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We introduce a data-driven framework to automatically identify interpretable and physically meaningful hyperelastic constitutive models from sparse data. Leveraging symbolic regression, our approach generates elegant hyperelastic models that achieve accurate data fitting with parsimonious mathematic formulas, while strictly adhering to hyperelasticity constraints such as polyconvexity/ellipticity. Our investigation spans three distinct hyperelastic models-invariant-based, principal stretch-based, and normal strain-based-and highlights the versatility of symbolic regression. We validate our new approach using synthetic data from five classic hyperelastic models and experimental data from the human brain cortex to demonstrate algorithmic efficacy. Our results suggest that our symbolic regression algorithms robustly discover accurate models with succinct mathematic expressions in invariant-based, stretch-based, and strain-based scenarios. Strikingly, the strain-based model exhibits superior accuracy, while both stretch-based and strain-based models effectively capture the nonlinearity and tension-compression asymmetry inherent to the human brain tissue. Polyconvexity/ellipticity assessment affirm the rigorous adherence to convexity requirements both within and beyond the training regime. However, the stretch-based models raise concerns regarding potential convexity loss under large deformations. The evaluation of predictive capabilities demonstrates remarkable interpolation capabilities for all three models and acceptable extrapolation performance for stretch-based and strain-based models. Finally, robustness tests on noise-embedded data underscore the reliability of our symbolic regression algorithms. Our study confirms the applicability and accuracy of symbolic regression in the automated discovery of isotropic hyperelastic models for the human brain and gives rise to a wide variety of applications in other soft matter systems. STATEMENT OF SIGNIFICANCE: Our research introduces a pioneering data-driven framework that revolutionizes the automated identification of hyperelastic constitutive models, particularly in the context of soft matter systems such as the human brain. By harnessing the power of symbolic regression, we have unlocked the ability to distill intricate physical phenomena into elegant and interpretable mathematical expressions. Our approach not only ensures accurate fitting to sparse data but also upholds crucial hyperelasticity constraints, including polyconvexity, essential for maintaining physical relevance.

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Different hyperelastic material models (Mooney-Rivlin, Yeoh, Gent, Arruda-Boyce and Ogden) are able to estimate Treloar's test data series containing uniaxial and biaxial tension and pure shear stress-strain characteristics of rubber. If the rubber behaviour is only determined for the specific load of the product, which, in the case of rubber bumpers, is the compression, the time needed for the laboratory test can be significantly decreased. The stress-strain characteristics of the uniaxial compression test of rubber samples were used to fit hyperelastic material models. Laboratory and numerical tests of a rubber bumper with a given compound and complex geometry were used to determine the accuracy of the material models. Designing rubber products requires special consideration of the numerical discretization process due to the nonlinear behaviours (material nonlinearity, large deformation, connections, etc.). Modelling considerations were presented for the finite element analysis of the rubber bumper. The results showed that if only uniaxial compression test data are available for the curve fitting of the material model, the Yeoh model performs the best in predicting the rubber product material response under compressive load and complex strain state.

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An equibiaxial tension test could be necessary to set up hyperelastic material constants for elastomers exactly. Unfortunately, very often, only uniaxial tension experimental data are available. It is possible to use only uniaxial data to compute hyperelastic constants for a hyperelastic model, but the prediction of behavior in different deformation modes (as is equibiaxial or pure shear) will not work correctly with this model. It is quite obvious that there is some relation between uniaxial and equibiaxial behavior for the elastomers. Thus, we could use uniaxial data to predict equibiaxial behavior. If we were able to predict (at least approximately) equibiaxial data, then we could create a hyperelastic model usable for the general prediction of any deformation mode of elastomer. The method of the appropriate processing of experimental data for such prediction is described in the article and is verified by the comparison with the experiment. The presented results include uniaxial and equibiaxial experimental data, the created average curve of both the deformation modes, and the predicted equibiaxial data. Using Student's t-test, a close coincidence of the real and predicted equibiaxial data was confirmed.

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Advanced numerical simulations of the mechanical behavior of human skin require thorough calibration of the material's constitutive models based on experimental ex vivo mechanical tests along with images of tissue microstructure for a variety of biomedical applications. In this work, a total of 14 human healthy skin samples and 4 additional scarred skin samples were experimentally analyzed to gain deep insights into the biomechanics of human skin. In particular, second harmonic generation (SHG) microscopy was used to extract detailed images of the distribution of collagen fibers, which were subsequently processed using a three-dimensional Fourier transform-based method recently proposed by the authors to quantify the distribution of fiber orientations. Mechanical tests under both biaxial and uniaxial loading were performed to calibrate the relevant mechanical parameters of two widely used constitutive models of soft fiber-reinforced biological tissues that account for non-symmetrical fiber dispersion. The calibration of the models allowed us to identify correlations between the mechanical parameters of the constitutive models considered. STATEMENT OF SIGNIFICANCE: Constitutive models for soft collagenous tissues can accurately reproduce the complex nonlinear and anisotropic mechanical behavior of skin. However, a comprehensive analysis of both microstructural and mechanical parameters is still missing for human skin. In this study, these parameters are determined by combining biaxial mechanical tests and SHG stacks of collagen fibers on ex vivo healthy human skin samples. The constitutive parameters are provided for two widely used hyperelastic models and enable accurate characterization of skin mechanical behavior for advanced numerical simulations.

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BACKGROUND AND OBJECTIVE: Tooth extraction is a common clinical procedure with biomechanical factors that can directly influence patient outcomes. Recent development in atraumatic extraction techniques have endeavoured to improve treatment outcomes, but the characterization of extraction biomechanics is sparse. An axisymmetric inverse finite element (FE) approach is presented to represent the biomechanics of vertical atraumatic tooth extraction in an ex-vivo swine model. METHODS: Geometry and boundary conditions from the model are determined to match the extraction of swine incisors in a self-aligning ex vivo extraction experiment. Material parameters for the periodontal ligament (PDL) model are determined by solving an inverse FE problem using clusters of data obtained from 10 highly-controlled mechanical experiments. A seven-parameter visco-hyperelastic damage model, based on an Arruda-Boyce framework, is used for curve fitting. Three loading schemes were fit to obtain a common set of material parameters. RESULTS: The inverse FE results demonstrate good predictions for overall force-time curve shape, peak force, and time to peak force. The fit model parameters are sufficiently consistent across all three cases that a coefficient-averaged model was taken that compares well to all three cases. Notably, the initial modulus ,u, converged across trials to an average value of 0.472 MPa with an average viscoelastic constant g of 0.561. CONCLUSIONS: The presented model is found to have consistent parameters across loading cases. The capability of this model to represent the fundamental mechanical characteristics of the dental complex during vertical extraction loading is a significant advancement in the modelling of extraction procedures. Future work will focus on verifying the model as a predictive design tool for assessing new loading schemes in addition to investigating its applications to subject-specific problems.

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The perineum is a layered soft tissue structure with mechanical properties that maintain the integrity of the pelvic floor. During childbirth, the perineum undergoes significant deformation that often results in tears of various degrees of severity. To better understand the mechanisms underlying perineal tears, it is crucial to consider the mechanical properties of the different tissues that make up the perineum. Unfortunately, there is a lack of data on the mechanical properties of the perineum in the literature. The objective of this study is to partly fill these gaps. Hence sow perineums were dissected and the five perineal tissues involved in tears were characterized by uniaxial tension tests: Skin, Vagina, External Anal Sphincter, Internal Anal Sphincter and Anal Mucosa. From our knowledge, this study is the first to investigate all these tissues and to design a testing protocol to characterize their material properties. Six material models were used to fit the experimental data and the correlation between experimental and predicted data was evaluated for comparison. As a result, even if the tissues are of different nature, the best correlation was obtained with the Yeoh and Martins material models for all tissues. Moreover, these preliminary results show the difference in stiffness between the tissues which indicates that they might have different roles in the structure. These obtained results will serve as a basis to design an improved experimental protocol for a more robust structural model of the porcine perineum that can be used for the human perineum to predict perineal tears.

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The gastrointestinal (GI) peristalsis is an involuntary wave-like contraction of the GI wall that helps to propagate food along the tract. Many GI diseases, e.g., gastroparesis, are known to cause motility disorders in which the physiological contractile patterns of the wall get disrupted. Therefore, to understand the pathophysiology of these diseases, it is necessary to understand the mechanism of GI motility. We present a coupled electromechanical model to describe the mechanism of GI motility and the transduction pathway of cellular electrical activities into mechanical deformation and the generation of intraluminal pressure (IP) waves in the GI tract. The proposed model consolidates a smooth muscle cell (SMC) model, an actin-myosin interaction model, a hyperelastic constitutive model, and a Windkessel model to construct a coupled model that can describe the origin of peristaltic contractions in the intestine. The key input to the model is external electrical stimuli, which are converted into mechanical contractile waves in the wall. The model recreated experimental observations efficiently and was able to establish a relationship between change in luminal volume and pressure with the compliance of the GI wall and the peripheral resistance to bolus flow. The proposed model will help us understand the GI tract's function in physiological and pathophysiological conditions.

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The personalisation of finite element models is an important problem in the biomechanical fields where subject-specific analyses are fundamental, particularly in studying soft tissue mechanics. The personalisation includes the choice of the constitutive law of the model's material, as well as the choice of the material parameters. In vivo identification of the material properties of soft tissues is challenging considering the complex behaviour of soft tissues that are, among other things, non-linear hyperelastic and heterogeneous. Hybrid experimental-numerical methods combining in vivo indentations and inverse finite element analyses are common to identify these material parameters. Yet, the uniqueness and the uncertainty of the multi-material hyperelastic model have not been evaluated. This study presents a sensitivity analysis performed on synthetic indentation data to investigate the identification uncertainties of the material parameters in a bi-material thigh phantom. Synthetic numerical data, used to replace experimental measurements, considered several measurement modalities: indenter force and displacement, stereo-camera 3D digital image correlation of the indented surface, and ultrasound B-mode images. A finite element model of the indentation was designed with either Ogden-Moerman or Mooney-Rivlin constitutive laws for both materials. The parameters' identifiability (i.e. the possibility of converging to a unique parameter set within an acceptable margin of error) was assessed with various cost functions formulated using the different synthetic data sets. The results underline the need for multiple experimental modalities to reduce the uncertainty of the identified parameters. Additionally, the experimental error can impede the identification of a unique parameter set, and the cost function depends on the constitutive law. This study highlights the need for sensitivity analyses before the design of the experimental protocol. Such studies can also be used to define the acceptable range of errors in the experimental measurement. Eventually, the impact of the evaluated uncertainty of the identified parameters should be further investigated according to the purpose of the finite element modelling.

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The protective and preservative role of apple skin in maintaining the integrity of the fruit is well-known, with its mechanical behaviour playing a pivotal role in determining fruit storage capacity. This study employs a combination of experimental and numerical methodologies, specifically utilising the digital image correlation (DIC) technique. A specially devised inverse strategy is applied to evaluate the mechanical behaviour of apple skin under uniaxial tensile loading. Three apple cultivars were tested in this work: Malus domestica Starking Delicious, Malus pumila Rennet, and Malus domestica Golden Delicious. Stress-strain curves were reconstructed, revealing distinct variations in the mechanical responses among these cultivars. Yeoh's hyperelastic model was fitted to the experimental data to identify the coefficients capable of reproducing the non-linear deformation. The results suggest that apple skin varies significantly in composition and structure among the tested cultivars, as evidenced by differences in elastic properties and non-linear behaviour. These differences can significantly affect how fruit is handled, stored, and transported. Thus, the insights resulting from this research enable the development of mathematical models based on the mechanical behaviour of apple tissue, constituting important data for improvements in the economics of the agri-food industry.

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The continuous softening behavior of the brain tissue, i.e., the softening in the primary loading path with an increase in deformation, is modeled in this work as a state of hyperelasticity up to the onset of failure. That is, the softening behavior is captured via a core hyperelastic model without the addition of damage variables and/or functions. Examples of the application of the model will be provided to extant datasets of uniaxial tension and simple shear deformations, demonstrating the capability of the model to capture the whole-range deformation of the brain tissue specimens, including their softening behavior. Quantitative and qualitative comparisons with other models within the brain biomechanics literature will also be presented, showing the clear advantages of the current approach. The application of the model is then extended to capturing the rate-dependent softening behavior of the tissue by allowing the parameters of the core hyperelastic model to evolve, i.e., vary, with the deformation rate. It is shown that the model captures the rate-dependent and softening behaviors of the specimens favorably and also predicts the behavior at other rates. These results offer a clear set of advantages in favor of the considered modeling approach here for capturing the quasi-static and rate-dependent mechanical properties of the brain tissue, including its softening behavior, over the existing models in the literature, which at best may purport to capture only a reduced set of the foregoing behaviors, and with ill-posed effects.

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The investigation examines the transference of stiffness from intervertebral discs (IVDs) to the lumbar body of the L1 vertebra and the interactions among adjacent tissues. A computational model of the vertebra was developed, considering parameters such as cortical bone thickness, trabecular bone elasticity, and the nonlinear response of the nucleus pulposus to external loading. A nonlinear dynamic analysis was performed, revealing certain trends: a heightened stiffness of the annulus fibrosus correlates with a significant reduction in the vertebral body's ability to withstand external loading. At a supplied displacement of 6 mm, the vertebra with a degenerative disc reached its yielding point, whereas the vertebrae with a healthy annulus fibrosus exhibited a strength capacity exceeding 20%. The obtained findings and proposed methodology are potentially useful for biomedical engineers and clinical specialists in evaluating the condition of the annulus fibrosus and predicting its influence on the bone components of the spinal system.

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A new modelling approach is employed in this work for application to the rate-dependent mechanical behaviour of the brain tissue, as an incompressible isotropic material. Extant datasets encompassing single- and multi-mode compression, tension and simple shear deformation(s) are considered, across a wide range of deformation rates from quasi-static to rates akin to blast loading conditions, in the order of 1000 s-1 . With a simple functional form and a reduced number of parameters, the model is shown to capture the considered rate-dependent behaviours favourably, including in both single- and multi-mode deformation fits, and over all range of deformation rates. The provided modelling results here are obtained from either first fitting the model to the quasi-static data, or/and predicting the behaviour at a different rate than those used for calibrating the model parameters. Given its simplicity, versatility, predictive capability and accuracy, the application of the utilised modelling framework in this work to the rate-dependent mechanical behaviour of the brain tissue is proposed.

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This paper is concerned with the growth-driven shape-programming problem, which involves determining a growth tensor that can produce a deformation on a hyperelastic body reaching a given target shape. We consider the two cases of globally compatible growth, where the growth tensor is a deformation gradient over the undeformed domain, and the incompatible one, which discards such hypothesis. We formulate the problem within the framework of optimal control theory in hyperelasticity. The Hausdorff distance is used to quantify dissimilarities between shapes; the complexity of the actuation is incorporated in the cost functional as well. Boundary conditions and external loads are allowed in the state law, thus extending previous works where the stress-free hypothesis turns out to be essential. A rigorous mathematical analysis is then carried out to prove the well-posedness of the problem. The numerical approximation is performed using gradient-based optimisation algorithms. Our main goal in this part is to show the possibility to apply inverse techniques for the numerical approximation of this problem, which allows us to address more generic situations than those covered by analytical approaches. Several numerical experiments for beam-like and shell-type geometries illustrate the performance of the proposed numerical scheme.

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The formulation of the entropic statistical theory and the related neo-Hookean model has been a major advance in the modeling of rubber-like materials, but the failure to explain some experimental observations such as the slope in Mooney plots resulted in hundreds of micromechanical and phenomenological models. The origin of the difficulties, the reason for the apparent need for the second invariant, and the reason for the relative success of models based on the Valanis-Landel decomposition have been recently explained. From that insight, a new micro-macro chain stretch connection using the stretch tensor (instead of the right Cauchy-Green deformation tensor) has been proposed and supported both theoretically and from experimental data. A simple three-parameter model using this connection has been suggested. The purpose of this work is to provide further insight into the model, to provide an analytical expression for the Gaussian contribution, and to provide a simple procedure to obtain the parameters from a tensile test using the Mooney space or the Mooney-Rivlin constants. From different papers, a wide variety of experimental tests on different materials and loading conditions have been selected to demonstrate that the simple model calibrated only from a tensile test provides accurate predictions for a wide variety of elastomers under different deformation levels and multiaxial patterns.

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In order to understand how the biomechanical properties of rabbit cornea change over time after corneal ablation, 21 healthy adult rabbits were used in this study, with the left eye as experimental side and the right eye as the control side. Firstly, a lamellar knife was used to remove a portion of the anterior corneal surface tissue (30%~50% of the original corneal thickness) from the left eye of each rabbit, as an animal model simulating corneal refractive surgery. Secondly, postoperative experimental rabbits were kept for one, three, or six months until being euthanized. Strip specimens were produced using their corneas in vitro to perform a uniaxial tensile test with an average loading-unloading rate of approximately 0.16 mm/s. Finally, the visco-hyperelastic material constitutive model was used to fit the data. The results showed that there was a significant difference in the viscoelastic parameters of the corneas between the experimental and the control eyes at the first and third postoperative months. There was a difference in tangential modulus between the experimental and the control eyes at strain levels of 0.02 and 0.05 at the third postoperative month. There was no significant difference in biomechanical parameters between the experimental and the control eyes at the sixth postoperative month. These results indicate that compared with the control eyes, the biomechanical properties of the experimental eyes vary over postoperative time. At the third postoperative month, the ratio of corneal tangential modulus between the experimental and the control eyes significantly increased, and then decreased. This work lays a preliminary foundation for understanding the biomechanical properties of the cornea after corneal refractive surgery based on rapid testing data obtained clinically.

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This work proposes a displacement-based finite element model for large strain analysis of isotropic compressible and nearly-incompressible hyperelastic materials. Constitutive law is written in terms of invariants of the right Cauchy-Green tensor; coupled and decoupled formulations of strain energy functions are presented, whereas a penalty function is used to impose an incompressibility constraint. Based on a total Lagrangian formulation, the nonlinear governing equations are thus obtained by employing the principle of virtual displacements. Analytic expression of both internal forces vector and tangent matrix of linear and high-order hexahedral finite elements are derived by adopting a three-dimensional formalism based on the Carrera Unified Formulation. Popular benchmark problems in hyperelasticity are analyzed to establish the capabilities of the present implementation of fully-nonlinear solid elements in the case of compressible and nearly-incompressible beams, cylindrical shells, and curved structures.

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The comprehension and modeling of the mechanical behavior of soft biological tissues are essential due to their clinical applications. This knowledge is essential for predicting tissue responses accurately and enhancing our ability to compute the behavior of biological structures and bio-prosthetic devices under specific loading conditions. The current research is centered on modeling the initiation and progression of soft tissues damage, which typically exhibit intricate anisotropic and nonlinear elastic characteristics. For this purpose, the following study presents a comparative analysis of the computational performance of two distinct damage modeling techniques. The first technique employs a well-established damage model, based on a piece-wise exponential damage function as proposed by Calvo et al. (Int J Numer Methods Eng 69:2036-2057, 2007. https://doi.org/10.1002/nme.1825 ). The second approach adopts a sigmoid function, as proposed by López-Campos et al. (Comput Methods Biomech Biomed Eng 23(6):213-223. https://doi.org/10.1080/10255842.2019.1710742 ). The aim of this study is to verify the validity of the López-Campos sigmoid-based damage model to be used in finite element simulation, the implementation of which is unknown. For this proposal, both models were implemented within a commercial Finite Element software package, and their responses to local and non-local damage algorithms were assessed in depth through two standard benchmark tests: a plate with a hole and a ball burst. The results of this study indicate that, for a wide range of cases, such as in-plane stresses, out-plane stresses, stress concentration and contact, all over large displacement conditions, the López-Campos damage model shows a good response to non-local algorithms achieving mesh independence and convergence in all these cases. The results obtained are in line with those obtained for the Calvo's damage model, showing, in addition, larger deformations under in-plane stress and stress concentration conditions and a lower number of iterations under out-plane stress and contact conditions. Consequently, the López-Campos' damage model emerges as a valuable and useful tool in the field of mechanical damage research in biological systems.

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A numerical investigation into the ultrasound-induced collapse of air bubbles near soft materials, utilizing a novel multi-material diffuse interface method (DIM) model with block-structured adaptive mesh refinement is presented. The present work expands from a previous five-equation DIM by incorporating Eulerian hyperelasticity. The model is applicable to any arbitrary number of interacting fluid and solid material. A single conservation law for the elastic stretch tensor enables tracking the deformations for all the solid materials. A series of benchmark cases are conducted, and the solution is found to be in excellent agreement against theoretical data. Subsequently, the ultrasound-induced bubble-tissue flow interactions are examined. The bubble radius was found to play a crucial role in dictating the stresses experienced by the tissue, underscoring its significance in medical applications. The results reveal that soft tissues primarily experience tensile forces during these interactions, suggesting potential tensile-driven injuries that may occur in relevant treatments. Moreover, regions of maximal tensile forces align with tissue elongation areas. It is documented that while early bubble dynamics remain relatively unaffected by changes in shear modulus, at later stages of the penetration processes and the deformation shapes, exhibit notable variations. Lastly, it is demonstrated that decreasing standoff distances enhances the interaction between bubbles and tissue, thereby increasing the stress levels in the tissue, although the behavior of the bubble dynamics remains largely unchanged.

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Fabricating helical scaffolds using electrospinning is a common approach for cardiac implantation, aiming to achieve properties similar to native tissue. However, this process requires multiple experimental attempts to select suitable electrospun properties and validate resulting mechanical responses. To overcome the time and cost constraints associated with this iterative procedure, Finite Element Analysis (FEA) can be applied using stable hyperelastic and viscoelastic models that describe the response of electrospun scaffolds under different conditions. In this study, we aim to create accurate simulations of the viscoelastic behavior of electrospun helical scaffolds. We fabricated helical fibers from Poly (3-caprolactone) (PCL) using the electrospinning process to achieve this. The electrospun samples were subjected to uniaxial deformation, and their response was modelled using existing hyperelastic and stress relaxation models. The simulations were built on experimental data for specific deformation speed and maximum strain conditions. The FEM results were evaluated by accounting for the stress-softening phenomenon, which significantly impacted the models. The electrospun scaffolds' predictions were performed in other than the initial experimental conditions to verify our simulations' accuracy and reliability.

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Simulating the mechanical behavior of cellular materials stands as a pivotal step in their practical application. Nonetheless, the substantial multitude of unit cells within these materials necessitates a considerable finite element mesh, thereby leading to elevated computational expenses and requisites for formidable computer configurations. In order to surmount this predicament, a novel and straightforward equivalent calculation method is proposed for the computation of mechanical properties concerning both random and ordered hyper-elastic cellular materials. By amalgamating the classical finite element approach with the distribution attributes of cells, the proposed equivalent calculation method adeptly captures the deformation modes and force-displacement responses exhibited by cell materials under tensile and shear loads, as predicted through direct numerical simulation. This approach reflects the deformation characteristics induced by micro-unit cells, elucidates an equivalent principle bridging cellular materials and equivalent materials, and substantially curtails exhaustive computational burdens. Ultimately, this method furnishes an equivalent computational strategy tailored for the engineering applications of cellular materials.