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Related Concept Videos

Castigliano's Theorem01:18

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Castigliano's theorem analyzes displacements and rotations in elastic structures. It relates the derivative of elastic strain energy to the applied forces or moments, allowing for the calculation of deformations. The theorem states that the partial derivative of the total strain energy of a system with respect to a specific load results in the displacement at the point where the load is applied. This principle applies to both forces and moments.
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Joints, also known as articulations, are classified based on their structural characteristics, i.e., based on whether the articulating surfaces of the adjacent bones are directly connected by fibrous connective tissue or cartilage, or whether the articulating surfaces contact each other within a fluid-filled joint cavity. These differences serve to divide the joints of the body into three structural classifications.
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When a rod is made of different materials or has various cross-sections, it must be divided into parts that meet the necessary conditions for determining the deformation. These parts are each characterized by their internal force, cross-sectional area, length, and modulus of elasticity. These parameters are then used to compute the deformation of the entire rod.
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Temperature Dependent Deformation01:12

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In a nonhomogeneous rod made up of steel and brass, restrained at both ends and subjected to a temperature change, several steps are involved in calculating the stress and compressive load. Due to the problem's static indeterminacy, one end support is disconnected, allowing the rod to experience the temperature change freely. Next, an unknown force is applied at the free end, triggering deformations in the rod's steel and brass portions. These deformations are then calculated and added...
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When a car’s weight and driving forces act on a tire, they impose an external load on the rubber material. This load is resisted internally by forces distributed throughout the tire structure, which are defined as stress. The resulting deformation of the rubber due to this stress is quantified as strain. The relationship between stress and strain governs how the tire deforms under load and is central to understanding its mechanical response during operation.Rubber exhibits a nonlinear...
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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.
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Related Experiment Video

Updated: Feb 24, 2026

Subject-specific Musculoskeletal Model for Studying Bone Strain During Dynamic Motion
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A differentiable variational model for structural self-contact and fracture.

Mirko Ciceri1, Charlie Aveline1, Dilaksan Thillaithevan1

  • 1Department of Aeronautics, Imperial College London, Exhibition Rd, South Kensington, London, SW7 2AZ UK.

Engineering with Computers
|February 23, 2026
PubMed
Summary

This study introduces a unified numerical model for structural self-contact and crack propagation. The new framework efficiently analyzes complex nonlinear behaviors, enabling advanced structural design.

Keywords:
FracturePhase fieldThird medium contact

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Area of Science:

  • Computational mechanics
  • Materials science
  • Structural engineering

Background:

  • Numerical modeling of structural self-contact and crack propagation is challenging due to discontinuous phenomena.
  • Traditional methods require explicit tracking of contact points and predefined crack sites, limiting analysis.
  • Existing models often necessitate separate treatments for contact and fracture, increasing complexity.

Purpose of the Study:

  • To develop a unified, numerically stable variational framework for modeling structural self-contact and crack propagation.
  • To overcome limitations of traditional methods by avoiding explicit contact point tracking and predefined crack initiation.
  • To create an efficient and differentiable numerical model for complex nonlinear structural analysis.

Main Methods:

  • Utilizing a hyperelastic third medium contact model for structural self-contact.
  • Representing fracture using a phase field approach within a unified variational formulation.
  • Embedding structures in a compressive-stiffening third medium to facilitate force transfer and model void behavior.

Main Results:

  • A novel, differentiable numerical model that efficiently captures both self-contact and crack propagation.
  • Demonstration of a unified framework that overcomes the need for pre-defined contact points and crack initiation sites.
  • Successful coupling of contact and fracture phenomena, including void material behavior.

Conclusions:

  • The developed framework provides an efficient tool for analyzing complex nonlinear structural behaviors.
  • The differentiable nature of the model allows for seamless integration into topology optimization.
  • Enables designers to leverage self-contact and material failure as functional design features for enhanced performance.