Solution for the effective bulk modulus of a medium with square pores

The effective bulk modulus K of brittle porous materials such as concrete is related to the matrix bulk modulus \(K_\mathrm{m}\) and is affected by factors such as the pore shape and Poisson’s ratio. Basing on Walsh functions and the complex variable function solution for the square pore, this paper derives a formula for calculating the effective bulk modulus of a medium with square pores under planar arrangement condition. Meanwhile, the effective bulk modulus of a model with a random pore distribution is calculated using a numerical method. The results of the numerical calculation and the formula are compared with the results of tests and numerical simulations from previous studies. The results show that there is a critical porosity for the theoretical formula; when the porosity is lower than the critical porosity, the results of the single-pore formula match the test and numerical results well, whereas when the porosity exceeds the critical porosity, the results of the multi-pore formula match the test and numerical results; when the porosity is greater than 0.45, the results of the numerical simulation deviate significantly from those of the theoretical formula. A verification of the numerical results versus the theoretical results shows that the square pore formula is an excellent representation of the relationship between the effective bulk modulus and the matrix bulk modulus for porous media such as concrete within limits.