2. Assumptions

Hypothesis 1 (H.P.P.)

The law is written for small disturbances.

Hypothesis 2 (partition of deformations)

In small deformations, the tensor of total deformations is broken down into several terms relating to the processes under consideration. With regard to the description of the various mechanisms of delayed deformation of concrete, it is assumed that the total deformation can be written as:

(2.1)\[\begin{split} \ mathrm {\ epsilon} =\ underset {\ begin {array} {c}\ text {deformation}\\\ text {elastic}\ end {array}} {\ underset {} {{\ mathrm {\ mathrm {\ epsilon}}}} ^ {epsilon}}}} ^ {e}}}} +\ underset {\ begin {array} {c}}\ text {creep}} {\ underset {} {}} {{\ mathrm {\ epsilon}}}} ^ {e}}} ^ {e}}} +\ underset {\ begin {array} {c}}\ text {creep}\\ text {creep}} {\ text {clean}\ end {array}} {\ underset {} {{\ mathrm {\ epsilon}}} ^ {\ mathit {fp}}}}} +\ underset {\ begin {array} {c}\ text {creep of}\\ flow of}\\\ text {desiccation}\\ text {dessiccation}}\ end {\ array}} {\ underset {}} {\ mathrm {\ epsilon}}\ text {creep of}\\ text {desiccation}\ end {array}}} {\ underset {} {\ mathrm {\ epsilon}} ^ {\ mathit}} ^ {\ mathit}} {fdess}}}} +\ underset {\ begin {array} {c} {c}\ text {withdrawal}\\\ text {endogenous}\ end {array}} {\ underset {} {{\ mathrm {\ epsilon}}}} ^ {\ text {epsilon}}}} ^ {\ text {epsilon}}} ^ {\ text {epsilon}}} ^ {\ text {epsilon}}} ^ {\ text {re}}}}} +\ underset {\ begin {array} {c}\ text {withdrawal} {c}\ text {withdrawal} {c}\ text {withdrawal of}\\\ text {dessicut}} location}\ end {array}} {\ underset {} {{\ mathrm {\ epsilon}}} ^ {\ mathit {rd}}}}}} +\ underset {\ begin {array} {c}\ text {deformation}\\ deformation}\\\ text {deformation}\\\ text {thermal}\\ text {thermal}\\ text {thermal}\\ text {thermal}\ end {array}}} ^ {\ mathit {th}}}}}\end{split}\]

In this document, we do not describe how to take into account different types of withdrawals (for this, see the documentation for code_aster [R7.01.12]), so that () is reduced to:

(2.2)\[ \ mathrm {\ epsilon} = {\ mathrm {\ epsilon}}} ^ {e} + {\ mathrm {\ epsilon}} ^ {\ mathit {fp}}} + {\ mathrm {\ epsilon}}} ^ {\ mathit {fdess}} + {\ mathrm {\ epsilon}} + {\ mathrm {\ epsilon}} + {\ mathrm {\ epsilon}}\]

Hypothesis 3 (decomposition of natural creep components)

In general, natural creep can be modelled by combining the elastic behavior of the solid and the viscous behavior of the fluid. For the law presented, proper creep is described as the combination of the elastic behavior of hydrates and aggregates and the viscous behavior of water.

In the case of model BETON_BURGER, it is assumed that natural creep can be decomposed into a process decoupling a spherical part and a deviatory part. The tensor of the total natural creep deformations is then written:

(2.3)\[\begin{split} {\ underline {\ underline {\ mathrm {\ epsilon}}}}}}} ^ {\ mathit {fp}} =\ underset {\ begin {array} {c}\ text {part}\\ text {spherical}\\ text {spherical}\ end {\ text {spherical}}\ end {\ array}}} {\ underset {\ array}}} {\ underset {}} {\ mathrm {\ epsilon}} ^ {\ text {fs}}\ cdot\ underset {\ array}}} {\ underset {}} {\ mathrm {\ epsilon}} ^ {\ text {fs}}\\\ text {fs}}\ cdot\ underset {\ array}} line {\ underline {1}}}}} +\ underset {\ begin {array} {c}\ text {part}\\ text {deviatory}\ end {array}} {\ underset {} {\ underline {\ underline {\ underline {\ underline {\ underline {\ underline {\ underline {\ underline {\ underline {\ underline {\ underline {\ underline {\ underline {\ underline {\ underline {\ underline {\ underline {\ underline {\ underline {\ underline {\ underline {\ underline {\ underline {\ underline {\ underline {\ underline {\ underline {\ underline {\ underline {\ underline {\end{split}\]

The stress tensor can be developed in a similar form:

\[\]

: label: eq-4

underline {underline {mathrm {sigma}}}} =underset {begin {array} {c}text {part}\ text {spherical}end {array}} {underset {}} {underset {}} {{mathrm {sigma}}} {{mathrm {sigma}}} ^ {s}cdotunderline {underline {1}}}}} {underset {}} {underset {}} {{}} {{mathrm {sigma}}} ^ {s}cdotunderline {underline {1}}}}} +underset {begin {array} {c}text {part}\text {deviatory}\ text {deviatory}end {array}} {underline {underline {underline {underline {mathrm {sigma}}}}}}} ^ {d}}}

Model BETON_BURGER assumes a total decoupling between the spherical and deviatory components of natural creep: the deformations induced by spherical stresses are purely spherical and the deformations induced by deviatory stresses are purely deviatory. On the other hand, cumulative viscous deformations have an effect on the viscous properties of the fluid, regardless of its origin (spherical or deviatoric). To take into account the effect of internal humidity, the deformations are multiplied by the internal relative humidity:

\[\]

: label: eq-5

{mathrm {epsilon}} ^ {s} =hcdot fleft ({mathrm {sigma}}} ^ {s}right)

Or \(h\) refers to internal relative humidity.

The condition () makes it possible to check a posteriori that the natural creep deformations are proportional to the relative humidity.