De-Pouplana et al 2016a - Revision history

Cinmemj: Cinmemj moved page Draft Samper 843551501 to De-Pouplana et al 2016a

2019-07-11T12:34:29Z

Cinmemj moved page Draft Samper 843551501 to De-Pouplana et al 2016a

Cinmemj at 12:33, 11 July 2019

2019-07-11T12:33:56Z

Cinmemj at 09:31, 12 February 2019

2019-02-12T09:31:02Z

Cinmemj at 09:29, 12 February 2019

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Cinmemj at 09:28, 12 February 2019

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Cinmemj at 09:27, 12 February 2019

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Cinmemj at 11:17, 25 January 2019

2019-01-25T11:17:48Z

Cinmemj at 11:17, 25 January 2019

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 This research was partially funded by the Advanced Grant project SAFECON of the European Research Council and project NUMEXAS of the FP7 of the European Commission.
-==REFERENCES==
+==References==
 <div id="cite-1"></div>

@@ Line 561: / Line 561: @@
 ==5 Mesh-adaptive technique==
-In this work we have used the error estimation and adaptive mesh refinement strategy presented by Oñate and co-workers <span id='citeF-5'></span> <span id='citeF-6'></span>[[#cite-5|[5, 6]], <span id='citeF-27'></span> <span id='citeF-28'> </span><span id='citeF-29'></span>[[#cite-27|27-29]]].
+In this work we have used the error estimation and adaptive mesh refinement strategy presented by Oñate and co-workers <span id='citeF-5'></span> <span id='citeF-6'></span>[[#cite-5|[5]], [[#cite-6|6]], <span id='citeF-27'></span> <span id='citeF-28'> </span><span id='citeF-29'></span>[[#cite-27|27-29]]].
 The general algorithm of non-linear adaptive FEM analysis implemented in this work is described as follows. After reaching the equilibrium state and updating the solution, an error estimation is performed in order to evaluate the error distribution over the mesh. Then, a remeshing criterion uses the information about error distribution and determines the required mesh density. From this analysis, we obtain a new spatial discretization using a mesh generator interface.

@@ Line 11: / Line 11: @@
 Continuum damage mechanics is a branch of continuum mechanics that describes the progressive loss of material integrity due to the propagation and coalescence of micro-cracks, micro-voids, and similar defects. These changes in the micro-structure lead to an irreversible material degradation, characterized by a loss of stiffness that can be observed on the macro-scale.
-The term “continuum damage mechanics” was first used by <span id='citeF-14'></span>[[#cite-14|[14]]], but the concept of damage was introduced by Kachanov in 1958 in the context of creep rupture <span id='citeF-16'></span>[[#cite-16|[16]]]. In that work Kachanov introduced the concept of effective stress, and by using continuum damage he solved problems related to creep in metals. <span id='citeF-31'></span>[[#cite-31|[31]]] gave the problem physical meaning by suggesting that the reduction of the sectional area was measured by means of the damage parameter. The thermodynamic formalism involved in the irreversible process of damage was developed by <span id='citeF-18'></span>[[#cite-18|[18]]]. Other important contributions on damage mechanics include: <span id='citeF-22'></span> [[#cite-22|[22]]], <span id='citeF-33'></span> [[#cite-33|33]], <span id='citeF-26'></span> [[#cite-26|26]], <span id='citeF-24'></span> <span id='citeF-25'></span> [[#cite-24|24]],[[#cite-25|25]], <span id='citeF-8'></span> <span id='citeF-9'></span> [[#cite-8|8]],[[#cite-9|9]]], to name but a few.
+The term “continuum damage mechanics” was first used by <span id='citeF-14'></span>[[#cite-14|[14]]], but the concept of damage was introduced by Kachanov in 1958 in the context of creep rupture <span id='citeF-16'></span>[[#cite-16|[16]]]. In that work Kachanov introduced the concept of effective stress, and by using continuum damage he solved problems related to creep in metals. <span id='citeF-31'></span>[[#cite-31|[31]]] gave the problem physical meaning by suggesting that the reduction of the sectional area was measured by means of the damage parameter. The thermodynamic formalism involved in the irreversible process of damage was developed by <span id='citeF-18'></span>[[#cite-18|[18]]]. Other important contributions on damage mechanics include: <span id='citeF-22'></span> [[#cite-22|[22]], <span id='citeF-33'></span> [[#cite-33|33]], <span id='citeF-26'></span> [[#cite-26|26]], <span id='citeF-24'></span> <span id='citeF-25'></span> [[#cite-24|24]], [[#cite-25|25]], <span id='citeF-8'></span> <span id='citeF-9'></span> [[#cite-8|8]], [[#cite-9|9]]], to name but a few.
 The behaviour of brittle or quasi-brittle materials such as concrete, rocks, mortar or other geo-materials is particularly difficult to predict. In those cases failure is preceded by a gradual development of a non-linear fracture process zone and a localization of strain. Realistic failure analysis of such quasi-brittle structures requires the consideration of progressive damage due to micro-cracking, modelled by a constitutive law with strain softening. This typically results in highly non-linear structural responses and so efficient non-linear solvers based on arc-length control are needed for the numerical simulations <span id='citeF-13'></span>[[#cite-13|[13]]].

@@ Line 11: / Line 11: @@
 Continuum damage mechanics is a branch of continuum mechanics that describes the progressive loss of material integrity due to the propagation and coalescence of micro-cracks, micro-voids, and similar defects. These changes in the micro-structure lead to an irreversible material degradation, characterized by a loss of stiffness that can be observed on the macro-scale.
-The term “continuum damage mechanics” was first used by <span id='citeF-14'></span>[[#cite-14|[14]]], but the concept of damage was introduced by Kachanov in 1958 in the context of creep rupture <span id='citeF-16'></span>[[#cite-16|[16]]]. In that work Kachanov introduced the concept of effective stress, and by using continuum damage he solved problems related to creep in metals. <span id='citeF-31'></span>[[#cite-31|[31]]] gave the problem physical meaning by suggesting that the reduction of the sectional area was measured by means of the damage parameter. The thermodynamic formalism involved in the irreversible process of damage was developed by <span id='citeF-18'></span>[[#cite-18|[18]]]. Other important contributions on damage mechanics include: <span id='citeF-22'></span> [[#cite-22|[22]]], <span id='citeF-33'></span> [[#cite-33|[33]]], <span id='citeF-26'></span> [[#cite-26|[26]]], <span id='citeF-24'></span> <span id='citeF-25'></span> [[#cite-24|[24]], [[#cite-25|25]]], <span id='citeF-8'></span> <span id='citeF-9'></span> [[#cite-8|[8]],[[#cite-9|9]]], to name but a few.
+The term “continuum damage mechanics” was first used by <span id='citeF-14'></span>[[#cite-14|[14]]], but the concept of damage was introduced by Kachanov in 1958 in the context of creep rupture <span id='citeF-16'></span>[[#cite-16|[16]]]. In that work Kachanov introduced the concept of effective stress, and by using continuum damage he solved problems related to creep in metals. <span id='citeF-31'></span>[[#cite-31|[31]]] gave the problem physical meaning by suggesting that the reduction of the sectional area was measured by means of the damage parameter. The thermodynamic formalism involved in the irreversible process of damage was developed by <span id='citeF-18'></span>[[#cite-18|[18]]]. Other important contributions on damage mechanics include: <span id='citeF-22'></span> [[#cite-22|[22]]], <span id='citeF-33'></span> [[#cite-33|33]], <span id='citeF-26'></span> [[#cite-26|26]], <span id='citeF-24'></span> <span id='citeF-25'></span> [[#cite-24|24]],[[#cite-25|25]], <span id='citeF-8'></span> <span id='citeF-9'></span> [[#cite-8|8]],[[#cite-9|9]]], to name but a few.
 The behaviour of brittle or quasi-brittle materials such as concrete, rocks, mortar or other geo-materials is particularly difficult to predict. In those cases failure is preceded by a gradual development of a non-linear fracture process zone and a localization of strain. Realistic failure analysis of such quasi-brittle structures requires the consideration of progressive damage due to micro-cracking, modelled by a constitutive law with strain softening. This typically results in highly non-linear structural responses and so efficient non-linear solvers based on arc-length control are needed for the numerical simulations <span id='citeF-13'></span>[[#cite-13|[13]]].

@@ Line 11: / Line 11: @@
 Continuum damage mechanics is a branch of continuum mechanics that describes the progressive loss of material integrity due to the propagation and coalescence of micro-cracks, micro-voids, and similar defects. These changes in the micro-structure lead to an irreversible material degradation, characterized by a loss of stiffness that can be observed on the macro-scale.
-The term “continuum damage mechanics” was first used by <span id='citeF-14'></span>[[#cite-14|[14]]], but the concept of damage was introduced by Kachanov in 1958 in the context of creep rupture <span id='citeF-16'></span>[[#cite-16|[16]]]. In that work Kachanov introduced the concept of effective stress, and by using continuum damage he solved problems related to creep in metals. <span id='citeF-31'></span>[[#cite-31|[31]]] gave the problem physical meaning by suggesting that the reduction of the sectional area was measured by means of the damage parameter. The thermodynamic formalism involved in the irreversible process of damage was developed by <span id='citeF-18'></span>[[#cite-18|[18]]]. Other important contributions on damage mechanics include: <span id='citeF-22'></span> [[#cite-22|[22]]], <span id='citeF-33'></span> [[#cite-33|[33]]], <span id='citeF-26'></span> [[#cite-26|[26]]], <span id='citeF-24'></span> <span id='citeF-25'></span> [[#cite-24|[24, 25]]], <span id='citeF-8'></span> <span id='citeF-9'></span> [[#cite-8|[8, 9]]], to name but a few.
+The term “continuum damage mechanics” was first used by <span id='citeF-14'></span>[[#cite-14|[14]]], but the concept of damage was introduced by Kachanov in 1958 in the context of creep rupture <span id='citeF-16'></span>[[#cite-16|[16]]]. In that work Kachanov introduced the concept of effective stress, and by using continuum damage he solved problems related to creep in metals. <span id='citeF-31'></span>[[#cite-31|[31]]] gave the problem physical meaning by suggesting that the reduction of the sectional area was measured by means of the damage parameter. The thermodynamic formalism involved in the irreversible process of damage was developed by <span id='citeF-18'></span>[[#cite-18|[18]]]. Other important contributions on damage mechanics include: <span id='citeF-22'></span> [[#cite-22|[22]]], <span id='citeF-33'></span> [[#cite-33|[33]]], <span id='citeF-26'></span> [[#cite-26|[26]]], <span id='citeF-24'></span> <span id='citeF-25'></span> [[#cite-24|[24]], [[#cite-25|25]]], <span id='citeF-8'></span> <span id='citeF-9'></span> [[#cite-8|[8]],[[#cite-9|9]]], to name but a few.
 The behaviour of brittle or quasi-brittle materials such as concrete, rocks, mortar or other geo-materials is particularly difficult to predict. In those cases failure is preceded by a gradual development of a non-linear fracture process zone and a localization of strain. Realistic failure analysis of such quasi-brittle structures requires the consideration of progressive damage due to micro-cracking, modelled by a constitutive law with strain softening. This typically results in highly non-linear structural responses and so efficient non-linear solvers based on arc-length control are needed for the numerical simulations <span id='citeF-13'></span>[[#cite-13|[13]]].
@@ Line 21: / Line 21: @@
 The first non-local models of this type, proposed in the 1960s, aimed at improving the description of elastic wave dispersions in crystals. Non-local elasticity was further developed by <span id='citeF-12'></span>[[#cite-12|[12]]] who later extended it to non-local elasto-plasticity <span id='citeF-11'></span>[[#cite-11|[11]]]. Subsequently, it was found that certain non-local formulations could act as efficient localization limiters with a regularizing effect on problems with strain localization <span id='citeF-30'></span>[[#cite-30|[30]]].
-Non-local models lead to smooth solutions with a continuous variation of strain. However, to resolve narrow bands of highly localized strains using the finite element method it is necessary to use sufficiently fine computational grids. Fortunately, the mesh must be fine only in the damage progression zone, while the remaining part of the structure can be reasonably well represented by a coarser mesh. In general, the localization pattern is not known in advance, and it is actually tedious to suitably construct refined meshes by hand. Thereby, the efficiency of the analysis can be greatly increased by means of an adaptive mesh refinement technique, which automates the whole process <span id='citeF-5'></span><span id='citeF-29'></span>[[#cite-5|[5, 29]]].
+Non-local models lead to smooth solutions with a continuous variation of strain. However, to resolve narrow bands of highly localized strains using the finite element method it is necessary to use sufficiently fine computational grids. Fortunately, the mesh must be fine only in the damage progression zone, while the remaining part of the structure can be reasonably well represented by a coarser mesh. In general, the localization pattern is not known in advance, and it is actually tedious to suitably construct refined meshes by hand. Thereby, the efficiency of the analysis can be greatly increased by means of an adaptive mesh refinement technique, which automates the whole process <span id='citeF-5'></span><span id='citeF-29'></span>[[#cite-5|[5]],[[#cite-29|29]]].
 In the present work we present a robust non-local isotropic damage model for quasi-brittle materials that works in a small deformation regime, along with an adaptive-mesh finite element technique that permits adapting the spatial discretization in an optimal manner.

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 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: left;" | <math>\boldsymbol{K}_{tan}  =\frac{\partial \boldsymbol{f}_{int}}{\partial \boldsymbol{a}}</math>
+| style="text-align: center;" | <math>\boldsymbol{K}_{tan}  =\frac{\partial \boldsymbol{f}_{int}}{\partial \boldsymbol{a}}</math>
 |-
-| style="text-align: left;" | <math>=\sum_pw_p\boldsymbol{B}_p^T(1-d_p)\boldsymbol{E}\boldsymbol{B}_p-\sum _pw_p\boldsymbol{B}_p^T\boldsymbol{E}\boldsymbol{B}_p\boldsymbol{a}{g'}_pt_p\frac{\partial \tilde{\varepsilon }_{eq,p}}{\partial \boldsymbol{a}} </math>
+| style="text-align: center;" | <math>=\sum_pw_p\boldsymbol{B}_p^T(1-d_p)\boldsymbol{E}\boldsymbol{B}_p-\sum _pw_p\boldsymbol{B}_p^T\boldsymbol{E}\boldsymbol{B}_p\boldsymbol{a}{g'}_pt_p\frac{\partial \tilde{\varepsilon }_{eq,p}}{\partial \boldsymbol{a}} </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (29)

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 | style="text-align: left;" | <math>\boldsymbol{K}_{tan}  =\frac{\partial \boldsymbol{f}_{int}}{\partial \boldsymbol{a}}</math>
 |-
-| style="text-align: left;" | <math>\hspace{2cm} =\sum_pw_p\boldsymbol{B}_p^T(1-d_p)\boldsymbol{E}\boldsymbol{B}_p-\sum _pw_p\boldsymbol{B}_p^T\boldsymbol{E}\boldsymbol{B}_p\boldsymbol{a}{g'}_pt_p\frac{\partial \tilde{\varepsilon }_{eq,p}}{\partial \boldsymbol{a}} </math>
+|\hspace{2cm}
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (29)