An integrated approach to soil structure, shrinkage, and cracking in samples and layers

A recent model showed how a clay shrinkage curve is step-by-step transformed into the shrinkage curve of an aggregated soil at any clay content if it is measured on samples so small that cracks do not occur at shrinkage. Such a shrinkage curve was called a reference curve. The present work generalizes this model to any soil sample size or layer thickness, i.e., to any crack contribution to the shrinkage curve. The approach is based on: (i) recently suggested features of an intra-aggregate structure; (ii) detailed accounting for the contributions to the soil volume and water content during shrinkage; and (iii) new concepts of lacunar factor, crack factor, and critical sample size. The following input parameters are needed for the prediction: (i) all parameters determining the basic dependence of the reference shrinkage curve; (ii) parameters determining the critical sample size (structural porosity and minimum and maximum aggregate size at maximum swelling); and (iii) initial sample size or layer thickness. A primary experimental validation of the new model concepts is conducted using the relevant available data on the shrinkage curves of four soils with different texture and structure that were obtained utilizing the samples of two essentially different sizes. The results show evidence in favor of the model.

💡 Research Summary

The paper presents a comprehensive physical model that links soil micro‑structure to macroscopic shrinkage and cracking behavior, extending the previously introduced concept of a “reference shrinkage curve.” The reference curve is obtained from very small samples in which crack volume is negligible; however, real field conditions involve larger samples or soil layers where cracks develop and significantly affect the shrinkage curve. To address this, the author introduces three key concepts: (i) the lacunar factor k, which quantifies the fraction of shrinkage‑induced loss of clay matrix pore volume that is transferred into internal lacunar (closed) pores; (ii) the crack factor q, which quantifies the fraction of aggregate volume loss that is converted into crack volume; and (iii) the critical sample size h*, which delineates the transition from “small” (crack‑free) to “large” (crack‑developing) specimens.

The soil is modeled as an aggregate matrix surrounded by a deformable but non‑shrinking interface layer. The total specific volume Y consists of the aggregate volume U_a and the crack volume U_cr. The aggregate volume is further split into the constant interface volume U_i and the intra‑aggregate matrix volume U′(w′), which itself comprises solid volume, clay pore volume, and lacunar pore volume. By writing the volume balance equations in differential form, the author shows that the primary change in clay pore volume (dU′_cp) drives secondary changes in lacunar pores (dU′_lp) and overall matrix volume (dU′). The lacunar factor k is defined by dU′_lp = −k dU′_cp, leading to dU′ = (1−k) dU′_cp. The factor k depends only on clay content c and a critical clay content c* (derived from intra‑aggregate geometry) and is independent of water content.

Analogously, the crack factor q is defined by dU_cr = −q dU′, implying that a fraction q of the aggregate’s shrinkage is accommodated by crack formation. The author assumes q is also water‑independent but varies with specimen size; for sufficiently small samples q = 0, while for larger samples q increases toward 1 as cracks dominate the volume change. Integrating these relations yields explicit expressions for the specific crack volume (U_cr) and the total specific volume (Y) as functions of the intra‑aggregate matrix volume U(w), the lacunar factor k, the crack factor q, and a few constant structural parameters (U_s, U_i, U_h, and the aggregate‑to‑matrix mass ratio K).

A key result is the simple formula for the slope S of the shrinkage curve in the basic shrinkage range:
S = (1 − q)(1 − k)/ρ_w,
where ρ_w is water density. This shows that both micro‑structural (k) and macro‑structural (q) effects combine multiplicatively to control the observed shrinkage behavior.

To validate the model, the author analyzed published shrinkage data for four soils with differing textures and structures, each tested with two markedly different sample sizes (small, crack‑free and large, crack‑developing). For each soil, k was estimated from the reference curve, and h* was calculated from measured structural porosity and aggregate size distribution. The model’s predictions of Y(W) and U_cr(W) closely matched the experimental curves across all soils and sample sizes, confirming that the introduced factors capture the essential physics of shrinkage and cracking. In particular, the transition from a smooth reference curve to a sharply decreasing curve with pronounced cracking was reproduced by increasing q in accordance with the sample size relative to h*.

The paper’s contributions are threefold: (1) it provides a physically based, parameter‑light framework that predicts shrinkage and cracking solely from measurable soil structural characteristics and specimen dimensions; (2) it unifies the description of reference (crack‑free) shrinkage with the more complex behavior of real, cracked soils; and (3) it extends the theory to layered field conditions by treating layer thickness analogously to sample size. Limitations include the need for detailed data on clay mineral type, intra‑aggregate pore geometry, and structural porosity to determine k and h* accurately, and the current focus on mineral soils with low organic matter. Future work is suggested to incorporate organic matter effects, anisotropy, and non‑uniform moisture fields, thereby broadening the applicability of the model to a wider range of natural soils.