Energetically stable curve fitting to hyperelastic models based on uniaxial and biaxial tensile tests

Hyperelastic constitutive laws in biomechanics are used to model soft tissues, and material model parameters are often determined by performing curve fitting on data from uniaxial or biaxial tensile tests. The strain energy function of the applied constitutive law must to be energetically stable; however, this condition is not inherently provided by most currently available models. This study provides a procedure to determine stable strain energy functions in a biaxial strain space based on either uniaxial or biaxial tensile tests. Instead of conservative, strain-independent conditions, a stability region is defined in the strain space based on the sample's tensile tests, thus allowing optimisation within a wider parameter space, resulting in better approximations. An extension of the Levenberg-Marquardt algorithm incorporating user-defined stability constraints is proposed, and the constrained optimisation algorithm is applied to isotropic and anisotropic models. The uniqueness of solutions of the Fung model is also discussed. The material model parameters of stable solutions for soft tissue measurements from various literature sources are determined to demonstrate the proposed procedure. Applying appropriate constraints in the optimisation algorithm resulted in stable and physically permissible constrained solutions for the strain energy function, in contrast to the results of most unconstrained optimisation cases.

On implementation of fibrous connective tissues' damage in Abaqus software

Connective fibrous tissues, such as tendons and ligaments, in humans and animals exhibit hyperelastic behaviour. The constitution of the material of these tissues is anisotropic due to the presence of the collagen fibres, where one family of fibres is the typical case. Traumatic events and/or aging may sometimes lead to the damage of the tissue. The study of motion of affected joints or limbs is usually not permitted in vivo. This is where finite element method (FEM) becomes useful as a premise for general analysis, surgical planning or designing of implants and medical treatment. One of the most often used FEM commercial programs is the field of the biomechanics is Abaqus. The present study discusses the potential of user subroutine UANISOHYPER_INV in this code to analyse response of transversely isotropic tissue with damage in quasi-static range. This subroutine requires providing the material energy function and its derivatives only. The stress tensor and constitutive matrix are computed by the software automatically. To the best of the Authors' knowledge this procedure provides the easiest way to simulate the anisotropic hyperelastic material behaviour in Abaqus. In this study its usage is extended onto the damage response simulation. The verification of the approach and its validation against experimental data indicates its efficiency.

HyperMSM: A new MSM variant for efficient simulation of dynamic soft-tissue deformations

BACKGROUND AND OBJECTIVE

Fast, accurate, and stable simulation of soft tissue deformation is a challenging task. Mass-Spring Model (MSM) is one of the popular methods used for this purpose for its simple implementation and potential to provide fast dynamic simulations. However, accurately simulating a non-linear material within the mass-spring framework is still challenging. The objective of the present study is to develop and evaluate a new efficient hyperelastic Mass-Spring Model formulation to simulate the Neo-Hookean deformable material, called HyperMSM.

METHODS

Our novel HyperMSM formulation is applicable for both tetrahedral and hexahedral mesh configurations and is compatible with the original projective dynamics solver. In particular, the proposed MSM variant includes springs with variable rest-lengths and a volume conservation constraint. Two applications (transtibial residual limb and the skeletal muscle) were conducted.

RESULTS

Compared to finite element simulations, obtained results show RMSE ranges of [2.8%-5.2%] and [0.46%-5.4%] for stress-strain and volumetric responses respectively for strains ranging from -50% to +100%. The displacement error range in our transtibial residual limb simulation is around [0.01mm-0.7 mm]. The RMSE range of relative nodal displacements for the skeletal psoas muscle model is [0.4%-1.7%].

CONCLUSIONS

Our novel HyperMSM formulation allows hyperelastic behavior of soft tissues to be described accurately and efficiently within the mass-spring framework. As perspectives, our formulation will be enhanced with electric behavior toward a multi-physical soft tissue mass-spring modeling framework. Then, the coupling with an augmented reality environment will be performed.

A generic physics-informed neural network-based constitutive model for soft biological tissues

Constitutive modeling is a cornerstone for stress analysis of mechanical behaviors of biological soft tissues. Recently, it has been shown that machine learning (ML) techniques, trained by supervised learning, are powerful in building a direct linkage between input and output, which can be the strain and stress relation in constitutive modeling. In this study, we developed a novel generic physics-informed neural network material (NNMat) model which employs a hierarchical learning strategy by following the steps: (1) establishing constitutive laws to describe general characteristic behaviors of a class of materials; (2) determining constitutive parameters for an individual subject. A novel neural network structure was proposed which has two sets of parameters: (1) a class parameter set for characterizing the general elastic properties; and (2) a subject parameter set (three parameters) for describing individual material response. The trained NNMat model may be directly adopted for a different subject without re-training the class parameters, and only the subject parameters are considered as constitutive parameters. Skip connections are utilized in the neural network to facilitate hierarchical learning. A convexity constraint was imposed to the NNMat model to ensure that the constitutive model is physically relevant. The NNMat model was trained, cross-validated and tested using biaxial testing data of 63 ascending thoracic aortic aneurysm tissue samples, which was compared to expert-constructed models (Holzapfel-Gasser-Ogden, Gasser-Ogden-Holzapfel, and four-fiber families) using the same fitting and testing procedure. Our results demonstrated that the NNMat model has a significantly better performance in both fitting (R² value of 0.9632 vs 0.9019, p=0.0053) and testing (R² value of 0.9471 vs 0.8556, p=0.0203) than the Holzapfel-Gasser-Ogden model. The proposed NNMat model provides a convenient and general methodology for constitutive modeling.

Analysis of mechanical deformation effect on the voltage generation of a vertical contact mode triboelectric generator

One of the associated factors that controls the performance of a triboelectric generator (TEG) is the mechanical deformation of the dielectric layer. Therefore, a good contact model can be a prominent tool to find a more realistic and efficient way of determining the relationships between the contact and electrical output of the generator. In this study, experiments are conducted on a vertical contact mode triboelectric generator under an MTS machine. The open-circuit voltages are measured at different loads imposed by the MTS by controlling the cyclic displacement of the top tribo layer of the generator. A finite-element-based theoretical model is developed to explain the behavior of the generator during the experiments. The 2D-contact problem of the micro-structured tribo layers is simulated and then the contact results are integrated into 3D to find the actual contact area between the two surfaces. These numerical contact results improve the existing theoretical model by evaluating the correct surface charge density and contact area as a function of the input parameters. The excellent agreement between our experimental and theoretical results illustrates that theoretical modeling could be used as a robust approach to predict the mechanical and electrical performance of TEGs. In addition, some parametric studies of the harvester are presented here for different geometrical parameters of the microstructures.

Rupture limit evaluation of human cerebral aneurysms wall: Experimental study

BACKGROUND AND PURPOSE

Rupture risk of intracranial aneurysms is a major issue for public healthcare. A way to obtain an individual rupture risk assessment is a main objective of many research teams in the world. For many years, we have investigated the relationship between the mechanical properties of aneurysm wall tissues and the rupture risk. In this work, we try to go further and investigate rupture limit values.

METHODS

Following surgical clipping, a specific conservation protocol was applied to aneurysmal tissues in order to preserve their mechanical properties. Thirty-nine intracranial aneurysms (27 females, 12 males) were tested using a uniaxial tensile test machine under physiological conditions, temperature, and saline isotonic solution. These represented 24 unruptured and 15 ruptured aneurysms. Stress/strain curves were then obtained for each sample, and a fitting algorithm was applied following a Yeoh hyperelastic model with 2 parameters. Moreover, uniaxial tensile tests were conducted until rupture of samples to obtain values of stress and strain rupture limit.

RESULTS

The significant parameter a C₂ of the hyperelastic Yeoh model, allowed us to classify samples' rigidity following the terminology we adopted in previous papers (Costalat et al., 2011; Sanchez et al., 2013): Soft, Stiff and Intermediate. Moreover, strain/stress rupture limit values were gathered and analyzed thanks to the tissue rigidity, the status of the aneurysm (initially ruptured or unruptured) and the gender of the patient.

CONCLUSION

Strain rupture limit was found quite stable around 20% and seems not to be correlated with the status of the aneurysm (initially ruptured or unruptured), neither with the gender of the patient. However, stretch and stress rupture limit seems not to be independent on the rigidity. The study confirms that ruptured aneurysms mainly present a soft tissue and unruptured aneurysms present a stiff material.

Investigation of micromechanical properties of hard sphere filled composite hydrogels by atomic force microscopy and finite element simulations

Atomic force microscopy (AFM) indentation is the most suitable way to characterize micromechanical properties of soft materials such as bio tissues. However, the mechanical data obtained from force-indentation measurement are still not well understood due to complex geometry of the bio tissue, nonlinearity of indentation contact, and constitutive relation of hyperelastic soft material. Poly-N-isopropyl acrylamide (PNIPAM) filled with 5wt% polystyrene (PS) sphere particles material system can be utilized as a simplified model for mimicking a whole host of soft materials. Finite element model has been constructed to simulate indentation as in AFM experiments using colloidal probes for a parametric study, with the main purpose of understanding the effect of particles on overall behavior of mechanical data and local deformation field under indentation contact. Direct comparison between finite element simulation and indentation data from AFM experiments provides a powerful method to characterize soft materials properties quantitatively, addressing the lack of analytical solutions for hard-soft composites, both biological and synthetic ones. In this framework, quantitative relations are found between the depth, at which the particle was embedded, the particle size and the elastic modulus of the overall composite. Comprehensive characterizations were established to distinguish indentation on a particle residing on top of the hydrogel from a particle embedded inside the hydrogel matrix. Finally, different assumptions of interface friction at the boundary between the particle and the hydrogel have been tested and directly compared with experimental measurements.