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Related Concept Videos

Generalized Hooke's Law01:22

Generalized Hooke's Law

963
The generalized Hooke's Law is a broadened version of Hooke's Law, which extends to all types of stress and in every direction. Consider an isotropic material shaped into a cube subjected to multiaxial loading. In this scenario, normal stresses are exerted along the three coordinate axes. As a result of these stresses, the cubic shape deforms into a rectangular parallelepiped. Despite this deformation, the new shape maintains equal sides, and there is a normal strain in the direction of the...
963
Relation between Poisson's ratio, Modulus of Elasticity and Modulus of Rigidity01:15

Relation between Poisson's ratio, Modulus of Elasticity and Modulus of Rigidity

270
Deformation occurs in axial and transverse directions when an axial load is applied to a slender bar. This deformation impacts the cubic element within the bar, transforming it into either a rectangular parallelepiped or a rhombus, contingent on its orientation. This transformation process induces shearing strain. Axial loading elicits both shearing and normal strains. Applying an axial load instigates equal normal and shearing stresses on elements oriented at a 45° angle to the load axis.
270
Elastic Curve from the Load Distribution01:16

Elastic Curve from the Load Distribution

180
The structural behavior of beams under distributed loads is critical for engineering analysis, which focuses on predicting how beams bend and react under such conditions. Different types of beams (e.g., cantilever, supported, or overhanging) behave differently under distributed load conditions.
For all beams, the analysis of the beam's reaction to distributed loads begins by understanding the relationship between a beam's load and the resulting shear forces and bending moments.
180

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Data-driven Tissue Mechanics with Polyconvex Neural Ordinary Differential Equations.

Vahidullah Tac1, Francisco Sahli Costabal2, Adrian B Tepole1,3

  • 1School of Mechanical Engineering, Purdue University, West Lafayette, IN, USA.

Computer Methods in Applied Mechanics and Engineering
|December 4, 2023
PubMed
Summary

Neural ordinary differential equations (N-ODEs) create physics-compliant data-driven material models. This approach ensures polyconvexity, outperforming traditional models for complex materials like skin in simulations.

Keywords:
Constitutive modelingMachine learningNonlinear finite elementsSkin mechanics

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Area of Science:

  • Computational Mechanics
  • Materials Science
  • Applied Mathematics

Background:

  • Data-driven methods offer advantages over traditional material modeling in computational mechanics.
  • Deep neural networks can learn complex material responses but often lack physics-based constraints like polyconvexity.
  • Polyconvexity is crucial for ensuring the existence of minimizers in elasticity boundary value problems.

Purpose of the Study:

  • To develop data-driven material models using neural ordinary differential equations (N-ODEs) that inherently satisfy polyconvexity.
  • To address the limitations of current data-driven approaches in enforcing essential mathematical requirements.
  • To create a general framework for modeling diverse materials, particularly soft biological tissues.

Main Methods:

  • Utilized neural ordinary differential equations (N-ODEs) to construct data-driven material models.
  • Leveraged properties of ODEs to generate monotonic functions approximating strain energy derivatives.
  • Ensured polyconvexity by guaranteeing the monotonicity of these derivatives with respect to deformation invariants.

Main Results:

  • The N-ODE material model successfully captured synthetic data from established material models.
  • Demonstrated superior performance compared to conventional models using experimental data from nonlinear, anisotropic skin tissue.
  • Successfully integrated the N-ODE model into finite element simulations for reconstructive surgery.

Conclusions:

  • N-ODE based data-driven material models automatically satisfy polyconvexity, a key physics requirement.
  • The proposed methodology shows significant potential for modeling complex materials, especially biological soft tissues.
  • This framework is expected to advance the application of data-driven methods in computational mechanics.