper.