3. Methods#
3.1. Three-dimensional linear thermal homogenization#
For homogenization in three-dimensional linear thermal, three fields of thermal correctors, each associated with three unit temperature gradients in each of the three spatial directions, homogeneous in the volume of the base cell \(C\) three dimensional.
As the base cell mesh is restricted to octant \((X>0,Y>0,Z>0)\), the conditions of material and geometric symmetry and periodicity impose the conditions of Dirichlet following:
The three elementary heat conduction problems on base cell \(C\), which provide the three corrector fields, written in variational form:
the space of eligible periodic fields \({V}_{\text{pér}}\) is restricted here to those verifying the Dirichlet conditions referred to in the table above.
By noting \(\underline{z}\) the position vector in the base cell \(C\), placed at the point macroscopic \({\underline{x}}_{0}\), the microscopic temperature field is then written as the sum of the macroscopic temperature field \({T}^{0}\) developed on the base cell via its gradient \({\underline{g}}^{0}={\underline{\nabla} T^0}\) and the contribution of the editors:
After calculating the three corrector fields, the spatial averaging is carried out over the cell base \(C\) of the calorific capacity, of the isotropic thermal conduction coefficient. These terms corresponding to the « law of mixtures », from which we will subtract the contribution of correctors to obtain the components of the homogenized conduction tensor. To do this, the calculation of the energies of dissipation by thermal conduction is carried out \({W}_{\text{ther}}\) associated with linear combinations of corrector fields thermal, which come as a deduction from the terms of the law of mixtures:
with:
The components of the orthotropic homogenized conduction tensor are expressed in the axes of base cell frame of reference:
- note
- The extra-diagonal components of this matrix are zero due to the symmetries of
base cell considered.
- If base cell \(C\) has geometric and material symmetry
additional, compared to a plane making an angle \(\theta =\pm \frac{\pi}{3},\pm \frac{\pi}{4}\) with plane \(X=0\), or an invariance by rotation \(\frac{2 \, \pi}{3}\) around the axis \(\mathit{OZ}\) for example, then we will necessarily have \({K}_{11}^{\text{hom}}={K}_{22}^{\text{hom}}\).
3.2. Homogenization in three-dimensional linear thermoelasticity#
For homogenization in three-dimensional linear thermoelasticity, six fields of elastic correctors, each associated with the six components of the symmetric tensor of order two of three-dimensional deformation homogeneous units \({\boldsymbol{\varepsilon}}^{0}\) in volume of the three-dimensional base cell, and a field of thermal expansion correctors, for a deformation equal to the expansion coefficient in each sub-domain of the base cell three dimensional.
As the base cell mesh is restricted to octant \((X>0,Y>0,Z>0)\), the conditions of material and geometric symmetry and periodicity, cf. [bib1], impose the following Dirichlet conditions:
Concealer |
Dirichlet conditions |
Wall considered |
Imposed deformation |
\({\chi }_{11}\), \({\chi }_{22}\), \({\chi }_{33}\), \({\chi }_{\text{dil}}\) |
\({U}_{x}=0\) |
\({x}_{\min}\) and \({x}_{\max}\) |
|
\({U}_{y}=0\) |
\({y}_{\min}\) and \({y}_{\max}\) |
||
\({U}_{z}=0\) |
\({z}_{\min}\) and \({z}_{\max}\) |
||
\({\chi }_{12}\) |
\({U}_{y}={U}_{z}=0\) |
\({x}_{\min}\) and \({x}_{\max}\) |
|
\({U}_{x}={U}_{z}=0\) |
\({y}_{\min}\) and \({y}_{\max}\) |
||
\({U}_{z}=0\) |
\({z}_{\min}\) and \({z}_{\max}\) |
+——————–+————————-+—————————————–++ |:math:`{\chi }_{13}`|\({U}_{y}={U}_{z}=0`|:math:`{x}_{\min}\) and \({x}_{\max}`|| + +-------------------------+-----------------------------------------++ | |:math:`{U}_{y}=0\) |:math:`{y}_{\min}` and :math:`{y}_{\max}`|| + +————————-+—————————————–++ | |:math:`{U}_{x}={U}_{y}=0`|\({z}_{\min}\) and :math:`{z}_{max}`|| +——————–+————————-+—————————————–++
+——————–+————————-+—————————————–++ |:math:`{\chi }_{23}`|\({U}_{x}=0\) |:math:`{x}_{\min}` and :math:`{x}_{\max}`|| + +————————-+—————————————–++ | |:math:`{U}_{x}={U}_{z}=0`|\({y}_{\min}\) and \({y}_{\max}`|| + +-------------------------+-----------------------------------------++ | |:math:`{U}_{x}={U}_{y}=0`|:math:`{z}_{\min}\) and :math:`{z}_{max}`|| +——————–+————————-+—————————————–++
In the case where there is a cavity (or several) within the base cell, we perform also the calculation of another field of elastic correctors \({\chi }_{p}^{\text{int}}\), under the action of a unit pressure exerted within these cavities, in order to have access complete to the constraints at the microscopic scale.
The six basic elasticity problems on base cell \(C\), which provide the six fields of correctors, can be written in variational form:
- left{
- begin {aligned}
&text {Find} {boldsymbol {chi}}} (x, y, z)in {mathbf {V}} _ {text {perr}}text {such as:}\ & {int} _ {C} {boldsymbol {varepsilon}} (boldsymbol {chi})cdotmathbf {A}cdotboldsymbol {varepsilon} (underline {v}), dV
=
{int} _ {C} {boldsymbol {varepsilon}}} ^ {0}cdotmathbf {A}cdotboldsymbol {varepsilon} (underline {v})(underline {v}),, dVqquadvarepsilon}), dVqquadforallunderline {v}}in {mathbf {V}}} _ {text {v}} _ {text {v}), dVqquadforallunderline {v}}
end {aligned} right.
the space of periodic eligible fields \({\mathbf{V}}_{\text{pér}}\) is limited here to those verifying the Dirichlet conditions reported in the table above.
The elementary elasticity problem on base cell \(C\), which provides the field of thermal expansion corrector, is written in variational form:
- left{
- begin {aligned}
&text {Find} {boldsymbol {chi}}} _ {text {dil}} (x, y, z)in {mathbf {V}} _ {text {perr}}}text {such as:}\ & {int} _ {C}boldsymbol {varepsilon} ({boldsymbol {chi}}} _ {text {dil}})cdotmathbf {A}cdotboldsymbol {varepsilon} (underline {v}}), dV
= - {int} _ {C}alpha (x, y, z),mathbf, z),mathbf {Id}cdotboldsymbol {varepsilon} (underline {v}),underline {v}), dVqquadqquadforallunderline {v}in {mathbf {V}}} _ {text {v}} _ {text {v}), dVqquadforallunderline {v}}
end {aligned} right.
It should be noted that if the thermal expansion coefficient \(\alpha\) is uniform throughout the base cell, then we’ll have \({\boldsymbol{\chi}}_{\text{dil}}=\alpha \, ({\chi }^{11}+{\chi }^{22}+{\chi }^{33}) \, {\mathbf{Id}}\).
By noting \({\underline{z}}\) the position vector in base cell \(C\), placed at macroscopic point \({\underline{x}}_{0}\), the microscopic field of displacement is written then as the sum of the macroscopic displacement field \({\underline{u}}^{0}\) expanded on the base cell via the associated deformation \({\boldsymbol{\varepsilon }}^{0}\) and contribution of the editors, keeping only the macroscopic term \({T}^{0}\) from the temperature on the base cell:
- underline {u} (underline {x} _ {0}, {underline {z}}))
= {underline {u}} ^ {0}} (underline {x} _ {0}) + {boldsymbol {varepsilon}}} ^ {0} ({underline {x}} _ {0})cdot {underline {z}} + sum_ {k, l} {varepsilon} _ {kl} ^ {0}} ({underline {x}} _ {0}) {chi} ^ {kl} ({underline {z}}) - ({T} ^ {0} ({underline {x}}} _ {0}) - {T} _ {text {ref}}), (alpha, (alpha, {alpha, {mathbf {Id}}}cdot {underline {z}}} + {boldsymbol {chi}}}) _ {text {chi}}} _ {text {dil}}} _ {text {dil}}} _ {text {dil}}} ({text {dil}}}) + {p} _ {text {app}}}, {boldsymbol {chi}} _ {p} ^ {text {int}}} ({{underline {z}}}}) ({{underline {z}}}})
And the field of microscopic constraints is deduced from this:
- boldsymbol {sigma} ({underline {x}}} _ {0}, {{underline {z}}}))
= mathbf {A}cdot {boldsymbol {boldsymbol {varepsilon}}} ^ {0}} ({underline {x}}} _ {0}) + sum _ {k, l} {varepsilon} _ {kl}} _ {kl} ^ {0}} ({underline {x}} _ {0})mathbf {A}cdot {boldsymbol {varepsilon}}} _ {varepsilon}}} _ {0})mathbf {A}cdot {boldsymbol {varepsilon}}}} ({boldsymbol {varepsilon}}}} ({boldsymbol {varepsilon}}}} ({boldsymbol {varepsilon}}}}) ({boldsymbol {varepsilon}}}} ({boldsymbol {varepsilon {0} ({underline {x}} _ {0}) - {0}) - {T} _ {text {ref}})cdotmathbf {A}cdot (alpha. {bf Id} + {boldsymbol {varepsilon}} ({boldsymbol {chi}}} _ {text {dil}}} ({{underline {z}}}))))) + {p} _ {text {varepsilon}}{text {app}}cdot {chi}} _ {p} ^ {text {int}}} ({{underline {z}}}))
After calculating the six corrector fields, the spatial averaging is carried out over the cell. base \(C\) of density (this term corresponds to the « law of mixtures »), Lamé coefficients of isotropic elasticity, and also of the stiffness of expansion \(3 \, \alpha \, K\). These terms correspond to the « law of mixtures », to which we go subtract the contribution of correctors to obtain the components of the elasticity tensor homogenized. To do this, the elastic deformation energies are calculated \({W}_{\text{élas}}\) associated with linear combinations of corrector fields elastics, which come as a deduction from the terms of the law of mixtures:
with:
The components of the orthotropic homogenized elasticity tensor are in the axes of the reference frame of the base cell:
This tensor is inverted to obtain the orthotropic elasticity modules and Poisson coefficients.
The material will be « transversely isotropic » if base cell \(C\) has symmetry additional geometric and material, compared to a plane making a angle \(\theta =\pm \frac{\pi}{3},\pm \frac{\pi}{4}\) with plane \(X=0\), or an invariance by rotation \(\frac{2 \, \pi}{3}\) around the \(\mathit{OZ}\) axis for example. There is then more than five elasticity coefficients that are independent a prima facie:
To be sure of this, we calculate the relative error on the expression for the stiffness coefficient elastic in distortion \({A}_{1212}^{\text{hom}}\) above: this is the object of the calculation of « ISOTRANS » which should be numerically null in this case.
We recall that in the homogeneous isotropic elastic case we have:
The components of the orthotropic homogenized thermal expansion tensor are in the axes from the base cell repository:
In the « transverse isotropic » case around the \(\mathit{OZ}\) axis, \({\alpha }_{11}^{\text{hom}}={\alpha }_{22}^{\text{hom}}\) will be checked.
Thus the homogenized linear thermoelastic macroscopic behavior relationship is written:
We recall that in the homogeneous isotropic thermoelastic case we have \(\boldsymbol{\alpha}=\alpha \mathbf{Id}\).
3.3. Homogenization and linear elasticity of Love-Kirchhoff plates in membrane-flexure#
For the homogenization in linear elasticity of Love-Kirchhoff plates under membrane-flexure, the following are calculated:
- three fields of elastic correctors, each associated with the three components of the tensor
symmetric deformation of membrane homogeneous units in the volume of three-dimensional base cell,
- three fields of elastic correctors, each associated with the three components of the tensor
symmetric deformation of order two homogeneous flexion units in the volume of three-dimensional base cell,
- a thermal expansion corrector field in a membrane, for a deformation equal to
expansion coefficient in each subdomain of the three-dimensional base cell.
As the base cell mesh is restricted to octant \((X>0,Y>0,Z>0)\), having adopted here as the direction in the thickness the \(\mathit{OZ}\) axis, the symmetry conditions material and geometric and periodicity impose the following Dirichlet conditions:
Concealer |
Dirichlet conditions |
Wall considered |
Imposed deformation |
\({\chi }_{11}^{m}\), \({\chi }_{22}^{m}\), \({\chi }_{\text{dil}}^{m}\) |
\({U}_{x}=0\) |
\({x}_{\min}\) and \({x}_{\max}\) |
|
\({U}_{y}=0\) |
\({y}_{\min}\) and \({y}_{\max}\) |
||
\({U}_{z}=0\) |
\({z}_{\min}\) and \({z}_{\max}\) |
||
\({\chi }_{12}^{m}\) |
\({U}_{y}=0\) |
\({x}_{\min}\) and \({x}_{\max}\) |
|
\({U}_{x}=0\) |
\({y}_{\min}\) and \({y}_{\max}\) |
||
\({U}_{z}=0\) |
\({z}_{\min}\) and \({z}_{\max}\) |
||
\({\chi }_{11}^{f}\), \({\chi }_{22}^{f}\) |
\({U}_{x}=0\) |
\({x}_{\min}\) and \({x}_{\max}\) |
|
\({U}_{y}=0\) |
\({y}_{\min}\) and \({y}_{\max}\) |
||
\({U}_{z}=0\) |
\({z}_{\min}\) and \({z}_{\max}\) |
||
\({\chi }_{12}^{f}\) |
\({U}_{y}=0\) |
\({x}_{\min}\) and \({x}_{\max}\) |
|
\({U}_{x}=0\) |
\({y}_{\min}\) and \({y}_{\max}\) |
||
\({U}_{z}=0\) |
\({z}_{\min}\) and \({z}_{\max}\) |
The three + three elementary elasticity problems on base cell \(C\), which provide the six fields of membrane-flexure correctors, can be written respectively under variational form:
- left{
- begin {aligned}
&text {Find} {chi} _ {alphabeta} _ {alphabeta}} ^ {m} (x, y, z)in {mathbf {V}} _ {text {perr}}}text {such as:}\ & {int} _ {C}varepsilon ({chi}} _ {alphabeta} ^ {m})cdotmathbf {A}cdotvarepsilon (mathrm {v}), dV = - {int} _ {C} {varepsilon} _ {varepsilon} _ {varepsilon}varepsilon (mathrm {v}),, dVqquadforallforallforallunderline {v}}in {mathbf {V}} _ {text {v}} _ {text {v}), dVqquadforallforallforallunderline {v}in {mathbf {V}} _ {text {v}} _ {text {v}), dVqquadforallforallforallunderline {v}}in {mathbf {V}
end {aligned} right.
and
- left{
- begin {aligned}
&text {Find} {chi} _ {alphabeta} _ {alphabeta}} ^ {m} (x, y, z)in {mathbf {V}} _ {text {perr}}}text {such as:}\ & {int} _ {C}boldsymbol {varepsilon} ({chi}} _ {alphabeta} ^ {m})cdotmathbf {A}cdotboldsymbol {varepsilon} (mathrm {v}), dV = {int} _ {C} {underline {z}}}cdot {boldsymbol {varepsilon}} _ {alphabeta} ^ {0}cdotmathbf {A}cdotboldsymbol {z}}}cdot {boldsymbol {varepsilon}} (mathrm {v})} _ {alphabeta} ^ {0}\\\\\\\\\{mathbf {V}} _ {text {perr}}
end {aligned} right.
…. math: .. begin{array}{c}text{Trouver}{chi }_{alpha beta }^{m}(x,y,z)in {V}_{text{pér}},text{tel que}mathrm{:}hfill \ {int }_{C}varepsilon ({chi }_{alpha beta }^{m})cdotmathbf{A}cdotvarepsilon (mathrm{v}), dV=-{int }_{C}{varepsilon }_{alpha beta }^{0}cdotmathbf{A}cdotvarepsilon (mathrm{v}), dVphantom{rule{6em}{0ex}}forall mathrm{v}in {V}_{text{pér}}\ text{Trouver}{~chi }_{alpha beta }^{f}(x,y,{underline{z}})in {V}_{text{pér}},text{tel que}mathrm{:}hfill \ {int }_{C}varepsilon ({chi }_{alpha beta }^{f})cdotmathbf{A}cdotvarepsilon (mathrm{v}), dV={int }_{C}{underline{z}}cdot{varepsilon }_{alpha beta }^{0}cdotmathbf{A}cdotvarepsilon (mathrm{v}), dVphantom{rule{6em}{0ex}}forall mathrm{v}in {V}_{text{pér}}end{array}
…. math: .. begin{array}{c}text{Trouver}{chi }_{alpha beta }^{m}(x,y,z)in {V}_{text{pér}},text{tel que}mathrm{:}hfill \ {int }_{C}varepsilon ({chi }_{alpha beta }^{m})cdotmathbf{A}cdotvarepsilon (mathrm{v}), dV=-{int }_{C}{varepsilon }_{alpha beta }^{0}cdotmathbf{A}cdotvarepsilon (mathrm{v}), dVphantom{rule{6em}{0ex}}forall mathrm{v}in {V}_{text{pér}}\ text{Trouver}{~chi }_{alpha beta }^{f}(x,y,{underline{z}})in {V}_{text{pér}},text{tel que}mathrm{:}hfill \ {int }_{C}varepsilon ({chi }_{alpha beta }^{f})cdotmathbf{A}cdotvarepsilon (mathrm{v}), dV={int }_{C}{underline{z}}cdot{varepsilon }_{alpha beta }^{0}cdotmathbf{A}cdotvarepsilon (mathrm{v}), dVphantom{rule{6em}{0ex}}forall mathrm{v}in {V}_{text{pér}}end{array}
the space of periodic eligible fields \({\mathbf{V}}_{\text{pér}}\) is limited here to those verifying the Dirichlet conditions reported in the table above.
After calculating the six corrector fields, the spatial averaging is carried out over the cell. basic density, Lamé coefficients of isotropic elasticity, and the mean Spatial on the base cell of the moment of order one according to the thickness of the Lamé coefficients isotropic elasticity.
Next, the elastic deformation energies associated with the combinations are calculated. linear fields of elastic membrane correctors, and similarly for flexure, which come from deduction of the terms of the law of mixtures. Because of the symmetry of the base cell by compared to the middle sheet, it is not useful to perform the linear intersecting combinations of fields of elastic membrane and flexure correctors.
with:
with:
where we noted \(h\) the total thickness of the base cell \(C\) in the direction of the \(\mathit{OZ}\) axis, \({\boldsymbol{\varepsilon}}^{0}\) the tensor of membrane strains and \({\boldsymbol{\kappa} }^{0}\) the macroscopic curvature variation tensor of the plate or shell.
The components of the homogenized orthotropic membrane and flexure elasticity tensor are in the axes of the frame of reference of the base cell:
It is recalled that the units of these elastic stiffness components are respectively: stress*length in membrane and stress*length to the cubed in bending.
We recall that in the homogeneous isotropic elastic case we have: