ProductsAbaqus/Standard A porous medium in Abaqus/Standard is considered to consist of a mixture of solid matter, voids that contain liquid and gas, and entrapped liquid attached to the solid matter. The mechanical behavior of the porous medium consists of the responses of the liquid and solid matter to local pressure and of the response of the overall material to effective stress. The assumptions made about these responses are discussed in this section. Liquid responseFor the liquid in the system (the free liquid in the voids and the entrapped liquid) we assume that ρwρ0w≈1+uwKw-εthw, where ρw is the density of the liquid, ρ0w is its density in the reference configuration, Kw(θ) is the liquid's bulk modulus, and εthw=3αw(θ-θ0w)-3αw|θI(θI-θ0w) is the volumetric expansion of the liquid caused by temperature change. Here αw(θ) is the liquid's thermal expansion coefficient, θ is the current temperature, θI is the initial temperature at this point in the medium, and θ0w is the reference temperature for thermal expansion. Both u/Kw and εthw are assumed to be small. Grains responseThe solid matter in the porous medium is assumed to have the local mechanical response under pressure ρgρ0g≈1+1Kg(suw+ˉp1-n-nt)-εthg, where Kg(θ) is the bulk modulus of this solid matter, s is the saturation in the wetting fluid, and εthg=3αg(θ-θ0g)-3αg|θI(θI-θ0g) is its volumetric thermal strain. Here αg(θ) is the thermal expansion coefficient for the solid matter and θ0g is the reference temperature for this expansion. |1-ρg/ρ0g| is assumed to be small. It is important to distinguish Kg and αg as properties of the solid grains material. The porous medium as a whole will exhibit a much softer (and generally nonrecoverable) bulk behavior than is indicated by Kg and will also show a different thermal expansion. These effects are partially structural, caused by the medium being made up of irregular grains in partial contact. They may also be caused by the system being only partially saturated, with the voids containing a mixture of relatively compressible gas and relatively incompressible liquid. Liquid entrapmentEntrapment of liquid is associated with specific materials that absorb liquid and swell into a “gel.” A simple model of this behavior is based on the idealization of this gel as a volume of individual spherical particles of equal radius ra. Tanaka and Fillmore (1979) show that, when a single sphere of such material is fully exposed to liquid, its radius change can be modeled as ra=rfa-∑NaNexp(-tτN), where rfa is the fully swollen radius approached as t→∞ and N, aN and τN are material parameters. Tanaka and Fillmore also show the first term in the series dominates, so the model can be simplified to ra=rfa-a1exp(-tτ1). This provides the rate form ˙ra=rfa-raτ1. When the gel particles are only partially exposed to liquid (in an unsaturated system), it seems reasonable to assume that the swelling rate will be lessened according to the level of saturation. Further, we assume that the gel will swell only when the saturation of the surrounding medium exceeds the effective saturation of the gel, 1-[(rfa)3-(ra)3]/[(rfa)3-(rdrya)3], where rdrya is the radius of a gel particle that is completely dry. We combine these into a simple, linear effect: ˙ra=rfa-raτ1⟨s-1+((rfa)3-(ra)3(rfa)3-(rdrya)3)⟩, where ⟨f⟩=f if f>0, ⟨f⟩=0 otherwise. The packing density and swelling may cause the gel particles to touch. In that case the surface available to absorb and entrap liquid is reduced until, if the gel particles occupy the entire volume except for solid material, liquid entrapment must cease altogether. With ka gel particles per unit reference volume, the maximum radius that the gel particles can achieve before they must touch (in a face center cubic arrangement) is rtadef=(n0J4√2ka)13, and the volume is entirely occupied with gel and solid matter when the effective gel radius is rsa=(34πn0Jka)13. The gel swelling behavior is, therefore, further modified to be ˙ra=rfa-raτ1⟨s-1+((rfa)3-(ra)3(rfa)3-(rdrya)3)⟩(1-⟨ra-rtarsa-rta⟩2). Thus, in an unstressed medium the entrapped liquid volume is assumed to be dVt=htdV0, where htdef=43π(r3a-(rdrya)3)ka, where ra(J,s) is defined by the integration of Equation 3. This entrapped liquid can be compressed by pressure so that, when the porous medium is under stress, we assume dVt=(1-uwKw+εthw)htdV0, and thus nt=dVtdV=htJ(1-uwKw+εthw). Combining this with Equation 1 and neglecting small terms compared to unity then provides Jρwρ0wnt≈ht. We assume that, in the initial state, the effective saturation of the gel is the same as the saturation of the surrounding medium: r0a=[(rfa)3-((rfa)3-(rdrya)3)(1-s0)]13. The constitutive behavior of the gel containing entrapped fluid is given by the elastic bulk relationship ˉpt=-Kw(ˉεvol-εthw), where ˉpt is the average pressure stress in the gel fluid and ˉεvol is its volumetric effective strain. Effective strainFrom Equation 2 we see that the volumetric strain represents that part of the total volumetric strain caused by pore pressure acting on the solid matter in the porous medium and by thermal expansion -suw/Kg+εthg of that solid matter. In addition, entrapment of liquid in the medium may cause an additional volume change ratio: 1+dVt-dV0tdV0=1+Jnt-n0t. Finally, εms(s) is a saturation driven moisture swelling strain that represents the volumetric swelling of the solid skeleton in partially saturated flow conditions. This moisture swelling can be isotropic or anisotropic. The remaining part of the strain in the medium, ˉεdef=ε+(13(suwKg-εthg)-13ln(1+Jnt-n0t))I-εms(s) is the strain that is assumed to modify the effective stress in the medium. That is, we assume ˉσ=ˉσ(history of ˉε, θ, state variables, etc). Specific constitutive models of this type are discussed in Mechanical Constitutive Theories. From this assumption, and using Equation 5, we can write the Jaumann rate of change of the effective stress in terms of the rate of change of the kinematic and pore liquid pressure variables as d∇(J(1-nt)ˉσ)=J(1-nt)ˉD:(dε+(s3Kg+uw3Kgdsduw+ht3Kw(1+Jnt-n0t))Iduw,-dεmsdsdsduwduw), where ˉD is defined for each particular model in Mechanical Constitutive Theories. Also, for the effective pressure stress of the fluid entrapped in the gel, d(JntˉptI)=JntKwII:(dε+(s3Kg+uw3Kgdsduw+ht3Kw(1+Jnt-n0t))Iduw-dεmsdsdsduwduw). Then, from Equation 3, dIN=∫V[βN:{(1-nt)ˉD+ntKwII}:dε-βN:((χ+uwdχduw)I-(s3Kg+uw3Kgdsduw+ht3Kw(1+Jnt-n0t)){(1-nt)ˉD+ntKwII}:I+dsduw{(1-nt)ˉD+ntKwII}:dεmsds)duw+σ:(dβN+2βN⋅dΩ)-βN:IχuwI:dε]dV. |