# Deviatoric stress

*δσ _{ij}* is 2

^{nd}order tensor represented by a 3x3 matrix of form:

σ_{11}-σ_{m} |
σ_{12} |
σ_{13} |

σ_{21} |
σ_{22}-σ_{m} |
σ_{23} |

σ_{31} |
σ_{32} |
σ_{33}-σ_{m} |

with
six independent stress parameters (*σ _{12} = σ_{21}, σ_{13} = σ_{31}, σ_{23} = σ_{32}*) and

*σ _{m} = (σ_{11} + σ_{22} + σ_{33})/3*

In
a triaxial test it is possible to arrange the stress components in a way that
the diagonal stress components *σ _{11}, σ_{22}* and

*σ*are oriented parallel to the applied Cartesian coordinate system. Consequently, these three stress components become the principal stresses

_{33}*σ*and

_{1}, σ_{2}*σ*and the off-diagonal components equal zero. Therefore, in a laboratory situation the deviatoric stress tensor

_{3}*[δσ*can be simplified to:

_{ij}]σ_{1}-σ_{m} |
_{0} |
_{0} |

_{0} |
σ_{2}-σ_{m} |
_{0} |

_{0} |
_{0} |
σ_{3}-σ_{m} |

According
to **Engelder (1994)** deviatoric stress is very often confused with other types of
stresses such as **differential stress** and **effective stress**. **Engelder (1994)** recommends
not to use the term “deviatoric stress” unless for fault slip problems.

# Differential stress

*ΔS* is the difference between the maximum principal stress *S _{1}*
and the minimum principal stress

*S*.

_{3}*ΔS = S_{1} - S_{3}*

is also the diameter of a Mohr circle.

# Effective stress

*σ _{ij}* (with

*i*=

*j*and the main principle stresses

*σ*

_{1}≥ σ_{2}≥ σ_{3}) is the difference between the applied external stress

*S*and the pore pressure

_{ij}*P*. In its simple form (

_{p}**Terzaghi 1923**):

*σ _{ij }= S_{ij }- δ_{ij}P_{p}*

it denies the importance of pore volume and
its compressibility by disregarding the **Biot coefficient** *α*. This equaition is applied generally in soil mechanics where full
efficacy of pore pressure (i.e. *α* =
1) can be assumed.

In case of solid rock things appear more
complicated and pore pressure efficacy never is at 100%. Consequently, the **Biot
coefficient** has to be considered and the exact form of the equation according to
**Nur & Byerlee 1971** is:

*σ _{ij }= S_{ij }- δ_{ij}αP_{p}*.

# Entry pressure

See "**Threshold pressure**"

# News

21.02.2018 10:12

## AAPG ACE 2018 in Salt Lake City (19.-23.05.2018)

Gesteinslabor will present brandnew results from the frontline of research at one of the key events for petroleum geologists.

Read more … AAPG ACE 2018 in Salt Lake City (19.-23.05.2018)

21.02.2018 08:45

## TSK 2018 in in Jena (21.-23.03.2018)

Gesteinslabor will take part at the distinguished conference dealing with tectonics, structural geology and crystalline rocks.

31.01.2018 12:06

## Spring meeting 2018 in Celle

Following a well established tradition, also in 2018 DGMK and ÖGEW will host their annual conference in the Congress Union Celle.