UHPFRC is an extraordinary material by any measure. Its ultra-dense matrix dramatically restricts chloride penetration, while embedded steel fibers give it compressive strengths above 150 megapascals and tensile strengths far beyond those of conventional concrete. Earlier experiments have shown that a 60-millimeter UHPFRC layer can more than triple the load capacity of a severely corroded beam, and that U-shaped jackets encasing a beam on three sides can dramatically boost stiffness. Yet a critical complication has gone largely unexamined: not all UHPFRC behaves the same way after it cracks. The French standard NF P18-470 formally separates the family into strain-softening, low strain-hardening, and high strain-hardening grades. Strain-hardening UHPFRC keeps gaining stress after first cracking, spawning networks of fine, distributed micro-cracks and a pseudo-ductile plateau. Strain-softening UHPFRC, though still mechanically superior to ordinary concrete, sheds stress progressively once past its peak, relying on fewer, wider cracks.

Which of these behaviors a given mix displays depends on fiber type, mix proportions, admixtures, and curing conditions, and the consequences for structural rehabilitation are far from academic. Prior experimental campaigns produced apparently contradictory results: one study found that strain-hardening UHPFRC enhanced the ductility of corroded beams, while another reported that strain-softening layers actually reduced it. Because most existing research on this strengthening technique has been experimental, and because numerical modeling of corrosion-induced degradation remains notoriously difficult—with researchers split over how to represent rust expansion, bond loss, and interfacial behavior—the field lacked a reliable computational tool to systematically separate these effects. The new study, published in Case Studies in Construction Materials, was designed to close that gap.

Lead author Zhukai Tang, together with Wai-Meng Quach, Ran Feng, and Zhiyuan Chen, built a nonlinear finite element framework in ABAQUS that tackles the three hardest problems head-on. First, nonlinear spring elements spaced at 50-millimeter intervals connect reinforcement nodes to the surrounding concrete, capturing the bond-slip deterioration that corrosion causes; the springs follow calibrated constitutive laws in which maximum bond stress falls with corrosion ratio according to established models for corroded reinforcement. Second, the interface between the UHPFRC jacket and the substrate concrete is governed by a traction-separation law, whose normal and shear strengths of 19.2 and 5.9 megapascals were extracted directly from slant-shear and double-shear interface tests performed by the team, with damage initiating at a plastic displacement of 0.241 millimeters. Third, the researchers implemented two distinct post-peak tensile constitutive models for UHPFRC, one strain-hardening and one strain-softening, carefully tuned so that both materials shared an identical tensile strength of 11.4 megapascas under the same steel fiber conditions, 15-millimeter-long fibers at 0.2 millimeters diameter and a 2 percent volume fraction. This clever control variable isolates the pure effect of post-cracking behavior on structural response.

The model itself was grounded in real experiments. The team validated it against 24 corroded, UHPFRC-strengthened beams from two published four-point bending campaigns. In the first, six strengthened beams and three unstrengthened controls were exposed to 360 days of wet-dry cycling in artificial seawater, producing 2 to 3 percent corrosion of the bottom reinforcement and stirrups before being retrofitted with 15, 30, or 45 millimeter layers of strain-hardening UHPFRC with a compressive strength of 153.7 megapascals. In the second, 18 beams were corroded by impressed current to controlled ratios of 10 to 23 percent in the bottom bars, then strengthened with strain-softening UHPFRC in bottom-face or U-shape configurations at thicknesses from 20 to 60 millimeters. Across all 24 specimens, the simulated-to-experimental load ratios ranged from 0.90 to 1.08, averaging 0.994 with a standard deviation of just 0.0504, an accuracy rare in corrosion modeling, where stochastic deterioration usually wrecks convergence.

With the model validated, the researchers ran 48 simulations of simply supported beams, sweeping UHPFRC thicknesses from 10 to 60 millimeters in 10-millimeter increments, comparing bottom-face and U-shape jacket configurations, varying corrosion ratios from 5 to 30 percent, and switching between the two tensile constitutive laws. The results are striking. For uncorroded beams, increasing the strain-hardening UHPFRC thickness from 10 to 60 millimeters raised peak load capacity by 8.2 to 79.6 percent with bottom-face strengthening, and by a remarkable 27.8 to 273 percent with U-shape jackets. The strain-softening variant followed similar trends, gaining 4.5 to 59.2 percent and 26.3 to 222.2 percent respectively. In the most extreme case, a 60-millimeter strain-hardening U-shape jacket boosted flexural capacity by 273 percent, yield load by 200.1 percent, and lifted the cracking load from 19.7 kilonewtons to 210.2 kilonewtons—more than a tenfold improvement in the load at which the beam first cracks.

The mechanism behind these gains is partly geometric and partly material. Thicker jackets increase the section depth and moment of inertia, while UHPFRC’s superior tensile capacity carries stress across cracked zones and unloads the corroded steel. But the fiber-bridging effect proved especially potent for crack control: fibers mechanically anchor micro-cracks, and energy dissipated during fiber pull-out delays the coalescence of macro-cracks. Crack resistance improved even faster with thickness than raw strength did. Failure modes also evolved with the jacket. Bottom-face strengthened beams shifted from tightly concentrated midspan flexural cracking toward a more uniform flexural-shear crack pattern as thickness grew, while U-shape beams tended toward a single dominant flexural crack at peak load, a signature of UHPFRC’s crack resistance concentrating deformation once fibers began pulling out. Interfacial contact stress analysis showed the highest stresses at the jacket ends, flagging those zones as debonding initiation sites, but the U-shape configuration’s side arms relieved end-of-span stress concentrations and made debonding significantly less likely than in bottom-face strengthening.

The most consequential finding, however, concerns the two tensile behaviors. Although both materials shared the same tensile strength, the performance gap widened with thickness: peak load differences between strain-hardening and strain-softening versions grew from 5.6 to 25.2 kilonewtons for bottom-face beams, and from 2.0 to 68.5 kilonewtons for U-shape beams, as thickness increased from 10 to 60 millimeters. Yield strength gaps followed the same trajectory. The strain-hardening material’s extended hardening phase means fibers slip over a longer strain range before pull-out or fracture, sustaining load even as micro-cracks accumulate. Corrosion amplified the contrast further: as corrosion ratios climbed to 30 percent, ductility of strain-softening-strengthened beams degraded much more sharply, because those beams lean heavily on the steel reinforcement for deformation capacity, precisely the component corrosion destroys. Strain-hardening jackets preserved ductility far better. Interestingly, strain-softening beams showed marginally higher initial stiffness, attributed to their stiffer pre-cracking tensile response—small consolation given their deficits elsewhere.

Beyond the simulations, the team distilled a closed-form theoretical model for predicting moment resistance of strengthened beams, accounting for the different stress blocks that strain-hardening and strain-softening UHPFRC produce across the cracked section. Validated against the 48 simulated beams plus test data from four independent experimental studies, the formulation achieved a mean prediction-to-test ratio of 1.012 with a standard deviation of 0.097, giving designers a practical hand-calculation tool for the first time that explicitly distinguishes post-peak material classes. The authors are candid about limitations: corrosion was modeled as uniform rather than pitting, stirrup corrosion was excluded, the flexural-shear mechanism transition was not captured, and the traction-separation law awaits full sensitivity analysis. The model is therefore best suited to moderate corrosion and monotonic loading, with non-uniform corrosion, coupled degradation, and long-term performance flagged for future work.

The practical message for infrastructure owners is clear and actionable. Bottom-face UHPFRC strengthening is the more economical option and still delivers large capacity gains, making it attractive for routine rehabilitation budgets. But when maximum strength, long-term corrosion resistance after retrofit, and preserved ductility are the priorities—as they usually are for critical marine structures—the U-shape jacket in a strain-hardening UHPFRC is the superior choice, and the study shows that specifying the material’s post-cracking class is not a detail but a decisive engineering parameter. As coastal concrete assets worldwide age into their corrosion-prone decades, tools like this model give engineers a quantitative basis for choosing not just whether to strengthen, but exactly how.

Subject of Research: Finite element analysis of corroded reinforced concrete beams strengthened with strain-hardening and strain-softening UHPFRC layers

Article Title: Flexural behavior of corrosion-damaged RC beams strengthened with strain-hardening and strain-softening UHPFRC: Numerical study

Article References: Tang, Z., Quach, W.-M., Feng, R., & Chen, Z. (2026). Flexural behavior of corrosion-damaged RC beams strengthened with strain-hardening and strain-softening UHPFRC: Numerical study. Case Studies in Construction Materials, 25, Article e06486. https://doi.org/10.1016/j.cscm.2026.e06486

Image Credits: AI Generated

DOI: 10.1016/j.cscm.2026.e06486

Keywords: UHPFRC, reinforced concrete, corrosion, finite element analysis, structural strengthening, flexural behavior, marine infrastructure, bond-slip, strain hardening, ductility, concrete durability, rehabilitation

Constitutive models are the connective tissue of solid mechanics. They are the mathematical rules that translate stress, strain, temperature, and strain rate into predictions of how a real object will bend, dent, stretch, or shatter. Every crash simulation of a car body, every structural analysis of a bridge, and every design study for a battery electrode depends on them. Hao Xu, a postdoctoral researcher at EIT and lead author of the study, explains that the traditional paradigm, while enormously successful, carries an inherent ceiling. Researchers derive a mathematical form based on physical intuition and then calibrate its parameters with data, but the predetermined equation structure restricts what the model can ultimately describe and predict. If the true material behavior involves a functional dependence the chosen formula cannot express, no amount of parameter fitting will recover it. The new framework, called GraphED, inverts the workflow: data comes first, and the equation emerges from it.

The central technical obstacle to such data-driven discovery is representation. Machine learning systems can only optimize what they can encode, and mathematical equations are notoriously awkward objects for algorithms to manipulate. Yuntian Chen, EIT associate professor and co-corresponding author of the study, describes the challenge as efficiently encoding both equation architectures and material-specific parameters into a searchable format for computational algorithms. Conventional symbolic regression approaches typically represent expressions as trees, with operators branching from a root node down to leaves. Trees, however, make it difficult to share structure across different materials and to embed tunable parameters cleanly. The EIT team’s solution is to abandon the tree entirely and encode each candidate equation as a directed graph. In this graph architecture, nodes correspond to mathematical operators and physical variables, while directed edges define the logical and computational connections between them.

The graph representation is more than a cosmetic change, because the edges themselves become carriers of physical meaning. An edge can hold a fixed physical constant, such as a universal exponent, or a tunable material-dependent parameter that varies from one alloy or polymer to another. This design allows GraphED to do two things simultaneously that earlier methods treated separately. It can identify a universal mathematical structure shared across diverse materials and experimental conditions, and it can adaptively calibrate personalized parameters for each individual material scenario. The framework then iterates: it generates candidate graph-structured equations, evaluates how well each one reproduces the experimental data, and optimizes both the topology of the graph and the values of its parameters. The output is a constitutive law that is physically consistent, mathematically compact, and, crucially, interpretable by a human engineer who can read the equation and understand what each term contributes.

That last quality, interpretability, is what separates this work from the black-box models that dominate much of modern machine learning. A neural network trained on material data may predict well, but it offers no equation, no insight, and no transferable knowledge. GraphED delivers the opposite: an explicit formula that a researcher can inspect, question, embed in existing simulation software, and carry into new contexts. The team validated this promise across three very different solid material systems, each chosen for its distinct mechanical character and its historical resistance to empirical modeling. The breadth of the test cases matters, because a discovery method that only works on one material family is a curiosity; one that generalizes across steels, metals, and elastomers is a tool.

The first test case was alloy steel, the workhorse of structural engineering, whose behavior under high strain rates and large plastic deformation is governed by strain-rate dependence and strain hardening. These are precisely the phenomena captured by the Johnson–Cook model, one of the most widely adopted constitutive formulations in impact and crashworthiness analysis. GraphED discovered explicit equations governing both strain-rate dependence and strain hardening directly from experimental data, and when the team integrated these data-driven equations into a complete constitutive model, the result delivered more precise mechanical predictions than the Johnson–Cook benchmark. For a field in which the incumbent model has been refined over decades, the demonstration that an autonomously discovered equation can beat it on predictive accuracy is a significant result.

The second material pushed the method into far harder territory. Lithium metal is a critical component of next-generation energy devices, including high-energy-density batteries, but its mechanical behavior is notoriously difficult to model. Its plastic flow is highly sensitive to both temperature and strain rate, and the interplay of these dependencies has made traditional empirical formulations unreliable. GraphED yielded concise plastic-flow constitutive equations that achieve superior agreement with experimental measurements compared with conventional empirical models. For battery designers, whose simulations of electrode deformation and dendrite-related failure depend on accurate mechanical descriptions, the ability to extract a trustworthy equation directly from data could shorten development cycles and improve the fidelity of electrochemical-mechanical coupled models.

The third validation targeted filled rubbers, a class of elastomers whose hyperelastic response changes with filler composition and temperature. Rubber components in tires, seals, and vibration isolators experience large, reversible deformations, and their stiffness depends on both what they are made of and how hot they are. GraphED captured a compact hyperelastic constitutive equation that maintains robust accuracy across varying material compositions and temperature conditions, meaning a single discovered formula could serve a family of rubber compounds rather than requiring a bespoke model for each formulation. Together, the three case studies span rate-dependent plasticity, temperature-sensitive metal plasticity, and finite-strain hyperelasticity, covering a remarkably wide slice of solid mechanics with one framework.

The researchers see the implications reaching well beyond computational mechanics. Dongxiao Zhang, Chair Professor at EIT and corresponding author of the study, notes that many complex material mechanical behaviors cannot be well described by existing empirical constitutive models, and that equation discovery via graph-based AI provides a powerful new paradigm to assist researchers in deriving rigorous mathematical descriptions when traditional model forms fall short. The phrase that matters is rigorous mathematical description: the goal is not a surrogate that mimics data, but a law, expressed in symbols, that captures the underlying behavior. Because the framework requires only experimental or observational data as input, its potential applications extend to any discipline seeking interpretable physical laws, from soft matter physics and biomechanics to geophysics and beyond, wherever data exists but the governing equation does not.

The study also signals a broader shift in how science may handle the equation-discovery step itself. Symbolic regression and equation learning have been growing fields, but the graph-based encoding developed here resolves a key bottleneck by enabling simultaneous optimization of equation topology and material parameterization, a combination that makes the search tractable for realistic, multi-material datasets. If the paradigm spreads, the role of the mechanician may evolve from proposing formulas to curating data, specifying physical constraints, and interpreting the equations that the algorithm returns. The authors declare no competing interest, and the research was published as a peer-reviewed contribution in Science Advances. For a field built on equations handed down through generations of researchers, the prospect of machines that read raw experimental data and hand back compact, readable laws of material behavior is a genuinely new way of doing physics, and the steels, lithium, and rubber that proved it are only the beginning.

Subject of Research: Graph-based AI equation discovery of constitutive laws in solid materials from experimental data

Article Title: AI discovers interpretable constitutive laws in solids directly from data

Article References: AI discovers interpretable constitutive laws in solids directly from data. (n.d.). Original publication

Image Credits: AI Generated

DOI: Not provided

Keywords: constitutive models, solid mechanics, graph-based equation discovery, symbolic regression, alloy steels, lithium metal, filled rubbers, machine learning, Science Advances, interpretable AI, strain hardening, hyperelasticity