Abstract: Alluvium sand–gravel (SG) foundations are among the typical foundation conditions for dams, and the mechanical behavior of the foundation materials directly determines the stress and deformation characteristics of the dam. These mechanical responses influence both the safety of dam deformation coordination and the effectiveness of long-term deformation control. However, due to the heterogeneous composition of sand–gravel mixed with fines such as mud, their stress–strain behavior under external loading remains complex and difficult to characterize using conventional soil mechanics models. To provide reliable technical support for the accurate estimation of dam deformation on alluvium foundations, this study focuses on the stress–strain characteristics of alluvium SG materials containing different mud mass fractions under triaxial compression. A series of consolidated drained triaxial shear tests were conducted in the laboratory. Specimens were prepared with controlled relative density to minimize variability, and confining pressures covering typical in-situ ranges were applied to simulate different stress conditions within a dam foundation. The mud mass fraction was considered a key influencing factor, and tests were carried out under various combinations of mud mass fraction and confining pressure. Through these experiments, the effects of mud mass fraction and confining pressure on stress–strain curves, nonlinear shear strength indices, and volumetric strain behavior were systematically analyzed. The test results indicate clear and consistent trends. With decreasing confining pressure and increasing mud mass fraction, specimens with the same relative density exhibited a reduction in peak strength, showing an increased tendency to yield at lower stress levels. The phenomenon of strain-softening becomes more pronounced under lower confining pressures, where specimens reach a peak stress and then experience a notable drop in strength with continued deformation. Regarding volumetric strain behavior, specimens with higher mud mass fractions show a greater propensity for shear contraction at the same confining pressure. At relatively low confining pressures, specimens initially exhibit dilatancy before transitioning into contraction, and the dilatancy effect is more prominent as the confining pressure decreases. These findings highlight the dual role of mud: while it weakens the structural strength of the gravel matrix, it also modifies the deformation mode from dilation to contraction, thereby influencing the shear resistance mechanism. Based on these test results, the applicability of the classical Duncan–Chang model to alluvium SG materials containing mud was assessed. Although the Duncan–Chang model can represent general nonlinear stress–strain behavior, its limitations become apparent when applied to alluvium SG materials exhibiting significant strain-softening and dilatancy effects. To overcome these shortcomings, an improved constitutive model is proposed in this study. The new model incorporates mechanisms that explicitly capture strain-softening and dilatancy, making it more suitable for describing the instantaneous deformation behavior of SG materials under triaxial shear conditions. A methodology for determining the model parameters is provided, and regression relationships between mud mass fraction and the key model parameters are established. These regression functions allow the model to dynamically adjust with varying mud fractions, enhancing its predictive capacity. Finally, the performance of the proposed model was evaluated using the laboratory tests presented in this paper as well as data from existing studies, indicating that the model can accurately reflect the stress–strain characteristics of alluvium SG specimens under triaxial shear conditions. This study provides a theoretical basis for the accurate estimation and coordinated control of dam deformation on alluvium SG foundations.