Pouplana Oñate 2018a - Revision history

Cinmemj: Cinmemj moved page Draft Samper 198709577 to Pouplana Oñate 2018a

2018-02-07T15:23:31Z

Cinmemj moved page Draft Samper 198709577 to Pouplana Oñate 2018a

Cinmemj at 15:22, 7 February 2018

2018-02-07T15:22:23Z

Cinmemj at 15:50, 6 February 2018

2018-02-06T15:50:33Z

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Cinmemj at 15:09, 6 February 2018

2018-02-06T15:09:59Z

Cinmemj at 15:05, 6 February 2018

2018-02-06T15:05:42Z

Cinmemj at 14:40, 6 February 2018

2018-02-06T14:40:32Z

Cinmemj at 14:34, 6 February 2018

2018-02-06T14:34:13Z

Cinmemj at 14:22, 6 February 2018

2018-02-06T14:22:07Z

Cinmemj at 14:13, 6 February 2018

2018-02-06T14:13:24Z

Cinmemj at 14:10, 6 February 2018

2018-02-06T14:10:55Z

@@ Line 68: / Line 68: @@
 It seems natural that a proper analysis of these kind of problems should require taking into account both parts, i.e. solid and fluid, for the derivation of the governing equations and the definition of the boundary conditions.
-Thereby, when solving this coupled problem, it will be necessary to capture the seepage of the fluid through the porous medium, and the deformations of the solid skeleton at any point of the domain. From the numerical perspective, one of the most widespread approaches is the ''displacement-pressure'' formulation, in which the main unknowns are the displacements of the solid skeleton <math display="inline">\boldsymbol{u}</math> displacement vector and the pore pressure.
+Thereby, when solving this coupled problem, it will be necessary to capture the seepage of the fluid through the porous medium, and the deformations of the solid skeleton at any point of the domain. From the numerical perspective, one of the most widespread approaches is the ''displacement-pressure'' formulation, in which the main unknowns are the displacements of the solid skeleton <math display="inline">\boldsymbol{u}</math> and the pore pressure <math display="inline">p</math>.
 However, in the limit of nearly incompressible pore fluid and small permeability, coupled poromechanics formulations suffer from instability problems. Finite elements exhibit locking in the pressure field and spurious oscillations appear due to the violation of the so-called Babuska-Brezzi conditions <span id='citeF-11'></span><span id='citeF-12'></span>[[#cite-11|[11,12]]]. The oscillations can be overcome by locally refining the mesh or by using shape functions of the displacement field one order higher than those of the pressure field. In practical applications this is, however, not feasible because of the increment in the computational cost. In this sense, stabilization methods have been found to be powerful tools to circumvent the Babuska-Brezzi conditions violation without compromising the efficiency of the computation.
@@ Line 295: / Line 295: @@
 |}
-where <math>E</math> is the Young's modulus and <math display="inline">\nu </math> Poisson's ratio is the Poisson's ratio.
+where <math>E</math> is the Young's modulus and <math display="inline">\nu </math> is the Poisson's ratio.
 In order to account for the coupling between the solid and fluid phases, the behaviour of a saturated porous medium is governed by the combination of two equations: the balance of momentum for the mixture solid-fluid, and the mass balance for the pore fluid.
@@ Line 312: / Line 312: @@
 |}
-where <math display="inline">b_i</math> is the body force per unit mass, <math display="inline">\ddot{u}_i</math> is the acceleration of the solid skeleton and <math display="inline">\rho </math> is the density of the solid-fluid mixture <math display="inline">\rho =\phi \rho _f+(1-\phi )\rho _s</math>, being <math display="inline">\phi </math> porosity the porosity of the soil, <math display="inline">\rho _f</math> fluid density the density of the fluid and <math display="inline">\rho _s</math> solid density the density of the solid.
+where <math display="inline">b_i</math> is the body force per unit mass, <math display="inline">\ddot{u}_i</math> is the acceleration of the solid skeleton and <math display="inline">\rho </math> is the density of the solid-fluid mixture <math display="inline">\rho =\phi \rho _f+(1-\phi )\rho _s</math>, being <math display="inline">\phi </math>  the porosity of the soil, <math display="inline">\rho _f</math> the density of the fluid and <math display="inline">\rho _s</math> the density of the solid.
 Let us introduce the general definition for the effective stress ([[#eq2.1|2.1]]) into the balance of momentum equation ([[#eq2.4|2.4]]). This gives
@@ Line 342: / Line 342: @@
 where <math display="inline">\zeta </math>  is the fluid mass content per unit volume and <math display="inline">q_i</math> represents the flow rate. It should be noted that Equation ([[#eq2.6|2.6]]) is in a linearized form of the general mass balance expression as the fluid density variation effect has been ignored.
-The mass balance equation is modified as follows. Let us first consider the constitutive equation presented in <span id='citeF-2'></span>[[#cite-2|[2]]] relating the fluid mass content per unit volume <math display="inline">\zeta </math> with the volumetric strain of the solid skeleton <math display="inline">\epsilon </math> volumetric strain of the solid skeleton and the pore pressure <math display="inline">p</math>.
+The mass balance equation is modified as follows. Let us first consider the constitutive equation presented in <span id='citeF-2'></span>[[#cite-2|[2]]] relating the fluid mass content per unit volume <math display="inline">\zeta </math> with the volumetric strain of the solid skeleton <math display="inline">\epsilon </math>  and the pore pressure <math display="inline">p</math>
 <span id="eq2.7"></span>
@@ Line 382: / Line 382: @@
 |}
-where <math display="inline">\mu </math> dynamic viscosity is the dynamic viscosity of the fluid and <math display="inline">k_{ij}</math> is the intrinsic permeability matrix of the porous medium.
+where <math display="inline">\mu </math>  is the dynamic viscosity of the fluid and <math display="inline">k_{ij}</math> is the intrinsic permeability matrix of the porous medium.
 Taking into account the relations from equations ([[#eq2.7|2.7]]) and ([[#eq2.9|2.9]]), the mass balance equation ([[#eq2.6|2.6]]) can be rewritten as
@@ Line 1,801: / Line 1,801: @@
 |}
-The above expression implies that <math display="inline">r(t)\ge r_y</math>, where <math display="inline">r_y</math><math>r_y</math>damage threshold is the so-called damage threshold, a material parameter that represents the value of strain at which damage starts. In this formula, <math display="inline">t</math> is not necessarily the physical time (it can be any monotonically increasing parameter controlling the loading process). The damage evolution law ([[#eq-3.5|3.5]]) is then rewritten in the form:
+The above expression implies that <math display="inline">r(t)\ge r_y</math>, where <math display="inline">r_y</math> is the so-called damage threshold, a material parameter that represents the value of strain at which damage starts. In this formula, <math display="inline">t</math> is not necessarily the physical time (it can be any monotonically increasing parameter controlling the loading process). The damage evolution law ([[#eq-3.5|3.5]]) is then rewritten in the form:
 <span id="eq-5.7"></span>
@@ Line 2,238: / Line 2,238: @@
 |}
-The parameter <math display="inline">\kappa </math><math>\kappa </math>compressive to tensile strength ratio<math>\sigma _{y_c}</math>compressive strength<math>\sigma _{y_t}</math>tensile strength in ([[#eq-3.26|3.26]]) is defined as the ratio between the compression elastic limit and the tension elastic limit, i.e.
+The parameter <math display="inline">\kappa </math> in ([[#eq-3.26|3.26]]) is defined as the ratio between the compression elastic limit and the tension elastic limit, i.e.
 <span id="eq-3.29"></span>
@@ Line 2,770: / Line 2,770: @@
 | style="width: 5px;text-align: right;white-space: nowrap;" | (3.54)
 |}
 And the internal state variable <math display="inline">r</math> is then the largest previously reached value of the non-local equivalent strain:
@@ Line 3,753: / Line 3,751: @@
 |}
-where <math display="inline">\delta _c</math> critical relative displacement of the interface is the critical relative displacement, i.e. it is the relative displacement at which the cohesive zone stops transmitting forces.
+where <math display="inline">\delta _c</math> is the critical relative displacement of the interface, i.e. it is the relative displacement at which the cohesive zone stops transmitting forces.
 Expression ([[#eq-4.14|4.14]]) implies that <math display="inline">\varrho _y \le \varrho (t) \le 1</math>, which shows that, in fact, the internal state variable <math display="inline">\varrho </math> can also be seen as the damage variable of the cohesive fracture zone.
@@ Line 3,769: / Line 3,767: @@
 |}
-with <math display="inline">\sigma _y</math> yield stress of the interface being the yield stress at the cohesive zone, i.e. the stress at which the cohesive zone starts damaging. Each local component of the effective stress is computed as follows:
+with <math display="inline">\sigma _y</math>  being the yield stress at the cohesive zone, i.e. the stress at which the cohesive zone starts damaging. Each local component of the effective stress is computed as follows:
 <span id="eq-4.17"></span>
@@ Line 3,824: / Line 3,822: @@
 |}
-where <math display="inline">\mu _F</math> friction coefficient of the interface is the friction coefficient, and <math display="inline">c_l</math> and <math display="inline">c_m</math> are the tangential cohesive stresses. They are computed as:
+where <math display="inline">\mu _F</math> is the friction coefficient, and <math display="inline">c_l</math> and <math display="inline">c_m</math> are the tangential cohesive stresses. They are computed as:
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 4,271: / Line 4,269: @@
 ====Crack growth====
-In order to determine the start of the crack growth, we estimate the damage at the crack tip <math display="inline">\mathbb{d}_t</math> and compare it with a prefixed threshold value, called propagation damage <math display="inline">\mathbb{d}_p</math><math>\mathbb{d}_p</math>fracture propagation damage.
+In order to determine the start of the crack growth, we estimate the damage at the crack tip <math display="inline">\mathbb{d}_t</math> and compare it with a prefixed threshold value, called propagation damage <math display="inline">\mathbb{d}_p</math>.
 Damage at the crack tip is defined here as a non-local measure of the damage inside the region of influence of the fracture <math display="inline">\Omega _f</math>. This measure follows the same idea as the non-local average performed in Chapter [[#3 Continuum Damage Mechanics|3]] for the equivalent strain (Eq ([[#eq-3.56|3.56]])). The damage at the tip is evaluated as:

@@ Line 2,671: / Line 2,671: @@
 |}
-where <math display="inline">Z(\boldsymbol{x},\boldsymbol{\chi })</math><math>Z</math>non-local weighting function is the non-local weighting function.
+where <math display="inline">Z(\boldsymbol{x},\boldsymbol{\chi })</math> is the non-local weighting function.
 In an infinite, isotropic and homogeneous medium, the weighting function depends only on the distance <math display="inline">\mathcal{D}_{x\chi }</math><math>\mathcal{D}</math>distance between the source point <math display="inline">\boldsymbol{\chi }</math>, and the receiver point <math display="inline">\boldsymbol{x}</math>. Thereby, we usually write <math display="inline">Z(\boldsymbol{x},\boldsymbol{\chi })=Z_0(\| \boldsymbol{x}-\boldsymbol{\chi }\| )=Z_0(\mathcal{D}_{x\chi })</math>, where <math display="inline">Z_0</math> is usually chosen as a non-negative function monotonically decreasing for <math display="inline">\mathcal{D}\ge 0</math>.
@@ Line 2,831: / Line 2,831: @@
 |}
-where <math display="inline">w_\chi </math><math>w</math>integration coefficient is a coefficient containing the product of the determinant of the Jacobian and the integration weight of Gauss point <math display="inline">\chi </math>, and <math display="inline">Z_{x,\chi }</math> is the aforementioned weight of non-local interaction between points <math display="inline">x</math> and <math display="inline">\chi </math>, computed as:
+where <math display="inline">w_\chi </math> is a coefficient containing the product of the determinant of the Jacobian and the integration weight of Gauss point <math display="inline">\chi </math>, and <math display="inline">Z_{x,\chi }</math> is the aforementioned weight of non-local interaction between points <math display="inline">x</math> and <math display="inline">\chi </math>, computed as:
 <span id="eq-3.57"></span>

@@ Line 3,666: / Line 3,666: @@
 |}
-with <math display="inline">t_I</math> joint width being the distance between the two faces of the interface (Figure [[#img-57|57]]). If the interface is closed, then we use a small value <math display="inline">t_I = t_{min}</math> minimum joint width to avoid dividing by zero.
+with <math display="inline">t_I</math> joint width being the distance between the two faces of the interface (Figure [[#img-57|57]]). If the interface is closed, then we use a small value <math display="inline">t_I = t_{min}</math>  to avoid dividing by zero.
 It is important to note that the terms in ([[#eq-4.7|4.7]]) are integrated over a quasi-zero-thickness interface element and thus:

@@ Line 1,859: / Line 1,859: @@
 |}
-which is the multidimensional generalization of ([[#eq-2.1|2.1]]), and where <math display="inline">\boldsymbol{\sigma }_e</math><math>\boldsymbol{\sigma }_e</math>effective/undamaged stress vector (damage mechanics) is the effective stress vector defined as:
+which is the multidimensional generalization of ([[#eq-3.1|3.1]]), and where <math display="inline">\boldsymbol{\sigma }_e</math> is the effective stress vector defined as:
 <span id="eq-3.11"></span>

@@ Line 1,846: / Line 1,846: @@
 |}
-<math>\boldsymbol{\varepsilon }</math> strain vector where <math display="inline">\boldsymbol{D}_e</math> is the elastic constitutive matrix of the intact material. One can clearly notice that ([[#eq-3.9|3.9]]) is a generalization of ([[#eq-3.4|3.4]]). Also, Equation ([[#eq-3.9|3.9]]) can alternatively be written as:
+where <math display="inline">\boldsymbol{D}_e</math> is the elastic constitutive matrix of the intact material. One can clearly notice that ([[#eq-3.9|3.9]]) is a generalization of ([[#eq-3.4|3.4]]). Also, Equation ([[#eq-3.9|3.9]]) can alternatively be written as:
 <span id="eq-3.10"></span>
@@ Line 1,888: / Line 1,888: @@
 | style="width: 5px;text-align: right;white-space: nowrap;" | (3.12)
 |}
 * The loading-unloading conditions ([[#eq-3.8|3.8]])

← Older revision	Revision as of 15:23, 7 February 2018
(No difference)

@@ Line 895: / Line 895: @@
 |}
-where <math display="inline">G</math><math>G</math>shear modulus is the shear modulus,
+where <math display="inline">G</math> is the shear modulus,
 <span id="eq-2.47"></span>

← Older revision		Revision as of 14:13, 6 February 2018
Line 572:		Line 572:
	\|}		\|}

−	~~<math>\boldsymbol{M}</math>mass matrix~~

	* Mass balance		* Mass balance

← Older revision		Revision as of 14:10, 6 February 2018
Line 478:		Line 478:
	\|}		\|}

−	<math>\boldsymbol{\sigma }'</math>effective stress vector (poromechanics)<math>\boldsymbol{\sigma }</math>total stress vector (poromechanics)/nominal stress vector (damage mechanics)<math>\boldsymbol{b}</math>body force per unit mass

	* Mass balance		* Mass balance