Constitutive models are fundamental to solid mechanics and materials science, underpinning the quantitative description and prediction of material responses under diverse loading conditions. Traditional phenomenological models, which are derived through empirical fitting, often lack generalizability and rely heavily on expert intuition and predefined functional forms. In this work, we propose a graph - based equation discovery framework for the automated discovery of constitutive laws directly from multisourc e experimental data. This framework expresses equations as directed graphs, where nodes represent operators and variables, edges denote computational relations, and edge features encode parametric dependencies . This enables the generation and optimization of free - form symbolic expressions with undetermined material - specific parameters . Through the proposed framework, we have discovered new constitutive models for strain - rate effects in alloy steel materials and the deformation behavior of lithium metal. Com pared with conventional empirical models, these new models exhibit compact analytical structures and achieve higher accuracy. The proposed graph - based equation discovery framework provides a generalizable and interpretable approach for data - driven scientific mode l ling, particularly in contexts where traditional empirical formulations are inadequate for representing complex physical phenomena. Keywords: Constitutive model, graph, equation discovery, solid mechanics, data - driven modelling . Introduction Constitutive laws serve as fundamental elements in solid mechanics, establishing the relationship between kinematic measures and static quantities to characterize material - specific behavior. Unlike conservation principles and kinematic relations, which are derived from first principles and regarded as axiomatic foundations, constitutive models encapsulate empirical descriptions of material responses to external stimuli . Accordingly, they are typically established through phenomenological approaches, guided by systematic experimentation and theoretical generalization, to characterize nonlinear behaviors across varying conditions ( 1) . The accuracy and generality of constitutive models are critical for the reliability of mechanical analysis, directly influencing both theoretical developments and practical applications in computational mechanics and materials engineering.

Process-structure-property relationships are fundamental in materials science and engineering and are key to the development of new and improved materials. Symbolic regression serves as a powerful tool for uncovering mathematical models that describe these relationships. It can automatically generate equations to predict material behaviour under specific manufacturing conditions and optimize performance characteristics such as strength and elasticity. The present work illustrates how symbolic regression can derive constitutive models that describe the behaviour of various metallic alloys during plastic deformation. Constitutive modelling is a mathematical framework for understanding the relationship between stress and strain in materials under different loading conditions. In this study, two materials (age-hardenable aluminium alloy and high-chromium martensitic steel) and two different testing methods (compression and tension) are considered to obtain the required stress-strain data. The results highlight the benefits of using symbolic regression while also discussing potential challenges.

Mechanistic microstructure-informed constitutive models for the mechanical response of polycrystals are a cornerstone of computational materials science. However, as these models become increasingly more complex - often involving coupled differential equations describing the effect of specific deformation modes - their associated computational costs can become prohibitive, particularly in optimization or uncertainty quantification tasks that require numerous model evaluations. To address this challenge, surrogate constitutive models that balance accuracy and computational efficiency are highly desirable. Data-driven surrogate models, that learn the constitutive relation directly from data, have emerged as a promising solution. In this work, we develop two local surrogate models for the viscoplastic response of a steel: a piecewise response surface method and a mixture of experts model. These surrogates are designed to adapt to complex material behavior, which may vary with material parameters or operating conditions. The surrogate constitutive models are applied to creep simulations of HT-9 steel, an alloy of considerable interest to the nuclear energy sector due to its high tolerance to radiation damage, using training data generated from viscoplastic self-consistent (VPSC) simulations. We define a set of test metrics to numerically assess the accuracy of our surrogate models for predicting viscoplastic material behavior, and show that the mixture of experts model outperforms the piecewise response surface method in terms of accuracy.

Rigid robots can be precise in repetitive tasks, but struggle in unstructured environments. Nature's versatility in such environments inspires researchers to develop biomimetic robots that incorporate compliant and contracting artificial muscles. Among the recently proposed artificial muscle technologies, electrohydraulic actuators are promising since they offer performance comparable to that of mammalian muscles in terms of speed and power density. However, they require high driving voltages and have safety concerns due to exposed electrodes. These high voltages lead to either bulky or inefficient driving electronics that make untethered, high-degree-of-freedom bio-inspired robots difficult to realize. Here, we present hydraulically amplified low voltage electrostatic (HALVE) actuators that match mammalian skeletal muscles in average power density (50.5 W kg-1) and peak strain rate (971 % s-1) at a driving voltage of just 1100 V. This driving voltage is approx. 5-7 times lower compared to other electrohydraulic actuators using paraelectric dielectrics. Furthermore, HALVE actuators are safe to touch, waterproof, and self-clearing, which makes them easy to implement in wearables and robotics. We characterize, model, and physically validate key performance metrics of the actuator and compare its performance to state-of-the-art electrohydraulic designs. Finally, we demonstrate the utility of our actuators on two muscle-based electrohydraulic robots: an untethered soft robotic swimmer and a robotic gripper. We foresee that HALVE actuators can become a key building block for future highly-biomimetic untethered robots and wearables with many independent artificial muscles such as biomimetic hands, faces, or exoskeletons.

Classically, the mechanical response of materials is described through constitutive models, often in the form of constrained ordinary differential equations. These models have a very limited number of parameters, yet, they are extremely efficient in reproducing complex responses observed in experiments. Additionally, in their discretized form, they are computationally very efficient, often resulting in a simple algebraic relation, and therefore they have been extensively used within large-scale explicit and implicit finite element models. However, it is very challenging to formulate new constitutive models, particularly for materials with complex microstructures such as composites. A recent trend in constitutive modeling leverages complex neural network architectures to construct complex material responses where a constitutive model does not yet exist. Whilst very accurate, they suffer from two deficiencies. First, they are interpolation models and often do poorly in extrapolation. Second, due to their complex architecture and numerous parameters, they are inefficient to be used as a constitutive model within a large-scale finite element model. In this study, we propose a novel approach based on the physics-informed learning machines for the characterization and discovery of constitutive models. Unlike data-driven constitutive models, we leverage foundations of elastoplasticity theory as regularization terms in the total loss function to find parametric constitutive models that are also theoretically sound. We demonstrate that our proposed framework can efficiently identify the underlying constitutive model describing different datasets from the von Mises family.

This paper investigates the structure-property relations of thin-walled lattices under dynamic longitudinal compression, characterized by their cross-sections and heights. These relations elucidate the interactions of different geometric features of a design on mechanical response, including energy absorption. We proposed a combinatorial, key-based design system to generate different lattice designs and used the finite element method to simulate their response with the Johnson-Cook material model. Using an autoencoder, we encoded the cross-sectional images of the lattices into latent design feature vectors, which were supplied to the neural network model to generate predictions. The trained models can accurately predict lattice energy absorption curves in the key-based design system and can be extended to new designs outside of the system via transfer learning.

Brain strain and strain rate are effective in predicting traumatic brain injury (TBI) caused by head impacts. However, state-of-the-art finite element modeling (FEM) demands considerable computational time in the computation, limiting its application in real-time TBI risk monitoring. To accelerate, machine learning head models (MLHMs) were developed, and the model accuracy was found to decrease when the training/test datasets were from different head impacts types. However, the size of dataset for specific impact types may not be enough for model training. To address the computational cost of FEM, the limited strain rate prediction, and the generalizability of MLHMs to on-field datasets, we propose data fusion and transfer learning to develop a series of MLHMs to predict the maximum principal strain (MPS) and maximum principal strain rate (MPSR). We trained and tested the MLHMs on 13,623 head impacts from simulations, American football, mixed martial arts, car crash, and compared against the models trained on only simulations or only on-field impacts. The MLHMs developed with transfer learning are significantly more accurate in estimating MPS and MPSR than other models, with a mean absolute error (MAE) smaller than 0.03 in predicting MPS and smaller than 7 (1/s) in predicting MPSR on all impact datasets. The MLHMs can be applied to various head impact types for rapidly and accurately calculating brain strain and strain rate. Besides the clinical applications in real-time brain strain and strain rate monitoring, this model helps researchers estimate the brain strain and strain rate caused by head impacts more efficiently than FEM.

In this paper, a tree based neural network viz. MARS (Friedman, 1991) for the modelling of the yield strength of a steel rolling plate mill is described. The inputs to the time series model are temperature, strain, strain rate, and interpass time and the output is the corresponding yield stress. It is found that the MARSbased model reveals which variable's functional dependence is nonlinear, and significant. The results are compared with those obta.ined

In this paper, a tree based neural network viz. MARS (Friedman, 1991) for the modelling of the yield strength of a steel rolling plate mill is described. The inputs to the time series model are temperature, strain, strain rate, and interpass time and the output is the corresponding yield stress. It is found that the MARSbased model reveals which variable's functional dependence is nonlinear, and significant. The results are compared with those obta.ined