Sorrenti 2025a - Revision history

Marherna: Marherna moved page Review 346206359192 to Sorrenti 2025a

2025-09-22T08:38:51Z

Marherna moved page Review 346206359192 to Sorrenti 2025a

Msorrenti94: Msorrenti94 moved page Draft Sorrenti 399626909 to Review 346206359192

2025-04-13T14:35:44Z

Msorrenti94 moved page Draft Sorrenti 399626909 to Review 346206359192

Msorrenti94 at 14:11, 13 April 2025

2025-04-13T14:11:09Z

Msorrenti94 at 14:10, 13 April 2025

2025-04-13T14:10:17Z

Msorrenti94 at 14:09, 13 April 2025

2025-04-13T14:09:08Z

Msorrenti94 at 13:54, 13 April 2025

2025-04-13T13:54:46Z

Msorrenti94: Created page with " =1. Introduction= Multilayered composite and sandwich structures have been widely used in different engineering fields, including aerospace, marine, civil, energy, and aut..."

2025-04-13T13:38:28Z

Created page with " =1. Introduction= Multilayered composite and sandwich structures have been widely used in different engineering fields, including aerospace, marine, civil, energy, and aut..."

Show changes

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-[[Image:Draft_Sorrenti_399626909-image2-c.png|600px]]
+[[Image:Draft_Sorrenti_399626909-image2-c.png|600px]] </div>
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-[[Image:Draft_Sorrenti_399626909-image3-c.png|600px]]
+[[Image:Draft_Sorrenti_399626909-image3-c.png|600px]] </div>
 <div id="_Ref195294239" class="center" style="width: auto; margin-left: auto; margin-right: auto;">

@@ Line 232: / Line 232: @@
 The en-RZT model of the sandwich plate is made of 80x40 elements along ''x<sub>1</sub>'' and ''x<sub>2</sub>'', respectively, with a total number of dofs 23247, nearly thirty times less than those present in the high-fidelity 3D Nastran model. Hereafter are presented the results concerning the static bending analysis in terms of through-the-thickness distributions of displacements and strains.
 [[Image:Draft_Sorrenti_399626909-image2-c.png|600px]]
@@ Line 239: / Line 240: @@
 According to <span id='cite-_Ref195294047'></span>[[#_Ref195294047|Figure 2]], the through-the-thickness distributions of the in-plane displacements are in good agreement with the 3D ones. Although there are some discrepancies, the percent error of the transverse displacement with respect to the average 3D value is -2.14 %. Higher errors are observed in the core layer due to the intrinsic limitation of the en-RZT linear displacement field. <span id='cite-_Ref195294239'></span>[[#_Ref195294239|Figure 3]] reports the through-the-thickness distribution of the in-plane strains. The en-RZT results are able to follow, in some cases with remarkable accuracy, the 3D strain distributions. Again, the differences that are present, especially in the core layer, are mainly due to the limitation of the current linear model for in-plane displacements, and the assumption of neglecting the transverse normal deformability.
 [[Image:Draft_Sorrenti_399626909-image3-c.png|600px]]

@@ Line 237: / Line 237: @@
 Figure 2: Through-the-thickness distribution of displacements in some plate’s points. </div>
-According to <span id='cite-_Ref195294047'></span>[[#_Ref195294047|Figure 2]], the through-the-thickness distributions of the in-plane displacements are in good agreement with the 3D ones. Although there are some discrepancies, the percent error of the transverse displacement with respect to the average 3D value is -2.14 %. Higher discrepancies are observed in the core layer due to the limitation of the en-RZT linear displacement field. <span id='cite-_Ref195294239'></span>[[#_Ref195294239|Figure 3]] reports the through-the-thickness distribution of the in-plane strains. The en-RZT results are able to follow, in some cases with remarkable accuracy, the 3D strain distributions. Some discrepancies, especially in the core layer, are due to the limitation of the current linear model.
+According to <span id='cite-_Ref195294047'></span>[[#_Ref195294047|Figure 2]], the through-the-thickness distributions of the in-plane displacements are in good agreement with the 3D ones. Although there are some discrepancies, the percent error of the transverse displacement with respect to the average 3D value is -2.14 %. Higher errors are observed in the core layer due to the intrinsic limitation of the en-RZT linear displacement field. <span id='cite-_Ref195294239'></span>[[#_Ref195294239|Figure 3]] reports the through-the-thickness distribution of the in-plane strains. The en-RZT results are able to follow, in some cases with remarkable accuracy, the 3D strain distributions. Again, the differences that are present, especially in the core layer, are mainly due to the limitation of the current linear model for in-plane displacements, and the assumption of neglecting the transverse normal deformability.
 [[Image:Draft_Sorrenti_399626909-image3-c.png|600px]]
@@ Line 244: / Line 244: @@
 Figure 3: Through-the-thickness distributions of in-plane strains.</div>
-A free vibration analysis on the anti-symmetric angle-ply sandwich rectangular plate has been conducted to evaluate the en-RZT predictive capability to compute its first 10 natural frequencies. The results are compared with those obtained using the 3D Nastran model and reported in <span id='cite-_Ref195294497'></span>[[#_Ref195294497|Figure 4]]. Additionally, for each mode, the percent error on the en-RZT frequency with respect to the 3D one is also reported. It is worth noting that the en-RZT shows a remarkable accuracy in predicting the frequency even at higher modes, with an error that does not exceed 2.3 %.
+A free vibration analysis on the anti-symmetric angle-ply sandwich rectangular plate has been conducted to evaluate the en-RZT predictive capability for dynamic problems. The first 10 natural frequencies of sandwich S are computed using the same model for static analysis and the results are compared with those obtained using the 3D Nastran model and reported in <span id='cite-_Ref195294497'></span>[[#_Ref195294497|Figure 4]]. The selected frequencies are for both flexural and torsional modes. Additionally, for each mode, the percent error on the en-RZT frequency with respect to the 3D one is also reported. It is worth noting that the en-RZT shows a remarkable accuracy in predicting the frequency even at higher modes, with an error that does not exceed 2.3 %. These results confirm the accuracy of the en-RZT model in making accurate predictions for static and dynamic problems at an affordable computational cost.
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-where <math display="inline">{{U}_{1}}^{(k)}</math>, <math display="inline">{{U}_{2}}^{(k)}</math> and <math display="inline">{U}_{3}</math> are the orthogonal components of the displacement vector along the <math display="inline">{x}_{i}</math> coordinate. In the en-RZT the transverse displacement is assumed constant along the thickness. The <math display="inline">{\mathit{\boldsymbol{u}}}^{T}=</math><math>⌊ \begin{matrix}{u}_{1}&{u}_{2}\end{matrix}⌋</math>  are the uniform in-plane displacements, <math display="inline">{\mathit{\boldsymbol{\theta }}}^{T}=</math><math>⌊ \begin{matrix}{\theta }_{1}&{\theta }_{2}\end{matrix}⌋</math>  the average bending rotation, and <math display="inline">{\mathit{\boldsymbol{\psi }}}^{T}=</math><math>⌊ \begin{matrix}{\psi }_{1}&{\psi }_{2}\end{matrix}⌋</math>  the zigzag rotations. The <math display="inline">{\mathit{\boldsymbol{\varphi }}}^{(k)}({x}_{3})</math> matrix is the set of the through-the-thickness piecewise continuous zigzag functions. They are characterized by the partial enforcement of the transverse shear continuity at the layer interfaces, and the vanish condition of these functions on the bottom and top external surfaces of the plate, i.e.  <math display="inline">{\mathit{\boldsymbol{\varphi }}}^{(1)}(-</math><math>h/2)={\mathit{\boldsymbol{\varphi }}}^{(N)}(+h/2)=\mathit{\boldsymbol{0}}</math>. According to Ref. [21] their expression reads:
+where <math display="inline">{{U}_{1}}^{(k)}</math>, <math display="inline">{{U}_{2}}^{(k)}</math> and <math display="inline">{U}_{3}</math> are the orthogonal components of the displacement vector along the <math display="inline">{x}_{i}</math> coordinate. In the en-RZT the transverse displacement is assumed constant along the thickness. The <math display="inline">{\mathit{\boldsymbol{u}}}^{T}=</math><math>\left[ \begin{matrix}{u}_{1}&{u}_{2}\end{matrix}\right]</math>  are the uniform in-plane displacements, <math display="inline">{\mathit{\boldsymbol{\theta }}}^{T}=</math><math>\left[ \begin{matrix}{\theta }_{1}&{\theta }_{2}\end{matrix}\right]</math>  the average bending rotation, and <math display="inline">{\mathit{\boldsymbol{\psi }}}^{T}=</math><math>\left[ \begin{matrix}{\psi }_{1}&{\psi }_{2}\end{matrix}\right]</math>  the zigzag rotations. The <math display="inline">{\mathit{\boldsymbol{\varphi }}}^{(k)}({x}_{3})</math> matrix is the set of the through-the-thickness piecewise continuous zigzag functions. They are characterized by the partial enforcement of the transverse shear continuity at the layer interfaces, and the vanish condition of these functions on the bottom and top external surfaces of the plate, i.e.  <math display="inline">{\mathit{\boldsymbol{\varphi }}}^{(1)}(-</math><math>h/2)={\mathit{\boldsymbol{\varphi }}}^{(N)}(+h/2)=\mathit{\boldsymbol{0}}</math>. According to Ref. [21] their expression reads:
 {| class="formulaSCP" style="width: 100%; text-align: center;"
@@ Line 71: / Line 71: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math display="inline">\begin{matrix}{\mathit{\boldsymbol{\epsilon }}}_{m}^{T}\left( \mathit{\boldsymbol{x}};t\right) =⌊ \begin{matrix}{u}_{1,1}\left( \mathit{\boldsymbol{x}};t\right) &{u}_{2,2}\left( \mathit{\boldsymbol{x}};t\right) &{u}_{1,2}\left( \mathit{\boldsymbol{x}};t\right) +{u}_{2,1}\left( \mathit{\boldsymbol{x}};t\right) \end{matrix}⌋ \\{\mathit{\boldsymbol{\epsilon }}}_{\theta }^{T}\left( \mathit{\boldsymbol{x}};t\right) =⌊ \begin{matrix}{\theta }_{1,1}\left( \mathit{\boldsymbol{x}};t\right) &{\theta }_{2,2}\left( \mathit{\boldsymbol{x}};t\right) &{\theta }_{1,2}\left( \mathit{\boldsymbol{x}};t\right) +{\theta }_{2,1}\left( \mathit{\boldsymbol{x}};t\right) \end{matrix}⌋ \end{matrix}</math>
+| <math display="inline">\begin{matrix}{\mathit{\boldsymbol{\epsilon }}}_{m}^{T}\left( \mathit{\boldsymbol{x}};t\right) =\left[ \begin{matrix}{u}_{1,1}\left( \mathit{\boldsymbol{x}};t\right) &{u}_{2,2}\left( \mathit{\boldsymbol{x}};t\right) &{u}_{1,2}\left( \mathit{\boldsymbol{x}};t\right) +{u}_{2,1}\left( \mathit{\boldsymbol{x}};t\right) \end{matrix}\right] \\{\mathit{\boldsymbol{\epsilon }}}_{\theta }^{T}\left( \mathit{\boldsymbol{x}};t\right) =\left[ \begin{matrix}{\theta }_{1,1}\left( \mathit{\boldsymbol{x}};t\right) &{\theta }_{2,2}\left( \mathit{\boldsymbol{x}};t\right) &{\theta }_{1,2}\left( \mathit{\boldsymbol{x}};t\right) +{\theta }_{2,1}\left( \mathit{\boldsymbol{x}};t\right) \end{matrix}\right] \end{matrix}</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (5)
@@ Line 81: / Line 81: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math display="inline">\begin{matrix}{\mathit{\boldsymbol{\epsilon }}}_{\psi }^{T}\left( \mathit{\boldsymbol{x}};t\right) =⌊ \begin{matrix}{\psi }_{1,1}\left( \mathit{\boldsymbol{x}};t\right) &{\psi }_{2,2}\left( \mathit{\boldsymbol{x}};t\right) &{\psi }_{1,2}\left( \mathit{\boldsymbol{x}};t\right) &{\psi }_{2,1}\left( \mathit{\boldsymbol{x}};t\right) \end{matrix}⌋ \\{\mathit{\boldsymbol{\Phi }}}^{(k)}({x}_{3})=\left[ \begin{matrix}{\varphi }_{11}^{(k)}({x}_{3})&0&0&{\varphi }_{12}^{(k)}({x}_{3})\\0&{\varphi }_{22}^{(k)}({x}_{3})&{\varphi }_{21}^{(k)}({x}_{3})&0\\{\varphi }_{21}^{(k)}({x}_{3})&{\varphi }_{12}^{(k)}({x}_{3})&{\varphi }_{11}^{(k)}({x}_{3})&{\varphi }_{22}^{(k)}({x}_{3})\end{matrix}\right] \end{matrix}</math>
+| <math display="inline">\begin{matrix}{\mathit{\boldsymbol{\epsilon }}}_{\psi }^{T}\left( \mathit{\boldsymbol{x}};t\right) =\left[ \begin{matrix}{\psi }_{1,1}\left( \mathit{\boldsymbol{x}};t\right) &{\psi }_{2,2}\left( \mathit{\boldsymbol{x}};t\right) &{\psi }_{1,2}\left( \mathit{\boldsymbol{x}};t\right) &{\psi }_{2,1}\left( \mathit{\boldsymbol{x}};t\right) \end{matrix}\right] \\{\mathit{\boldsymbol{\Phi }}}^{(k)}({x}_{3})=\left[ \begin{matrix}{\varphi }_{11}^{(k)}({x}_{3})&0&0&{\varphi }_{12}^{(k)}({x}_{3})\\0&{\varphi }_{22}^{(k)}({x}_{3})&{\varphi }_{21}^{(k)}({x}_{3})&0\\{\varphi }_{21}^{(k)}({x}_{3})&{\varphi }_{12}^{(k)}({x}_{3})&{\varphi }_{11}^{(k)}({x}_{3})&{\varphi }_{22}^{(k)}({x}_{3})\end{matrix}\right] \end{matrix}</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (6)
@@ Line 112: / Line 112: @@
-where <math display="inline">{{\mathit{\boldsymbol{\sigma }}}_{p}^{(k)}}^{T}\left( \mathit{\boldsymbol{x}},{x}_{3};t\right) =</math><math>⌊ \begin{matrix}{\sigma }_{11}^{(k)}&{\sigma }_{22}^{(k)}&{\tau }_{12}^{(k)}\end{matrix}⌋</math>  and <math display="inline">{{\mathit{\boldsymbol{\tau }}}_{t}^{(k)}}^{T}\left( \mathit{\boldsymbol{x}},{x}_{3};t\right) =</math><math>⌊ \begin{matrix}{\tau }_{13}^{(k)}&{\tau }_{23}^{(k)}\end{matrix}⌋</math>  are the in-plane and transverse shear stress components, respectively. Moreover, <math display="inline">{\breve{\mathit{\boldsymbol{C}}}}_{p}^{(k)}=</math><math>\left[ {\breve{C}}_{ij}\right] \left( i,j=1,2,6\right)</math>  and <math display="inline">{\breve{\mathit{\boldsymbol{C}}}}_{t}^{(k)}=</math><math>\left[ {\breve{C}}_{\alpha \beta }\right] \left( \alpha ,\beta =4,5\right)</math>  are, respectively, the in-plane reduced and transverse shear elastic stiffness coefficients of the ''k<sup>th </sup>''layer expressed in the plate axes.
+where <math display="inline">{{\mathit{\boldsymbol{\sigma }}}_{p}^{(k)}}^{T}\left( \mathit{\boldsymbol{x}},{x}_{3};t\right) =</math><math>\left[ \begin{matrix}{\sigma }_{11}^{(k)}&{\sigma }_{22}^{(k)}&{\tau }_{12}^{(k)}\end{matrix}\right]</math>  and <math display="inline">{{\mathit{\boldsymbol{\tau }}}_{t}^{(k)}}^{T}\left( \mathit{\boldsymbol{x}},{x}_{3};t\right) =</math><math>\left[ \begin{matrix}{\tau }_{13}^{(k)}&{\tau }_{23}^{(k)}\end{matrix}\right]</math>  are the in-plane and transverse shear stress components, respectively. Moreover, <math display="inline">{\breve{\mathit{\boldsymbol{C}}}}_{p}^{(k)}=</math><math>\left[ {\breve{C}}_{ij}\right] \left( i,j=1,2,6\right)</math>  and <math display="inline">{\breve{\mathit{\boldsymbol{C}}}}_{t}^{(k)}=</math><math>\left[ {\breve{C}}_{\alpha \beta }\right] \left( \alpha ,\beta =4,5\right)</math>  are, respectively, the in-plane reduced and transverse shear elastic stiffness coefficients of the ''k<sup>th </sup>''layer expressed in the plate axes.
 The governing equations and variationally consistent boundary conditions can be derived using the d’Alembert principle. It reads:
@@ Line 129: / Line 129: @@
 In Eq. <span id='cite-ZEqnNum297760'></span>[[#ZEqnNum297760|(9)]], denoting with <math display="inline">\delta</math>  the variational operator, appear (from left to the right) the virtual variation of the inertia forces, the virtual variation of the internal work and the virtual variation of the external works.
-For general plate problems, a robust and efficient finite element formulation can be used. The kinematic variables of a generic quadrilateral element (e), i.e.  <math display="inline">{{\mathit{\boldsymbol{d}}}^{(e)}}^{T}=</math><math>{⌊ \begin{matrix}{u}_{1}&{u}_{2}&w&{\theta }_{1}&{\theta }_{2}&{\psi }_{1}&{\psi }_{2}\end{matrix}⌋ }^{(e)}</math>, can be approximated by using the anisoparametric constrained interpolation strategy. This procedure, originally developed for Mindlin-based elements, has also been applied for RZT-based formulation to manage the shear-locking problem. In this scheme, the transverse displacement is interpolated using polynomials with one degree higher than those used for the other kinematic variables. The additional mid-edge nodes required for this interpolation are then condensed out by enforcing the following condition along the element edges (''s''  is the local edge coordinate), i.e. <math display="inline">{\gamma }_{s3,s}-</math><math>{\psi }_{n,s}=0</math>. It results:
+For general plate problems, a robust and efficient finite element formulation can be used. The kinematic variables of a generic quadrilateral element (e), i.e.  <math display="inline">{{\mathit{\boldsymbol{d}}}^{(e)}}^{T}=</math><math>{\left[ \begin{matrix}{u}_{1}&{u}_{2}&w&{\theta }_{1}&{\theta }_{2}&{\psi }_{1}&{\psi }_{2}\end{matrix}\right] }^{(e)}</math>, can be approximated by using the anisoparametric constrained interpolation strategy. This procedure, originally developed for Mindlin-based elements, has also been applied for RZT-based formulation to manage the shear-locking problem. In this scheme, the transverse displacement is interpolated using polynomials with one degree higher than those used for the other kinematic variables. The additional mid-edge nodes required for this interpolation are then condensed out by enforcing the following condition along the element edges (''s''  is the local edge coordinate), i.e. <math display="inline">{\gamma }_{s3,s}-</math><math>{\psi }_{n,s}=0</math>. It results:
 {| class="formulaSCP" style="width: 100%; text-align: center;"
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 The en-RZT model of the sandwich plate is made of 80x40 elements along ''x<sub>1</sub>'' and ''x<sub>2</sub>'', respectively, with a total number of dofs 23247, nearly thirty times less than those present in the high-fidelity 3D Nastran model. Hereafter are presented the results concerning the static bending analysis in terms of through-the-thickness distributions of displacements and strains.
+[[Image:Draft_Sorrenti_399626909-image2-c.png|600px]]
-{|
-|-
-| [[Image:Draft_Sorrenti_399626909-image2.png|198px]]
-| [[Image:Draft_Sorrenti_399626909-image3.png|center|198px]]
-|}
- [[Image:Draft_Sorrenti_399626909-image4.png|198px]]
 <div id="_Ref195294047" class="center" style="width: auto; margin-left: auto; margin-right: auto;">
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 According to <span id='cite-_Ref195294047'></span>[[#_Ref195294047|Figure 2]], the through-the-thickness distributions of the in-plane displacements are in good agreement with the 3D ones. Although there are some discrepancies, the percent error of the transverse displacement with respect to the average 3D value is -2.14 %. Higher discrepancies are observed in the core layer due to the limitation of the en-RZT linear displacement field. <span id='cite-_Ref195294239'></span>[[#_Ref195294239|Figure 3]] reports the through-the-thickness distribution of the in-plane strains. The en-RZT results are able to follow, in some cases with remarkable accuracy, the 3D strain distributions. Some discrepancies, especially in the core layer, are due to the limitation of the current linear model.
+[[Image:Draft_Sorrenti_399626909-image3-c.png|600px]]
-{|
-|-
-| [[Image:Draft_Sorrenti_399626909-image5.png|198px]]
-| [[Image:Draft_Sorrenti_399626909-image6.png|center|198px]]
-|}
- [[Image:Draft_Sorrenti_399626909-image7.png|198px]]
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-  [[Image:Draft_Sorrenti_399626909-image8-c.png|426px]] </div>
+  [[Image:Draft_Sorrenti_399626909-image4-c.png|426px]] </div>
 <div id="_Ref195294497" class="center" style="width: auto; margin-left: auto; margin-right: auto;">

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