Muscle is a hierarchical composite material, and that architecture is the key to the new framework. At the smallest level sit myofibrils, the contractile protein threads built from repeating sarcomeres, each containing Z-disks, A-bands, and H-bands. Myofibrils bundle together into muscle fibers, wrapped in a thin collagenous sheath called the endomysium. Fibers in turn group into bundles encased in a thicker connective tissue layer, the perimysium. Connective tissues form a continuous network that reinforces the muscle and transmits force between fibers and bundles. Because each level contributes to the whole, the team hypothesized that macroscopic texture could be quantitatively predicted if the mechanical properties of the microscopic components and their volume fractions were known.

Testing that hypothesis required measuring mechanics at scales where no conventional test can operate. The researchers turned to atomic force microscopy, or AFM, a technique in which an ultrasharp silicon nitride tip, with a radius of just a few nanometers, is pressed against a sample while the force of contact is recorded. Fitting the resulting force-indentation curves with the Derjaguin-Muller-Toporov contact model, which accounts for both elastic deformation and van der Waals adhesion, yields the local Young’s modulus. The team applied this to myofibrils isolated from pork tenderloin, beef tenderloin, and chicken breast, as well as to extracted endomysium and perimysium, measuring both dried and hydrated samples. Hydration mattered enormously: dried myofibrils showed moduli in the gigapascal range, while wet samples dropped by one to two orders of magnitude into the tens of megapascals, a reminder that data from dried biological materials can have limited relevance to real food.

The nanoscale maps revealed striking patterns. In raw myofibrils, stiffness followed a consistent spatial order across all three species: the Z-disk was stiffest, the A-band intermediate, and the H-band softest, echoing earlier AFM studies on rabbit and rat muscle. Species differences were also clear. In the hydrated raw state, pork myofibrils were the stiffest at roughly 50 megapascals, beef followed at about 47, and chicken was softest at around 34. Cooking reshuffled the dried-state landscape, erasing the species-specific sarcomere structures as proteins denatured and aggregated, but the wet cooked myofibrils preserved the same hierarchy, with pork highest at about 75 megapascals and chicken lowest at about 45. The collagenous membranes told their own story: the perimysium was consistently slightly stiffer than the endomysium, and both stiffened after cooking, likely through heat-induced collagen denaturation and aggregation.

With the microscopic inputs in hand, the team built a two-level analytical model based on the classical Reuss series scheme from composite mechanics. Under transverse compression, a single muscle fiber was treated as myofibrils embedded in an endomysium matrix, and a fiber bundle as equivalent fibers in series with the perimysium. The effective modulus of each level is the volume-fraction-weighted combination of its phases, with the phase fractions measured from scanning electron microscope images of fiber cross-sections. Running the numbers, the model predicted bundle moduli that ranked pork highest, beef intermediate, and chicken lowest, and predicted a significant increase after cooking, trends that mirror the myofibril data feeding into it. Notably, the predicted bundle modulus correlated strongly with the myofibril and endomysium moduli but not with the perimysium modulus, suggesting that in this low-connective-tissue muscle system, the bundle stiffness is governed primarily by the fibers themselves.

The validation step came from macroscopic compression tests on cubes of raw and cooked meat using a texture analyzer, with the compression axis carefully aligned perpendicular to the muscle fibers to match the orientation of the nanoscale measurements. The stress-strain curves were strongly nonlinear, stiffening as strain increased, so the team fitted them with two constitutive models. A combined logarithmic-polynomial hyperelastic model, originally developed for liver tissue, captured the general shape but showed systematic deviations, particularly for chicken. A logistic model, adapted from work on passive spinal muscle mechanics, performed far better, with coefficients of determination above 0.997 for every sample. Its parameters carried clear physical meaning: an initial modulus reflecting the relaxed response of the microscopic components, a hardening increment describing nonlinear stiffening, a hardening rate, and a critical strain at which stiffening accelerates.

The correlation analysis tied the scales together. The logistic model’s initial modulus correlated strongly with the linear modulus measured from the first five percent of strain, and both correlated with the multiscale model’s predicted bundle modulus and, through it, with the myofibril and endomysium properties. In other words, the initial stiffness of a piece of meat is largely written into its myofibrils. The nonlinear parameters told a different story. Beef showed the largest hardening increment, which the authors attribute to its denser network of mature collagen crosslinks and the pronounced nonlinear hardening behavior of the perimysium observed in the AFM force curves, potentially explaining why beef is chewier than pork or chicken. Bound water content correlated positively with the initial stiffness parameters, while free water correlated negatively with the hardening rate, hinting that water acts as a lubricant that smooths the structural response under compression.

The authors are careful about scope. The AFM-derived moduli of the isolated connective tissues represent effective properties of collagen-rich fractions rather than absolute in situ values, and the constitutive models describe rate-independent behavior at a single loading speed rather than the full viscoelastic response of muscle. Validation was performed on three relatively lean, low-connective-tissue muscles, so extending the framework to collagen-rich cuts such as tendon-laden muscles remains future work. The link between micro- and macroscale is also framed as a parametric correlation rather than a direct numerical equivalence, with microscopic stiffness serving as the physical foundation for macroscopic structural stiffness rather than substituting for it one-to-one.

Even with those caveats, the implications are broad. The study delivers a quantitative, bottom-up route from nanoscale protein mechanics to the texture a consumer actually perceives, clarifying which hierarchical structures dominate which aspects of the mechanical response. For the booming plant-based meat industry, where replicating the fibrous architecture and anisotropic mechanics of real muscle remains the central challenge, the framework offers something like a design blueprint: a way to specify the stiffness and volume fraction of the protein and matrix phases needed to hit a target bite. More broadly, it demonstrates that with enough care at the nanoscale, even something as familiar and complex as the texture of dinner can be reduced to physics you can measure, model, and ultimately engineer.

Subject of Research: Multiscale mechanical modeling linking AFM nanomechanics of myofibrils and connective tissue to the macroscopic compressive texture of meat

Article Title: Predicting the compression mechanical properties of muscle from microscale indentation mechanics using multiscale mechanical models

Article References: Zhao, C., Jiang, R., Zhang, Z., Jia, J., Nishinari, K., & Yang, N. (2026). Predicting the compression mechanical properties of muscle from microscale indentation mechanics using multiscale mechanical models. Current Research in Food Science, Article 101577. https://doi.org/10.1016/j.crfs.2026.101577

Image Credits: AI Generated

DOI: 10.1016/j.crfs.2026.101577

Keywords: meat texture, atomic force microscopy, myofibrils, multiscale modeling, connective tissue, food science, compression mechanics, plant-based meat, collagen, constitutive models, Reuss model, nanoindentation

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