DEFORMATION, FINITE AN INFINITESIMAL

Infinitesimal strain is the step by step strain that leads to finite strain ellipsoid. The total difference between the initial and final strain is the finite strain. The infinitesimal strain is all the strains that the object has gone through to get to the finite strain. While finite strain is the end position all the positions used to get to the final position is the infinitesimal strain. The product and squares of any infinitesimal strain can be ignored

At any stage during deformation, strain may be represented by two strain ellipses (or ellipsoids in 3D), a finite strain ellipsean infinitesimal (or incremental) strain ellipse. During progressive deformation. Infinitesimal strain is the strain taking place at a given time (what is happening) Finite strain is the strain that has taken place up to that time (what has happened). Any finite strain state is built by addition of gradual infinitesimally small increments of strain to build up the next finite strain state. The diagram below shows a sequence like the series of figures which could be likened to a series of stills from a continuous motion picture. If we introduce a circle at the 5th stage of deformation, it will be deformed into an ellipse which is known as the INFINITESIMAL OR INCREMENTAL STRAIN ELLIPSE. The incremental strains when added on to the 5th stage finite strain state then produce the next finite strain state or the 6th state. In reality the process can be far too complex than that shown in the figure.

Difference between __finite strain__
and __infinitesimal strain__:

Finite strain compares only the initial and former states.

An increment of natural strain ei = dl
/ l' , where l' is length at beginning of increment, but not the original
length, since the previous strain increment changed l'. The natural strain is
the sum of all the increments. If you integrate from an initial length to a
final length, then __by definition__ you end up with the natural strain = ln
( final length/ original length).

Substituting the fact that stretch = S = final length/ orginal length = (1 + e), where e is the finite strain equivalent, then the natural strain = ln ( 1+finite strain).

**example**:
if **e**i = .1 and initial length lo = 1

strain increment |
length of line |
finite elongation |
natural strain |

0.1 |
11 |
0.1 |
0.09531018 |

0.1 |
12.1 |
0.21 |
0.19062036 |

0.1 |
13.31 |
0.331 |
0.28593054 |

0.1 |
14.641 |
0.4641 |
0.38124072 |

0.1 |
16.105 |
0.61051 |
0.4765509 |

0.1 |
17.716 |
0.77156 |
0.57186108 |

0.1 |
19.487 |
0.94872 |
0.66717126 |

0.1 |
21.436 |
1.14359 |
0.76248144 |

0.1 |
23.579 |
1.35795 |
0.85779162 |

Partly from http://maps.unomaha.edu/Maher/GEOL3300/week6/rheology.html