Oller-Aramayo et al 2017a - Revision history

Cinmemj: Cinmemj moved page Draft Samper 893761715 to Oller-Aramayo et al 2017a

2018-06-05T10:26:13Z

Cinmemj moved page Draft Samper 893761715 to Oller-Aramayo et al 2017a

Cinmemj at 10:19, 5 June 2018

2018-06-05T10:19:14Z

Cinmemj at 10:04, 5 June 2018

2018-06-05T10:04:16Z

Cinmemj at 10:00, 5 June 2018

2018-06-05T10:00:42Z

Cinmemj at 09:59, 5 June 2018

2018-06-05T09:59:05Z

Cinmemj at 09:58, 5 June 2018

2018-06-05T09:58:00Z

Cinmemj at 09:55, 5 June 2018

2018-06-05T09:55:35Z

Cinmemj at 09:52, 5 June 2018

2018-06-05T09:52:11Z

Cinmemj at 09:49, 5 June 2018

2018-06-05T09:49:35Z

Cinmemj at 09:47, 5 June 2018

2018-06-05T09:47:49Z

@@ Line 1,492: / Line 1,492: @@
 |-
 |  style="border-left: 1pt solid black;border-bottom: 1pt solid black;border-right: 1pt solid black;vertical-align: bottom;"|'''Steel'''
-|  style="border-bottom: 1pt solid black;border-right: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">1.8</span>
+|  style="border-bottom: 1pt solid black;border-right: 1pt solid black;text-align: center;vertical-align: bottom;"|1.8
-|  style="border-bottom: 1pt solid black;border-right: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">217</span>
+|  style="border-bottom: 1pt solid black;border-right: 1pt solid black;text-align: center;vertical-align: bottom;"|217
-|  style="border-bottom: 1pt solid black;border-right: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">25.324,00</span>
+|  style="border-bottom: 1pt solid black;border-right: 1pt solid black;text-align: center;vertical-align: bottom;"|25.324,00
-|  style="border-bottom: 1pt solid black;border-right: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">2,21E-03</span>
+|  style="border-bottom: 1pt solid black;border-right: 1pt solid black;text-align: center;vertical-align: bottom;"|2,21E-03
 |-
 |  style="border-left: 1pt solid black;border-bottom: 1pt solid black;border-right: 1pt solid black;vertical-align: bottom;"|'''Comp. ±45º'''
-|  style="border-bottom: 1pt solid black;border-right: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">6×0.3=1.8</span>
+|  style="border-bottom: 1pt solid black;border-right: 1pt solid black;text-align: center;vertical-align: bottom;"|6×0.3=1.8
-|  style="border-bottom: 1pt solid black;border-right: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">40</span>
+|  style="border-bottom: 1pt solid black;border-right: 1pt solid black;text-align: center;vertical-align: bottom;"|40
-|  style="border-bottom: 1pt solid black;border-right: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">13.397,00</span>
+|  style="border-bottom: 1pt solid black;border-right: 1pt solid black;text-align: center;vertical-align: bottom;"|13.397,00
-|  style="border-bottom: 1pt solid black;border-right: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">1,65E-03</span>
+|  style="border-bottom: 1pt solid black;border-right: 1pt solid black;text-align: center;vertical-align: bottom;"|1,65E-03
 |-
 |  style="border-left: 1pt solid black;border-bottom: 1pt solid black;border-right: 1pt solid black;vertical-align: bottom;"|'''Comp. 0º/45º'''
-|  style="border-bottom: 1pt solid black;border-right: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">6×0.3=1.8</span>
+|  style="border-bottom: 1pt solid black;border-right: 1pt solid black;text-align: center;vertical-align: bottom;"|6×0.3=1.8
-|  style="border-bottom: 1pt solid black;border-right: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">40</span>
+|  style="border-bottom: 1pt solid black;border-right: 1pt solid black;text-align: center;vertical-align: bottom;"|40
-|  style="border-bottom: 1pt solid black;border-right: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">10.622,00</span>
+|  style="border-bottom: 1pt solid black;border-right: 1pt solid black;text-align: center;vertical-align: bottom;"|10.622,00
-|  style="border-bottom: 1pt solid black;border-right: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">2,73E-03</span>
+|  style="border-bottom: 1pt solid black;border-right: 1pt solid black;text-align: center;vertical-align: bottom;"|2,73E-03
 |-
 |  style="border-left: 1pt solid black;border-bottom: 1pt solid black;border-right: 1pt solid black;vertical-align: bottom;"|'''Comp. 0º/90º'''
-|  style="border-bottom: 1pt solid black;border-right: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">6×0.3=1.8</span>
+|  style="border-bottom: 1pt solid black;border-right: 1pt solid black;text-align: center;vertical-align: bottom;"|6×0.3=1.8
-|  style="border-bottom: 1pt solid black;border-right: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">40</span>
+|  style="border-bottom: 1pt solid black;border-right: 1pt solid black;text-align: center;vertical-align: bottom;"|40
-|  style="border-bottom: 1pt solid black;border-right: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">12.342,00</span>
+|  style="border-bottom: 1pt solid black;border-right: 1pt solid black;text-align: center;vertical-align: bottom;"|12.342,00
-|  style="border-bottom: 1pt solid black;border-right: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">2,42E-03</span>
+|  style="border-bottom: 1pt solid black;border-right: 1pt solid black;text-align: center;vertical-align: bottom;"|2,42E-03
 |}</div>

@@ Line 1,214: / Line 1,214: @@
 The numerical simulations have been carried out using a linear rectangular finite element mesh of 4 nodes with a total of 343 elements, 392 nodes and 766 degrees of freedom for the micro model and 291 elements, 336 nodes and 644 degrees of freedom for the macro model. Figure 17 shows the geometry, material assignment, meshes and boundary conditions used for each case.
-{| style="width: 100%;border-collapse: collapse;"
+{| style="width: 100%;border-collapse: collapse;font-size:85%;"
 |-
 |  style="border: 1pt solid black;vertical-align: top;"|
-|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">'''Material 1 Matrix of the'''</span>
+|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|'''Material 1 Matrix of the'''
-<span style="text-align: center; font-size: 75%;">'''Concrete'''</span>
+'''Concrete'''
-|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">'''Material 2'''</span>
+|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|'''Material 2'''
-<span style="text-align: center; font-size: 75%;">'''Steel'''</span>
+'''Steel'''
-<span style="text-align: center; font-size: 75%;">'''Reinforcement'''</span>
+'''Reinforcement'''
-|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">'''Material 3'''</span>
+|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|'''Material 3'''
-<span style="text-align: center; font-size: 75%;">'''Matrix-Reinforcement'''</span>
+'''Matrix-Reinforcement'''
-<span style="text-align: center; font-size: 75%;">'''Interface'''</span>
+'''Interface'''
 |-
-|  style="border: 1pt solid black;font-size: 75%;"|'''Type of material behavior'''
+|  style="border: 1pt solid black;"|'''Type of material behavior'''
-|  style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">Mohr-Coulomb</span>
+|  style="border: 1pt solid black;text-align: center;"|Mohr-Coulomb
-<span style="text-align: center; font-size: 75%;">Isotropic Elasto-plastic</span>
+Isotropic Elasto-plastic
-<span style="text-align: center; font-size: 75%;">Model</span>
+Model
-|  style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">Isotropic  Elastic</span>
+|  style="border: 1pt solid black;text-align: center;"|Isotropic  Elastic
-<span style="text-align: center; font-size: 75%;">Model</span>
+Model
-|  style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">Kachanov  Damage</span>
+|  style="border: 1pt solid black;text-align: center;"|Kachanov  Damage
-<span style="text-align: center; font-size: 75%;">Model</span>
+Model
 |-
-|  style="border: 1pt solid black;font-size: 75%;"|'''Young Modulus [kp/cm<sup>2</sup>]'''
+|  style="border: 1pt solid black;"|'''Young Modulus [kp/cm<sup>2</sup>]'''
-|  style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">3,5×10<sup>5</sup></span>
+|  style="border: 1pt solid black;text-align: center;"|3,5×10<sup>5</sup>
-|  style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">2,1×10<sup>6</sup></span>
+|  style="border: 1pt solid black;text-align: center;"|2,1×10<sup>6</sup>
-|  style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">3,5×10<sup>5</sup></span>
+|  style="border: 1pt solid black;text-align: center;"|3,5×10<sup>5</sup>
 |-
-|  style="border: 1pt solid black;font-size: 75%;"|'''Poisson Coefficient'''
+|  style="border: 1pt solid black;"|'''Poisson Coefficient'''
-|  style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">0,2</span>
+|  style="border: 1pt solid black;text-align: center;"|0,2
-|  style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">0,0</span>
+|  style="border: 1pt solid black;text-align: center;"|0,0
-|  style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">0,0</span>
+|  style="border: 1pt solid black;text-align: center;"|0,0
 |-
-|  style="border: 1pt solid black;font-size: 75%;"|'''Internal Friction'''
+|  style="border: 1pt solid black;"|'''Internal Friction'''
-|  style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">30º</span>
+|  style="border: 1pt solid black;text-align: center;"|30º
-|  style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">-</span>
+|  style="border: 1pt solid black;text-align: center;"|-
-|  style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">30º</span>
+|  style="border: 1pt solid black;text-align: center;"|30º
 |-
-|  style="border: 1pt solid black;font-size: 75%;"|'''Compression Strength [kp/cm<sup>2</sup>]'''
+|  style="border: 1pt solid black;"|'''Compression Strength [kp/cm<sup>2</sup>]'''
-|  style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">200</span>
+|  style="border: 1pt solid black;text-align: center;"|200
 |  style="border: 1pt solid black;text-align: center;"|2000
 |  style="border: 1pt solid black;text-align: center;"|20
 |-
-|  style="border: 1pt solid black;font-size: 75%;"|'''Tension Strength [kp/cm<sup>2</sup>]'''
+|  style="border: 1pt solid black;"|'''Tension Strength [kp/cm<sup>2</sup>]'''
-|  style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">20</span>
+|  style="border: 1pt solid black;text-align: center;"|20
-|  style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">2000</span>
+|  style="border: 1pt solid black;text-align: center;"|2000
-|  style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">20</span>
+|  style="border: 1pt solid black;text-align: center;"|20
 |-
 |  style="border: 1pt solid black;"|'''G<sub>f</sub> , G<sub>c </sub>[kp/cm]'''
-|  style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">0,25  ,   26,0</span>
+|  style="border: 1pt solid black;text-align: center;"|0,25  ,   26,0
 |  style="border: 1pt solid black;text-align: center;"|
-<span style="text-align: center; font-size: 75%;">* </span>
-|  style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">2,0  ,  2,0</span>
+|  style="border: 1pt solid black;text-align: center;"|2,0  ,  2,0
 |-
-|  style="border: 1pt solid black;font-size: 75%;"|'''Behavior law after the '''
+|  style="border: 1pt solid black;"|'''Behavior law after the '''
 '''Yield point'''
-|  style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">Line function </span>
+|  style="border: 1pt solid black;text-align: center;"|Line function
-<span style="text-align: center; font-size: 75%;">with softening</span>
+with softening
 |  style="border: 1pt solid black;text-align: center;"|
-<span style="text-align: center; font-size: 75%;">* </span>
-|  style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">Exponential function</span>
+|  style="border: 1pt solid black;text-align: center;"|Exponential function
-<span style="text-align: center; font-size: 75%;">with softening</span>
+with softening
 |}

@@ Line 1,231: / Line 1,231: @@
 <span style="text-align: center; font-size: 75%;">'''Interface'''</span>
 |-
-|  style="border: 1pt solid black;"|'''Type of material behavior'''
+|  style="border: 1pt solid black;font-size: 75%;"|'''Type of material behavior'''
 |  style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">Mohr-Coulomb</span>
@@ Line 1,244: / Line 1,244: @@
 <span style="text-align: center; font-size: 75%;">Model</span>
 |-
-|  style="border: 1pt solid black;"|'''Young Modulus [kp/cm<sup>2</sup>]'''
+|  style="border: 1pt solid black;font-size: 75%;"|'''Young Modulus [kp/cm<sup>2</sup>]'''
 |  style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">3,5×10<sup>5</sup></span>
@@ Line 1,255: / Line 1,255: @@
 |-
-|  style="border: 1pt solid black;"|'''Poisson Coefficient'''
+|  style="border: 1pt solid black;font-size: 75%;"|'''Poisson Coefficient'''
 |  style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">0,2</span>
 |  style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">0,0</span>
 |  style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">0,0</span>
 |-
-|  style="border: 1pt solid black;"|'''Internal Friction'''
+|  style="border: 1pt solid black;font-size: 75%;"|'''Internal Friction'''
 |  style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">30º</span>
 |  style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">-</span>
 |  style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">30º</span>
 |-
-|  style="border: 1pt solid black;"|'''Compression Strength [kp/cm<sup>2</sup>]'''
+|  style="border: 1pt solid black;font-size: 75%;"|'''Compression Strength [kp/cm<sup>2</sup>]'''
 |  style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">200</span>
 |  style="border: 1pt solid black;text-align: center;"|2000
 |  style="border: 1pt solid black;text-align: center;"|20
 |-
-|  style="border: 1pt solid black;"|'''Tension Strength [kp/cm<sup>2</sup>]'''
+|  style="border: 1pt solid black;font-size: 75%;"|'''Tension Strength [kp/cm<sup>2</sup>]'''
 |  style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">20</span>
 |  style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">2000</span>
@@ Line 1,281: / Line 1,281: @@
 |  style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">2,0  ,  2,0</span>
 |-
-|  style="border: 1pt solid black;"|'''Behavior law after the '''
+|  style="border: 1pt solid black;font-size: 75%;"|'''Behavior law after the '''
 '''Yield point'''

@@ Line 1,297: / Line 1,297: @@
 <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 Table 3 – Mechanical properties used in the micro and macro model.</div>
 The micro model consists of three materials: matrix, fiber-matrix interface zone and reinforcement. The macro model is made up of one composite material consisting of three component materials: reinforced fiber, matrix, and fiber-matrix interface. Table 3 shows the mechanical properties of the materials used in the micro model. The mechanical properties of the composite material phases in the macro model are identical to the corresponding ones in the matrix and reinforcement of the micro model.

@@ Line 1,131: / Line 1,131: @@
 |  style="border: 2pt solid black;vertical-align: top;"|24%
 |}
 <sup>(*)</sup>Note: Adopted equal to matrix elastic modulus
@@ Line 1,137: / Line 1,136: @@
 <div id="_Ref16306933" class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 Table 2 – Fiber carbon properties for the macromodel and micromodel.</div>
 The micromodel is made up of three materials: reinforced fiber, epoxy matrix and fiber-matrix interface. Epoxy matrix and interface material are considered isotropic and homogenous and have mechanical properties that coincide with the mechanical properties of the macromodel components. In Figure 11 the micromodel materials distribution are shown schematically.

← Older revision	Revision as of 10:26, 5 June 2018
(No difference)

@@ Line 953: / Line 953: @@
 Material degradation -or damage- in a simple continuum material component due to a dissipative process can be simulated by means a local damage formulation [16,36,37,40,49]. This model is used at each simple matrix material embedded in the composite, inducing a stiffness degradation and strength reduction in the entire laminate.
-The isotropic damage formulation is based on a scalar internal variable  <math display="inline">d^d</math> that represents the level of elastic degradation at each simple component material. This variable is bounded between 0 and 1, being zero for an undamaged and one for a completely damaged state of a single component material. The local damage variable  <math display="inline">d^d\mbox{ }</math> is used to link the real stress tensor <math display="inline">{\boldsymbol\sigma} </math> with the effective undamaged stress tensor  <math display="inline">{\boldsymbol\sigma }_0</math> . Therefore, the relation between the damaged stress and the strain in the matrix component included in each layer depends on the internal damage variable  <math display="inline">d^d</math> and the elastic constitutive tensor  <math>{\mathbb{C}}_0\mbox{ }</math>,
+The isotropic damage formulation is based on a scalar internal variable  <math display="inline">d^d</math> that represents the level of elastic degradation at each simple component material. This variable is bounded between 0 and 1, being zero for an undamaged and one for a completely damaged state of a single component material. The local damage variable  <math display="inline">d^d\mbox{ }</math> is used to link the real stress tensor <math display="inline">{\boldsymbol\sigma} </math> with the effective undamaged stress tensor  <math display="inline">{\boldsymbol\sigma }_0</math>. Therefore, the relation between the damaged stress and the strain in the matrix component included in each layer depends on the internal damage variable  <math display="inline">d^d</math> and the elastic constitutive tensor  <math>{\mathbb{C}}_0</math>,
 {| class="formulaSCP" style="width: 100%;width: 100%;text-align: center;"
@@ Line 979: / Line 979: @@
-being  <math>F^d\left({\boldsymbol\sigma }_0;{\bf q}^d\right)</math> the damage threshold function,  <math>f^d\left({\boldsymbol\sigma }_0\right)</math> the scalar equivalent stress function and  <math>c\left(d^d\right)</math> the uniaxial strength evolution depending of the internal damage variable  <math display="inline">d^d</math> .
+being  <math>F^d\left({\boldsymbol\sigma }_0;{\bf q}^d\right)</math> the damage threshold function,  <math>f^d\left({\boldsymbol\sigma }_0\right)</math> the scalar equivalent stress function and  <math>c\left(d^d\right)</math> the uniaxial strength evolution depending of the internal damage variable  <math display="inline">d^d</math>.
 In the same form of the plastic damage model, this formulation allows the damage onset and evolution using any isotropic limit criteria already defined in literature (von Mises, Mohr Coulomb, Drucker Prager, etc.)  [26,48], and the anisotropic behavior is included by means the mapping spaces theory previously defined in section 5.4.

@@ Line 884: / Line 884: @@
-where  <math display="inline">h_c({\boldsymbol\sigma} ,d^p,c)</math> is a scalar function of the current state of the stress-free variable  <math display="inline">{\boldsymbol\sigma} </math> and of the internal variables  <math display="inline">d^p</math> and  <math display="inline">c</math> . The expression used for the evolution law of the internal variable of cohesion, or uniaxial strength, is obtained from the following expression for  <math display="inline">h_c\mbox{ }</math> ,
+where  <math display="inline">h_c({\boldsymbol\sigma} ,d^p,c)</math> is a scalar function of the current state of the stress-free variable  <math display="inline">{\boldsymbol\sigma} </math> and of the internal variables  <math display="inline">d^p</math> and  <math display="inline">c</math> . The expression used for the evolution law of the internal variable of cohesion, or uniaxial strength, is obtained from the following expression for  <math display="inline">h_c\mbox{ }</math>,
 {| class="formulaSCP" style="width: 100%;width: 100%;text-align: center;"
@@ Line 910: / Line 910: @@
-such that  <math display="inline">\aleph \mbox{ }</math> is a coefficient depending on the criterion of the discontinuity threshold and represents a scalar factor between cohesion and the uniaxial stress of compression [36,37,40]. For example, for Tresca and von-Mises its value is  <math display="inline">\aleph =1\mbox{ }</math> , for Mohr Coulomb  <math display="inline">\aleph =2\mbox{ }\sqrt{R_0}\mbox{ }</math> where  <math display="inline">R^0=\left[f_C^0/f_T^0\right]={\left[{\sigma }_C(d^p=0)/{\sigma }_T(d^p=0)\right]}^{}</math> is the relation between compression-tension uniaxial strengths, for Drucker-Prager inscribed in the surface of Mohr-Coulomb  <math display="inline">\aleph =6\mbox{ }cos\varphi /(sin\varphi -3)\mbox{ }</math> and for Drucker-Prager circumscribed in the surface of Mohr-Coulomb  <math display="inline">\aleph =6\mbox{ }cos\varphi /(3\mbox{ }sin\varphi -</math><math>3)\mbox{ }</math> . Thus, for any discontinuity threshold criterion, this coefficient must be defined.
+such that  <math display="inline">\aleph \mbox{ }</math> is a coefficient depending on the criterion of the discontinuity threshold and represents a scalar factor between cohesion and the uniaxial stress of compression [36,37,40]. For example, for Tresca and von-Mises its value is  <math display="inline">\aleph =1\mbox{ }</math>, for Mohr Coulomb  <math display="inline">\aleph =2\mbox{ }\sqrt{R_0}\mbox{ }</math> where  <math display="inline">R^0=\left[f_C^0/f_T^0\right]={\left[{\sigma }_C(d^p=0)/{\sigma }_T(d^p=0)\right]}^{}</math> is the relation between compression-tension uniaxial strengths, for Drucker-Prager inscribed in the surface of Mohr-Coulomb  <math display="inline">\aleph =6\mbox{ }cos\varphi /(sin\varphi -3)\mbox{ }</math> and for Drucker-Prager circumscribed in the surface of Mohr-Coulomb  <math display="inline">\aleph =6\mbox{ }cos\varphi /(3\mbox{ }sin\varphi -</math><math>3)\mbox{ }</math>. Thus, for any discontinuity threshold criterion, this coefficient must be defined.
 The function  <math display="inline">c_T(d^p)</math> (see Figure 9) can be obtained explicitly and represents the cohesion evolution during a uniaxial simple tension test. The relation between cohesion and uniaxial tension stress is given by the following expression,
@@ Line 925: / Line 925: @@
-For Tresca and von-Mises its value is  <math display="inline">R^0/\aleph =1\mbox{ }</math> , for Mohr Coulomb  <math display="inline">R^0/\aleph =\sqrt{R_0}/2\mbox{ }</math> , for Drucker-Prager inscribed in the Mohr-Coulomb surface  <math display="inline">R_0/\aleph =(3+3\mbox{ }sin\varphi )/6\mbox{ }cos\varphi \mbox{ }</math> and for Drucker-Prager circumscribed in the Mohr-Coulomb surface  <math display="inline">R/\aleph =(3+sin\varphi )/6\mbox{ }cos\varphi \mbox{ }</math> .
+For Tresca and von-Mises its value is  <math display="inline">R^0/\aleph =1\mbox{ }</math>, for Mohr Coulomb  <math display="inline">R^0/\aleph =\sqrt{R_0}/2\mbox{ }</math> , for Drucker-Prager inscribed in the Mohr-Coulomb surface  <math display="inline">R_0/\aleph =(3+3\mbox{ }sin\varphi )/6\mbox{ }cos\varphi \mbox{ }</math> and for Drucker-Prager circumscribed in the Mohr-Coulomb surface  <math display="inline">R/\aleph =(3+sin\varphi )/6\mbox{ }cos\varphi \mbox{ }</math> .
 Some materials strength curves in simple tension and compression, obtained in uniaxial experimental tests, have similar shapes, in other words, it can be stated that the scale relationship between them is a constant during the whole quasi-static process and is given by
@@ Line 946: / Line 946: @@
 '''Figure 9'''. Transformation of the uniaxial strength measured in the lab into the uniaxial strength used in the plastic damage model</div>
 ==5.6 Local damage (elastic degradation) model for a component material==

@@ Line 782: / Line 782: @@
 {| style="text-align: center; margin:auto;width: 100%;"
 |-
-|  <math display="inline">\displaystyle\frac{d {\bf q}^p}{dt}={\dot{\bf q}}^p=\left\{\begin{array}{c}
+|  <math>\displaystyle\frac{d {\bf q}^p}{dt}={\dot{\bf q}}^p=\left\{\begin{array}{c}
 {\dot{\boldsymbol\epsilon }}^P\\
 {\dot{d}}^p\\
@@ Line 800: / Line 800: @@
-where  <math display="inline">\textbf{h}_p</math> and  [[Image:Draft_Samper_893761715-image150.png|12px]] are a second-degree tensor and a scalar function respectively that will be defined later, and which depend on the current state of the free variable  <math display="inline">{\boldsymbol\epsilon }^e</math> and the rest of the internal variables  <math display="inline">{\bf q}^p</math>. As observed in this equation, the main internal variable is the plastic deformation  <math display="inline">{\boldsymbol\epsilon }^p</math>, and the others are obtained from it. The plastic consistency factor  [[Image:Draft_Samper_893761715-image154.png|12px]] is obtained from the consistency condition of the plastic yield function [36,37,40].
+where  <math display="inline">\textbf{h}_p</math> and  [[Image:Draft_Samper_893761715-image150.png|12px]] are a second-degree tensor and a scalar function respectively that will be defined later, and which depend on the current state of the free variable  <math display="inline">{\boldsymbol\epsilon }^e</math> and the rest of the internal variables  <math display="inline">{\bf q}^p</math>. As observed in this equation, the main internal variable is the plastic deformation  <math display="inline">{\boldsymbol\epsilon }^p</math>, and the others are obtained from it. The plastic consistency factor <math display="inline">\lambda</math>  is obtained from the consistency condition of the plastic yield function [36,37,40].
 * '''A secant and tangent constitutive equation, '''defined as the classic plasticity,
@@ Line 809: / Line 809: @@
 {| style="text-align: center; margin:auto;width: 100%;"
 |-
-|  <math display="inline">\begin{array}{c}
+|  <math>\begin{array}{c}
 {\boldsymbol\sigma} =\mathbb{C}{:}\left({\boldsymbol\epsilon} -{\boldsymbol\epsilon }^p\right)\\
 \\

@@ Line 706: / Line 706: @@
 ===5.5 Local plastic-damage model for a component material===
-The theory of plasticity provides a suitable physical-mathematical framework to formulate the behavior of metallic materials subjected to loading. From the extension of its main basic principles and the reinterpretation of its main variables, the ''Plastic Damage Model'' has emerged as one of the more general plastic constitutive model [36,37,40]. This plastic model is based on the ''plastic damage variable''  <math>d^p\mbox{ }</math> formulated as an internal variable representing a unit normalized dissipated plastic energy ranging from  <math>0\leq d^p\leq 1\mbox{ }</math> . For  <math>d^p=0\mbox{ }\mbox{ }</math> there is no plastic damage and for  <math>d^p=1\mbox{ }</math> the limit of total plastic damage in a solid point is defined. The latter state can be interpreted as a total loss of strength in a point of the solid produced for the accumulated plastic effect. A brief presentation of this model will be given in this section. For further details, check references [36,37,40]
+The theory of plasticity provides a suitable physical-mathematical framework to formulate the behavior of metallic materials subjected to loading. From the extension of its main basic principles and the reinterpretation of its main variables, the ''Plastic Damage Model'' has emerged as one of the more general plastic constitutive model [36,37,40]. This plastic model is based on the ''plastic damage variable''  <math>d^p\mbox{ }</math> formulated as an internal variable representing a unit normalized dissipated plastic energy ranging from  <math>0\leq d^p\leq 1\mbox{ }</math>. For  <math>d^p=0\mbox{ }\mbox{ }</math> there is no plastic damage and for  <math>d^p=1\mbox{ }</math> the limit of total plastic damage in a solid point is defined. The latter state can be interpreted as a total loss of strength in a point of the solid produced for the accumulated plastic effect. A brief presentation of this model will be given in this section. For further details, check references [36,37,40]
 In short, for a plastic mechanical process with no stiffness degradation, the plastic damage model uses the following set of plastic internal variables <math display="inline">{\bf q}^p=\lbrace {\boldsymbol\epsilon }^p,{\boldsymbol\alpha }^p\rbrace =\lbrace {\boldsymbol\epsilon }^p,d^p,c\rbrace \mbox{ }</math> , and its evolution laws will be presented as part of the main equations governing the model.
@@ Line 745: / Line 745: @@
-being  <math>F^P\left({\boldsymbol\sigma} ;{\bf q}^p\right)\mbox{ }</math> the plastic damage threshold function,  <math>G^P\left({\boldsymbol\sigma} ;{\bf q}^p\right)\mbox{ }</math> the plastic potential,  <math>c\left(d^p\right)</math> the cohesion or uniaxial strength evolution, depending on internal plastic damage variable  <math>d^p</math> ,  <math>f^P\left({\boldsymbol\sigma} \right)\mbox{ }</math> and  <math>g^P\left({\boldsymbol\sigma}\right)\mbox{ }</math> are two scalar functions of tensorial arguments called yield function and plastic potential respectively, and can be represented by any classical limit criteria (von Mises, Mohr Coulomb, Drucker Prager, etc.)  [26,34,40]
+being  <math>F^P\left({\boldsymbol\sigma} ;{\bf q}^p\right)\mbox{ }</math> the plastic damage threshold function,  <math>G^P\left({\boldsymbol\sigma} ;{\bf q}^p\right)\mbox{ }</math> the plastic potential,  <math>c\left(d^p\right)</math> the cohesion or uniaxial strength evolution, depending on internal plastic damage variable  <math>d^p</math>,  <math>f^P\left({\boldsymbol\sigma} \right)\mbox{ }</math> and  <math>g^P\left({\boldsymbol\sigma}\right)\mbox{ }</math> are two scalar functions of tensorial arguments called yield function and plastic potential respectively, and can be represented by any classical limit criteria (von Mises, Mohr Coulomb, Drucker Prager, etc.)  [26,34,40]
-The uniaxial strength evolution  <math>c\left(d^P\right)\mbox{ }</math> is assumed as a scaled magnitude regarding an ultimate strength to uniaxial compression of the composite  <math>{\sigma }_C^{}\mbox{ }</math> (stress discontinuity threshold), that is the stress level for which the volumetric deformation  <math display="inline">{\epsilon }_V\mbox{ }</math> reaches its maximum value. Therefore, the initial uniaxial strength, is defined as  <math>c^0\propto {\sigma }_C^{}\mbox{ }</math> for  <math>d^p=0\mbox{ }</math> , setting the initial position of the yield criterion and the final uniaxial strength of the material totally deteriorated as,  <math>c^u=0\mbox{ }</math> for  <math>d^p=1\mbox{ }</math> , defining the final position of the yield criterion (see Figure 9).
+The uniaxial strength evolution  <math>c\left(d^P\right)\mbox{ }</math> is assumed as a scaled magnitude regarding an ultimate strength to uniaxial compression of the composite  <math>{\sigma }_C^{}\mbox{ }</math> (stress discontinuity threshold), that is the stress level for which the volumetric deformation  <math display="inline">{\epsilon }_V\mbox{ }</math> reaches its maximum value. Therefore, the initial uniaxial strength, is defined as  <math>c^0\propto {\sigma }_C^{}\mbox{ }</math> for  <math>d^p=0\mbox{ }</math> , setting the initial position of the yield criterion and the final uniaxial strength of the material totally deteriorated as,  <math>c^u=0\mbox{ }</math> for  <math>d^p=1\mbox{ }</math>, defining the final position of the yield criterion (see Figure 9).
-Unlike the classic plasticity formulation with isotropic hardening, the cohesion or uniaxial strength in this case is not a simple function of the plastic hardening variable <math display="inline">c(d^p)\mbox{ }</math> , but is an internal variable that depends on the evolution of the elasto-plastic process governed by its evolution equation.
+Unlike the classic plasticity formulation with isotropic hardening, the cohesion or uniaxial strength in this case is not a simple function of the plastic hardening variable <math display="inline">c(d^p)\mbox{ }</math>, but is an internal variable that depends on the evolution of the elasto-plastic process governed by its evolution equation.
 * A '''plastic strain, plastic damage, and cohesion internal variables,'''
@@ Line 800: / Line 800: @@
-where  <math display="inline">\textbf{h}_p</math> and  [[Image:Draft_Samper_893761715-image150.png|12px]] are a second-degree tensor and a scalar function respectively that will be defined later, and which depend on the current state of the free variable  <math display="inline">{\boldsymbol\epsilon }^e</math> and the rest of the internal variables  <math display="inline">{\bf q}^p</math> . As observed in this equation, the main internal variable is the plastic deformation  <math display="inline">{\boldsymbol\epsilon }^p</math> , and the others are obtained from it. The plastic consistency factor  [[Image:Draft_Samper_893761715-image154.png|12px]] is obtained from the consistency condition of the plastic yield function [36,37,40].
+where  <math display="inline">\textbf{h}_p</math> and  [[Image:Draft_Samper_893761715-image150.png|12px]] are a second-degree tensor and a scalar function respectively that will be defined later, and which depend on the current state of the free variable  <math display="inline">{\boldsymbol\epsilon }^e</math> and the rest of the internal variables  <math display="inline">{\bf q}^p</math>. As observed in this equation, the main internal variable is the plastic deformation  <math display="inline">{\boldsymbol\epsilon }^p</math>, and the others are obtained from it. The plastic consistency factor  [[Image:Draft_Samper_893761715-image154.png|12px]] is obtained from the consistency condition of the plastic yield function [36,37,40].
 * '''A secant and tangent constitutive equation, '''defined as the classic plasticity,