Flores et al 2018c - Revision history

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← Older revision	Revision as of 15:01, 29 October 2018
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	+	Published in ''Comput. Methods Appl. Mech. Engrg.'', Vol. 194, pp. 907-932, 2005; <br />
	+	doi: 10.1016/j.cma.2003.08.012

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 The first author is a member of the scientific staff of the Science Research  Council of Argentina (CONICET). The support provided by grants of CONICET and  Agencia Córdoba Ciencia S.E. is gratefully acknowledged.
-==BIBLIOGRAPHY==
+==Bibliography==
 <div id="cite-1"></div>

@@ Line 347: / Line 347: @@
 |}
-These constitutive equations are integrated using a standard return algorithm.  The following Mises-Hill<sup><span id='citeF-11'></span>[[#cite-11|[11]]]</sup> yield function with non-linear isotropic  hardening is adopted
+These constitutive equations are integrated using a standard return algorithm.  The following Mises-Hill <span id='citeF-11'></span>[[#cite-11|[11]]] yield function with non-linear isotropic  hardening is adopted
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 701: / Line 701: @@
 |}
-In the original BST element<sup><span id='citeF-4'></span><span id='citeF-10'></span>[[#cite-4|[4,10]]]</sup> the gradient <math display="inline">\mathbf{\varphi  }_{\alpha }^{I}</math> was computed as the average of the linear approximations over  the two adjacent elements. In the new version, the gradient is  computed at each side <math display="inline">I</math> from the quadratic interpolation
+In the original BST element <span id='citeF-4'></span><span id='citeF-10'></span>[[#cite-4|[4,10]]] the gradient <math display="inline">\mathbf{\varphi  }_{\alpha }^{I}</math> was computed as the average of the linear approximations over  the two adjacent elements. In the new version, the gradient is  computed at each side <math display="inline">I</math> from the quadratic interpolation
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 793: / Line 793: @@
 |}
-The variations of '''t'''<math display="inline">_{3}</math> can be computed as shown in the  original BST element <sup><span id='citeF-10'></span>[[#cite-10|[10]]]</sup>.
+The variations of '''t'''<math display="inline">_{3}</math> can be computed as shown in the  original BST element <span id='citeF-10'></span>[[#cite-10|[10]]].
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 1,289: / Line 1,289: @@
 ==7 Numerical examples in the linear range==
-In this section, a summary of the results obtained in the assessment of the  proposed element in the linear range are presented. In this first part of the  numerical experiments, results have been obtained with a static/dynamic code  that uses a implicit solution of the discretized equilibrium equations. In the examples,  the original BST element<sup><span id='citeF-10'></span>[[#cite-10|[10]]]</sup> is compared with the enhanced version here proposed, denoted by CBST when a numerical integration is  performed with three points, and  CBST1 when only one integration point is  used. Note that one integration quadrature is equivalent to averaging  the metric tensors computed  at each side.
+In this section, a summary of the results obtained in the assessment of the  proposed element in the linear range are presented. In this first part of the  numerical experiments, results have been obtained with a static/dynamic code  that uses a implicit solution of the discretized equilibrium equations. In the examples,  the original BST element <span id='citeF-10'></span>[[#cite-10|[10]]] is compared with the enhanced version here proposed, denoted by CBST when a numerical integration is  performed with three points, and  CBST1 when only one integration point is  used. Note that one integration quadrature is equivalent to averaging  the metric tensors computed  at each side.
 ===7.1 Membrane patch test===
@@ Line 1,515: / Line 1,515: @@
 |}
-In Figure [[#img-7|7]].b the displacements of the points under the loads  have been plotted versus the number of nodes used in the discretization. Due  to the orientation of the meshes chosen, the displacement of the point under the  inward load is not the same as the displacement under the outward load, so in  the figure an average (the absolute values) has been used. Results  obtained with other elements have been included for comparison: two membrane  locking free elements namely the original linear BST and a transverse  shear-deformable quadrilateral (QUAD); a transverse shear deformable quadratic  triangle (TRIA)<sup><span id='citeF-15'></span>[[#cite-1|[1]]]</sup> that is vulnerable to locking and an  assumed strain version of this triangle (TRIA-1)<sup><span id='citeF-1'></span>[[#cite-1|[1]]]</sup> that  alleviates membrane locking but does not eliminate it.
+In Figure [[#img-7|7]].b the displacements of the points under the loads  have been plotted versus the number of nodes used in the discretization. Due  to the orientation of the meshes chosen, the displacement of the point under the  inward load is not the same as the displacement under the outward load, so in  the figure an average (the absolute values) has been used. Results  obtained with other elements have been included for comparison: two membrane  locking free elements namely the original linear BST and a transverse  shear-deformable quadrilateral (QUAD); a transverse shear deformable quadratic  triangle (TRIA) <span id='citeF-15'></span>[[#cite-1|[1]]] that is vulnerable to locking and an  assumed strain version of this triangle (TRIA-1)<sup><span id='citeF-1'></span>[[#cite-1|[1]]]</sup> that  alleviates membrane locking but does not eliminate it.
 From the plotted results it can be seen that the  CBST element presents  membrane locking in bending dominated problems with initially curved  geometries. This locking is much less severe than in a standard quadratic  triangle and even than for the assumed strain version used for comparison. When only  one integration point is used (CBST1 element) membrane locking disappears.
@@ Line 1,521: / Line 1,521: @@
 ==8 Non linear numerical examples==
-Results for examples with  geometric and material non-linearities are presented next using the CBST1 element. Due to the features of  the modelized problems, with strong nonlinearities associated to instabilities  and frictional contact conditions, a code with explicit integration of the dynamic  equilibrium equations has been used<sup><span id='citeF-14'></span>[[#cite-14|[14]]]</sup>. This code allows to  obtain pseudo-static solutions throught dynamic relaxation.
+Results for examples with  geometric and material non-linearities are presented next using the CBST1 element. Due to the features of  the modelized problems, with strong nonlinearities associated to instabilities  and frictional contact conditions, a code with explicit integration of the dynamic  equilibrium equations has been used <span id='citeF-14'></span>[[#cite-14|[14]]]. This code allows to  obtain pseudo-static solutions throught dynamic relaxation.
 ===8.1 Inflation of a sphere===
-The example is the inflation of a spherical shell under internal pressure. An  incompressible Mooney-Rivlin constitutive material have been considered.  The Ogden parameters are <math display="inline">N=2</math>, <math display="inline">\alpha _{1}=2</math>, <math display="inline">\mu _{1}=40</math>, <math display="inline">\alpha  _{2}=-2</math>, <math display="inline">\mu _{2}=-20</math>. Due to the simple geometry an analytical solution  exists<sup><span id='citeF-15'></span>[[#cite-15|[15]]]</sup> (with <math display="inline">\gamma =R/R^{\left( 0\right) }</math>):
+The example is the inflation of a spherical shell under internal pressure. An  incompressible Mooney-Rivlin constitutive material have been considered.  The Ogden parameters are <math display="inline">N=2</math>, <math display="inline">\alpha _{1}=2</math>, <math display="inline">\mu _{1}=40</math>, <math display="inline">\alpha  _{2}=-2</math>, <math display="inline">\mu _{2}=-20</math>. Due to the simple geometry an analytical solution  exists <span id='citeF-15'></span>[[#cite-15|[15]]] (with <math display="inline">\gamma =R/R^{\left( 0\right) }</math>):
 {| class="formulaSCP" style="width: 100%; text-align: left;"
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 ===8.3 Deep drawing of a rolled sheet===
-The element performance in problems involving large strains and  anisotropic plastic behaviour is assesed in a benchmark proposed in a recent  NUMISHEET<sup><span id='citeF-175'></span>[[#cite-17|[17]]]</sup>> meeting. This is the deep  drawing of a circular mild steel sheet with a spherical punch of radius <math display="inline">50 mm</math> (<math display="inline">R_{p}</math>). The original radius of the sheet (<math display="inline">R_{s}</math>) is <math display="inline">100 mm</math>  (<math display="inline">\frac{R_{L}}{R_{P}}=\beta=2</math>). The drawing depth is (<math display="inline">DD</math>) is 85 mm  (<math display="inline">\frac{DD}{R_{P}}=1.7</math>) and the force over the blank holder is <math display="inline">80 kN</math>. The  constitutive material of the rolled sheet is characterized by three tensile  tests along three different directions respect to rolling direction (<math display="inline">RD</math>) (Table  [[#table-4|4]]). Note that the modellization requires to treat  the material as transversally anisotropic, as this improves the deep  drawability and if not considered leads to an early localized necking of the  material. The yield function originally proposed by Hill<sup><span id='citeF-11'></span>[[#cite-11|[11]]]</sup> has been chosen for both associative and non-associative  potential functions. For elastic-plastic problems a numerical integration along  the thickness direction is necessary. This is accomplished here using four integration points.
+The element performance in problems involving large strains and  anisotropic plastic behaviour is assesed in a benchmark proposed in a recent  NUMISHEET <span id='citeF-175'></span>[[#cite-17|[17]]] meeting. This is the deep  drawing of a circular mild steel sheet with a spherical punch of radius <math display="inline">50 mm</math> (<math display="inline">R_{p}</math>). The original radius of the sheet (<math display="inline">R_{s}</math>) is <math display="inline">100 mm</math>  (<math display="inline">\frac{R_{L}}{R_{P}}=\beta=2</math>). The drawing depth is (<math display="inline">DD</math>) is 85 mm  (<math display="inline">\frac{DD}{R_{P}}=1.7</math>) and the force over the blank holder is <math display="inline">80 kN</math>. The  constitutive material of the rolled sheet is characterized by three tensile  tests along three different directions respect to rolling direction (<math display="inline">RD</math>) (Table  [[#table-4|4]]). Note that the modellization requires to treat  the material as transversally anisotropic, as this improves the deep  drawability and if not considered leads to an early localized necking of the  material. The yield function originally proposed by Hill<sup><span id='citeF-11'></span>[[#cite-11|[11]]]</sup> has been chosen for both associative and non-associative  potential functions. For elastic-plastic problems a numerical integration along  the thickness direction is necessary. This is accomplished here using four integration points.
@@ Line 1,647: / Line 1,647: @@
 |}
-The reported results are associated to three different meridiands: '''A'''  at 90<math display="inline">^{0}</math> of the rolling direction, '''B''': at 45<math display="inline">^{0}</math> of the rolling  direction and '''C''': in the rolling direction. The numerical results are  compared with a set of experimental values reported by Thyssen Krupp Stahl AG  (who proposed the benchmark and supplied the sheet samples)<span id='citeF-17'></span>[[#cite-17|[17]]]. It must be noted  that there was a great dispersion in the experimental data send to NUMISHEET,  so it is difficult to extract conclusions from a unique comparison. In any case the  experimental data supplied  seems to be consistent enought to have an  idea if the numerical model gives reasonable results.
+The reported results are associated to three different meridiands: '''A'''  at 90<math display="inline">^{0}</math> of the rolling direction, '''B''': at 45<math display="inline">^{0}</math> of the rolling  direction and '''C''': in the rolling direction. The numerical results are  compared with a set of experimental values reported by Thyssen Krupp Stahl AG  (who proposed the benchmark and supplied the sheet samples) <span id='citeF-17'></span>[[#cite-17|[17]]]. It must be noted  that there was a great dispersion in the experimental data send to NUMISHEET,  so it is difficult to extract conclusions from a unique comparison. In any case the  experimental data supplied  seems to be consistent enought to have an  idea if the numerical model gives reasonable results.
 The first element of comparison associated to the planar anisotropy, are the  different displacements (draw in) of the points in the boundary of the sheet. Table [[#table-5|5]] shows the values obtained for the three reference meridians chosen.

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 The Mises-Hill yield function has the advantage of simplicity and it allows, as a  first approximation, to treat rolled thin metal sheets with  planar and transversal anisotropy.
-For the case of rubbers, the Ogden<sup><span id='citeF-12'></span>[[#cite-12|[12]]]</sup> model extended to  the compressible range is considered. The material behaviour is characterized  by the strain energy density per unit undeformed volume of the form
+For the case of rubbers, the Ogden <span id='citeF-12'></span>[[#cite-12|[12]]] model extended to  the compressible range is considered. The material behaviour is characterized  by the strain energy density per unit undeformed volume of the form
 {| class="formulaSCP" style="width: 100%; text-align: left;"

@@ Line 9: / Line 9: @@
 ==1 Introduction==
-Development of efficient and robust shell finite elements for modelling large-scale  industrial applications has been a field of constant research in the last  thirty years. The continuining efforts of many scientists has focused in the direction of deriving simple elements (preferably triangles) applicable to large scale computation, typical of practical shell problems. Some recent successful shell triangles are reported in [1,2]. For a comprehensive review of shell elements see [3]. From the theoretical point of view Kirchhoff hypothesis are  enough for the simulation of the essential aspects of most shells problems,  including structures in civil, mechanical and naval engineering, sheet metal  forming, crash-worthiness analysis and others. Due the well-known difficulties  associated to <math display="inline">C^{1}</math> continuity, “thick shell” approaches that need <math display="inline">C^{0}</math>  continuity only have been extensively used, although they are computationally  more expensive due to the number of degrees of freedom necessary to obtain thin   solutions with similar precision. The development of shell elements without  transversal shear strains then collides with the need for C<math display="inline">^{1}</math> continuity, which is not  simple to satisfy in general conditions and has led to the development of  non-conforming approaches. Among the many options of this kind, the use of shell elements with only  displacements as degrees of freedom (rotation-free) is very attractive from  the computational point of view.
+Development of efficient and robust shell finite elements for modelling large-scale  industrial applications has been a field of constant research in the last  thirty years. The continuining efforts of many scientists has focused in the direction of deriving simple elements (preferably triangles) applicable to large scale computation, typical of practical shell problems. Some recent successful shell triangles are reported in <span id='citeF-1'></span>[[#cite-1|[1,2]]]. For a comprehensive review of shell elements see <span id='citeF-3'></span>[[#cite-3|[3]]]. From the theoretical point of view Kirchhoff hypothesis are  enough for the simulation of the essential aspects of most shells problems,  including structures in civil, mechanical and naval engineering, sheet metal  forming, crash-worthiness analysis and others. Due the well-known difficulties  associated to <math display="inline">C^{1}</math> continuity, “thick shell” approaches that need <math display="inline">C^{0}</math>  continuity only have been extensively used, although they are computationally  more expensive due to the number of degrees of freedom necessary to obtain thin   solutions with similar precision. The development of shell elements without  transversal shear strains then collides with the need for C<math display="inline">^{1}</math> continuity, which is not  simple to satisfy in general conditions and has led to the development of  non-conforming approaches. Among the many options of this kind, the use of shell elements with only  displacements as degrees of freedom (rotation-free) is very attractive from  the computational point of view.
 Few shell elements exists in the literature with only translations as degrees of  freedom. They are generally based on non-conforming approaches. A state of the  art can be found in Ref. <span id='citeF-4'></span>[[#cite-4|[4]]]. Recently Cirak y  Ortiz <span id='citeF-5'></span>[[#cite-5|[5]]] have developed a conforming thin shell element,  but it departs in some aspects from the standard finite element formulation. The  use of rotation-free shell elements in large scale simulations of sheet metal forming processes  is growing <span id='citeF-6'></span>[[#cite-6|[6]]]-<span id='citeF-9'></span>[[#cite-9|[9]]], specially in  explicit integration codes.

← Older revision	Revision as of 14:54, 29 October 2018
(No difference)

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 The first author is a member of the scientific staff of the Science Research  Council of Argentina (CONICET). The support provided by grants of CONICET and  Agencia Córdoba Ciencia S.E. is gratefully acknowledged.
-===BIBLIOGRAPHY===
+==BIBLIOGRAPHY==
 <div id="cite-1"></div>
@@ Line 1,714: / Line 1,714: @@
 <div id="cite-2"></div>
-'''[2]''' J.H. Argyris, M. Papadrakakis, C. Aportolopoulun and S. Koutsourelakis. The TRIC element. Theoretical and numerical investigation. ''Comput. Meth. Appl. Mech. Engrg.'', '''182''', 217&#8211;245, 2000.
+'''[[#citeF-2|[2]]]''' J.H. Argyris, M. Papadrakakis, C. Aportolopoulun and S. Koutsourelakis. The TRIC element. Theoretical and numerical investigation. ''Comput. Meth. Appl. Mech. Engrg.'', '''182''', 217&#8211;245, 2000.
 <div id="cite-3"></div>
@@ Line 1,729: / Line 1,729: @@
 <div id="cite-7"></div>
-'''[7]''' M. Brunet and F. Sabourin, “Prediction of necking and  wrinkles with a simplified shell element in sheet forming”, ''Int.  Conf. of Metal Forming Simulation in Industry'', Vol. II, pp. 27&#8211;48, B.  Kröplin (Ed.), 1994.
+'''[[#citeF-7|[7]]]''' M. Brunet and F. Sabourin, “Prediction of necking and  wrinkles with a simplified shell element in sheet forming”, ''Int.  Conf. of Metal Forming Simulation in Industry'', Vol. II, pp. 27&#8211;48, B.  Kröplin (Ed.), 1994.
 <div id="cite-8"></div>
-'''[8]''' G. Rio, B. Tathi and H. Laurent, “A new efficient finite  element model of shell with only three degrees of freedom per node.  Applications to industrial deep drawing test”, in ''Recent Developments  in Sheet Metal Forming Technoloy'', Ed. M.J.M. Barata Marques, 18th IDDRG  Biennial Congress, Lisbon, 1994.
+'''[[#citeF-8|[8]]]''' G. Rio, B. Tathi and H. Laurent, “A new efficient finite  element model of shell with only three degrees of freedom per node.  Applications to industrial deep drawing test”, in ''Recent Developments  in Sheet Metal Forming Technoloy'', Ed. M.J.M. Barata Marques, 18th IDDRG  Biennial Congress, Lisbon, 1994.
 <div id="cite-9"></div>

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 ===7.2 Bending patch test (torsion)===
 <div id='img-4'></div>
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 | colspan="1" | '''Figure 4:''' Patch test for uniform  torsion
 |}
 ===7.3 Cook's membrane problem===

@@ Line 1,292: / Line 1,292: @@
 ===7.1 Membrane patch test===
 <div id='img-3'></div>
@@ Line 1,300: / Line 1,302: @@
 | colspan="1" | '''Figure 3:''' Membrane patch  test
 |}
 ===7.2 Bending patch test (torsion)===

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   \mathbf{t}_{3}\otimes \mathbf{\tilde{\varphi }}_{^{\prime }2} \end{array}  \right] \right\}</math>
 |-
-| style="text-align: center;" | <math>   \sum _{I=1}^{3}\left[ \begin{array}{cc} \bar{N}_{^{\prime }1}^{I} & \bar{N}_{^{\prime }2}^{I} \end{array}  \right] \left[ \begin{array}{c}[c]{cc} M_{11} & M_{12}\\
+| style="text-align: center;" | <math>   \sum _{I=1}^{3}\left[ \begin{array}{cc} \bar{N}_{^{\prime }1}^{I} & \bar{N}_{^{\prime }2}^{I} \end{array}  \right] \left[ \begin{array}{cc} M_{11} & M_{12}\\
   M_{12} & M_{22} \end{array}  \right] \sum _{K=1}^{4}\left[ \begin{array}{c} N_{^{\prime }1}^{K\left( I\right) }\\
   N_{^{\prime }2}^{K\left( I\right) } \end{array}  \right] \Delta \mathbf{u}^{K\left( I\right) }</math>