Mathematical Modelling of Residual-Stress Based Volumetric Growth in Soft Matter

Area of Science:

• Mechanics of Materials
• Biomechanical Engineering
• Continuum Mechanics

Background:

• Biological growth is inherently linked to the development of residual stresses.
• These stresses contribute to heterogeneity and anisotropy across all scales of growing materials.
• Understanding mechanical regulation is key to comprehending growth patterns.

Purpose of the Study:

• To develop a theoretical model for stress-mediated growth based on the unloaded configuration.
• To analyze growth without assuming a fictitious stress-free grown state.
• To provide a framework applicable to experimental scenarios like arterial opening-angle measurements.

Main Methods:

• Formulation of a growth model using residual stress or deformation gradient as the growth variable.
• Analysis based on the unloaded configuration of the growing material.
• Application of the model to a spherically symmetric thick-walled shell under incompressibility constraints.

Main Results:

• The proposed model successfully analyzes stress-mediated growth without requiring a stress-free reference configuration.
• The framework directly relates to experimental measurements of residual stress, such as the arterial opening-angle.
• An initial illustration demonstrates the model's utility in a constrained thick-walled shell scenario.

Conclusions:

• The novel theoretical framework enables the study of stress-mediated growth by focusing on the unloaded configuration.
• This approach simplifies the analysis of heterogeneous and anisotropic growth phenomena.
• The model has direct implications for experimental biomechanics and materials science research.