By Yozo Matsushima
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Extra info for Differentiable Manifolds (1972)
7. With the same relative density, the stiffness of hexagonal honeycomb is far less than the other two kinds of lattices, so the plastic dissipation is a less value at the beginning. Large deformation occurs as the sketching goes further, the topology of hexagonal honeycomb transfers to an approximate rectangle. Therefore, the rising velocity of its plastic dissipation increases quickly. The Kagome lattice shows slight advantage in plastic dissipation compared with the triangular honeycomb. 4 Conclusions The distinction between the tensile and compressive yield strengths of the solid material can inﬂuence the yield surfaces of the lattice structures in both the shape and size.
Besides the generalized Young-Laplace Eq. (1), we need a surface constitutive equation to solve boundary-value problems with the surface stress effect. For an elastically isotropic surface, it is (Duan and Karihaloo, 2007) τ = λs (trε s )1 + 2µs ε s − ∆Tαs κs 1 ε s = 12 D ⊗ u + D ⊗ uT , D ⊗ u = P · (∇s ⊗ u) (3) where P = I(2) −n⊗n, I(2) is the second-order identity tensor in a three dimensional space, ε s is the surface strain tensor, λs and µs are the surface moduli, κs = 2(λs + µs ), αs is the coefﬁcient of thermal expansion (CTE) of the surface, 1 is the secondorder unit tensor in two-dimensional space, and ∆T is the temperature difference.
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Differentiable Manifolds (1972) by Yozo Matsushima