PSI - Issue 80

Haolin Li et al. / Procedia Structural Integrity 80 (2026) 23–30

25

Author name / Structural Integrity Procedia 00 (2023) 000–000

3

Fig. 1. Periodic domain in homogenisation problems.

where ¯ F denotes the applied average strain and ∂ Ω represents the domain boundary, V is the volume of the domain. The first relation in Eq. (4) prescribes the macro strain applied to the cell, and the second imposes homogeneous Neumann boundary conditions. The objective of the cell problem is to obtain the homogenised e ff ective first Piola–Kirchho ff stress:

1 V Ω

¯ P =

P ( x ) dV , ∀ x ∈ Ω

(4)

2.2. Hyperelasticity model

This work implements the large-deformation formulation via the Complete Lagrangian formulation, by introducing hyperelasticity. The functional associated with Eq. 1 defining the potential energy is: H ( X ) = Ω Ψ ( X ( x )) dV (5) where X denotes the deformed configuration: X ( x ) = x + u ( x ) (6) and Ψ represents the strain energy. Minimising Eq. (5) is equivalent to solving the strong-form PDE in Eq. (1). In this work, the strain energy Ψ is based on the Neo-Hookean hyperelastic model:

2

¯ I 1 − 3 +

C 10

1 D 1

( J − 1) 2 ,

(7)

Ψ =

where

• C 10 is a material constant that governs the deviatoric (shear) response of the material, related to the shear modulus by µ = 2 C 10 . • D 1 is a material constant that governs the volumetric response (compressibility), related to the bulk modulus by K = 2 D 1 . • ¯ I 1 = J − 2 / 3 I 1 is the first deviatoric invariant of the right Cauchy–Green tensor, • I 1 = tr( C ), with C = F T F the right Cauchy–Green tensor, • J = det( F ) is the Jacobian of the deformation gradient F .