EQUATIONS OF STATE FOR POROUS BODIES

Danish Scientific Journal No102, 2025 95 EQUATIONS OF STATE FOR POROUS BODIES Abbasov S. Doctor of Philosophy in Engineering, Associate Professor of the Department of Mechanics Azerbaijan State Oil and Industry University Ismayilov M. PhD Candidate, Lecturer at Department of Mechanics Azerbaijan State Oil and Industry University https://doi.org/10.5281/zenodo.17741002 Abstract The mechanical characteristics of porous bodies depending on the mechanical characteristics of its phases are consederend. The reasons for the failure of Hooke's law for porous bodies, as well as the formation of a Hysteresis loop during unloading, are clarified. The general form of equations of state for porous bodies is obtained. A concrete form of state equations for porous bodies is obtained based on the fact that the tensile diagram of a porous elastic body is similar to the tensile diagram of an elastoplastic non-porous body. Keywords. Porous body, diagram, deformation, stress, equations of state, elasticity, plasticity. Introduction. Many natural bodies are porous: soils, rocks, wood, leather, bone, soft tissues of animals and humans, as well as artificial materials: construction (concrete, brick), food (bread), artificial leather, ceramics, metal parts obtained by powder metallurgy, etc. Porous soil is the top layer of soil, which serves as the basis of agriculture. Even this simple enumeration shows the huge role played by porous media in people's lives. The characteristic feature of all these materials is the ability to accumulate liquid in itself and allow it to move under the action of external forces. At least three of the most important aspects of our life are directly dependent on the movement of liquids through porous media. First of all, this is the movement of liquids through porous biomaterials in living organisms. The movement of moisture plays the same role in the soil. Ultimately, it is the water that filters or seeps into the soil that brings nutrients to plants and serves as the basis of nutrition for all living things. Finally, the main sources of energy of the XXI century - oil and gas are extracted from deep underground layers. The accumulation of oil and gas in these porous layers-collectors theory and the main technologies of extraction (production) are governed by the laws of the filtration and serve as one of the main sources of its tasks. The most important quantitative characteristic of porous bodies is their porosity m, which is defined as the fraction of the volume of the body that reaches the pores, or the volume of pores per unit volume of the material. Usually, closed isolated pores are ignored and only interconnected flow pores are taken into account. They form a pore space - a complex branched and irregular network of pores. The porosity of most materials is within 0.1-0.4. We took the typical value of m = 0.25 for many rocks, we find that the volume of pores in 1 m3 of rock ~ 0.25 m3 = 250 l. When it comes to rocks - oil and gas reservoirs or layers saturated with fresh water in desert areas, porosity is the main parameter, since it determines the reserves of the field, that is, the amount of liquid in this layer [1]. Setting task. Let′s consider a porous body the pores of which are filled with liquid. Such a medium is considered twophase. As it is known, the instantaneous modulus of elasticity of such a porous body is determined by the following formula [1]: 𝐸𝑟=𝑚𝑠𝐸𝑠+𝑚𝑙𝐸𝑙 𝑚 (1) Where 𝐸𝑟 is the reduced modulus of elasticity, 𝑚𝑠 - the mass of the skeleton, 𝑚 - the mass of a porous body saturated with liquid, 𝑚𝑙 - is the mass of the liquid in the pores, 𝐸𝑠 - is the Young's modulus of the skeleton, Considering that the liquids in the pores are under pressure, then 𝐸𝑙 - the Young's modulus of the liquid under compression. When a porous body is under the action of external influences, it deforms and the liquid filling the pores begins to move from pores with higher pressures to pores with low pressures, i.e. in the direction of the pressure gradient - P in the pores. Consequently, 𝑚𝑙 the mass of liquid in individual pores changes its value, i.e. 𝐸𝑟 is a function of pressure. With a slow change in external influences, the fluid has time to leave the skeleton of the body and 𝑚𝑙 decreases. If we assume that only 𝑚𝑙 changes over time, then 𝑚𝑙, as it can be seen from (1), the instantaneous reduced modulus of elasticity decreases. Indeed, from (1) 𝐸𝑟′=𝑚𝑠(𝐸𝑙−𝐸𝑠) (𝑚𝑠+𝑚𝑙)2∙𝑚𝑙′ (2) Where the prime means the derivative with respect to time. If we consider that for many porous bodies, 𝐸𝑙>𝐸𝑠 c and over time 𝑚𝑙 decreases i.e. 𝑚𝑙′ <0, then 𝐸𝑟′<0. This means that over time, the reduced modulus of elasticity decreases at points where the liquid decreases. Then the tension diagram between normal stress σ and relative strain ε in uniaxial tension will look like in Figure 1, i.e. the curve will be convex upwards (When stretched, the pores are deformed and their volume decreases. Therefore, part of the liquid leaves the pores and the pressure in it drops). Thus, under slow loading, Hooke's law is not valid for porous bodies, i.e. there is no rectilinear section on the 𝜎~𝜀 diagram (Fig. 1). 96 Danish Scientific Journal No102, 2025 Fig. 1. Tension diagram of a porous body (solid line). Figure 2 shows the 𝜎~𝜀 diagram for porous bodies under slow tension followed by unloading, and Figure 3 shows the same diagram under tension with a high reward rate. The curvilinearity of the diagram in Figure 2 is explained by the fact that, under slow loading, the liquid in the pores has time to leave its skeleton. In Figure 3, the loading process is fast, therefore, the fluid in the pores does not have time to leave its skeleton, so the reduced modulus of elasticity remains constant. Fig. 2 Slew streching Fig.3 high speed streching Thus, when the studying stress-strain state of a porous body is studied, different equations of state should be used depending on the loading rate. Figure 2 shows the σ~ε diagram for porous bodies under slow tension followed by unloading, and Figure 3 presents the same diagram under tension with a high reward rate. The curvilinearity of the diagram in Figure 2 is explained by the fact that, under slow loading, the liquid in the pores has time to leave its skeleton. In Figure 3, the loading process is fast, therefore, the fluid in the pores does not have time to leave its skeleton, so the reduced modulus of elasticity remains constant. Fig.4. Tensile diagram for an elastic-plastic, non-porous body 𝑶 𝜺 𝜺𝒎 𝝈𝒎 𝝈 𝑨 𝑨 𝑩 𝑩 𝜺 𝜺 𝑶 𝑶 𝝈 𝝈 𝑶 𝑩 𝑨 𝜺 𝝈 𝑪 Danish Scientific Journal No102, 2025 97 Thus, when the stress-strain state of a porous body is studied, different equations of state should be used depending on the loading rate. The diagram in Figure 2 differs from the diagram in Figure 4 in that, the deformations in Figure 2 disappear completely, when unloading, i.e. this diagram is for non-linearly elastic bodies, and in figure 4, residual deformations take place. The OABC area represents the work lost in the deformation. Most of this work, as experimental studies snow, goes into heat, but in the absence of heat transfer causes very slight (for deformation 𝜀=4%− about 2°) heating of the test sample [2]. It is known that plastic deformations arise due to tangential stresses; they do not occur under all-round compression [4]. When tangential stresses act, internal friction occurs during plastic deformation leading to heating of the sample, heat exchange with the environment occurs, and work that is equivalent to the amount of heat released due to internal friction is lost. When a porous material is deformed, internal friction practically does not occur, because the liquids and gases that fill the pores play the role of alubricating oil, thereby the work is not lost and it is converted into the potential energy of elastic deformation. Hysteresis loop in figure 2 arise due to the fact that during unloading, the Young's modulus of the material decreases. This is because after the removal of the external load, the sample restores its original shape due to the accumulated potential energy of elastic deformation, and the experiments are carried out not in a liquid medium, but in some rooms where the sample is surrounded by air, so the material instead of the lost liquid sucks in air. Considering that for air 𝐸𝑙>𝐸𝑠, when the sample restores its original shape, the pores are filled with air and 𝑚𝑙′ >0, then from (2) 𝐸𝑟′<0. This means that after starting unloading, over time, the reduced modulus of elasticity decreases and the diagram curve becomes a downward concavity. It should be noted that the area of the Hysteresis loop in this case is not lost work, it is formed due to a change in mass during deformation in other words, the loading and unloading diagrams in Figure 2 refer to different bodies. Thus, the equation 𝜎=𝑓 (𝜀), the loading diagram branches, can represent both plastic and non-linear elastic deformation of a rod made of porous material. Equations of state for porous elastic bodies or for non-linear elastic bodies In [3], the following equations of state for elastoplastic bodies were obtained according to the basic law of plasticity (In uniaxial tension, true residual strains are directly proportional to residual stresses). 𝜀𝑖𝑗 𝑝=1−𝑘0 2(1 𝐸−1 𝐸)[(1−𝜈)𝜎𝑖𝑗−𝜈𝐽𝑖𝑗𝑔𝑖𝑗] (3) Where 𝐸 is the modulus of elasticity, 𝐸 is the modulus of plasticity, 𝐸=𝜎о𝑠/𝜀о𝑠, 𝜎о𝑠 and 𝜀о𝑠, are, respectively, the residual stress and residual deformation after unloading in uniaxial tension, 𝜈 - is Poisson's ratio. 𝑘0=√2𝜎𝑇2 𝐽12+2(1+𝜈)𝐽2−1 (4) Where 𝜎𝑇 - is the yield strength, 𝜈 - is the Poisson's ratio of the material, 𝐽1 and 𝐽2 - respectively, the first and second invariants of the stress tensor. If to compare (3) for 𝑘𝑖 and 𝐵 we get: 𝐵=1−𝑘0 2(1 𝐸−1 𝐸)(1+𝜈); 𝑘1 =−1−𝑘0 2(1 𝐸−1 𝐸)3𝜈 (5) From (4) and (5) it is seen that, 𝑘𝑖 and 𝐵, in addition to the stress tensor invariants, also depend on Young's modulus - 𝐸, plasticity modulus - 𝐸 and yield strength - 𝜎𝑇 (Poisson's ratio can be considered constant). Figure 4, it shows that for porous bodies there are no modulus of elasticity, no yield strength. Therefore, in the equations of state (3), the parameters 𝐸,𝐸,𝜎𝑇 should be expressed in terms of other parameters. To this end, we write the equations of state (3) for one-dimensional tension. In this case 𝜎11=𝜎; 𝜎12=𝜎13=𝜎23=𝜎22=𝜎33=0; 𝐼1 =𝜎; 𝐼2=0;𝑘0=√𝜎𝑇2−𝜎2 𝜎 Taking into account these relations, from (3) we obtain: 𝜀𝑝=12(𝜎−√2𝜎𝑇2−𝜎2)(1 𝐸−1 𝐸) (6) It was proved in [4] that the material can be strengthened by a factor of √2. If to denote the maximum stress and strain through 𝜎𝑚 and 𝜀𝑚, respectively, and substituting 𝜎=𝜎𝑚=√2𝜎𝑇 from (6) we have: 𝜀𝑚=𝜎𝑚 2 (1 𝐸−1 𝐸) (7) From (7) 12(1 𝐸 −1 𝐸)=𝜀𝑚 𝜎𝑚 (8) Taking into account equality 𝜎𝑚=√2𝜎𝑇 and (8) from (3) and (4) we get: 𝜀𝑖𝑗 𝑝=(1−𝑘0)𝜀𝑚 𝜎𝑚[(1+𝜈)𝜎𝑖𝑗−𝜈𝐽1𝑔𝑖𝑗] (9) 𝑘0=√2𝜎𝑚 2 𝐽12+2(1+𝜈)𝐽2−1 (10) Comparing (9) with (3) for 𝑘1 and В we have: 𝑘1=−3𝜈(1−𝑘0)𝜀𝑚 𝜎𝑚 𝐵=(1+𝜈)(1−𝑘0)𝜀𝑚 𝜎𝑚 Figure 1 shows: 𝜀𝑚 𝜎𝑚=1 𝐸𝑠е𝑐 (11) Where 𝐸𝑠е𝑐secant modulus of elasticity. Taking into account (11), from (9) we obtain: 𝜀𝑖𝑗=(1−𝑘0)(1+𝜈 𝐸𝑠е𝑐 𝜎𝑖𝑗−𝜈𝐼1 𝐸𝑠е𝑐𝑔𝑖𝑗) (12) Where 𝑘0 is determined by equality (10). 𝜎𝑚 and 𝜀𝑚 in (9) (10) and 𝐸𝑠е𝑐, which enters system (12) can always be determined from a uniaxial tension experiment. Thus, we have obtained the equations of state for porous bodies when the stress state changes slowly, i.e. 98 Danish Scientific Journal No102, 2025 loading is quasi-static. In a particular case, under dynamic loading 𝑘0=0, 𝐸𝑠е𝑐=𝐸𝑟 and (13) looks like: 𝜀𝑖𝑗=1+𝜈 𝐸𝑟𝜎𝑖𝑗−3𝜈 𝐸𝑟𝜎𝑔𝑖𝑗 (13) System (13) is Hooke's law for linear elastic bodies. 𝐸𝑟in (13) is determined by equality. In a particular case, with uniaxial tension, from (12) we have: 𝜀=(1−√𝜎𝑚 2 𝜎2−1) 𝜎 𝐸𝑠е𝑐 =(𝜎 −√𝜎𝑚 2−𝜎2)1 𝐸𝑠е𝑐)1 𝐸𝑠е𝑐 (14) As it can be seen from (14), at 𝜎=0, 𝜀<0, and it should be 𝜀=0. This contradiction is due to the fact that within the limits of endurance, any deformable material, including porous material, obeys Hooke's law, i.e. σ in (14) should change beyond endurance limits i.e. in the interval 𝜎𝑏≤𝜎≤𝜎𝑚. Where 𝜎𝑏 is the endurance limit of the material. This can also be seen from the diagram of shale extension [5]. Figure 1 shows that under fast loading, (blue line, 𝑡=0) the diagram is almost straight and similar to the tension diagram when Hooke's law works. Under slow loading (green line, 𝑡=∞) the curve is convex upwards. In the general case, the components of the stress tensor −𝜎𝑖𝑗, included in (12) and (13) should be located in the stress space, between the endurance and yield surfaces, the equations of which in the stress space have the form, respectively [4]: 𝐼12+2(1+𝜈)𝐼2=𝜎𝑏2 (15) 𝐼12+2(1+𝜈)𝐼2=𝜎𝑇2 (16) The endurance surface is a surface in the stress space that contains the origin of the coordinate system, the axes of which arethe stress tensor components. It should be noted that when the loading path is inside the endurance surface, the deformation process obeys Hooke's law. Thus, the physical relations for porous bodies have the form: 𝜀𝑖𝑗 = { 1+𝜈 𝐸𝑟𝜎𝑖𝑗−𝜈𝐼1 𝐸𝑟𝑔𝑖𝑗 𝑤ℎ𝑒𝑛 𝐼12+2(1+𝜈)𝐼2≤𝜎𝑏2 (1−𝑘0)𝜀𝑚 𝜎𝑚((1+𝜈)𝜎𝑖𝑗−𝜈𝐼1𝑔𝑖𝑗) 𝑤ℎ𝑒𝑛 𝜎𝑏2<𝐼12+2(1+𝜈)𝐼2<𝜎𝑇2 It should be noted that the endurance limit - 𝜎𝑏, for porous bodies is defined as the reduced endurance limit of the phases that form the porous body. In a particular case, if the porous body is two-phase, the reduced endurance limit is determined as follows: 𝜎𝑟𝑏=𝑚𝑠𝜎𝑠𝑓+𝑚𝑙𝜎𝑙𝑓 𝑚 Where 𝜎𝑠𝑓 and 𝜎𝑙𝑓 are skeletal and fluid endurance limits, respectively. Considering that the fluids in the pores are under pressure, then 𝜎𝑙𝑓, in the equation is the endurance limit of the fluid under compression. Fig.5. Schematic dependence of strain on stress for shale Conclusions The general form of the equations of state for porous bodies is obtained. Based on the fact that the tension diagram of a porous elastic body is similar to the tension diagram of an elastic-plastic non-porous body, a specific form of the equations of state for porous bodies is obtained. References: 1. Gulgazli A.S., Kheyrabadi G.S. "The Effect of Porosity on the Properties of Materials." LAP LAMBERT Academic Publishing, Saarbrücken, Germany, 2014, p. 130. 2. Kachanov L.M. Fundamentals of the Theory of Plasticity. Moscow: Nauka 1969, p. 420. 3. Gulgazli A.S. Plasticity and Creep under Repeated Loading. LAP LAMBERT Academic Publishing, Saarbrücken, Germany, 2012, p. 176. 4. Gulgazli A.S. Energy State of Plasticity. Applied Mechanics and Technical Physics. 2018. v. 59, no. 2, p. 142-148. 5. Handbook of Physical Constants of Rocks/Ed. S. Clark. Moscow, Mir, 1969. 544 p. Danish Scientific Journal No102, 2025 99 6. Rabotnov Yu.N. Creepiness of Structural Elements. Moscow: Nauka 1966, p. 780 7. Vakhidov M.A., Karimov O.M., Eyvazova Z.E., 2008. Oil and Gas Production Technique. Baku: Azernashr, p. 458.