\[\newcommand{\vector}[1]{\underline{#1}} \newcommand{\vectorZero}{\vector{0}} \newcommand{\tensTwo}[1]{\boldsymbol{#1}} \newcommand{\tensTwoZero}{\tensTwo{0}} \newcommand{\tensTwoUnit}{\tensTwo{I}} \newcommand{\tensFour}[1]{\mathbb{#1}} \newcommand{\inverse}[1]{{#1}^{-1}} \newcommand{\transpose}[1]{{#1}^{T}} \newcommand{\inverseTranspose}[1]{ {#1}^{-T}} \newcommand{\vectorCmpCO}[2]{#1_{\left(#2\right)}} \newcommand{\vectorCmpCT}[2]{#1^{\left(#2\right)}} \newcommand{\tensTwoCmpCO}[3]{#1_{\left(#2 #3\right)}} \newcommand{\tensTwoCmpCT}[3]{#1^{\left(#2 #3\right)}} \newcommand{\tensTwoInva}[2]{I^{#1}_{#2}} \newcommand{\tensTwoDevia}[1]{\boldsymbol{\tilde{#1}}} \newcommand{\tensFourCmpCO}[5]{#1_{\left(#2 #3 #4 #5\right)}} \newcommand{\tensFourCmpCT}[5]{#1^{\left(#2 #3 #4 #5\right)}} \newcommand{\domain}{\Omega} \newcommand{\domainRefe}{\domain^{0}} \newcommand{\domainCurr}{\domain^{t}} \newcommand{\bound}{\partial\domain} \newcommand{\boundRefe}{\bound^{0}} \newcommand{\boundCurr}{\bound^{t}} \newcommand{\boundN}{\bound_{\textrm{N}}} \newcommand{\boundNRefe}{\boundN^{0}} \newcommand{\boundNCurr}{\boundN^{t}} \newcommand{\boundD}{\bound_{\textrm{D}}} \newcommand{\boundDRefe}{\boundD^{0}} \newcommand{\boundDCurr}{\boundD^{t}} \newcommand{\normal}{\vector{n}} \newcommand{\normalRefe}{\normal^{0}} \newcommand{\normalCurr}{\normal^{t}} \newcommand{\posi}{\vector{x}} \newcommand{\posiRefe}{\posi^{0}} \newcommand{\posiCurr}{\posi^{t}} \newcommand{\disp}{\vector{u}} \newcommand{\dispVirt}{\delta\vector{v}} \newcommand{\funcTransfor}{\vector{\varphi}^{t}} \newcommand{\gradTransfor}{\tensTwo{F}} \newcommand{\jacobTransfor}{J} \newcommand{\posiIncr}{\vector{dx}} \newcommand{\posiIncrRefe}{\posiIncr^{0}} \newcommand{\posiIncrCurr}{\posiIncr^{t}} \newcommand{\strainCmp}{\varepsilon} \newcommand{\strain}{\tensTwo{\strainCmp}} \newcommand{\EGLCmp}{E} \newcommand{\ECGDroiteCmp}{C} \newcommand{\ECGGaucheCmp}{B} \newcommand{\ELOGCmp}{{E}_{ln}} \newcommand{\EGL}{\tensTwo{\EGLCmp}} \newcommand{\ECGDroite}{\tensTwo{\ECGDroiteCmp}} \newcommand{\ECGGauche}{\tensTwo{\ECGGaucheCmp}} \newcommand{\ELOG}{\tensTwo{\ELOGCmp}} \newcommand{\EDilCmp}{C^{\star}} \newcommand{\EDil}{\tensTwo{\EDilCmp}} \newcommand{\divTensTwo}[1]{\vector{\nabla} {\cdot} #1 } \newcommand{\divVector}[1]{\vector{\nabla} {\cdot} #1 } \newcommand{\gradScal}[1]{\vector{\nabla} \times #1 } \newcommand{\gradVector}[1]{\vector{\nabla} \times #1 } \DeclareMathOperator{\rand}{rand} \DeclareMathOperator{\round}{round} \DeclareMathOperator{\trace}{trace} \newcommand{\metric} {\tensTwo{g}} \newcommand{\vectorBaseCV}[1]{\vector{g_{#1}}} \newcommand{\vectorBaseCT}[1]{\vector{g^{#1}}} \newcommand{\metricRefe} {\metric^{t}} \newcommand{\metricCurr} {\metric^{0}} \newcommand{\discVect}[1]{\lbrace #1 \rbrace} \newcommand{\discVectLigne}[1]{\langle #1 \rangle} \newcommand{\discVectZero}{\discVect{0}} \newcommand{\discVectCmp}[2]{\lbrace #1 \rbrace_{(#2)}} \newcommand{\discVectLigneCmp}[2]{\langle #1 \rangle_{(#2)}} \newcommand{\discMatr}[1]{\left[ #1 \right]} \newcommand{\discMatrZero}{\discMatr{0}} \newcommand{\discMatrCmp}[3]{\left[ #1 \right]_{(#2#3)}} \newcommand{\onQuadPoint}[2]{{#1}_{#2}} \newcommand{\onNode}[2]{{#1}^{#2}} \newcommand{\quadOrder}{k_{Q}} \newcommand{\nbQuadPoint}{n_Q} \newcommand{\quadWeight}[1]{\omega_{#1}} \newcommand{\quadPointIndex}{i_{\textrm{pg}}} \newcommand{\nodeIndex}{i_{\textrm{no}}} \newcommand{\pres}{p} \newcommand{\presRefe}{\pres^{0}} \newcommand{\presCurr}{\pres^{t}} \newcommand{\stressCmp}{\sigma} \newcommand{\stress}{\tensTwo{\stressCmp}} \newcommand{\stressPKTwoCmp}{S} \newcommand{\stressPKTwo}{\tensTwo{\stressPKTwoCmp}} \newcommand{\work}[1]{W^{#1}} \newcommand{\enerInterne}{\Psi_i} \newcommand{\yieldStress}{\sigmaCmp_{Y}} \newcommand{\youngModulus}{E} \newcommand{\youngModulusCplx}{E^{\star}} \newcommand{\poissonCoef}{\nu} \newcommand{\poissonCoefCplx}{\nu^{\star}} \newcommand{\shearModulus}{G} \newcommand{\bulkModulus}{K} \newcommand{\modulusTangent}{\tensFour{K}} \newcommand{\anglDila}{\psi} \newcommand{\anglFric}{\varphi} \newcommand{\cohesion}{c} \newcommand{\presCapi}{\pres_{c}} \newcommand{\presEau}{\pres_{w}} \newcommand{\presAir}{\pres_{a}} \newcommand{\shapeFunc}{\Phi} \newcommand{\shapeDFunc}{B} \newcommand{\measLine}{\, \mathrm{d} l} \newcommand{\measDomain}{\, \mathrm{d} \domain} \newcommand{\measBound}{\, \mathrm{d} \Gamma} \newcommand{\normEucl}[1]{\Vert{#1} \Vert } \newcommand{\derivee}[2]{#1_{, #2}} \newcommand{\soundSpeed}{c} \newcommand{\soundSpeedComp}{c_{P}} \newcommand{\soundSpeedCisa}{c_{S}} \newcommand{\posiTang}{\posi_{\tau}} \newcommand{\posiNorm}{{x}_{n}} \newcommand{\dispTang}{\disp_{\tau}} \newcommand{\dispNorm}{{u}_{n}} \newcommand{\basisVector}[1]{\vector{e}_{#1}} \newcommand{\sigmVector}{\vector{t}} \newcommand{\sigmVectorTang}{\sigmVector_{\tau}} \newcommand{\sigmVectorNorm}{t_{n}} \newcommand{\lameLambda}{\lambda} \newcommand{\lameMu}{\mu}\]

2. Continued problem writing

2.1. Kinematic elements in large transformations

We consider a solid \(\domain\) subject to great deformations (see figure). Let \(\gradTransfor\) be the gradient tensor of the transformation, \(\phi\) making the initial configuration pass \(\domainRefe\) to the current deformed configuration \(\domainCurr\). Note \(\posiRefe\) the position of a point in \(\domainRefe\) and \(\posiCurr\) the position of this same point after deformation in \(\domainCurr\). \(\disp\) is then the displacement between the two configurations. So we have:

(2.1)\[ \ PosiCurr =\ PosiRefe +\ disp\]

The transformation gradient tensor is written as:

(2.2)\[ \ GradTransfor = \ frac {\ partial\ PosiCurr} {\ partial\ PosiRefe} = \ TensTwoUnit +\ gradVector {\ disp}\]

2.2. Virtual work of external pressure forces

We consider a pressure \(\presRefe\) normal to the surface in the reference configuration. This pressure is written \(\presCurr\) in the current configuration.

In the current configuration, the virtual work of external pressure efforts \(\work{\pres}\) is simply written (see r3_03_04_Configurations):

(2.3)\[ \ work {\ pres} (\ disp) = {\ int} _ {\ bound NCurr (\ disp)} {-\ pres\,\normalCurr\ cdot\ dispVirt\ measBound}\]

In addition, it is now assumed that the pressure value does not depend explicitly of the movement but only of the material point of application:

\[\]

: label: eq-4

PreCurr (PosiCurr) = PreCurr (FuncTransfor (PosiRefe)) = PresRefe (PosiRefe)

The follower side of the force comes from the dependence of the*normal* on displacement. In this case, it is then possible to express the virtual work of pressure efforts. in the reference configuration (change of variable in the integral):

\[\]

: label: eq-5

work {pres} (disp) = {int} _ {bound NRefe (disp)} {-preRefe,detgradTransfor,left [inverseTranspose {gradTransfor}cdotnormalReferight]cdotdispVirtleft (FuncTransforleft (FuncTransforleft (PosiReferight)right)measBound}

In practice, the formula (2.3) will be used to calculate the work of the pressure forces. However, the formula eq-5 is best suited to a derivation with respect to the displacement, which we will see the need for in the next paragraph.

2.3. Variation of the virtual work of external pressure forces

In order to solve the problem of balance of the structure by a Newton method, we are led to express the variation in the virtual work of external pressure forces by relationship to displacement, similar to what was done for the virtual work of domestic efforts in [R5.03.01]. Since the integration domain is fixed in expression eq-5, the derivation under the sign sum is lawful (cf. [Bib2] _):

(2.4)\[ \ frac {\ partial\ work {\ pres}\ left (\ disp\ right)} {\ partial\ disp} = {\ int} _ {\ bound NRefe (\ disp)} {-\ presRefe\}} {-\ presRefe\,\ frac {\ partial\ disp}\ left [\ det\ gradTransfor\,\ inverseTranspose {\ gradTransfor}\ right]\ cdot\ gradTransfor}\ right]\ cdot\ delta\ disp\ cdot\normalRefe\ cdot\ dispVirt\ measBound}\]

We decided to choose the current configuration as the reference configuration, for which \(\gradTransfor=\tensTwoUnit\). This choice leads to a simple expression of the derivative of the term in square brackets:

(2.5)\[\begin{split} \ frac {\ partial} {\ partial\ disp}\ left [\ det\ gradTransfor\,\ inverseTranspose {\ gradTransfor}\ right]\ cdot\ delta\ disp = \ divVector {\ delta\ disp}\\,\ tenStwounit -\ gradVector {\ delta\ disp}\end{split}\]

Finally, the variation in the virtual work of external pressure forces is written in the current configuration:

(2.6)\[ \ frac {\ partial\ work {\ pres} (\ pres} (\ disp)} {\ partial\ disp}\ cdot\ delta\ disp\ cdot\ dispVirt = {\ int} _ {\ bound NCurr (\ disp)} {-\ preCurr\,\ left [\ divVector {\ delta\ disp}\,\ tenStwounit -\ gradVector {\ delta\ disp}\ right]\ cdot\ delta\ disp}\ right]\ cdot\normalCurr\ cdot\ dispVirt\ measBound}\]

In the expression (2.6) there is still a difficulty. In fact, it is expected to obtain a quantity that is essentially on the surface while the integrand reveals terms of normal derivation on the surface. In other words, it is necessary to know the expression of virtual displacements not only on the surface of the domain but also inside this one (in a neighborhood near the surface to be able to express normal derivatives). This disadvantage is not trivial since in order to calculate the elementary terms due to surface forces, skin elements are used for which normal variation does not make sense.

2.4. Adoption of a curvilinear surface parameterization

To remedy the problem mentioned above, you should seek to express the (2.5) relationship. using area quantities only. For this, differential geometry elements are used, [Bia1] _, whose notations are adopted (in particular, we adopt the convention of summation of repeated indices where the Greek indices take the values \(1\) and \(2\) while the Latin indices take the values \(1\) to \(3\)).

: name: R3_03_04_Curvy Line Parameter

Curvilinear parameterization of the vicinity of the surface subjected to pressure

Let \(({\theta }^{1},{\theta }^{2})\) be an acceptable configuration of the surface. To describe the volume consisting of a neighborhood of this surface, a third variable, \({\theta }^{3}\), is added to it, which measures the progression according to the unit normal \(n\) in \(({\theta }^{1},{\theta }^{2})\). So we have (see r3_03_04_ParametrageCurviligne):

(2.7)\[ \ vector {OM}\ left ({\ theta} ^ {1}, {\ theta} ^ {2}, {\ theta} ^ {3}\ right) = \ vector {OS}\ left ({\ theta} ^ {1}, {\ theta} ^ {2}) + {\ theta} ^ {3}\,\normalCurr ({\ theta} ^ {1}, {\ theta} ^ {1}, {\ theta} ^ {2}\ right)\]

With this choice of configuration, the natural covariant base \((\vectorBaseCV{1},\vectorBaseCV{2},\vectorBaseCV{3})\) is written:

(2.8)\[ \ vectorBaseCV {i} = \ frac {\ partial\ vector {OM}} {\ partial {\ theta} ^ {i}}\]

While the metric tensor \(\metric\) is equal to:

(2.9)\[\begin{split} \ metric = \ VectorBaseCV {i}\ times\ VectorBaseCV {j} = \ left [\ begin {array} {ccc} \ tenStwocmpco {\ metric} {1} {1} {1} &\ tenstWocmpco {\ metric} {1} {2} & 0\\ \ TensTwoCmpco {\ metric} {2} {2} {1} &\ TenStwocmpco {\ metric} {2} {2} & 0\\ 0& 0& 1\ end {array}\ right]\end{split}\]

In this curvilinear setting, the expression for integrand (2.6) is:

(2.10)\[ {-\ pres\,\ TensTwoCmpco {\ metric} {\ metric} {i} {j}\,\ VectorCmpct {\normal} {i}\, \ left [ \ derivative {\ vectorCmpct {\ delta\ disp} {k}} {k}\,\ vectorCmpct {\ dispVirt} {j} - \ derivative {\ vectorCmpct {\ delta\ disp} {j}} {j}} {k}\,\ vectorCmpct {\ dispVirt} {k} \ right]}\]

This term is simplified considerably. Indeed, we can already note that when \(j=k\), the term between brackets is zero. Furthermore, in the adopted curvilinear system, the contravariant components of \(\normal\) are:

(2.11)\[ \ VectorCMPCT {\normal} {1} =0,\ quad\ VectorCMPCT {\normal} {2} =0\ quad\ VectorCMPCT {\normal} {\normal} {3} =1\]

Finally, taking into account the particular form of \(\metric\) (i.e. \(\tensTwoCmpCO{\metric}{1}{3}=0\), \(\tensTwoCmpCO{\metric}{2}{3}=0\), and \(\tensTwoCmpCO{\metric}{3}{3}=1\)), the variation of work (2.6) is simply written:

(2.12)\[ \ frac {\ partial\ work {\ pres} (\ pres} (\ disp)} {\ partial\ disp}\ cdot\ delta\ disp\ cdot\ dispVirt = {\ int} _ {\ bound NCurr (\ disp)} { -\ presCurr\, \ left [ \ derivative {\ vectorCmpct {\ delta\ disp} {\ delta\ disp} {\ alpha}} {\ alpha}\,\ vectorCMPCT {\ dispVirt} {3} - \ derivative {\ vectorCmpct {\ delta\ disp} {3}} {\ alpha}\,\ vectorCmpct {\ dispVirt} {\ dispVirt} {\ alpha} \ right] \ measBound}\]

On this expression, it can be seen that only surface differential operators are involved (covariant derivation with respect to \({\theta}^{1}\) and \({\theta }^{2}\) only), which is indeed the aim sought. By introducing the contravariant base \((\vectorBaseCT{1},\vectorBaseCT{2},\vectorBaseCT{3}=\normal)\), also called dual base and which is expressed from the covariant base by:

(2.13)\[ vectorBasect {i} = {\ left [\ metric^ {ij}\ right]} ^ {-1}\, vectorBaseCV {j}\]

Curvilinear components can be dispensed with:

(2.14)\[ \ frac {\ partial\ work {\ pres} (\ pres} (\ disp)} {\ partial\ disp}\ cdot\ delta\ disp\ cdot\ dispVirt = {\ int} _ {\ bound NCurr (\ disp)} { -\ presCurr\, \ left [ \ left (\ frac {\ partial\ delta\ disp} {\ partial {\ theta} ^ {\ alpha}}\ cdot\ VectorBasect {\ alpha}\ disp} {\ right) \, \ left (\ DispVirt\ cdot\normal\ right) - \ left (\ frac {\ partial\ delta\ disp} {\ partial {\ theta} ^ {\ alpha}}\ cdot\normal\ disp}} {\ partial\ delta\ disp} {\ partial\ disp} {\ partial {\ theta} ^ {\ alpha}}\ cdot\normal\ \, \ left (\ dispVirt\ cdot\ VectorBasect {\ alpha}\ right) \ right]\ measBound}\]

From now on, the expression (2.14) will be used to calculate the variation in the virtual work of pressure forces.

2.5. Special case of a structure subjected to constant internal or external pressure

In the particular case of a constant pressure in a cavity (see r3_03_04_Pression), it is shown that the pressure forces derive from a potential \(\Xi\) which is nothing other than the product of the pressure by the volume of the cavity. This result extends to the case of a structure immersed in a fluid at constant pressure.

We write this potential:

(2.15)\[ \ Xi = \ pres\, {\ int} _ {\ bound NCurr (\ disp)} {\ measDomain} = \ pres\, {\ int} _ {\ bound NRefe (\ disp)} {\ det\ gradTransfor\ measDomain}\]

Again, the current configuration is chosen as the reference configuration. The variation of \(\Xi\) then leads to the virtual work of external pressure efforts:

(2.16)\[ \ frac {\ partial\ Xi} {\ partial\ disp}\ cdot\ dispVirt = \ pres\, {\ int} _ {\ domainCurr (\ disp)} {\ divVector (\ dispVirt)\,\ measDomain} = - {\ int} _ {\ bound NCurr (\ disp)} {\ pres\,\ dispVirt\ cdot\normal\ measBound} = \ work {\ pres}\,\ dispVirt\]

In this specific case, the variation in virtual work is also the second variation in potential \(\Xi\), that is to say a symmetric bilinear form:

(2.17)\[ \ frac {\ partial {W} ^ {\ pres} (\ presp)} {\ partial\ disp)} {\ partial\ disp}\ cdot\ dispVirt=\ frac {{\ partial} ^ {2} ^ {2}\ Xi (\ disp)} {\ partial\ disp)} {\ partial\ disp}} {\ partial\ disp} ^ {2}}\ cdot\ delta\ dispVirt\]