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I have been recommended to read a book about nonlinear continuum mechanics, but when I have taken a look at the table of contents, I found myself familiar with most of the concepts in the book since I have taken before a basic course in continuum mechanics.

So I can't find a big difference between linear and nonlinear continuum mechanics except for hyper elasticity concept (probably).

Could someone explain to me the major difference between them?

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Continuum mechanics has two branches - solid mechanics, and fluid mechanics.

The subbranches of solid mechanics are Elasticity (linear) and Plasticity (nonlinear); the subbranches of fluid mechanics are Newtonian Fluids (linear) and Non-Newtonian Fluids (nonlinear). Depending on the branch of study, the nonlinear continuum mechanics involves matters with large displacement, or acting chaotically.

The nonlinear subbranches of solid and fluid mechanics form the base for Rheology.

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Although this is probably a gross oversimplification, I regard nonlinear continuum mechanics as a superset of linear continuum mechanics.

To my experience, the basic difference is in the form of constitutive equations. I.e in the case of linear continuum mechanics you have the stress strain relationship

$$\sigma = E \epsilon$$

where E is a constant and therefore proportional.

In the case of nonlinear mechanics, the stress strain constitutive equation is not limited to the proportional relationship above.

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    $\begingroup$ Nonlinear mechanics primarily deals with what happens when there are large deformations (large stretches and rotations). Material nonlinearities typically appear under these conditions and therefore also have to be considered. $\endgroup$ Mar 29 at 18:13

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