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## 6.1FEM problem formulations

This section gives a short theoretical reminder of supported FEM problems. The selection of the formulation for each element group is done through the material and element properties.

### 6.1.1  3D elasticity

Elements with a p_solid property entry with a non-zero integration rule are described under p_solid. They correspond exactly to the *b elements, which are now obsolete. These elements support 3D mechanics (DOFs .01 to .03 at each node) with full anisotropy, geometric non-linearity, integration rule selection, ... The elements have standard limitations. In particular they do not (yet)

• have any correction for shear locking found for high aspect ratios
• have any correction for dilatation locking found for nearly incompressible materials

With m_elastic subtypes 1 and 3, p_solid deals with 3D mechanics with strain defined by

{
 єx єy єz γyz γzx γxy
} [
 N,x 0 0 0 N,y 0 0 0 N,z 0 N,z N,y N,z 0 N,x N,y N,x 0
] {
 u v w
}     (6.1)

where the engineering notation γyz=2єyz, ... is used. Stress by

{
 σx σy σz σyz σzx σxy
} =[
 d1,1 N,x+d1,5 N,z+d1,6 N,y d1,2 N,y+d1,4 N,z+d1,6 N,x d1,3 N,z+d1,4 N,y+d1,5 N,x d2,1 N,x+d2,5 N,z+d2,6 N,y d2,2 N,y+d2,4 N,z+d2,6 N,x d2,3 N,z+d2,4 N,y+d2,5 N,x d3,1 N,x+d3,5 N,z+d3,6 N,y d3,2 N,y+d3,4 N,z+d3,6 N,x d3,3 N,z+d3,4 N,y+d3,5 N,x d4,1 N,x+d4,5 N,z+d4,6 N,y d4,2 N,y+d4,4 N,z+d4,6 N,x d4,3 N,z+d4,4 N,y+d4,5 N,x d5,1 N,x+d5,5 N,z+d5,6 N,y d5,2 N,y+d5,4 N,z+d5,6 N,x d5,3 N,z+d5,4 N,y+d5,5 N,x d6,1 N,x+d6,5 N,z+d6,6 N,y d6,2 N,y+d6,4 N,z+d6,6 N,x d6,3 N,z+d6,4 N,y+d6,5 N,x
] {
 u v w
}

Note that the strain states are {єx єy єz γyz γzx γxy} which may not be the convention of other software. In particular volume elements inherited from MODULEF order shear stresses differently σxy, σyz, σzx (these elements are obtained by setting p_solid integ value to zero. In fe_stress the stress reordering can be accounted for by the definition of the proper TensorTopology matrix.

For isotropic materials

D=[

 E(1−ν) (1+ν)(1−2ν)
[
1
 ν 1−ν
 ν 1−ν

 ν 1−ν
1
 ν 1−ν

 ν 1−ν
 ν 1−ν
]
0
[
 G 0 0 0 G 0 0 0 G
]
]     (6.2)

with at nominal G=E/(2(1+ν)).

For orthotropic materials, the compliance is given by

{є} = [D]−1{σ}=[
1/E1
 ν21 E2
 ν31 E3
00
 ν12 E1
1/E2
 ν32 E3
00
 ν13 E1
 ν23 E2
1/E300
000
 1 G23
00
0000
 1 G31

00000
 1 G12

]
{
 σx σy σz σyz σzx σxy
}
(6.3)

For constitutive law building, see p_solid.

### 6.1.2  2D elasticity

With m_elastic subtype 4, p_solid deals with 2D mechanical volumes with strain defined by (see q4p constants)

{
 єx єy γxy
} =[
 N,x 0 0 N,y N,y N,x
] {
 u v
}     (6.4)

and stress by

{
 σ єx σ єy σ γxy
} =[
 d1,1 N,x+d1,3 N,y d1,2 N,y+d1,3 N,x d2,1 N,x+d2,3 N,y d2,2 N,y+d2,3 N,x d3,1 N,x+d3,3 N,y d3,2 N,y+d3,3 N,x
] {
 u v
}     (6.5)

For isotropic plane stress (p_solid form=1), one has

D=
 E 1−ν2
[
1ν0
ν10
00
 1−ν 2

]     (6.6)

For isotropic plane strain (p_solid form=0), one has

D=
 E(1−ν (1+ν)(1−2ν)
[
1
 ν 1−ν
0
 ν 1−ν
10
00
 1−2ν 2(1−ν)

]     (6.7)

### 6.1.3  Acoustics

With m_elastic subtype 2, p_solid deals with 2D and 3D acoustics (see flui4 constants) where 3D strain is given by

{
 p,x p,y p,z
} =[
 N,x N,y N,z
] {
 p
}     (6.8)

This replaces the earlier flui4 ... elements.

### 6.1.4  Classical lamination theory

Both isotropic and orthotropic materials are considered. In these cases, the general form of the 3D elastic material law is

 σ11 σ22 σ33 τ23 τ13 τ12

 C11 C12 C13 0 0 0 C22 C23 0 0 0 C33 0 0 0 C44 0 0 (s) C55 0 C66

 є11 є22 є33 γ23 γ13 γ12

(6.9)

Plate formulation consists in assuming one dimension, the thickness along x3, negligible compared with the surface dimensions. Thus, vertical stress σ33=0 on the bottom and upper faces, and assumed to be neglected throughout the thickness,

σ33=0 ⇒ є33=−
 1 C33

C13є11+C23є22
,
(6.10)

and for isotropic material,

σ33=0 ⇒ є33=−
 ν 1−ν

є1122
.
(6.11)

By eliminating σ33, the plate constitutive law is written, with engineering notations,

 σ11 σ22 σ12 σ23 σ13

=

 Q11 Q12 0 0 0 Q12 Q22 0 0 0 0 0 Q66 0 0 0 0 0 Q44 0 0 0 0 0 Q55

 є11 є22 γ12 γ23 γ13

.
(6.12)

The reduced stiffness coefficients Qij (i,j = 1,2,4,5,6) are related to the 3D stiffness coefficients Cij by

Qij=

Cij−
 Ci3Cj3 C33
if i,j=1,2,
Cij if i,j=4,5,6.

(6.13)

The reduced elastic law for an isotropic plate becomes,

 σ11 σ22 τ12

 E (1−ν2)

1ν
ν1
00
 1−ν 2

 є11 є22 γ12

,
(6.14)

and

 τ23 τ13

 E 2(1+ν)

 1 0 0 1

 γ23 γ13

.
(6.15)

Under Reissner-Mindlin's kinematic assumption the linearized strain tensor is

є=

u1,1+x3β1,1
 1 2
(u1,2+u2,1+x31,22,1))
 1 2
1+w,1)
u2,2+x3β2,2
 1 2
2+w,2)
(s)

.
(6.16)

So, the strain vector is written,

є

=

 є11m+x3κ11 є22m+x3κ22 γ12m+x3κ12 γ23 γ13

,
(6.17)

with єm the membrane, κ the curvature or bending, and γ the shear strains,

єm=

 u1,1 u2,2 u1,2+u2,1

,  κ=

 β1,1 β2,2 β1,2+β2,1

,  γ=

 β2+w,2 β1+w,1

,
(6.18)

Note that the engineering notation with γ12=u1,2+u2,1 is used here rather than the tensor notation with є12=(u1,2+u2,1)/2 . Similarly κ121,22,1, where a factor 1/2 would be needed for the tensor.

The plate formulation links the stress resultants, membrane forces Nαβ, bending moments Mαβ and shear forces Qα3, to the strains, membrane єm, bending κ and shearing γ,

 N M Q

=

 A B 0 B D 0 0 0 F

 єm κ γ

.
(6.19)

The stress resultants are obtained by integrating the stresses through the thickness of the plate,

Nα β=
 ht hb
σα β dx3,   Mα β=
 ht hb
x3 σα β dx3,   Qα 3=
 ht hb
σα 3 dx3,
(6.20)

with α, β = 1, 2.

Therefore, the matrix extensional stiffness matrix [A], extension/bending coupling matrix [B], and the bending stiffness matrix [D] are calculated by integration over the thickness interval [hb ht]

Aij=
 ht hb
Qij dx3,
Bij=
 ht hb
x3 Qij dx3,

Dij=
 ht hb
x32 Qij dx3,
Fij=
 ht hb
Qij dx3
(6.21)

An improvement of Mindlin's plate theory with tranverse shear consists in modifying the shear coefficients Fij by

 Hij=kijFij, (6.22)

where kij are correction factors. Reddy's 3rd order theory brings to kij=2/3. Very commonly, enriched 3rd order theory are used, and kij are equal to 5/6 and give good results. For more details on the assessment of the correction factor, see [31].
For an isotropic symetric plate (hb=−ht=h/2), the in-plane normal forces N11, N22 and shear force N12 become

 N11 N22 N12

 Eh 1−ν2

1ν0
10
(s)
 1−ν 2

 u1,1 u2,2 u1,2+u2,1

,
(6.23)

the 2 bending moments M11, M22 and twisting moment M12

 M11 M22 M12

 Eh3 12(1−ν2)

1ν0
10
(s)
 1−ν 2

 β1,1 β2,2 β1,2+β2,1

,
(6.24)

and the out-of-plane shearing forces Q23 and Q13,

 Q23 Q13

 Eh 2(1+ν)

 1 0 0 1

 β2+w,2 β1+w,1

.
(6.25)

One can notice that because the symmetry of plate, that means the reference plane is the mid-plane of the plate (x3(0)=0) the extension/bending coupling matrix [B] is equal to zero.

Using expression (6.21) for a constant Qij, one sees that for a non-zero offset, one has

 Aij=h[Qij]    Bij=x3(0)h [Qij]     Cij= (x3(0)2h+h3/12) [Qij]     Fij=h[Qij] (6.26)

where is clearly appears that the constitutive matrix is a polynomial function of h, h3, x3(0)2h and x3(0)h. If the ply thickness is kept constant, the constitutive law is a polynomial function of 1,x3(0),x3(0)2.

### 6.1.5  Piezo-electric volumes

The strain state associated with piezoelectric materials is described by the six classical mechanical strain components and the electrical field components. Following the IEEE standards on piezoelectricity and using matrix notations, S denotes the strain vector and E denotes the electric field vector (V/m) :

{
 S E
} = {
 єx єy єz γyz γzx γxy Ex Ey Ez
} =[
 N,x 0 0 0 0 N,y 0 0 0 0 N,z 0 0 N,z N,y 0 N,z 0 N,x 0 N,y N,x 0 0 0 0 0 −N,x 0 0 0 −N,y 0 0 0 −N,z
] {
 u v w φ
}     (6.27)

where φ is the electric potential (V).

The constitutive law associated with this strain state is given by

{
 T D
} = [
 CE −eT e єS
]{
 S E
}     (6.28)

in which D is the electrical displacement vector (a density of charge in Cb/m2), T is the mechanical stress vector (N/m2). CE is the matrix of elastic constants at zero electric field (E=0, short-circuited condition, see section 6.1.1 for formulas (there CE is noted D). Note that using −E rather than E makes the constitutive law symmetric.

Alternatively, one can use the constitutive equations written in the following manner :

{
 S D
} = [
 sE dT d єT
]{
 T E
}     (6.29)

In which sE is the matrix of mechanical compliances, [d] is the matrix of piezoeletric constants (m/V=Cb/N):

[d] = [
 d11 d12 d13 d14 d15 d16 d21 d22 d23 d24 d25 d26 d31 d32 d33 d34 d35 d36
]     (6.30)

Matrices [e] and [d] are related through

 [e] = [d] [ CE]     (6.31)

Due to crystal symmetries, [d] may have only a few non-zero elements.

Matrix [єS] is the matrix of dielectric constants (permittivities) under zero strain (constant volume) given by

S] = [
 є11S є12S є13S є21S є22S є23S є31S є32S є33S
]     (6.32)

It is more usual to find the value of єT (Permittivity at zero stress) in the datasheet. These two values are related through the following relationship :

 [єS]= [єT] − [d] [e]T     (6.33)

For this reason, the input value for the computation should be [єT].
Also notice that usually relative permittivities are given in datasheets:

єr =
 є є0
(6.34)

є0 is the permittivity of vacuum (=8.854e-12 F/m)

The most widely used piezoelectric materials are PVDF and PZT. For both of these, matrix [єT] takes the form

T] = [
 є11T 0 0 0 є22T 0 0 0 є33T
]     (6.35)

For PVDF, the matrix of piezoelectric constants is given by

[d] = [
 0 0 0 0 0 0 0 0 0 0 0 0 d31 d32 d33 0 0 0
]     (6.36)

and for PZT materials :

[d] = [
 0 0 0 0 d15 0 0 0 0 d24 0 0 d31 d32 d33 0 0 0
]     (6.37)

### 6.1.6  Geometric non-linearity

The following gives the theory of large transformation problem implemented in OpenFEM function of_mk_pre.c Mecha3DInteg.
The principle of virtual work in non-linear total Lagrangian formulation for an hyperelastic medium is

 Ω0
(ρ0 u″, δ v) +
 Ω0
S : δ e =
 Ω0
f . δ v   ∀ δ v     (6.38)

with p the vector of initial position, x = p +u the current position, and u the displacement vector. The transformation is characterized by

 Fi,j = I + ui,j = δij+{N,j}T{qi}     (6.39)

where the N,j is the derivative of the shape functions with respect to cartesian coordinates at the current integration point and qi corresponds to field i (here translations) and element nodes. The notation is thus really valid within a single element and corresponds to the actual implementation of the element family in elem0 and of_mk. Note that in these functions, a reindexing vector is used to go from engineering ({e11 e22 e33 2e23 2e31 2e12}) to tensor [eij] notations ind_ts_eg=[1 6 5;6 2 4;5 4 3];e_tensor=e_engineering(ind_ts_eg);. One can also simplify a number of computations using the fact that the contraction of a symmetric and non symmetric tensor is equal to the contraction of the symmetric tensor by the symmetric part of the non symmetric tensor.

One defines the Green-Lagrange strain tensor e=1/2(FTFI) and its variation

 deij = ⎛ ⎝ FT dF ⎞ ⎠ Sym = ⎛ ⎝ Fki {N,j}T{qk} ⎞ ⎠ Sym     (6.40)

Thus the virtual work of internal loads (which corresponds to the residual in non-linear iterations) is given by

 Ω
S : δ e =
 Ω
{δ qk}T{N,jFki Sij     (6.41)

and the tangent stiffness matrix (its derivative with respect to the current position) can be written as

KG=
 Ω
Sij δ uk,i ul,j +
 Ω
de :
 ∂2 W ∂ e2
: δ e     (6.42)

which using the notation ui,j = {N,j}T{qi} leads to

KGe=
 Ω
{δ qm} {N,l

Fmk
 ∂2 W ∂ e2
ijkl Fni + Slj

{N,j} {dqn}     (6.43)

The term associated with stress at the current point is generally called geometric stiffness or pre-stress contribution.

In isotropic elasticity, the 2nd tensor of Piola-Kirchhoff stress is given by

S = D:e(u) =
 ∂2 W ∂ e2
:e(u) =  λ Tr(eI + 2µ e      (6.44)

the building of the constitutive law matrix D is performed in p_solid BuildConstit for isotropic, orthotropic and full anisotropic materials. of_mk_pre.c nonlin_elas then implements element level computations. For hyperelastic materials ∂2 W/∂ e2 is not constant and is computed at each integration point as implemented in hyper.c.

For a geometric non-linear static computation, a Newton solver will thus iterate with

[K(qn)]{qn+1qn} =   R(qn) =
 Ω
f . dv −
 Ω0
S(qn) : δ e      (6.45)

where external forces f are assumed to be non following.

### 6.1.7  Thermal pre-stress

The following gives the theory of the thermoelastic problem implemented in OpenFEM function of_mk_pre.c nonlin_elas.
In presence of a temperature difference, the thermal strain is given by [eT] = [α] (TT0), where in general the thermal expansion matrix α is proportional to identity (isotropic expansion). The stress is found by computing the contribution of the mechanical deformation

 S = C:(e − eT) =  λ Tr(e) I + 2µ e − (C:[α])(T−T0)      (6.46)

This expression of the stress is then used in the equilibrium (6.38), the tangent matrix computation(6.42), or the Newton iteration (6.45). Note that the fixed contribution ∫Ω0 (−C:eT) : δ e can be considered as an internal load of thermal origin.

The modes of the heated structure can be computed with the tangent matrix.

An example of static thermal computation is given in ofdemos ThermalCube.

### 6.1.8  Hyperelasticity

The following gives the theory of the thermoelastic problem implemented in OpenFEM function hyper.c (called by of_mk.c MatrixIntegration).
For hyperelastic media S=∂ W/∂ e with W the hyperelastic energy. hyper.c currently supports Mooney-Rivlin materials for which the energy takes one of following forms

 W = C1(J1−3) + C2(J2−3) + K(J3−1)2,     (6.47)
 W = C1(J1−3) + C2(J2−3) + K(J3−1) − (C1 + 2C2 + K)ln(J3),     (6.48)

where (J1,J2,J3) are the so-called reduced invariants of the Cauchy-Green tensor

 C=I+2e,     (6.49)

linked to the classical invariants (I1,I2,I3) by

J1=I1 I
 1 3

3
,   J2=I2 I
 2 3

3
,    J3=I
 1 2

3
,     (6.50)

where one recalls that

I1=tr C,    I2=
 1 2
[(tr C)2tr C2],    I3=det C.     (6.51)

Note : this definition of energy based on reduced invariants is used to have the hydrostatic pressure given directly by p=−K(J3−1) (K “bulk modulus”), and the third term of W is a penalty on incompressibility.

Hence, computing the corresponding tangent stiffness and residual operators will require the derivatives of the above invariants with respect to e (or C). In an orthonormal basis the first-order derivatives are given by:

 ∂ I1 ∂ Cij
= δij,
 ∂ I2 ∂ Cij
= I1δijCij,
 ∂ I3 ∂ Cij
= I3 Cij−1,     (6.52)

where (Cij−1) denotes the coefficients of the inverse matrix of (Cij). For second-order derivatives we have:

 ∂2 I1 ∂ Cij∂ Ckl
= 0,
 ∂2 I2 ∂ Cij∂ Ckl
= −δikδjlijδkl,
 ∂2 I3 ∂ Cij∂ Ckl
= Cmn єikmєjln,     (6.53)

where the єijk coefficients are defined by

 єijk =0 when 2 indices coincide =1 when (i,j,k) even permutation of (1,2,3) =−1 when (i,j,k) odd permutation of (1,2,3)
(6.54)

Note: when the strain components are seen as a column vector (“engineering strains”) in the form (e11,e22,e33,2e23,2e31,2e12)′, the last two terms of (6.53) thus correspond to the following 2 matrices

 0 1 1 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 0 0 0 −1/2 0 0 0 0 0 0 −1/2 0 0 0 0 0 0 −1/2

,     (6.55)

 0 C33 C22 −C23 0 0 C33 0 C11 0 −C13 0 C22 C11 0 0 0 −C12 −C23 0 0 −C11/2 C12/2 C13/2 0 −C13 0 C12/2 −C22/2 C23/2 0 0 −C12 C13/2 C23/2 −C33/2

.     (6.56)

We finally use chain-rule differentiation to compute

S =
 ∂ W ∂ e
=
 ∑ k

 ∂ W ∂ Ik

 ∂ Ik ∂ e
,     (6.57)
 ∂2 W ∂ e2
=
 ∑ k

 ∂ W ∂ Ik

 ∂2 Ik ∂ e2
 ∑ k
 ∑ l

 ∂2 W ∂ Ik∂ Il

 ∂ Ik ∂ e
 ∂ Il ∂ e
.     (6.58)

Note that a factor 2 arise each time we differentiate the invariants with respect to e instead of C.

The specification of a material is given by specification of the derivatives of the energy with respect to invariants. The laws are implemented in the hyper.c EnPassiv function.

### 6.1.9  Gyroscopic effects

Written by Arnaud Sternchuss ECP/MSSMat.

In the fixed reference frame which is galilean, the eulerian speed of the particle in x whose initial position is p is

 ∂x ∂ t
 ∂u ∂ t
+Ω∧(p+u)

and its acceleration is

 ∂2x ∂ t2
=
 ∂2u ∂ t2
+
 ∂Ω ∂ t
∧(p+u)+2Ω
 ∂u ∂ t
+ΩΩ∧(p+u)

Ω is the rotation vector of the structure with

Ω=

 ωx ωy ωz

in a (x,y,z) orthonormal frame. The skew-symmetric matrix [Ω] is defined such that

[Ω] = [
 0 −ωz ωy ωz 0 −ωx −ωy ωx 0
]

The speed can be rewritten

 ∂x ∂ t
 ∂u ∂ t
+[Ω](p+u

and the acceleration becomes

 ∂2x ∂ t2
=
 ∂2u ∂ t2
+
 ∂[Ω] ∂ t
(p+u)+2[Ω]
 ∂u ∂ t
+[Ω]2(p+u)

In this expression appear

• the acceleration in the rotating frame ∂2u/∂ t2,
• the centrifugal acceleration ag=[Ω]2(p+u),
• the Coriolis acceleration ac=∂[Ω]/∂ t(p+u)+2[Ω]∂u/∂ t.

S0e is an element of the mesh of the initial configuration S0 whose density is ρ0. [N] is the matrix of shape functions on these elements, one defines the following elementary matrices

[Dge] =
 S0e
2ρ0 [N][Ω] [NdS0e   gyroscopic coupling
[Kae] =
 S0e
ρ0 [N]
 ∂[Ω] ∂ t
[N]  dS0e   centrifugal acceleration
[Kge] =
 S0e
ρ0 [N][Ω]2 [N]  dS0e   centrifugal softening/stiffening
(6.59)

### 6.1.10  Centrifugal follower forces

This is the embryo of the theory for the future implementation of centrifugal follower forces.

δ Wω
 Ω
ρ ω2 R(x) δ vR,     (6.60)

where δ vR designates the radial component (in deformed configuration) of δv. One assumes that the rotation axis is along ez. Noting nR = 1/R {x1  x2   0}T, one then has

 δ vR= nR·δ v.     (6.61)

Thus the non-linear stiffness term is given by

dδ Wω= −
 Ω
ρ ω2 (dR δ vR + R dδ vR).     (6.62)

One has dR=nR· dx(= dxR) and dδ vR = dnR·δ v, with

dnR=−
 dR R
nR +
 1 R
{dx1  dx2   0}T.

Thus, finally

dδ Wω= −
 Ω
ρ ω2 (du1 δ v1 + du2 δ v2).     (6.63)

Which gives

 du1 δ v1 + du2 δ v2= {δ qα}T {N}{N}T {d qα},     (6.64)

with α=1,2.

### 6.1.11  Poroelastic materials

The poroelastic formulation comes from [32], recalled and detailed in [33].

Domain and variables description:

 Ω Poroelastic domain ∂Ω Bounding surface of poroelastic domain n Unit external normal of ∂Ω u Solid phase displacement vector uF Fluid phase displacement vector uF = φ/ρ22ω2∇ p − ρ12/ρ22u p Fluid phase pressure σ Stress tensor of solid phase σt Total stress tensor of porous material σt=σ−φ(1+Q/R)pI

Weak formulation, for harmonic time dependence at pulsation ω:

 Ω
σ(u) : є(δ u)  dΩ − ω2
 Ω
ρ  u.δ u  dΩ −
 Ω
 φ α
∇ p.δ u  dΩ
−
 Ω
φ

1+
 Q R

p∇.δ u  dΩ −
 ∂Ω
t(u).n).δ u  dS =0   ∀ δ u
(6.65)

 Ω
 φ2 αρoω2
∇ p.∇δ p  dΩ −
 Ω
 φ2 R
p δ p  dΩ −
 Ω
 φ α
u.∇ δ p  dΩ
−
 Ω
φ

1+
 Q R

δ p∇. u  dΩ −
 ∂Ω
φ(uFu).n δ p  dS = 0  ∀ δ p
(6.66)

Matrix formulation, for harmonic time dependence at pulsation ω:

[
K−ω2MC1C2
C1TC2T
 1 ω2
FKp
] {
 u p
} = {
 Fst Ff
}     (6.67)

where the frequency-dependent matrices correspond to:

 Ω
σ(u):є(δ udΩ
⇒δ uT K u

 Ω
ρ  u.δ u dΩ
⇒δ uT M u

 Ω
 φ2 αρo
∇ p.∇δ p
⇒δ pT Kp p

 Ω
 φ2 R
p δ p
⇒δ pT F p

 Ω
 φ α
∇ p.δ u  dΩ
⇒δ uT C1 p

 Ω
φ

1+
 Q R

p∇.δ u  dΩ
⇒δ uT C2 p

 ∂Ω
t(u).n).δ u  dS
⇒δ uT Fst

 ∂Ω
φ(uFu).n δ p  dS
⇒δ pT Ff
(6.68)

N.B. if the material of the solid phase is homogeneous, the frequency-dependent parameters can be eventually factorized from the matrices:

[
(1+iηs)K−ω2ρM
 φ α
C1− φ

1+
 Q R

C2
 φ α
C1T−φ

1+
 Q R

C2T
 1 ω2
 φ2 R
F−
 φ2 αρo
Kp
] {
 u p
} = {
 Fst Ff
}     (6.69)

where the matrices marked with bars are frequency independant:

K=(1+iηs)KMM
C1=
 φ α
C1
C2

1+
 Q R

C2
F=
 φ2 R
F
Kp=
 φ2 αρo
Kp
(6.70)

Material parameters:

 φ Porosity of the porous material σ Resistivity of the porous material α∞ Tortuosity of the porous material Λ Viscous characteristic length of the porous material Λ′ Thermal characteristic length of the skeleton ρ Density of the skeleton G Shear modulus of the skeleton ν Poisson coefficient of the skeleton ηs Structural loss factor of the skeleton ρo Fluid density γ Heat capacity ratio of fluid (=1.4 for air) η Shear viscosity of fluid (=1.84×10−5 kg m−1 s−1 for air)

Constants:

 Po=1,01× 105 Pa Ambient pressure Pr=0.71 Prandtl number

Poroelastic specific (frequency dependent) variables:

 ρ11 Apparent density of solid phase ρ11 = (1−φ)ρ−ρ12 ρ22 Apparent density of fluid phase ρ22 = φρo−ρ12 ρ12 Interaction apparent density ρ12=−φρo(α∞−1) ρ Effective density of solid phase ρ = ρ11 − (ρ12)2/ρ22 ρ11 Effective density of solid phase ρ11 = ρ11 + b/ iω ρ22 Effective density of fluid phase ρ22 = ρ22 + b/ iω ρ12 Interaction effective density ρ12 = ρ12 − b/ iω b Viscous damping coefficient b = φ2σ√1 + i 4α∞2ηρoω/ σ2Λ2φ2 γ Coupling coefficient γ = φ(ρ12/ρ22 − Q/R) Q Elastic coupling coefficient Biot formulation Q=1−φ−Kb/Ks/1−φ−Kb/Ks+φKs/Kfφ Ks Approximation from Kb/Ks<<1 Q=(1−φ)Kf R Bulk modulus of air in fraction volume Biot formulation R=φ2Ks/1−φ−Kb/Ks+φKs/Kf Approximation from Kb/Ks<<1 R=φKf Kb Bulk modulus of porous material in vacuo Kb= 2G(1+ν)/ 3(1−2ν) Ks Bulk modulus of elastic solid est. from Hashin-Shtrikman's upper bound Ks=1+2φ/1−φKb Kf Effective bulk modulus of air in pores Kf= Po/ 1 − γ −1/ γ α ′ α ′ Function in Kf (Champoux-Allard model) α ′ = 1 + ωT/ 2iω(1+ iω/ ωT)1/2 ωT Thermal characteristic frequency ωT= 16η/ PrΛ′2ρo

• coupling conditions with poroelastic medium, elastic medium, acoustic medium
• dissipated power in medium

### 6.1.12  Heat equation

This section is based on an OpenFEM contribution by Bourquin Frédéric and Nassiopoulos Alexandre from Laboratoire Central des Ponts et Chaussées.

The variational form of the Heat equation is given by

 Ω
(ρ c θ)(vdx +
 Ω
 ∂Ω
αθ v  dγ =

 Ω
f v   dx +
 ∂Ω
(g+α θextv  dγ

∀ v ∈ H1(Ω)
(6.71)

with

• ρ the density, c the specific heat capacity.
• K the conductivity tensor of the material. The tensor K is symmetric, positive definite, and is often taken as diagonal. If conduction is isotropic, one can write K=k(x)Id where k(x) is called the (scalar) conductivity of the material.
• Acceptable loads and boundary conditions are
• Internal heat source f
• Prescribed temperature (Dirichlet condition, also called boundary condition of first kind)
 θ=θext    on    ∂Ω     (6.72)
modeled using a DofSet case entry.
• Prescribed heat flux g (Neumann condition, also called boundary condition of second kind)
=g    on    ∂Ω     (6.73)
leading to a load applied on the surface modeled using a FVol case entry.
• Exchange and heat flux (Fourier-Robin condition, also called boundary condition of third kind)
+α(θ−θext)=g    on    ∂Ω     (6.74)

leading to a stiffness term (modeled using a group of surface elements with stiffness proportional to α) and a load on the associated surface proprotional to g+αθext (modeled using FVol case entries).

#### Test case

One considers a solid square prism of dimensions Lx,Ly, Lz in the three directions (Ox), (Oy) and (Oz) respectively. The solid is made of homogeneous isotropic material, and its conductivity tensor thus reduces to a constant k.

The faces, Γi (i=1..6, ∪i=16 Γi = ∂ Ω), are subject to the following boundary conditions and loads

• f=40 is a constant uniform internal heat source
• Γ1  (x=0) : exchange & heat flux (Fourier-Robin) given by α=1,g1=α θext + α f Lx2/2k=25
• Γ2  (x=Lx) : prescribed temperature : θ(Lx,y,z)=θext=20
• Γ3  (y=0), Γ4  (y=Ly), Γ5  (z=0), Γ6  (z=Lz): exchange & heat flux g+α θext =αθextf/2 k (Lx2x2)+g1=25−x2/20

The problem can be solved by the method of separation of variables. It admits the solution

θ(x,y,z) =−
 f 2 k
x2 + θext +
 f Lx2 2k
=
 g(x) α
= 25 −
 x2 20

The resolution for this example can be found in demo/heat_equation.