Cyclic loading of a heterogeneous non-linear poroelastic material

TL;DR

This study develops a one-dimensional, nonlinear poroelastic model for a saturated heterogeneous material under uniaxial cyclic loading, combining a neo-Hookean solid skeleton with porosity-dependent Kozeny–Carman permeability and Darcy flow. Heterogeneity is introduced as Gaussian perturbations in stiffness or permeability, and the system is solved in a Lagrangian framework to compare responses driven by applied stress versus displacement. Key findings show that spatially localized stiffness reductions enhance local strain and can alter flux distributions, with diffusion and localization governed by the poroelastic timescale and loading frequency; permeability heterogeneity mainly modulates flux and diffusion with weaker effects on overall strain. The work provides mechanistic insight into how continuous material heterogeneity shapes cyclic poroelastic responses, with potential implications for tendon pathology and other hydrated soft tissues, and offers a tractable basis for extending to more complex, higher-dimensional models.

Abstract

Cyclic loading is a common feature in poroelastic systems, the material response depending non-trivially on the exact form of boundary conditions, pore structure, and mechanical properties. The situation becomes more complex when heterogeneity is introduced in the properties of the poroelastic material, yet heterogeneity too is common in physical poroelastic structures. In this paper, we analyse the behaviour of a soft porous material in response to a uniaxial cyclic stress or displacement, with a focus on understanding how this response is affected by continuous heterogeneity in the stiffness or permeability. Our work is motivated by observed altered material properties of the diseased tendon, but the framework we develop and analyse is generically applicable. We construct a one-dimensional non-linear poroelastic model, assuming Darcy flow through the pores of the solid skeleton which we assume has neo-Hookean elasticity. The system is driven by an applied uniaxial cyclic stress or a uniaxial cyclic displacement at one boundary. Heterogeneity in the stiffness or permeability profile is imposed via a Gaussian bump function. By exploring a range of loading frequencies together with magnitudes and locations of heterogeneity, we characterise the effect of heterogeneity on the response of the material, and show that the response of the system to an applied stress is qualitatively distinct from the response to an applied displacement. Our analysis of this simple model provides a foundation for understanding how heterogeneity affects the poroelastic response to cyclic loading.

Figures (13)

• Figure 1: We consider a poroelastic material of length $L_0$ which is pulled at $Z=0$ (red arrow) where it is also subject to ambient fluid pressure $p_A$, and fixed at $Z=L_0$ with no deformation or flow. The blue arrows represent fluid flow. The material is shown in the reference configuration.
• Figure 2: Material damage is modelled with an inverted Gaussian $f$ of magnitude $d$ at $Z=l$
• Figure 3: Strain (first and third rows) and flux (second and fourth rows) plotted for 9 equally spaced values of cycle $38\pi/\omega \leq t \leq 40\pi/\omega$, for uniform stiffness (first column) and locally decreased stiffness with $d=0.35$ (columns 2-4). The first two rows are in response to an applied stress ($A_l=0.2$, $\omega=10$) and the second two rows are in response to an applied displacement ($A_d=0.1$, $\omega=10$). The applied stress $s^*(t)$ and displacement $A(t)$, and their respective time derivatives $\Dot{s}^*(t)$, $\Dot{a}(t)$ are displayed as insets in the first column. We differentiate between the loading phase ($\Dot{s}^*>0$, $\Dot{a}<0$, dark blue to light blue) and unloading phase ($\Dot{s}^*<0$,$\Dot{a}>0$, light red to dark red). The end of the cycle is in dotted red.
• Figure 4: Time integrated stress and flux response for uniform stiffness (light orange) and locally decreased stiffness (dark orange) against $Z$
• Figure 5: Cumulative strain (left) and flux (right) against $Z$ under AS ($A_l=0.2$, $\omega=10$, top row) and AD ($A_d=0.1$, $\omega=10$, bottom row), for homogeneous stiffness (dotted grey line) and for damaged stiffness with $d = 0.35$ and varying location $l = 0.25,0.5,0.75$.