Mechanics of Time-Dependent Materials最新文献

{"title":"Dynamic compression failure and constitutive modeling of fully solid-waste cemented tailings backfill under impact loading","authors":"Jianan Miao,&nbsp;Dongdong Ma,&nbsp;Rongrong Zhang,&nbsp;Haoyu Pei","doi":"10.1007/s11043-026-09887-z","DOIUrl":"10.1007/s11043-026-09887-z","url":null,"abstract":"<div><p>The growing emphasis on low-carbon and sustainable mining has intensified interest in alternative cementitious materials for cemented tailings backfill (CTB). However, the dynamic mechanical behavior of CTB incorporating fully solid-waste cementitious material remains insufficiently characterized. In this study, a fully solid-waste cementitious material composed of ground granulated blast-furnace slag, desulfurization gypsum, and fly ash was developed and applied to CTB. Using the glue-bone ratio, mass concentration, and average strain rate as variables, the dynamic compression tests were performed with a split Hopkinson pressure bar system to investigate the impact response of fully solid-waste cemented tailings backfill (FSWCTB). The strain-rate-dependent stress-strain behavior curves, dynamic compressive strength and failure morphology were analyzed, and the applicability of the Z-W-T constitutive model to FSWCTB was further evaluated. Furthermore, nuclear magnetic resonance and scanning electron microscopy coupled with energy dispersive spectroscopy were used to elucidate the microstructural features and hydration mechanisms of the FSWCTB material. Results indicated that (1) the dynamic compressive strength of FSWCTB increased progressively with higher glue-bone ratio, mass concentration, and average strain rate. (2) when the average strain rate increased to 415 s⁻¹, the failure mode of the FSWCTB specimens transitioned from localized collapse to global fragmentation. (3) The increase in glue-bone ratio and mass concentration led to the formation of a denser microstructure, which was beneficial for the development of backfill strength. The findings provide fundamental insights into the dynamic performance of FSWCTB and support its sustainable application in underground mining engineering.</p></div>","PeriodicalId":698,"journal":{"name":"Mechanics of Time-Dependent Materials","volume":"30 2","pages":""},"PeriodicalIF":2.3,"publicationDate":"2026-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148237000","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"材料科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Learning memory kernels in second-order Volterra models: a data-driven approach for viscoelastic systems","authors":"Yacine Khaldi,&nbsp;Meriem Belaifa,&nbsp;Amir Benzaoui","doi":"10.1007/s11043-026-09890-4","DOIUrl":"10.1007/s11043-026-09890-4","url":null,"abstract":"<div><p>Modeling viscoelastic vibration remains a persistent challenge due to memory effects, coupled dynamics, and the difficulty of linking data-driven models with physical interpretability. Conventional approaches, such as Prony-series identification or fractional calculus formulations, capture relaxation behavior but rely on handcrafted parameterization and are sensitive to numerical discretization. To address these limitations, we propose a physics-informed neural framework that integrates a causal temporal convolutional encoder with a Neural Prony State-Space Layer (NPSSL), a differentiable realization of the linear Volterra operator that generalizes the classical Prony model—and three supervised heads for multi-step displacement forecasting, kernel-family classification, and Prony-amplitude regression. The NPSSL explicitly parameterizes relaxation dynamics as a discrete-time hereditary memory with nonnegative amplitudes, while a physics residual term enforces consistency with the governing second-order oscillator equation balancing forcing, inertia, stiffness, and viscoelastic stress. The proposed framework is first evaluated on synthetic benchmarks covering single-Maxwell, multi-exponential Prony, stretched-exponential, and power-law kernels, where it achieves near-perfect kernel classification, sub-percent forecasting error, and accurate recovery of relaxation spectra. Normalized physics residuals remain approximately unbiased on synthetic data with unit-scale dispersion, confirming stable physical alignment across solvers. To further evaluate generalization in real-world scenarios, the model is validated using the experimental Linear Particle Chain Impact Damper dataset. The NPSSL accurately replicates the observed displacement dynamics, achieving a <span>(R^{2} approx 0.73 )</span> and exhibiting well-behaved residuals. This illustrates that integrating a differentiable Volterra-type memory into a neural architecture facilitates both interpretability and precision, connecting classical hereditary mechanics with real-world dissipative vibration phenomena.</p></div>","PeriodicalId":698,"journal":{"name":"Mechanics of Time-Dependent Materials","volume":"30 2","pages":""},"PeriodicalIF":2.3,"publicationDate":"2026-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148236843","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"材料科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Thermal performance analysis of a chemically reactive functional materials under rotational flow effect","authors":"Rashid Ayub,&nbsp;Shahzad Ahmad,&nbsp;Mushtaq Ahmad,&nbsp;Shajar Abbas,&nbsp;Rashid Jan,&nbsp;Imran Shakir,&nbsp;Tatyana Orlova,&nbsp;Ibrahim Mahariq","doi":"10.1007/s11043-026-09883-3","DOIUrl":"10.1007/s11043-026-09883-3","url":null,"abstract":"<div><p>This study investigates the rotational flow behavior and thermal transport characteristics of a chemically reactive viscous fluid subjected to a uniform magnetic field, with potential applications in advanced energy and thermal management systems. A fractional modeling framework based on the constant proportional Caputo derivative is employed to capture nonlocal and memory-dependent diffusion effects in the transport processes. The fluid flows over an infinite vertical plate undergoing arbitrary motion with Newtonian heating conditions, while the ambient temperature and concentration are maintained at <span>(T_{infty })</span> and <span>(C_{infty })</span>, respectively. Through suitable similarity transformations, the governing momentum, energy, and concentration equations are converted into a dimensionless form and solved analytically to obtain closed-form expressions for the velocity, temperature, and concentration fields. The influence of key dimensionless parameters is examined in detail. The results indicate that an increase in the fractional order parameter enhances the momentum and thermal boundary layers, producing approximately 15–<span>(25%)</span> enhancement in the velocity distribution, 10–<span>(20%)</span> increase in the temperature field, and about 12–<span>(18%)</span> growth in the concentration profile near the plate. Furthermore, the magnetic interaction parameter suppresses the velocity field due to the presence of the Lorentz force, whereas the chemical reaction parameter significantly modifies the concentration boundary layer thickness. These results highlight the effectiveness of fractional calculus in accurately describing memory-dependent transport phenomena and demonstrate the important role of governing dimensionless parameters in optimizing the performance of modern thermal systems.</p></div>","PeriodicalId":698,"journal":{"name":"Mechanics of Time-Dependent Materials","volume":"30 2","pages":""},"PeriodicalIF":2.3,"publicationDate":"2026-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148173386","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"材料科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Size dependent dynamics of a perforated Timoshenko nanobeam made of viscoelastic material with rotationally restrained ends","authors":"Hayrullah Gun Kadioglu,&nbsp;Mustafa Ozgur Yayli","doi":"10.1007/s11043-026-09885-1","DOIUrl":"10.1007/s11043-026-09885-1","url":null,"abstract":"<div><p>In this study, the free vibration behavior of perforated viscoelastic nonlocal Timoshenko beams has been investigated using a semi analytic method based on the Fourier series and the Stokes transformation. First, the geometric structure of the perforated beam was defined; then, by combining the Kelvin–Voigt viscoelastic model with the theory of nonlocal elasticity, the equations of motion describing the system were derived. During the solution phase, the boundary conditions of the beam were reduced to an eigenvalue problem using a semi analytic approach, with spring elements restricting rotation at both ends, and high-accuracy results have been obtained under various boundary conditions. The findings indicate that an increase in the nonlocal scale parameter has a softening effect on structural stiffness, thereby reducing vibration characteristics. Additionally, it is observed that an increase in the number of holes enhances discontinuity along the beam, leading to significant changes in system behavior. It is determined that the viscoelastic damping effect becomes more dominant, particularly under rigid boundary conditions, but weakens as the perforation density increases. The study also reveals that the void ratio, shear deformation, and boundary conditions are in strong interaction with one another. The proposed method offers an effective and reliable methodological approach for the dynamic analysis of perforated nanostructures, thanks to both its high convergence performance and its ability to model different boundary conditions within the same formulation.</p></div>","PeriodicalId":698,"journal":{"name":"Mechanics of Time-Dependent Materials","volume":"30 2","pages":""},"PeriodicalIF":2.3,"publicationDate":"2026-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s11043-026-09885-1.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148009904","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"材料科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Creep behavior and modeling of limestone powder using a fractal derivative and damage mechanics approach","authors":"Senjie Qian,&nbsp;Feng Ji,&nbsp;Haijun Zheng,&nbsp;Zhenyu Wang,&nbsp;Meng Guan,&nbsp;Zhihua Ji","doi":"10.1007/s11043-026-09882-4","DOIUrl":"10.1007/s11043-026-09882-4","url":null,"abstract":"<div><p>In southwestern China, limestone is the primary raw material for cement production, and its extraction and processing generate large amounts of limestone powder waste. This powder is commonly used in high embankments and slope engineering projects. A clear understanding of its long-term deformation behavior under self-weight loading is therefore essential for evaluating slope stability. In this study, the creep behavior of limestone powder is investigated using a series of triaxial consolidated undrained creep tests conducted under different confining pressures. This study proposes a novel fractal viscoelastic accelerated creep model (F-VEAP model). The model extends the linear Newtonian viscoelastic formulation to a nonlinear fractal Newtonian viscoelastic framework by introducing a fractal derivative, <span>(beta )</span>, thereby capturing nonlinear steady-state creep behavior. An exponentially accelerated term derived from damage mechanics is incorporated to represent the accelerated creep stage. Based on model parameter identification, the F-VEAP model is shown to effectively capture the full-stage creep behavior of limestone powder. Results of the parameter sensitivity analysis further indicate that individual model parameters distinctly influence creep behavior and can serve as a useful basis for distinguishing different creep stages. Comparative analyses with the classical Burgers model and the fractional-order derivative-based FFC model demonstrate that the proposed model provides an improved representation of the observed creep behaviour. The proposed framework offers a useful basis for analysing the long-term deformation characteristics of limestone powder and similar fill materials in slope engineering and may provide reference for the deformation assessment of cement-based fill materials; however, its broader applicability still requires further validation.</p></div>","PeriodicalId":698,"journal":{"name":"Mechanics of Time-Dependent Materials","volume":"30 2","pages":""},"PeriodicalIF":2.3,"publicationDate":"2026-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147938577","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"材料科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Delayed thermal pressure generation in hydrating mine backfill: modeling and implications","authors":"Linfeng Guo,&nbsp;Mohammadamin Jafari,&nbsp;Yong Wang,&nbsp;Yang Zhou,&nbsp;Gongda Lu","doi":"10.1007/s11043-026-09884-2","DOIUrl":"10.1007/s11043-026-09884-2","url":null,"abstract":"<div><p>Recycling cement-amended tailings, a typical time-dependent material due to multiphysics evolution, back into subsurface cavities has gained increasing popularity for the reduced aquifer contamination and enhanced mine profitability brought by the waste management technology. Despite the competitive socioeconomic benefits delivered via sustainable development, the recurring thermal pressure anomalies observed during stoppage of deposition have still raised growing concerns over the current practice of backfilling. In this paper, we propose a novel strategy for modeling the delayed thermal pressure generation caused by exothermic hydration, and the multiphysics nature of the anomalous pressure build-up has been replicated computationally for the first time. By further scrutinizing the influence of temperature configuration and mix design, we have hence identified practically the favourable settings for mobilizing abnormal pressure development. Our investigation suggests that the growing temperature of deep underground mines would pose unprecedented geotechnical challenges to the traditional concept of waste tailings recycling. Meanwhile, albeit exercised ubiquitously as a rule-of-thumb for enhancing mechanical backfill performance, increasing cement usage might still relax the temperature threshold for significant pressure generation due to release of greater chemical energy. Nevertheless, compared to chilly and warm backfill susceptible to exogenically-induced and spontaneous anomalies, respectively, operating at moderately higher temperatures could rather enable the best resistance to the potential thermal pressure build-up. This study not only provides new insight into the unexpected thermal pressure anomalies observed in field backfilling operations, but also has important implications for understanding the complex time-dependence in multiphysics evolution of cementitious materials under non-isothermal conditions.</p></div>","PeriodicalId":698,"journal":{"name":"Mechanics of Time-Dependent Materials","volume":"30 2","pages":""},"PeriodicalIF":2.3,"publicationDate":"2026-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147938134","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"材料科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Temporal–spatial deformation and failure mechanisms of roof rock beams in roadways under continuous mining and continuous backfilling","authors":"Zhaolong Li,&nbsp;Haoyu Dou,&nbsp;Renliang Shan,&nbsp;Jinbo Miao,&nbsp;Xiao Tong,&nbsp;Ruifeng Huang","doi":"10.1007/s11043-026-09881-5","DOIUrl":"10.1007/s11043-026-09881-5","url":null,"abstract":"<div><p>Continuous mining and continuous backfilling (CMCB) provides an effective approach to coordinating resource extraction and environmental protection. However, as CMCB progresses, the load-bearing structure of the backfill–surrounding rock system evolves continuously, while the stage-dependent deformation and failure mechanisms of the roof remain unclear. Taking a representative panel as the engineering background, this study systematically investigates the stage-dependent mechanical response of the roof through theoretical analysis, numerical simulation, and field validation. An elastic foundation beam model is established to derive the roof deflection equation and identify the key controlling factors. The second invariant of the stress deviator (<span>(J_{2})</span>) and roof displacement are employed to characterize plastic failure and deformation instability. The results indicate that a uniform stress field is most favorable for maintaining roof stability. In contrast, a vertically dominated stress field promotes <span>(J_{2})</span> concentration and rock failure, whereas a horizontally dominated stress field is more likely to induce large-deformation instability. The influence of backfill on <span>(J_{2})</span> evolution and deformation control is more pronounced under horizontally dominated stress conditions. Furthermore, the backfill degree governs the onset of roof support, with higher values enabling earlier load transfer and more effective suppression of roof deformation. Although increasing backfill strength enhances load-bearing capacity, its contribution becomes limited once a critical threshold is exceeded. Based on these findings, targeted optimization measures are proposed for the backfilled roadway, providing guidance for the design and application of CMCB.</p></div>","PeriodicalId":698,"journal":{"name":"Mechanics of Time-Dependent Materials","volume":"30 2","pages":""},"PeriodicalIF":2.3,"publicationDate":"2026-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147937786","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"材料科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Time-to-failure prediction and early warning method for red sandstone failure-instability based on damage acceleration behavior and acoustic emission technique","authors":"Ansen Gao,&nbsp;Xinyuan Jing,&nbsp;Chengzhi Qi,&nbsp;Kuan Jiang,&nbsp;Jinglong Li,&nbsp;Chunlai Wang,&nbsp;Genshui Wu,&nbsp;Zhen Wei","doi":"10.1007/s11043-026-09880-6","DOIUrl":"10.1007/s11043-026-09880-6","url":null,"abstract":"&lt;div&gt;&lt;p&gt;The failure of rock material is an instability process caused by progressive accumulation of internal damage. The damage state and evolution pattern of rocks are crucial for early warning of its failure time. Red sandstone was selected as the study object, this study quantitatively revealed its damage acceleration behavior and the key transitional points of the evolution stages under uniaxial compression. By integrating acoustic emission (AE) signals to construct damage variables, a method for predicting and early warning of rocks failure time was established. The results demonstrated that during the critical instability stage, red sandstone exhibited a clear damage acceleration phenomenon, which accompanied by the rapid initiation and propagation of tensile cracks. The key threshold points of the damage response could be determined based on the stress state and damage variables of the specimens. At the first key damage point B, the damage variable defined by AE ringing counts was approximately 0.28 for specimen a, and 0.31 for specimen b. The damage threshold at this first critical point reached about 30% of the peak damage variable. Subsequently, the point C corresponded to the damage acceleration point, which served as a critical node distinguishing different stages of damage evolution. After this point, its damage growth began to accelerate, and the damage response intensified noticeably. This study further proposed a quantitative method for identifying the damage acceleration point. The method demonstrated good robustness when the parameter &lt;span&gt;(alpha )&lt;/span&gt; ranged from 2 to 4. The coefficient of variation for the stress level corresponding to the acceleration point was less than 8%. Moreover, it exhibited highly consistent relative stress levels across different specimens, validating its effectiveness as a universal precursor to failure. A stress-damage evolution model based on the Boltzmann function was developed. It accurately described the entire damage process of red sandstone. The coefficient of determination R&lt;sup&gt;2&lt;/sup&gt; for the fitted pre-peak stress curve reached 99%. Based on the damage acceleration onset point and the critical point, the initial and critical damage warning thresholds for rocks failure were established. Then, the linear regression was performed on the data between the damage acceleration point and the failure point. It enabled high-precision prediction of the failure time of rocks. For the tested specimens, the prediction errors for the damage acceleration response time were 1.24% and 2.83%, respectively, and both errors fell within 3%. This meets the accuracy requirements for damage identification and instability warning in engineering rock masses. This study established an integrated analytical method for rock failure behaviors, and it combined the damage evolution modeling, the damage acceleration threshold identification, and the instability time prediction. It provided a theoretical basis for the ","PeriodicalId":698,"journal":{"name":"Mechanics of Time-Dependent Materials","volume":"30 2","pages":""},"PeriodicalIF":2.3,"publicationDate":"2026-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147937934","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"材料科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Synergistic effects of SBS polymer degradation and binder stiffening on the rheological properties of SBS-modified asphalt during RTFO aging","authors":"Alok Sharma,&nbsp;G. D. Ransinchung R.N,&nbsp;Sham S. Ravindranath","doi":"10.1007/s11043-026-09879-z","DOIUrl":"10.1007/s11043-026-09879-z","url":null,"abstract":"<div><p>During pavement construction, styrene–butadiene–styrene (SBS) modified asphalt binders (SBS-ABs) undergo short-term thermo-oxidative aging, during which SBS macromolecules degrade concurrently with the stiffening of the asphalt binder. In this paper, the coupled influence of these mechanisms on the rheological behavior of SBS-ABs is investigated over a wide range of angular frequencies and SBS dosages. Rheological results show that at low angular frequencies (<span>(omega )</span> &lt; 1 rad/s), rolling thin film oven (RT-FO) aged SBS-ABs exhibit substantially reduced viscoelastic properties compared with unaged binders at higher service temperatures, including decreases of approximately 50–70% in complex viscosity (<span>(eta )</span>*), storage modulus (G′), and complex modulus (G*). In addition, RTFO-aged binders display phase angle increases of approximately 15–30°, reflecting deterioration of the SBS-maltene-rich network. The strong dependence on slower <span>(omega )</span> arises from the sluggish mobility of SBS molecules and the presence of SBS-maltene-rich network in the asphalt binder, and the effect of RT-FO aging is found to be most pronounced at SBS dosages between 3 and 5 wt.%. These results demonstrate that the synergistic effects of SBS polymer degradation and binder stiffening can be effectively characterized only when rheological measurements span a broad frequency range (0.01–10 rad/s) and SBS content (2–7 wt.%). The findings provide practical guidance for SBS-modified binder formulation and performance evaluation during construction-related aging.</p></div>","PeriodicalId":698,"journal":{"name":"Mechanics of Time-Dependent Materials","volume":"30 2","pages":""},"PeriodicalIF":2.3,"publicationDate":"2026-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147829240","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"材料科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Time-dependent strain-softening Hoek-Brown constitutive model for progressive failure analysis of coal pillars","authors":"Prudhvi Raju Gadepaka,&nbsp; Sonu,&nbsp;Ashok Jaiswal","doi":"10.1007/s11043-026-09878-0","DOIUrl":"10.1007/s11043-026-09878-0","url":null,"abstract":"<div><p>The present study proposes a time-dependent Hoek-Brown strain softening constitutive model to assess the failure analysis of a coal mass. A methodology for implementing the proposed time-dependent constitutive model in the 3D FDM has been proposed. The constitutive model has three sets of parameters: crack initiation strength properties (<span>(m_{c})</span> and <span>(s_{c})</span>), ultimate peak strength parameters (<span>(m_{p})</span> and <span>(s_{p})</span>), and residual strength parameters (<span>(m_{r})</span> and <span>(s_{r})</span>). Failed cases of bord and pillar mines from the Witbank Coalfield were utilised to estimate the strength properties. The Strength properties of the coal were deduced using back-analysis techniques. A constant tributary area load was applied from the model’s top during the simulation. The simulation results were obtained in terms of the average axial strain with time. The simulation results showed that the average strain of the pillar increased exponentially during the simulation stage of age before failure. The pillar model is considered failed after getting an un-equilibrium condition. At that time, a sharp increase in axial strain was observed. Stable cases from the Witbank coal fields were also simulated with age to validate the deduced strength parameters. The stable cases results showed a negligible increment in axial strain, confirming the pillar’s stability. The simulation results showed that the pillar failure begins at the surface and progresses deeper into the core, ultimately forming an hourglass shape. The pillar strength formula with respect to the age of the pillar has also been proposed based on the statistical analysis.</p></div>","PeriodicalId":698,"journal":{"name":"Mechanics of Time-Dependent Materials","volume":"30 2","pages":""},"PeriodicalIF":2.3,"publicationDate":"2026-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147738600","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"材料科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}