Otero et al 2018a - Revision history

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← Older revision		Revision as of 11:56, 28 October 2019
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 To show the advantage of using the NLS proposed, in comparison with a full classical FE2, the calculation times are presented herein. A full FE2 solves the microstructural problem for every integration point of the structure, and for every time step.  This procedure does not distinguish between linear range and non-linear range.
-Tables [[#table-25|25]] and SFSTimeTable present the calculation times required to solve the shown example in the same desktop computer, an Intel CoreTM i7-2600 CPU @ 3.40GHz with 8GB of RAM. The tables show the times used by the FE2 and by the FE2 with the NLS incorporated. The speed ratio column has the relation times between both methodologies.
+Tables [[#table-25|25]] and [[#table-26|26]] present the calculation times required to solve the shown example in the same desktop computer, an Intel CoreTM i7-2600 CPU @ 3.40GHz with 8GB of RAM. The tables show the times used by the FE2 and by the FE2 with the NLS incorporated. The speed ratio column has the relation times between both methodologies.
 Table [[#table-25|25]] has the total real times necessary to complete the numerical test, up to d = 0.115mm, for all mesh configurations.  The speed ratio variable shows that the advantage of using the developed strategies increases specially when the size of the  macrostructure's mesh increases. This is an expected result because, in larger meshes, the proportion between linear and non-linear elements becomes also larger.

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 =Abstract=
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 =5 Validation and numerical results=
-In this chapter, the validation of the proposed constitutive model using data from the literature is presented.  Then, a numerical example is shown using the model calibrated. The basic formulation of the different constitutive models used for the simple materials can be seen in the Appendix [[#13 Constitutive models|13]].
+In this chapter, the validation of the proposed constitutive model using data from the literature is presented.  Then, a numerical example is shown using the model calibrated. The basic formulation of the different constitutive models used for the simple materials can be seen in the Appendix [[#A Constitutive models|A]].
 ==5.1 Validation of the elastic response==
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 ===5.3.4 Visco-elastic analysis===
-One of the main improvements shown by CNTs reinforced composites is their good damping response and energy dissipation, which makes them very useful for impact or vibration absorption purposes  <span id='citeF-69'></span><span id='citeF-79'></span><span id='citeF-80'></span>[[#cite-69|[69,79,80]]]. For this reason, in the following is analyzed the visco-elastic performance of the numerical model developed under such conditions. In order to obtain a viscous response of the composite it is necessary to use a visco-elastic model to characterize its components.  The visco-elastic model used for the matrix and the interface zone is the generalized Maxwell model (see Figure [[#img-17|17]]) <span id='citeF-3'></span>[[#cite-3|[3]]], which is already implemented in the PLCd code (see Section [[#13.4 Visco-elastic constitutive model|13.4]] in Appendix [[#13 Constitutive models|13]]).  To conduct the visco-elastic analysis, the MWCNTs are considered to have a linear elastic behavior.
+One of the main improvements shown by CNTs reinforced composites is their good damping response and energy dissipation, which makes them very useful for impact or vibration absorption purposes  <span id='citeF-69'></span><span id='citeF-79'></span><span id='citeF-80'></span>[[#cite-69|[69,79,80]]]. For this reason, in the following is analyzed the visco-elastic performance of the numerical model developed under such conditions. In order to obtain a viscous response of the composite it is necessary to use a visco-elastic model to characterize its components.  The visco-elastic model used for the matrix and the interface zone is the generalized Maxwell model (see Figure [[#img-17|17]]) <span id='citeF-3'></span>[[#cite-3|[3]]], which is already implemented in the PLCd code (see Section [[#A.4 Visco-elastic constitutive model|A.4]] in Appendix [[#A Constitutive models|A]]).  To conduct the visco-elastic analysis, the MWCNTs are considered to have a linear elastic behavior.
 <div id='img-17'></div>
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 The restrictions of degrees of freedom on the boundary domain can be accounted through several methods such as elimination of redundant unknowns, penalty methods and Lagrange multipliers <span id='citeF-105'></span>[[#cite-105|[105]]].  The last two methods have the disadvantage that they lead to ill-conditioned stiffness matrix or to an increase of the number of degrees of freedom of the problem.
-To avoid these two disadvantages, here it is proposed solving the RVE by an elimination of redundant unknowns <span id='citeF-142'></span>[[#cite-142|[142]]]. The constraint condition for the ''Periodic boundary fluctuations'' is shown in Section [[#9.5.2 Microscopic BVP|9.5.2]], where it is possible to observe the redundant boundary unknowns of the BVP. In these boundary constraint expressions it is possible identify master unknowns (the unknowns to solve) and slave unknowns.  In Appendix [[#14.1 Microscopic Kinematic relationships|14.1]] the master-slave kinematic relationships are derived for the ''Periodic boundary fluctuations'' condition.
+To avoid these two disadvantages, here it is proposed solving the RVE by an elimination of redundant unknowns <span id='citeF-142'></span>[[#cite-142|[142]]]. The constraint condition for the ''Periodic boundary fluctuations'' is shown in Section [[#9.5.2 Microscopic BVP|9.5.2]], where it is possible to observe the redundant boundary unknowns of the BVP. In these boundary constraint expressions it is possible identify master unknowns (the unknowns to solve) and slave unknowns.  In Appendix [[#B.1 Microscopic Kinematic relationships|B.1]] the master-slave kinematic relationships are derived for the ''Periodic boundary fluctuations'' condition.
 ====9.6.1.1 Linear implementation====
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 |}
-where <math display="inline">\Omega ^{h}_{\mu }</math> denotes the discretized domain of the RVE, <math display="inline">\mathbf{B}</math> is the global strain-displacement matrix (see Appendices [[#14.3 Derivatives of the shape functions|14.3]] and [[#14.4 Derivation of displacement field, deformation gradient and gradient of the deformation gradient tensor|14.4]]), <math display="inline">\bar{\mathbf{w}}</math> is the global vector of nodal displacement fluctuation, <math display="inline">\boldsymbol{\alpha }</math> is a set of internal variables and <math display="inline">\hbox{V}^{h}_{\Omega _{\mu }}</math> is the set of finite-dimensional nodal displacement vectors associated with the FE discretization ''h'' of the RVE domain <math display="inline">{\Omega _{\mu }}</math>. If a linear case is considered, the microscopic stress tensor is
+where <math display="inline">\Omega ^{h}_{\mu }</math> denotes the discretized domain of the RVE, <math display="inline">\mathbf{B}</math> is the global strain-displacement matrix (see Appendices [[#B.3 Derivatives of the shape functions|B.3]] and [[#B.4 Derivation of displacement field, deformation gradient and gradient of the deformation gradient tensor|B.4]]), <math display="inline">\bar{\mathbf{w}}</math> is the global vector of nodal displacement fluctuation, <math display="inline">\boldsymbol{\alpha }</math> is a set of internal variables and <math display="inline">\hbox{V}^{h}_{\Omega _{\mu }}</math> is the set of finite-dimensional nodal displacement vectors associated with the FE discretization ''h'' of the RVE domain <math display="inline">{\Omega _{\mu }}</math>. If a linear case is considered, the microscopic stress tensor is
 {| class="formulaSCP" style="width: 100%; text-align: left;"
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 |}
-where, <math display="inline">\mathbf{K}_{r}</math> is the reduced global stiffness matrix, <math display="inline">\bar{\mathbf{u}}_{r}</math> the reduced global nodal displacements vector and <math display="inline">\mathbf{RHS}</math> is the Right-Hand Side global vector which is obtained as a result of the applied boundary restriction.  Section [[#14.2.1 Linear implementation|14.2.1]] of Appendix [[#14.2 Elimination of the slave degrees of freedom|14.2]] shows how the elimination of redundant unknowns is addressed to obtain the expression given by [[#eq-9.98|9.98]].
+where, <math display="inline">\mathbf{K}_{r}</math> is the reduced global stiffness matrix, <math display="inline">\bar{\mathbf{u}}_{r}</math> the reduced global nodal displacements vector and <math display="inline">\mathbf{RHS}</math> is the Right-Hand Side global vector which is obtained as a result of the applied boundary restriction.  Section [[#B.2.1 Linear implementation|B.2.1]] of Appendix [[#B.2 Elimination of the slave degrees of freedom|B.2]] shows how the elimination of redundant unknowns is addressed to obtain the expression given by [[#eq-9.98|9.98]].
 ====9.6.1.2 Non-linear implementation====
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 is the global tangent stiffness matrix of the RVE.
-Like in the linear case, [[#eq-9.104|9.104]] does not include the boundary constraint over the RVE boundary because of the ''Periodic boundary fluctuations'' condition. If this condition is applied using an elimination of redundant unknowns method, the expression given by [[#eq-9.104|9.104]] can be rewritten as (see Section [[#14.2.2 Non-linear implementation|14.2.2]] of Appendix [[#14.2 Elimination of the slave degrees of freedom|14.2]])
+Like in the linear case, [[#eq-9.104|9.104]] does not include the boundary constraint over the RVE boundary because of the ''Periodic boundary fluctuations'' condition. If this condition is applied using an elimination of redundant unknowns method, the expression given by [[#eq-9.104|9.104]] can be rewritten as (see Section [[#B.2.2 Non-linear implementation|B.2.2]] of Appendix [[#B.2 Elimination of the slave degrees of freedom|B.2]])
 {| class="formulaSCP" style="width: 100%; text-align: left;"
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 ===11.2.3 Numerical homogenized tangent constitutive tensor===
-The proposed NLS has been implemented in PLCd <span id='citeF-9'></span>[[#cite-9|[9]]], a parallel finite element code that works with 3D solid geometries (see Appendix [[#14.5 FEM implementation in PLCd code|14.5]]). In the code, a Newton-Raphson scheme is adopted to solve the non-linear problem. To facilitate the convergence of the whole problem, the tangent constitutive tensor at the integration point is necessary to obtain the global tangent stiffness matrix.
+The proposed NLS has been implemented in PLCd <span id='citeF-9'></span>[[#cite-9|[9]]], a parallel finite element code that works with 3D solid geometries (see Appendix [[#B.5 FEM implementation in PLCd code|B.5]]). In the code, a Newton-Raphson scheme is adopted to solve the non-linear problem. To facilitate the convergence of the whole problem, the tangent constitutive tensor at the integration point is necessary to obtain the global tangent stiffness matrix.
-A perturbation method is used to obtain a numerical approximation of the homogenized tangent constitutive tensor of the RVE in the integration point.  The method implemented is analogous to the one proposed by Martinez et al. in <span id='citeF-33'></span>[[#cite-33|[33]]] (see Section&nbsp;[[#14.5.2 Numerical tangent constitutive tensor|14.5.2]] in Appendix&nbsp;[[#14.5 FEM implementation in PLCd code|14.5]]). Being the only difference that in current procedure the perturbations must be applied on the RVE instead of applying them to a constitutive equation. The <math display="inline">n</math> small perturbations (<math display="inline">\delta\bar{\boldsymbol{\varepsilon }}_{j}</math>) are applied to the homogenized or structural strain vector. The RVE is solved <math display="inline">n</math> times and as result gives the <math display="inline">n</math> stress vector <math display="inline">\delta^{j}\bar{\boldsymbol{\sigma }}</math>.  Therefore, the <math display="inline">j</math> columns of the tangent constitutive tensor for the RVE can be obtained  as
+A perturbation method is used to obtain a numerical approximation of the homogenized tangent constitutive tensor of the RVE in the integration point.  The method implemented is analogous to the one proposed by Martinez et al. in <span id='citeF-33'></span>[[#cite-33|[33]]] (see Section&nbsp;[[#B.5.2 Numerical tangent constitutive tensor|B.5.2]] in Appendix&nbsp;[[#B.5 FEM implementation in PLCd code|B.5]]). Being the only difference that in current procedure the perturbations must be applied on the RVE instead of applying them to a constitutive equation. The <math display="inline">n</math> small perturbations (<math display="inline">\delta\bar{\boldsymbol{\varepsilon }}_{j}</math>) are applied to the homogenized or structural strain vector. The RVE is solved <math display="inline">n</math> times and as result gives the <math display="inline">n</math> stress vector <math display="inline">\delta^{j}\bar{\boldsymbol{\sigma }}</math>.  Therefore, the <math display="inline">j</math> columns of the tangent constitutive tensor for the RVE can be obtained  as
 {| class="formulaSCP" style="width: 100%; text-align: left;"

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 == APPENDICES ==
-=A Constitutive models=
+=Appendix A. Constitutive models=
 The formulation developed is Section [[#4.2 Formulation of the composite constitutive model|4.2]] require that all composite components must fulfill the expression given by [[#eq-4.1|4.1]].  Therefore, it is possible to use any constitutive law to describe the mechanical performance of the different components.  In this appendix are defined the models that are used in this monograph to characterize the different composite component defined.

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 |}
-=14 Computational implementation=
+=Appendix B. Computational implementation=
 ==14.1 Microscopic Kinematic relationships==
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 |}
-===References===
+==References==
 <div id="cite-1"></div>

← Older revision	Revision as of 13:40, 13 March 2018
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 <math>G_{12nt}=G_{13nt}=\bar{G}_{nt}=85</math>[GPa], <math display="inline">G_{23nt}=G_{iz}=2.47</math> [GPa],
-<math>\nu _{12nt}=\nu _{13nt}=\nu _{23nt}=\nu _{nt}=0.2</math>
+<math>\nu _{12nt}=\nu _{13nt}=\nu _{23nt}=\nu _{nt}=0.2</math>,
+<math>\nu _{ij}=\frac{E_{i}}{E_{j}}\nu _{ji}\quad \Rightarrow \quad \nu _{21nt}=\nu _{31nt}=0.008\quad \nu _{32nt}=0.2</math>.
 The composite simulated in the analysis has a 3% weight of CNTs, which represent a 1.94% of volume fraction.  However, measurements made with X-rays show an apparent 5% weight.  This difference is obtained because the MWCNTs have a higher apparent diameter than the original one.  Therefore, the parameter <math display="inline">\frac{b}{r_{nt}}</math> is estimated assuming that this extra 2% weight in the measurement is the coating polymer around the nanotubes and then, it is the interface component considered. Taking this into account, the interface component has an approximate volume fraction of 1.31% and <math display="inline">\frac{b}{r_{nt}}=0.3</math>.