Chapter 3
Continuum Mechanics and Nonlinear
Elasticity
The theory of elasticity is a subset of the discipline of continuum mechanics,
which treats fluids and solids as continuous collections of particles. The focus is on
macroscopic behavior, which is assumed to represent an average of the microscopic
activities of individual particles.
The terms macroscale and microscale are relative. For tissue, a cell is considered
micro; for cells, a molecule is micro. Multiscale models can be used to link
behavior at two or more scales. Consider, for example, stretching of a sheet of cells
(epithelium). On the macro scale, we can ignore individual cells and treat the tissue
as a continuous membrane. Tissue-level stresses computed in the membrane would
represent the average of the stresses in the microscopic constituents at each point.
To compute stresses experienced by the cytoskeleton and cell nucleus of a cell, we
could then construct a detailed model of the cell, apply the computed tissue-level
stresses to the boundaries, and solve the micro problem using continuum mechanics
or a molecular-level biophysical analysis.
In this book, we consider primarily the mechanics of elastic solids, which
are composed of idealized materials that deform without loss of energy. During
loading, energy is stored in the deforming material in the form of strain energy.
When unloaded, the material returns to its original state as the strain energy is
recovered. In the real world, no material is perfectly elastic. This is especially
true for biological materials, which lose energy as they deform through internal
friction between its various constituents. These components can include proteins,
fluid, and other cellular and extracellular structures. In general, tissues behave as
viscoelastic materials with characteristics of both fluids and solids. These effects
can be significant, but they often are neglected and are beyond the scope of this
book.
© Springer Nature Switzerland AG 2020
L. A. Taber, Continuum Modeling in Mechanobiology,
https://doi.org/10.1007/978-3-030-43209-6_3
51
Continuum Mechanics and Nonlinear
Elasticity
The theory of elasticity is a subset of the discipline of continuum mechanics,
which treats fluids and solids as continuous collections of particles. The focus is on
macroscopic behavior, which is assumed to represent an average of the microscopic
activities of individual particles.
The terms macroscale and microscale are relative. For tissue, a cell is considered
micro; for cells, a molecule is micro. Multiscale models can be used to link
behavior at two or more scales. Consider, for example, stretching of a sheet of cells
(epithelium). On the macro scale, we can ignore individual cells and treat the tissue
as a continuous membrane. Tissue-level stresses computed in the membrane would
represent the average of the stresses in the microscopic constituents at each point.
To compute stresses experienced by the cytoskeleton and cell nucleus of a cell, we
could then construct a detailed model of the cell, apply the computed tissue-level
stresses to the boundaries, and solve the micro problem using continuum mechanics
or a molecular-level biophysical analysis.
In this book, we consider primarily the mechanics of elastic solids, which
are composed of idealized materials that deform without loss of energy. During
loading, energy is stored in the deforming material in the form of strain energy.
When unloaded, the material returns to its original state as the strain energy is
recovered. In the real world, no material is perfectly elastic. This is especially
true for biological materials, which lose energy as they deform through internal
friction between its various constituents. These components can include proteins,
fluid, and other cellular and extracellular structures. In general, tissues behave as
viscoelastic materials with characteristics of both fluids and solids. These effects
can be significant, but they often are neglected and are beyond the scope of this
book.
© Springer Nature Switzerland AG 2020
L. A. Taber, Continuum Modeling in Mechanobiology,
https://doi.org/10.1007/978-3-030-43209-6_3
51