One step further in biomechanical models in palaeontology: a nonlinear finite element analysis review

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Paleontology and Evolutionary Science

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Introduction

Search methodology

Discovering all the fea elements: solids, shells, plates, beams, springs, and trusses

Non-linearities in fea models

  1. Material non-linearity: When a material is non-linear, the strain it experiences is not proportional to the stress applied i.e., the material does not conform to Hooke’s Law. This situation occurs in plastic or hyperelastic materials where the relationship between stress and strain does not follow a lineal proportion.

  2. Large deformation non-linearity: The so-called finite strain theory, large strain theory, or large deformation theory is used when strains are large enough to invalidate the assumptions of the small strain theory, which is the theory commonly used in linear elastic problems. In this case, the deformed and undeformed configurations of the body under analysis are notably different, requiring a clear distinction between them in the formulation that, consequently, also affects the relation between stress and strain in the constitutive equation. This theory is common in elastomers and soft tissues and needs to be used when modelling hyperelastic materials.

  3. Large displacement non-linearity: Also called as geometrical non-linearity, assumes small strains but large rotations and displacements. In the geometrically linear case, the forces are applied in the undeformed geometry when solving the model whereas in the geometrically nonlinear cases, the applied forces depend on the deformed upcoming geometry. It involves an iterative solution accounting for the displacements and needs to be considered when analysing buckling.

  4. Non-linear contacts: Separate surfaces of two bodies are in contact without overlapping in such a way that they become mutually tangential. Depending on the relationship between these two surfaces, contacts that allow the separation in the perpendicular direction require a nonlinear solution because there are unknowns at the start of the solving process i.e., where and which force is applied.

Non-linear materials: hyperelasticity and plasticity

Non-linearities in geometry: buckling

Non-linearities in contacts

  1. Bonded contacts: when separation and sliding is not allowed. It is a linear contact.

  2. No-separation contact: when separation is not allowed but sliding in the tangential plane is allowed. It is a linear contact.

  3. Frictionless contact: when separation and sliding is allowed. It is a non-linear contact.

  4. Rough contact: when separation is allowed but sliding in the tangential plane is not allowed. It is a non-linear contact.

  5. Frictional contact: when separation is allowed but sliding in the tangential plane is controlled by a friction coefficient. It is a non-linear contact.

Summary: ideas for palaeontologists

Non-linear soft tissues

Plasticity in retrodeformations

Buckling in slender bones

Bone grouping using contacts

Models with shells, plates, beams, springs, and trusses

Additional Information and Declarations

Competing Interests

The author declares that they have no competing interests.

Author Contributions

Jordi Marcé-Nogué conceived and designed the experiments, performed the experiments, analyzed the data, prepared figures and/or tables, authored or reviewed drafts of the article, conceived the text, and approved the final draft.

Data Availability

The following information was supplied regarding data availability:

This is a literature review; there is no raw data or code.

Funding

This work was supported by the Serra-Hunter (URV) and the CERCA programme (ICP) from the Generalitat de Catalunya and the research project PID2020-117118GB-I00 funded by MCIN/AEI/10.13039/501100011033. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

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