김경종
13 KiB
line
| H | B_r |
|---|---|
| H_c | Low |
| H | High |
Figure 26.5.3–2 Remanence and coercivity in permanent magnets.
\mathbf {H} = \hat {H} \left(\left| \mathbf {B} \right|\right) \left(\frac {\mathbf {B}}{\left| \mathbf {B} \right|}\right) - \mathbf {H} _ {c}
for nonlinear isotropic - response.
Input File Usage: To specify permanent magnetization with underlying linear magnetic permeability:
*MAGNETIC PERMEABILITY
*PERMANENT MAGNETIZATION
direction of magnetization in the global system
magnitude of coercivity
To specify permanent magnetization with underlying nonlinear magnetic permeability (nonlinear response of the left top portion of the hysteresis curve):
*MAGNETIC PERMEABILITY, NONLINEAR
*NONLINEAR BH
input - by shifting the response to the right by H _ { c }
*PERMANENT MAGNETIZATION
direction of magnetization in the global system
magnitude of coercivity
Abaqus/CAE Usage: Permanent magnetization is not supported in Abaqus/CAE.
Elements
Magnetic material behavior is active only in electromagnetic elements (see “Choosing the appropriate element for an analysis type,” Section 27.1.3).
26.6 Pore fluid flow properties
• “Pore fluid flow properties,” Section 26.6.1
• “Permeability,” Section 26.6.2
• “Porous bulk moduli,” Section 26.6.3
• “Sorption,” Section 26.6.4
• “Swelling gel,” Section 26.6.5
• “Moisture swelling,” Section 26.6.6
26.6.1 PORE FLUID FLOW PROPERTIES
Abaqus/Standard allows specific properties to be defined for a fluid-filled porous material. This type of porous medium is considered in a coupled pore fluid diffusion/stress analysis (“Coupled pore fluid diffusion and stress analysis,” Section 6.8.1). The following properties are available:
• Permeability: Permeability defines the relationship between the flow rate of a liquid through a porous medium and the gradient of the piezometric head of that fluid (see “Permeability,” Section 26.6.2).
• Porous bulk moduli: The bulk moduli of the solid grains and of the fluid in a porous medium are defined such that their compressibility is considered in an analysis (see “Porous bulk moduli,” Section 26.6.3).
• Sorption: Sorption defines the absorption/exsorption behavior of a porous material under partially saturated flow conditions (see “Sorption,” Section 26.6.4).
• Swelling gel: The swelling gel model is used to simulate the growth of gel particles that swell and trap wetting liquid in a partially saturated porous medium (see “Swelling gel,” Section 26.6.5).
• Moisture swelling: Moisture swelling defines the saturation-driven volumetric swelling of a porous medium’s solid skeleton under partially saturated flow conditions (see “Moisture swelling,” Section 26.6.6).
Thermal expansion
For porous media such as soils or rock, the thermal expansion of both the solid grains and the permeating fluid can be defined. See “Thermal expansion” in “Coupled pore fluid diffusion and stress analysis,” Section 6.8.1, for more details.
26.6.2 PERMEABILITY
Products: Abaqus/Standard Abaqus/CFD Abaqus/CAE
References
• “Pore fluid flow properties,” Section 26.6.1
• “Material library: overview,” Section 21.1.1
• *PERMEABILITY
• “Defining permeability” in “Defining a fluid-filled porous material,” Section 12.12.3 of the Abaqus/CAE User’s Guide, in the HTML version of this guide
Overview
Permeability is the relationship between the volumetric flow rate per unit area of a particular wetting liquid through a porous medium and the gradient of the effective fluid pressure. It can be specified in Abaqus/Standard and Abaqus/CFD.
Permeability in Abaqus/Standard:
• must be specified for a wetting liquid for an effective stress/wetting liquid diffusion analysis (see “Coupled pore fluid diffusion and stress analysis,” Section 6.8.1);
• is defined, in general, by Forchheimer’s law, which accounts for changes in permeability as a function of fluid flow velocity; and
• can be isotropic, orthotropic, or fully anisotropic and can be given as a function of void ratio, saturation, temperature, and field variables.
Permeability in Abaqus/CFD:
• must be specified for porous media flows (see “Incompressible fluid dynamic analysis,” Section 6.6.2); and
• can be isotropic and specified as a function of porosity only or can be specified through the Carman-Kozeny permeability-porosity relation.
Permeability in Abaqus/Standard
Permeability is defined for pore fluid flow.
Forchheimer’s law
According to Forchheimer’s law, high flow velocities have the effect of reducing the effective permeability and, therefore, “choking” pore fluid flow. As the fluid flow velocity reduces, Forchheimer’s law approximates the well-known Darcy’s law. Darcy’s law can, therefore, be used directly in Abaqus/Standard by omitting the velocity-dependent term in Forchheimer’s law.
Forchheimer’s law is written as
\mathbf {f} (1 + \beta \sqrt {\mathbf {v} _ {w} \cdot \mathbf {v} _ {w}}) = - \frac {k _ {s}}{\gamma_ {w}} \mathbf {k} \cdot \left(\frac {\partial u _ {w}}{\partial \mathbf {x}} - \rho_ {w} \mathbf {g}\right),
where
| $\mathbf{f} = sn\mathbf{v}_w$ | is the volumetric flow rate of wetting liquid per unit area of the porous medium (the effective velocity of the wetting liquid); |
| $s = \frac{dV_w}{dV_v}$ | is the fluid saturation ( $s = 1$ for a fully saturated medium, $s = 0$ for a completely dry medium); |
| $n = \frac{dV_v}{dV}$ | is the porosity of the porous medium; |
| $e = \frac{dV_v}{(dV_g + dV_t)}$ | is the void ratio; |
| $dV_w$ | is the wetting fluid volume in the medium; |
| $dV_v$ | is the void volume in the medium; |
| $dV_g$ | is the volume of grains of solid material in the medium; |
| $dV_t$ | is the volume of trapped wetting liquid in the medium; |
| $dV$ | is the total volume of the medium; |
| $\mathbf{v}_w$ | is the fluid velocity; |
| $\beta(e)$ | is a “velocity coefficient,” which may be dependent on the void ratio of the material; |
| $k_s(s)$ | is the dependence of permeability on saturation of the wetting liquid such that $k_s = 1.0$ at $s = 1.0$ ; |
| $\rho_w = \gamma_w/g$ | is the density of the fluid; |
| $\gamma_w$ | is the specific weight of the wetting liquid; |
| $g$ | is the magnitude of the gravitational acceleration; |
| $\mathbf{k}(e, \theta, f_\beta)$ | is the permeability of the fully saturated medium, which can be a function of void ratio ( $e$ , common in soil consolidation problems), temperature ( $\theta$ ), and/or field variables ( $f_\beta$ ); |
| $u_w$ | is the wetting liquid pore pressure; |
| $\mathbf{x}$ | is position; and |
| $\mathbf{g}$ | is the gravitational acceleration. |
Permeability definitions
Permeability can be defined in different ways by different authors; caution should, therefore, be used to ensure that the specified input data are consistent with the definitions used in Abaqus/Standard.
Permeability in Abaqus/Standard is defined as
\overline {{\mathbf {k}}} = \frac {k _ {s}}{(1 + \beta \sqrt {\mathbf {v} _ {w} \cdot \mathbf {v} _ {w}})} \mathbf {k},
so that Forchheimer’s law can also be written as
\mathbf {f} = - \frac {\overline {{\mathbf {k}}}}{\gamma_ {w}} \cdot \left(\frac {\partial u _ {w}}{\partial \mathbf {x}} - \rho_ {w} \mathbf {g}\right).
The fully saturated permeability, , is typically obtained from experiments under low fluid velocity conditions. can be defined as a function of void ratio, e, (common in soil consolidation problems) and/or temperature, . The void ratio can be derived from the porosity, n, using the relationship e = n / ( 1 - n ) . Up to six variables may be needed to define the fully saturated permeability, depending on whether isotropic, orthotropic, or fully anisotropic permeability is to be modeled (discussed below).
Alternative definition of permeability
Some authors refer to the definition of permeability used in Abaqus/Standard, \overline { { \mathbf { k } } } ( \mathrm { u n i t s ~ o f ~ L T ^ { - 1 } } ) , as the “hydraulic conductivity” of the porous medium and define the permeability as
\widehat {\mathbf {K}} = \frac {\nu}{g} \frac {k _ {s}}{(1 + \beta \sqrt {\mathbf {v} _ {w} \cdot \mathbf {v} _ {w}})} \mathbf {k} = \frac {\nu}{g} \overline {{\mathbf {k}}},
where is the kinematic viscosity of the wetting liquid (the ratio of the liquid’s dynamic viscosity to its mass density), g is the magnitude of the gravitational acceleration, and \widehat { \bf K } has dimensions L ^ { 2 } (or Darcy). If the permeability is available in this form, it must be converted such that the appropriate values of are used in Abaqus/Standard.
Specifying the permeability
Permeability in Abaqus/Standard can be isotropic, orthotropic, or fully anisotropic. For non-isotropic permeability a local orientation (see “Orientations,” Section 2.2.5) must be used to specify the material directions.
Isotropic permeability
For isotropic permeability in Abaqus/Standard define one value of the fully saturated permeability at each value of the void ratio.
Input File Usage: *PERMEABILITY, TYPE=ISOTROPIC
Abaqus/CAE Usage: Property module: material editor: Other→Pore Fluid→Permeability: Type: Isotropic
Orthotropic permeability
For orthotropic permeability in Abaqus/Standard define three values of the fully saturated permeability ( k _ { 1 1 } , k _ { 2 2 } , \mathrm { a n d } k _ { 3 3 } ) at each value of the void ratio.
Input File Usage: *PERMEABILITY, TYPE=ORTHOTROPIC
Abaqus/CAE Usage: Property module: material editor: Other→Pore Fluid→Permeability: Type: Orthotropic
Anisotropic permeability
For fully anisotropic permeability in Abaqus/Standard define six values of the fully saturated permeability ( k _ { 1 1 } , k _ { 1 2 } , k _ { 2 2 } , k _ { 1 3 } , k _ { 2 3 } , and k _ { 3 3 } ) at each value of the void ratio.
Input File Usage: *PERMEABILITY, TYPE=ANISOTROPIC
Abaqus/CAE Usage: Property module: material editor: Other→Pore Fluid→Permeability: Type: Anisotropic
Velocity coefficient
Abaqus/Standard assumes that \beta = 0 . 0 by default, meaning that Darcy’s law is used. If Forchheimer’s law is required ( \beta > 0 . 0 ) , \beta ( e ) must be defined in tabular form.
Input File Usage: *PERMEABILITY, TYPE=VELOCITY
This must be a repeated use of the *PERMEABILITY option for the same material, since \mathbf { k } ( e , \theta ) must also be defined.
Abaqus/CAE Usage: Property module: material editor: Other→Pore Fluid→Permeability: Suboptions→Velocity Dependence
Saturation dependence
In Abaqus/Standard you can define the dependence of permeability, \overline { { \mathbf { k } } } , on saturation, s, by specifying k _ { s } . Abaqus/Standard assumes by default that k _ { s } = s ^ { 3 } for s < 1 . 0 ; k _ { s } = 1 . 0 for s \geq 1 . 0 . The tabular definition of k _ { s } ( s ) must specify k _ { s } = 1 . 0 for s \geq 1 . 0 .
Input File Usage: *PERMEABILITY, TYPE=SATURATION
This must be a repeated use of the *PERMEABILITY option for the same material, since \mathbf { k } ( e , \theta ) must also be defined.
Abaqus/CAE Usage: Property module: material editor: Other→Pore Fluid→Permeability: Suboptions→Saturation Dependence
Specific weight of the wetting liquid
In Abaqus/Standard the specific weight of the fluid, \gamma _ { w } , , must be specified correctly even if the analysis does not consider the weight of the wetting liquid (i.e., if excess pore fluid pressure is calculated).
Input File Usage: *PERMEABILITY, TYPE=type, SPECIFIC=
The SPECIFIC parameter must be defined in conjunction with the fully saturated *PERMEABILITY option for a given medium.
Abaqus/CAE Usage: Property module: material editor: Other→Pore Fluid→Permeability: Specific weight of wetting liquid: \gamma _ { w }