Cardoso et al 2019a - Revision history

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@@ Line 999: / Line 999: @@
 |}
-where <math display="inline">\mathbf{e}_q</math> is the <math display="inline">q^{th}</math> standard basis vector. The Figures [[#img-10|10]] and [[#img-11|11]] illustrates the original and the normalized simplices with the corresponding node numeration for 2D and 3D respectively. <div id='img-10'></div>
+where <math display="inline">\mathbf{e}_q</math> is the <math display="inline">q^{th}</math> standard basis vector. Figures [[#img-10|10]] and [[#img-11|11]] illustrate the original and the normalized simplices with the corresponding node numeration for 2D and 3D respectively. <div id='img-10'></div>
 {| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-

@@ Line 3,013: / Line 3,013: @@
 =Appendix B. Polynomial shape functions for time=
-The analytical solution [[#eq-5.11|5.11]] of equation [[#eq-5.2|5.2]] is used to generate numerical schemes for time discretization, these approximations has a similar structure but distinct coefficients that depend on the  shape function <math display="inline">{P}</math> used for time variation of stress state. In this appendix we propose two families of polynomial functions in order to get a continuous stress state in contiguous time steps, such polynomial functions meet the condition of <math display="inline">\dot{{C}}_{P}^{\Delta t}=\Delta t</math> for producing stable schemes in terms of total energy.
+The analytical solution [[#eq-5.11|(5.11)]] of equation [[#eq-5.2|(5.2)]] is used to generate numerical schemes for time discretization, these approximations has a similar structure but distinct coefficients that depend on the  shape function <math display="inline">{P}</math> used for time variation of stress state. In this appendix we propose two families of polynomial functions in order to get a continuous stress state in contiguous time steps, such polynomial functions meet the condition of <math display="inline">\dot{{C}}_{P}^{\Delta t}=\Delta t</math> for producing stable schemes in terms of total energy.
 The first polynomial family is defined by
@@ Line 3,032: / Line 3,032: @@
 {|  class="floating_tableSCP wikitable" style="text-align: left; margin: 1em auto;min-width:50%;"
-|+ style="font-size: 75%;" |<span id='table-2'></span>Table. 2 First few polynomials generated with equation [[#eq-10.1|10.1]] and its respective convolutions <math>{C}_{P}(t)</math>.
+|+ style="font-size: 75%;" |<span id='table-2'></span>Table. 2 First few polynomials generated with equation [[#eq-B.1|(B.1)]] and its respective convolutions <math>{C}_{P}(t)</math>.
 |- style="border-top: 2px solid;"
 | style="text-align: center;border-left: 2px solid;border-right: 2px solid;" | <math display="inline">p</math>
@@ Line 3,057: / Line 3,057: @@
 |[[Image:Draft_Samper_795975371_8969_monograph-picture-658a8e.png|600px|Curves for first few polynomials generated with equation [[#eq-10.1|10.1]]]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure 50:''' Curves for first few polynomials generated with equation [[#eq-10.1|10.1]]
+| colspan="1" | '''Figure 50:''' Curves for first few polynomials generated with equation [[#eq-B.1|(B.1)]]
 |}
@@ Line 3,077: / Line 3,077: @@
 {|  class="floating_tableSCP wikitable" style="text-align: left; margin: 1em auto;min-width:50%;"
-|+ style="font-size: 75%;" |<span id='table-3'></span>Table. 3 First few polynomials generated with equation [[#eq-B.2|B.2]] and its respective convolutions <math>{C}_{P}(t)</math>.
+|+ style="font-size: 75%;" |<span id='table-3'></span>Table. 3 First few polynomials generated with equation [[#eq-B.2|(B.2)]] and its respective convolutions <math>{C}_{P}(t)</math>.
 |- style="border-top: 2px solid;"
 | style="text-align: center;border-left: 2px solid;border-right: 2px solid;" | <math display="inline">p</math>
@@ Line 3,098: / Line 3,098: @@
 |[[Image:Draft_Samper_795975371_3913_monograph-picture-552cf0.png|600px|Curves for first few polynomials generated with equation [[#eq-B.2|B.2]]]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure 51:''' Curves for first few polynomials generated with equation [[#eq-B.2|B.2]]
+| colspan="1" | '''Figure 51:''' Curves for first few polynomials generated with equation [[#eq-B.2|(B.2)]]
 |}

@@ Line 2,858: / Line 2,858: @@
 =Appendix A. Analytical solution for time=
-In order to get an accurate approximation of the temporal derivative in the interval <math display="inline">t \in [0, \Delta t]</math>, we replace the stress state function of time <math display="inline">S_i(t):\mathbb{R}\rightarrow \mathbb{R}^{\hbox{dim}}</math> into the differential equation [[#eq-5.2|5.2]],
+In order to get an accurate approximation of the temporal derivative in the interval <math display="inline">t \in [0, \Delta t]</math>, we replace the stress state function of time <math display="inline">S_i(t):\mathbb{R}\rightarrow \mathbb{R}^{\hbox{dim}}</math> into the differential equation [[#eq-5.2|(5.2)]],
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 2,870: / Line 2,870: @@
 |}
-stress function is defined in equation [[#eq-5.4|5.4]]. Reordering terms we have
+stress function is defined in equation [[#eq-5.4|(5.4)]]. Reordering terms we have
 <span id="eq-A.2"></span>
@@ Line 2,896: / Line 2,896: @@
 |}
-In order to solve [[#eq-A.2|A.2]] by means of Laplace Transform
+In order to solve [[#eq-A.2|(A.2)]] by means of Laplace Transform
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 2,908: / Line 2,908: @@
 |}
-we change from time domain in equation [[#eq-A.2|A.2]] to frequency domain
+we change from time domain in equation [[#eq-A.2|(A.2)]] to frequency domain
 <span id="eq-A.5"></span>
@@ Line 2,933: / Line 2,933: @@
 |}
-and the Laplace transform of acceleration term includes initial conditions [[#eq-A.3|A.3]]
+and the Laplace transform of acceleration term includes initial conditions [[#eq-A.3|(A.3)]]
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 2,945: / Line 2,945: @@
 |}
-We can rewrite equation [[#eq-A.5|A.5]] as
+We can rewrite equation [[#eq-A.5|(A.5)]] as
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 3,009: / Line 3,009: @@
 |}
-Finally, analytical solution of [[#eq-A.2|A.2]] is equation [[#eq-A.9|A.9]] and it is completely dependent of the shape function <math display="inline">{P}</math>. This solution is used for building an accurate numerical squeme for discretizing time.
+Finally, analytical solution of [[#eq-A.2|(A.2)]] is equation [[#eq-A.9|(A.9)]] and it is completely dependent of the shape function <math display="inline">{P}</math>. This solution is used for building an accurate numerical squeme for discretizing time.
 =Appendix B. Polynomial shape functions for time=

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 |}
-Plane strain is assumed in a quasi-static analysis using 100 finite increments and smoothness <math display="inline">c=1</math>.  We perform two separate analysis in distinct meshes, the first one has an average size <math display="inline">\Delta \mathbf{x}= 12~mm </math> (2849 discrete volumes) and the second one has <math display="inline">\Delta \mathbf{x}= 7.5~mm</math> (7255 discrete volumes). Figure [[#img-40|40]] illustrates damage field obtained with our numerical approach, and its corresponding displacement deformated with a factor x100. The differences between both discretizations can be appreciated in this Figure. In contrast with most solutions obtained with damage models based on Finite Element Method, we get asymmetrical crack morphology induced by discrete volumes shape. <div id='img-40'></div>
+Plane strain is assumed in a quasi-static analysis using 100 finite increments and smoothness <math display="inline">c=1</math>.  We perform two separate analysis in distinct meshes, the first one has an average size <math display="inline">\Delta \mathbf{x}= 12~mm </math> (2849 discrete volumes) and the second one has <math display="inline">\Delta \mathbf{x}= 7.5~mm</math> (7255 discrete volumes). Figure [[#img-40|40]] illustrates damage field obtained with our numerical approach, and its corresponding displacement deformated with a factor x100. The differences between both discretizations can be appreciated in this figure. In contrast with most solutions obtained with damage models based on Finite Element Method, we get asymmetrical crack morphology induced by discrete volumes shape. <div id='img-40'></div>
 {| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-

@@ Line 1,853: / Line 1,853: @@
 ==5.2 Analytical solution==
-Using the asumption in [[#eq-5.4|5.4]], we get the analytical solution of the  equation [[#eq-5.2|(5.2)]] for the interval <math display="inline">t\in [0,\Delta t]</math> by means of the Laplace transform (see appendix [[#A Analytical solution for time|A]] for details),
+Using the asumption in [[#eq-5.4|(5.4)]], we get the analytical solution of the  equation [[#eq-5.2|(5.2)]] for the interval <math display="inline">t\in [0,\Delta t]</math> by means of the Laplace transform (see appendix [[#A Analytical solution for time|A]] for details),
 <span id="eq-5.5"></span>

← Older revision	Revision as of 11:46, 31 May 2019
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 =4 Second equation of motion=
-In this chapter we will focus on the numerical treatment of the second equation of motion  [[#eq-2.36.b|2.36.b]], this equation describes the damage mechanics within the physical system by considering the potential energy produced by tensile stress.
+In this chapter we will focus on the numerical treatment of the second equation of motion  [[#eq-2.36.b|(2.36.b)]], this equation describes the damage mechanics within the physical system by considering the potential energy produced by tensile stress.
 As discussed in the mathematical formulation, the damage field is a smooth approximation of the fracture surface, a benefit of this approach is that fracture morphology is completely defined by the solution of this equation and we do not have to track the crack propagation with auxiliary checking procedures neither to check for new crack nucleations. However, it is important to be aware about the effects over the stress field produced by the scale length parameter <math display="inline">h</math> which controls the smoothness of damage field solution. We observe that a length parameter proportional to the average size of control volumes, denoted <math display="inline">\Delta \mathbf{x}</math>, produces accurate results, these mesh size is taken as

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 =Bibliography=
 <div id=cite-1></div>
 [[#citeF-1|[1]]] A.K. Slone and K. Pericleous and C. Bailey and M. Cross. (2002) Dynamic fluid-structure interaction using finite volume unstructured mesh procedures, Volume 80. Elsevier. Computers and Structures 371-390
@@ Line 3,279: / Line 3,280: @@
 <div id=cite-58></div>
 [[#citeF-58|[58]]] T. N. Bittencourt. (1996) Quasi-automatic simulation of crack propagation for 2D LEFM problems, Volume 55. Elsevier. Engineering Fracture Mechanics 321-334