The difference between textile materials and traditional engineering materials.
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The textile materials include various raw fiber materials which are used in textile and various products processed from textile fibers, such as the one-dimensional yarn, thread, rope, and so on; two-dimensional and shape-based fabrics, textile nets, flakes, and so on; and three-dimensional and form-based clothing, braids, utensils, and its reinforced composites. The basic textile processing process is shown in Figure 1.
\nBasic process of textile processing.
As shown in Table 1, the textile materials are essentially different from traditional engineering materials; moreover, textile materials are flexible, easy to change their shape, and generally light weight; these characters can largely compensate the defects of engineering materials. According to the form, textile materials can be divided into fibers, yarns, flat fabrics, and three-dimensional fabrics. The fibers are spun to form the yarns, which are then weaved into the fabrics by weaving technology or knitting technology. In addition, the nonwoven fabrics are formed directly by winding the fibers.
\nTraditional engineering materials | \nTransitional material | \nTextile materials | \n
---|---|---|
Rigidity | \nFlexible | \nFlexible | \n
Homogeneous state | \nSolid state | \nDiscontinuous state | \n
Dense, non-permeable | \nDense or porous | \nPorous | \n
Smooth surface | \nSoft, fillable loose structure | \nSurface texture | \n
No buckling state | \nFlexion or non-buckling | \nMultiple buckling state | \n
The difference between textile materials and traditional engineering materials.
The shape of fibers is flexible and elongate, with length (Figure 2) and diameter ratio (Figure 3) of more than 103. Theoretically speaking, the fibers have round and slender bodies with the continuous homogeneous internal structure. But actually, they have a wide variety of cross-sectional shapes, and section shape changing along the length, heterogeneous internal structure, with the porosity form. According to the source of the fibers, they can be divided into natural fibers and chemical fibers. Fibers such as cotton, hemp, silk, and wool are the natural fibers with the longest history.
\nThe longitudinal morphology of natural fibers. (a) Wool fiber; (b) Cotton fiber; (c) Silk fiber; (d) Hemp fiber.
The cross-sectional morphology of natural fibers. (a) Wool fiber; (b) Cotton fiber; (c) Silk fiber; (d) Hemp fiber.
The yarn is an elongated body having a certain strength and toughness, in which the fibers are arranged in parallel and are cohered or entangled by twisting or other methods. The yarn is an intermediate product of textile processing. A number of short fibers or filaments are arranged in an approximately parallel state and twisted in the axial direction to form an elongated object having a certain strength and linear density, which is called the “yarn.” The strand of two or more single yarns is called the “thread” (Figure 4).
\nThe morphology of one kind of the yarn. (a) The appearance of the yarn and (b) The distribution of fibers in the yarn.
A two-dimensional object having thin thickness, large length, and wide width formed by interweaving and interlacing textile fibers and yarns by a certain method is the flat fabric. There are a variety of fabrics (as shown in Figure 5); they could have various materials, forms, colors, structures, and formation methods.
\nThe type of the fabric. (a) Woven fabric; (b) Knitted fabric and (c) Nonwoven fabric.
According to the forming methods, the fabrics can be divided into woven fabric, knitted fabric, and nonwoven fabric. The woven fabric is composed by warp and weft yarns arranged perpendicularly to each other according to some organization rules. The knitted fabric is formed by the yarns bent into a loop. Nonwoven fabrics are reinforced by oriented or randomly arranged fiber webs.
\nTraditional textile materials are mostly dielectric materials and are important electrical insulation materials. The electromagnetic properties of textile materials include electrical conductivity, dielectric properties, electrostatic and magnetic properties.
\nThe electrical conductivity of textile materials is expressed as specific resistance. There are usually three representations: volume specific resistance, mass specific resistance and surface specific resistance (Table 2).
\nType of fiber | \nlg ρm | \nn | \nlg K | \n
---|---|---|---|
Cotton | \n6.8 | \n11.4 | \n16.6 | \n
Ramie | \n7.5 | \n12.3 | \n18.6 | \n
Silk | \n9.8 | \n17.6 | \n26.6 | \n
Wool | \n8.4 | \n15.8 | \n26.2 | \n
Washed wool | \n9.9 | \n14.7 | \n26.6 | \n
Viscose fiber | \n7.0 | \n11.6 | \n19.6 | \n
Acetate fiber | \n11.7 | \n10.6 | \n20.1 | \n
Acrylic | \n8.7 | \n— | \n— | \n
Acrylic (degreasing) | \n14 | \n— | \n— | \n
Polyester | \n8.0 | \n— | \n— | \n
Polyester (degreasing) | \n14 | \n— | \n— | \n
Mass specific resistance of textile materials.
According to the law of resistance, the resistance R of the conductor is proportional to the length L of the conductor, inversely proportional to the cross-sectional area S, and related to material properties. That is,
\nwhere \n
For textile materials, the cross-sectional area or volume is not easy to measure; so, we usually use the mass specific resistance \n
where \n
where n and K are experimental constants.
\nThe dielectric constant of dried fiber is 2–5 at the frequent of 50 or 60 Hz. The dielectric constant of the liquid water is 20 and the adsorbed water is 80. The dielectric constants of common textile fibers measured at the frequency of 1 kHz and the relative humidity of 65% are shown in Table 3.
\nFiber | \nDielectric constant (\n | \nFiber | \nDielectric constant (\n | \n
---|---|---|---|
Cotton | \n18 | \nAcetate | \n4.0 | \n
Wool | \n5.5 | \nNylon staple fiber | \n3.7 | \n
Viscose fiber | \n8.4 | \nNylon yarn | \n4.0 | \n
Viscose wire | \n15 | \nPolyester staple fiber (deoiled) | \n2.3 | \n
Acetate staple fiber | \n3.5 | \nPolyester staple fiber | \n4.2 | \n
Acrylic staple fiber (deoiled) | \n2.8 | \n— | \n— | \n
The dielectric constant of common textile fibers.
The high moisture regain of cotton and viscose leads to its high dielectric constant.
As the relative dielectric constant of water is several tens of times larger than that of the dry textile material, the dielectric constant of the fiber is different when the moisture regain or moisture content of the textile material is different. The presence of frequency, temperature, and impurities also changes the dielectric constant of the materials.
\nA physical process in which a dielectric converts a portion of electrical energy into thermal energy under the action of an electric field is known as dielectric loss. The magnitude of the dielectric loss is related to the applied electric field frequency, electric field strength, fiber constant, and dielectric loss angle. In unit time, the heat energy P produced per unit volume of fiber is
\nwhere P is the power consumed by the electric field (W/cm3); f is the frequency of the applied electric field (Hz); E is the external electric field strength (V/cm); and \n
The dielectric constant of dry textile material generally is 2–5, for which \n
The specific resistance of textile materials with dielectric properties is generally high, especially for synthetic fibers with low hygroscopicity, such as polyester and acrylic fibers. Under normal atmospheric conditions, the mass specific resistance is as high as 1013 Ω cm/cm2 or more. In textile processing, the contact and friction between fibers or between fibers and machine parts tends to trigger charge transfer and static electricity generation. During the production process, static electricity will cause fiber hairiness, hairiness increase, filament winding mechanism, breakage, etc. In the course of the taking process, static electricity will cause clothes to stick and absorb dust.
\nAlthough the phenomenon of static electricity leads to many hazards during textile processing, the electrostatic properties of textile materials can also benefit to some processing technology, such as electrospinning and electrostatic flocking.
\nOrdinary textile materials are anti-magnets, which are negative. The magnetic susceptibility of some textile materials is shown in Table 4.
\nMaterial | \nMagnetic susceptibility (χ) | \nMaterial | \nMagnetic susceptibility (χ) | \n
---|---|---|---|
Ethylene | \n−10.3 × 10−6 | \nPolyester | \n−6.53 × 10−6 | \n
Polypropylene | \n−10.1 × 10−6 | \nNylon 66 | \n−9.55 × 10−6 | \n
Fluorine | \n−47.8 × 10−6 | \n\n | \n |
Magnetic susceptibility of some fibers.
The magnetic properties of textile materials are not as much as those of electrical properties, but they are gradually being valued by people to develop various types of magnetic fibers and textiles. For example, magnetic powders such as iron, cobalt, nickel, and ferrite are added to a spinning solution, and fibers having magnetic properties are obtained by wet spinning.
\nCommon textile materials have dielectric properties and electrostatic phenomena, but the electromagnetic parameter of them has not reached the order of magnitude of metals or semiconductors. Therefore, they generally do not own any electromagnetic function.
\nElectromagnetic textile materials are a new type of functional textile materials obtained from fibers or yarns with good electrical and magnetic properties through textile processing technology or by applying the materials with metallic properties to common textile material. Meanwhile, electromagnetic textile materials have unique structure of textile materials and the electromagnetic properties of the metal materials [1, 2].
\nOrdinary fibers are generally made of nonconductive and non-magnetic polymer materials. To obtain the functionalization of textile materials, special materials must be introduced during the preparation process. Textile materials include fibers, yarns, and fabrics. Therefore, electromagnetic functionalization of fibers, yarns, and fabrics can be achieved by spinning, weaving, and finishing.
\nIn the spinning process for fibers, metal fibers, carbon/graphite fibers, or intrinsically conductive polymer materials having intrinsic electromagnetic function may be used to take place of the ordinary fiber materials in whole or in part. It is possible to add the powder having electromagnetic properties to the spinning solution in the blending way during the spinning process.
\nIn the spinning process for yarns, electromagnetic fibers such as metal fibers and magnetic fibers can be added to the ordinary fibers through different ways to combine, producing the electromagnetic yarn. Metal fibers have low elongation and poor toughness; so, they are not suitable to be used alone for weaving. They are often used to form the yarn containing metal fiber with ordinary textile fibers by blending, enveloping, etc.
\nIn the weaving process, electromagnetically functionalized yarns can be directly woven. The common yarns can be interlaced into fabrics with the electromagnetically functionalized yarns.
\nThe finishing process is suitable for fibers, yarns, and fabrics. For the fiber or the yarn that has been formed and does not have electromagnetic function, the surface of it may be coated with a metal coating or magnetic powder by electroless plating, electroplating, magnetron sputtering, or other ways. For ordinary fabrics without electromagnet properties, the surface can be treated by finishing, such as the electroplating, electroless plating, or embroidery to make it electromagnetic.
\nThe electrostatic phenomenon of cellulose fibers in the processing process is not obvious; but the electrostatic interference of protein fibers is pretty serious. Although the wool fiber has high equilibrium moisture regain, its mass specific resistance is the highest in the natural fiber. The resistivity of synthetic fibers such as polyester, nylon, acrylic, and polypropylene, which are generally high in moisture regain, is as high as 1014 Ω cm, and the accumulation of electrostatic charge is obvious.
\nThe material is excited by various energies, causing the electrons to escape from the nucleus. The electrons overcome the binding of the nucleus, and the minimum energy required to escape from the surface of the material is called the work function. Different materials or the same material in different states have different work function. The generation and accumulation of electric charge causes the substance to carry static electricity, and the one that acquires the electron exhibits the negative electric property, and the one that loses the electron exhibits the positive electric property, which generate the electrostatic phenomenon.
\nThe resistivity of conventional textile materials is up to 1010 Ω cm or more, and the generated charge is not easily dissipated, resulting in very serious electrostatic phenomenon. Therefore, the antistatic properties of textile materials have become an important property having a great influence on the processing of textile materials and the use of textiles.
\nThe antistatic technology of textile materials includes the preparation of antistatic fibers, the preparation of conductive yarns, and the conductive treatment of textiles.
\nFor textile materials with higher mass specific resistance, surfactants are often added to fibers in fiber factories, which absorb water molecules from the environment and reduce static interference in the yarns. The hydrophobic end of the surfactant molecule is adsorbed on the surface of the fiber; the hydrophilic group is pointed to the outer space [3]. Then, the fiber forms the polar surface and adsorbs water molecules in the air. The surface resistivity of the fiber is reduced, and the charge dissipation is accelerated. The method is simple and easy to make; however, the antistatic effect is poor in durability, and the surfactant is volatile and less resistant to washing.
\nIn order to prepare relatively durable antistatic fiber, the methods are following: (1) Adding the surfactant to a fiber-forming polymer during blend spinning; (2) adding the hydrophilic group by block copolymerization; and (3) adding the hydrophilic group by graft modification in a fiber-forming polymer. These can make the fibers obtain durable hygroscopicity and antistatic properties.
\nIn addition, there are also another methods, including fixing the surfactant to the surface of the fiber with a binder and crosslinking the surfactant on the surface of the fiber to form a film. The effect is similar to applying an antistatic varnish onto the surface of the plastic.
\nAntistatic fibers are usually blended with ordinary fibers, and a higher content of antistatic fibers is required to achieve a more feasible antistatic effect. The specific ratio between antistatic fibers and ordinary fibers should be based on the resistivity of the ordinary fibers used, the final use environment, and requirements of the products.
\nThe electrical resistivity of the conductive fiber is smaller than that of the antistatic fiber, and it has a more significant antistatic effect. And during the blend fabrics with the same antistatic effects, the amount of conductive fiber added is much smaller than that of the antistatic fiber [4]. As long as a few thousandths to a few percent of the conductive yarn is added, the fabric can attain antistatic requirements. So with the widespread use of organic conductive fibers [5, 6], the field of application of antistatic fibers has been gradually reduced.
\nElectromagnetic shielding is a technical measure to prevent or suppress the transmission of electromagnetic energy by using a shield. The shield used can weaken the electromagnetic field strength generated by the field source in the electromagnetic space protection zone. There are two main purposes for shielding: one is to limit the field source electromagnetic energy leaking out from the area that needs protection and the other is to prevent the external electromagnetic field energy entering into the area protected.
\nThe shielding effectiveness equals to the ratio of electric field strength \n
Arching method is usually used to measure shielding effectiveness. The schematic diagram is shown in Figure 6, which can characterize the shielding effectiveness by measuring the power of the receiving antennas and transmitting antennas.
\nSchematic diagram of arching measuring method for shielding effectiveness.
Woven fabric is composed of warp and weft yarns interlaced vertically with each other according to a certain regularity. It is well known that ordinary fiber yarns are transparent to the EM wave, while yarn containing metal fibers is conductive. As metal fiber yarns are closely spaced, continuous conductive paths can be established easily. For an electromagnetic shielding fabric composed of metal fiber yarns or coated with a metal layer or a functional layer, it has a remarkable and typical mesh structure [7], as shown in Figure 7. Figure 7(a) is a schematic view of the structure of the metal fiber-containing yarn fabric and Figure 7(b) is a structural model diagram of the metal fiber yarn extracted in Figure 7(a). The geometric parameters in the figure correspond to the parameters in the fabric: h is the buckling wave height, similar to the thickness of the fabrics in the data; l is the length of the organization cycle; and a is the arrangement cycle spacing of the metal yarns in the fabric.
\nThe grid structure of electromagnetic shielding fabric. (a) Typical mesh structure (b) The structure of the metal fiber-containing yarn fabric.
The metal fiber yarn fabric is considered as a periodic grid structure model composed of conductive yarns [8], as shown in Figure 8. Assume that one parallel periodic array is composed of the warp yarn (as shown in Figure 8(b)) and the other is composed of the weft yarn (as shown in Figure 8(c)). The two wire arrays are directly cascaded at a certain orientation angle. The contact impedances between the wires of the two arrays are assumed to be negligible, due to the EM coupling between the two grid arrays. In particular, the grid structure model is only composed of metal fiber yarns, that is, the effect of ordinary yarns in fabric is not considered.
\nGrid structure model. (a) Grid structure; (b) Period parallel array 1 and (c) Period parallel array 2.
The two parallel periodic arrays (as shown in Figure 1) have wire orientation \n
The periodic grid is regarded as a stratified medium made of two periodic parallel arrays at a certain angle. The transmission matrix was established, and the SE for different polarization incident waves was calculated by analyzing the propagation of EM fields passing through the wire mesh. It can be used for the calculation of the SE of isotropic and anisotropic metal wire mesh structures for different polarization.
\nThe grid is excited by a plane wave having normal incidence, and the incident EM wave can be decomposed into \n
Schematic wire mesh illuminated by a plane wave with normal incidence and transverse magnetic (TM) or transverse electric (TE) polarization.
\n\n
where \n
As shown in Figure 10, the global coordinate system (x, y, z) and the local one (\n
The periodic array in the local coordinate system and the global coordinate system.
Assuming that the diameters of the metallic yarns and periodic spacing are small compared to the wavelength, then the parallel array can be modeled by a homogeneous thin anisotropic sheet with thickness a. Under the condition of the EM wave having normal incidence, the following relations can be written among the propagating field components expressed in the local coordinate system and averaged over the wire array period, on each side of the sheet.
\nin which J is the current density (A/m2) flowing inside the wire and all quantities are considered to be averaged over the wire array period. The current density J can be related to the tangential average electric field components along the wires. According to the following impedance condition,
\nin which the expressions of impedance \n
where \n
Combining Eqs. (10) and (11) yield
\nAccording to Eqs. (7)–(9) and (14), the boundary condition describing the relation among the EM field components tangential to the conductive grid is obtained, and its matrix expression is
\nin which \n
In the global coordinate system, the transformation matrix of the parallel array \n
in which the transformation matrix \n
The grid array transmission matrix \n
where \n
The total thickness of the wire grid is \n
in which the effective shunt admittance \n
Therefore, the matrix of the boundary condition for the metal grid is given as
\nThe shielding factors of the metal grid of against \n
in which \n
The electric field component on the back faces of the grid expressed in matrix form is given by
\nin which
\nIn the case where the incident wave is the \n
Similarly, for a \n
The shielding factors are, respectively
\nThe SE against \n
For isotropic material, the transmitted EM power due to incident \n
For anisotropic materials, the total incident and transmitted power are computed as the average of the two polarizations
\nFrom Eq. (6), we obtain
\nFor the grid that is isotropic in the \n
In combination with the mesh structure of the electromagnetic shielding fabric, the key factors affecting the electromagnetic shielding effectiveness SE are the structural parameters and material parameters of fabric [9, 10]. Some key 1parameters are as following: the periodic spacing of metal yarns, the conductivity of the yarns (decided by the type of yarns, the type of metal fibers, the content of metal fibers), the diameters of the yarns, the arrangement method, the connection of intersections, the direction of incidence of electromagnetic field, and the frequency.
\nSamples in which copper filaments are arranged in parallel at different intervals have different SE. As shown in Figure 11, the arrangement periodic intervals of copper filaments are 1, 2, 3, 4, and 5 mm, respectively. The SE of 10–14 GHz is 17–20, 7–12, 5–10, 2–5, and 2–4 dB, respectively. It can be seen that as the periodic spacing increases, the shielding effectiveness is significantly reduced.
\nThe shielding effectiveness at different intervals.
Metal yarn arrangement interval has an important effect on SE of fabrics. The metal yarn periodic spacing is related to these parameters such as fabric density and tightness. And as the fabric tightness and density increase, the spacing of the metal fibers reduces, the electromagnetic wave transmission decreases, and the SE increases.
\nThe woven fabric is interwoven from the yarns of two systems that are perpendicular to each other. Therefore, it is possible to introduce functional fibers into the parallel structure in only one system, or functional yarns are introduced to both systems to form a grid structure.
\nThe stainless steel core spun yarn and the blended yarn were arranged in parallel as a sample with a spacing of 2 mm, and the distance of the grid structure sample is 2 mm in both vertical and horizontal directions. The shielding effectiveness is shown in Figure 12. It can be seen that the SE of the yarn model of the parallel arrangement structure is the same to the grid arrangement structure, and as the frequency increases, the SE gradually decreases.
\nThe shielding effectiveness of metal fiber yarns in parallel and grid arrangement.
Separating the yarns of the horizontal and vertical systems in the sample with a thin insulating plate is seen as a nonconducting state. The bare copper wire is arranged in two states with conduction and nonconduction at the intersection, and the periodic intervals of the grid samples are 1, 2, 3, 4, and 5 mm, respectively, the shielding effectiveness is shown in Figure 13. It can be seen that under the same periodic spacing, the shielding effectiveness curves of the copper wire mesh model samples almost coincide in the two states of conduction and nonconduction.
\nThe shielding effectiveness of metal fiber yarns in parallel and grid arrangement.
The material parameters mainly include the way of forming the metal yarns, the material of the metal fibers, and the content of the metal fibers.
\nMetal monofilaments can be used to make fabrics after they have been formed into yarns by a certain yarn forming method. For metal filaments, core yarns and twisted yarns are the common yarns. However, for metal staple fibers, blended yarns are the commonly used yarn. The type of yarns affects the electromagnetic parameters of the yarns and fabrics, resulting in the difference in SE.
\nStainless steel filaments, core-spun yarns, blended yarns, and twisted yarns composed of stainless steel/cotton with a stainless steel content of 30% are arranged in a grid sample with a periodic spacing of 2 mm. The SE is shown in Figure 14. At the same spacing, the blended yarns have the best shielding effectiveness, while the shielding effectiveness of stainless steel filaments, core spun yarns, and twisted yarns are equivalent. At the frequency of 8–16 GHz, the SE of the former is about 7 dB higher than that of the latter, and both decrease with increasing frequency.
\nThe shielding effectiveness of different types of yarns.
The metal fibers used in the fabric are different, and the different electrical conductivity of the metal may affect the electrical resistivity of the yarns and the fabrics. For example, the electrical conductivity of copper fiber is 5.8 × 107 S/m, and the electrical conductivity of aluminum is 3.54 × 107 S/m. When the metal fiber content, linear density, and fabric specification parameters are the same, the SE of copper fiber fabrics are better than that of aluminum fiber fabrics.
\nThe five grid samples with completely nonconducting period of 2 mm is shown in Figure 15, for which each grid sample of five different materials consisting of stainless steel bare wire (the diameter is 35 μm), core spun yarn and blended yarn of stainless steel/cotton (containing the stainless steel content of 30%), silver-plated nylon filament (the diameter is 50 μm), and bare copper wire (the diameter is 80 μm). The SE of the samples made of, respectively, stainless steel blended yarns, silver-plated filaments, and bare copper wires are substantially equal and higher that of the samples of core spun yarn and stainless steel bare wire.
\nShielding effectiveness of different materials with grid period spacing of 2 mm in order.
The two blended yarns with the stainless steel content of 20 and 30% are woven in both warp and weft directions, and the SE of the obtained sample in the range of 1–18 GHz are shown in Figure 16. It can be seen that except for the frequency range of 10 and 12–14 GHz, the SE of the fabric with 30% stainless steel content is 5 dB higher than that of the fabric with 20% stainless steel content, and the difference of SE in other frequency bands is unobvious. The content of stainless steel fibers has a certain effect on the shielding effectiveness of the fabric, but after reaching a certain level, the difference is not significant.
\nThe shielding effectiveness of the fabrics with different content of stainless steel.
It is assumed that the fibers are evenly distributed in the yarn. As the content of the metal fibers increases, the shielding effectiveness of the fabrics will increase, but when it is increased to a certain extent, the bending stiffness and flexural modulus of the yarns will increase, and the porosity among fibers during the fabric will increase. So the SE of the fabrics becomes slow down or even lower. Considering the cost, the content of stainless steel is generally 20–30% for fabrics containing stainless steel fibers.
\nWhen electromagnetic waves radiate into the macroscopic object, which will causing induced electric charges and currents of the object, then the electromagnetic wave radiated into the object will be scattered into various directions. This process is called electromagnetic scattering [11, 12]. The electromagnetic scattering fabric is an electromagnetic functional material with the specific design structure, which makes the electromagnetic waves incident on the target are no longer reflected back along the way of the reflection of mirror, but radiated out into different directions. Thereby, it can reduce the radiated electromagnetic waves in the direction of propagation, and can make the human body and military targets invisible for certain direction Radar.
\nThe textile technology is relatively mature in the preparation of three-dimensional structural fabrics. Thus, it is very feasible to design the three-dimensional structure of metallized fabrics which have good scattering properties for incident electromagnetic waves.
\nAs shown in Figure 17, the composite materials of the three-dimensional periodic structure can be simplified into two-phase dielectric materials when studying the transmission process of electromagnetic waves in the three-dimensional structure.
\nDielectric column periodic three-dimensional structure.
When a simple harmonic uniform plane wave is incident on the three-dimensional structure, the three-dimensional coordinate system XYZ shown in Figure 18 is selected. The XOY plane of the coordinate system coincides with the lower surface of the object, and the Z axis is perpendicular to the interface of the upper and lower surfaces. Electromagnetic waves are incident from the upper surface with the incident angle θ, and reflected waves and transmitted waves are generated at the interface of the upper surface. The transmitted waves enter the periodic three-dimensional structure, and after being attenuated in the three-dimensional structure, the reflected waves and the transmitted waves are again generated at the lower surface interface.
\nThe schematic diagram of electromagnetic wave incident three-dimensional structure.
Electromagnetic waves are reflected at the interface of different medium, which conforms to the law of reflection, as shown in Figure 19. The concave-convex natural surfaces can be broken down into a series of planar elements with small-sized geometries, which is called roughness. The roughness of the scattering surface is very important in the surface scattering.
\nThe law of electromagnetic wave reflection.
If the surface is smooth, the incident energy would form two plane waves after interacting with the surface. One is a surface-reflected wave whose angle with the normal is the same as the angle of incidence, and the direction is opposite, as shown in Figure 20. The other is refracted or transmitted waves with downward surface.
\nThe reflection of electromagnetic waves on smooth reflective surfaces.
If the surface is rough, the incident energy interacts with the surface and then radiates and shoots in all directions, becoming a scattering field, as shown in Figure 21.
\nThe reflection of electromagnetic waves on rough reflective surfaces.
The fibers having electromagnetic properties are scattered as the fluff of the fabric on the surface, or are consolidated as a U-shaped structural unit on the fabric to form the fluff, and thereby a velvet structure fabrics with good radar wave scattering property is obtained.
\nThe woven velvet fabric for decoration and its reflection coefficient is displayed in Figure 22. It can be seen from the test results that the tested structural unit achieves attenuation of 5 dB in the bandwidth of 10 GHz, and the peak value reaches −30 dB. This is mainly due to the angle between the metal fluff of the structural unit and the plane of the sample. When electromagnetic waves are incident onto the sample, those metal fluffs with a certain angle in the plane have a certain scattering of the incident electromagnetic waves, which reduces the energy received by the receiving antenna, so that the reflection coefficient is reduced.
\nThe woven velvet fabric for decoration and its reflection coefficient.
The silver fiber spacer fabric is prepared on the warp knitting machine, of which the silver-plated filaments with a fineness of 83dtex are used in the middle layer, and the upper and lower surfaces are all made of polyester fibers, as shown in Figure 23. The silver-plated fibers make the radar waves absorption, reflection, and multiple reflections happen in the intermediate layer.
\nThe silver-plated fiber spacer fabric and its reflection coefficient.
The silver-plated fiber spacer fabric has a significant resonance peak, which should be related to the thickness of the intermediate layer of the fabric, and indicates that the shielding effect on the radar wave is not mainly due to the reflection radar wave mechanism. The reflectivity of silver-plated fiber spacer fabric is generally inferior to that of velvet fabrics, but its resonance peak can reach −30 dB, and when the reflection coefficient is below −5 dB, it has a wide bandwidth, even up to 18 GHz.
\nThe cut fabrics are obtained by cutting fabrics containing metal fibers or metallized fibers into different shapes. The planar fabrics are formed into the three-dimensional structure through some support, and the cut flower units of fabrics become scattering units for radar waves, which are a kind of flexible, lightweight, wide-band radar stealth fabric.
\nThe stainless steel/polyester/cotton blend fabric with a stainless steel content of 20% is cut as shown in Figure 24. The reflection coefficient of the fabric in the three-dimensional state and the state in which the fabric is flattened is as shown in Figure 25.
\nCut flower structure fabric. (a) Three-dimensional cut flower fabric and (b) Flat cut flower fabric.
The reflection coefficient of cut flower structure fabric. (a) The reflection coefficient of three-dimensional cut flower fabric and (b) The reflection coefficient of flat cut flower fabric.
From the test results, it can be found that the reflection coefficient of the flat structural unit and the uneven structural unit are very large in a wide frequency range, and the coefficient of the uneven cut flower fabric can reach −10 dB at 2 GHz, and the flat cut flower fabric is −5 dB. In the test results, mainly because of the antenna used in the test, the results of the test in the frequency bands less than 3 GHz and greater than 17 GHz are not regular enough. The irregularity of the structural unit produces a strong scattering for electromagnetic waves, making the reflection coefficient smaller. However, the main difference in the structural unit of unevenness and flatness is the difference in the position of the resonance peak that appears. This is because when the mesh structure becomes flat, the size of the unit structure becomes small, so that the resonance peak shifts toward the higher frequency.
\nBy adopting the method of embedding the heat shrinkable yarns, the textured structure containing metal fibers or metallized fibers can be obtained, which imparts good electromagnetic wave scattering properties to the fabric.
\nThe stainless steel fibers and the cotton fibers in a ratio of 40/60 were blended into a yarn of 116dtex linear density, and high heat-shrinkage polyester yarns with a shrinkage ratio of 53.7% in the boiling water and a linear density of 167dtex are embedded in the warp and weft directions. Mixed yarns and polyester yarns are woven into a plain fabric with a square weight of 127 g/m2 (as shown in Figure 26). The fabrics are treated at different temperatures to obtain fabrics with different concave and convex structures, as shown in Figure 27.
\nThe original plain weave.
Stainless steel fabrics with concave structure embedded with high heat shrinkage wires.
It can be seen from Figure 27 that the fabrics have different degrees of unevenness at different heat processing temperatures. The higher the treatment temperature, the more obvious the uneven structure; the lower the treatment temperature, the smaller the uneven structure. At 58°C, the fabric has a smaller degree of shrinkage.
\nAs Figure 28 shows, in the range of 2–18 GHz, for the fabrics containing the heat shrinkage yarns in both directions, as the processing temperature is lowered, the degree of the uneven structure of the fabrics is reduced, the unit size of the concave and convex structure becomes larger, which make the fabrics have poor scattering performance for radar waves. Besides, the reflection coefficients are becoming more and more higher and the difference is obvious. At a frequency of 14 GHz, the reflection coefficients of fabrics having heat treatment temperatures of 97, 75, 65, and 58°C and untreated fabrics, respectively, are −39, −27, −24, −10, and −4 dB. It can be seen from the shrinkage structure at different temperatures that the degree of wrinkles of the fabrics after treatment at 97°C is significantly higher than that of the wrinkles treated at other temperatures. The unevenness of the fabric structure causes the electromagnetic waves to form the diffuse reflection in the structure; meanwhile, the electromagnetic wave scattering forms multiple absorptions on the adjacent two intersecting slopes.
\nComparison of reflection coefficients under different conditions.
Frequency selective surface (FSS) is an infinitely large periodic array structure that is one-dimensional, two-dimensional, etc. It is mainly divided into two types: patch type and aperture type, which have frequency selective characteristic for the propagation of electromagnetic wave in space. The patch type can totally reflect electromagnetic waves of a specific frequency, and the aperture type can transmit all electromagnetic waves of a specific frequency.
\nThe textiles are light, soft, and flexible in processing. Relying on the media of textile materials, textile processing technology will qualify the textile products to attain the filtering characteristics and light, soft, and other characteristics, which can be applied in more fields [13]. Flexible periodic array structure prepared by textile processing technology is called frequency selective fabric (FSF). According to the filtering characteristics, the frequency selective fabric can be divided into four frequency response characteristics: high pass, low pass, band pass, and band stop, as shown in Figure 29.
\nThe four frequency response characteristics of the frequency selective surface. (a) High pass; (b) low pass; (c) band stop; and (d) band pass.
The researches mainly focus on the preparation of frequency selective fabrics with high-precision two-dimensional periodic structure produced by different textile processing techniques, which can be roughly divided into four categories [14].
Continuous conductive yarns form a periodic structure in the fabric. The continuous carbon fibers are directly woven into a square or rectangular periodic structure, as shown in Figure 30. Since the conductive carbon fibers are continuously present in the fabric, the structure is actually a conductive grid formed by the conductive yarns in the woven fabric. This structure is more suitable for the preparation of isotropic electromagnetic shielding fabrics, not a true frequency selective periodic structure, which is confirmed by the absence of resonance peaks in the test curves reported in the article.
The cut commercialized conductive material unit is directly bonded to the nonconductive fabric substrate, as shown in Figure 31.
Depositing conductive materials on the surface of fabrics by screen printing, inkjet printing, and other textile finishing techniques can form the conductive structural unit, as shown in Figure 32.
The high conductive yarns are formed into a periodic structural unit by textile weaving processing techniques such as weaving, weft knitting, embroidery, and so on, as shown in Figure 33.
Continuous conductive yarns form a periodic structure in the fabric. (a) Woven in a square structure and (b) Woven in a square structure rectangle.
The conductive material is adhered to the substrate to form a periodic structure.
Screen printing and inkjet printing form a periodic structure. (a) The screen printing and (b) The inkjet printing.
The periodic structures produced by knitting and embroidery processes. (a) The samples of woven fabrics and (b) The samples of knitted fabrics.
In China, the team that studies the periodic structure of textile materials is mainly a joint research group composed of Professor Meiwu Shi in textile materials and Professor Qun Wang in electromagnetic materials. Based on preliminary sample preparation, theoretical simulation analysis, and the preliminary experimental results and research ideas of special electromagnetic functional textile materials, in the aspect of 2D FSF, various types of bandpass, band-stop filter fabrics, etc. have been prepared by weaving, electroless plating, embroidery, transfer printing, and so on. Through experiments, the effects of cell shape and dimensional changes, periodic spacing, and dielectric materials on transmission and reflection coefficients have been studied.
\nThe preparation of frequency selective surface for flexible materials mainly includes screen printing, laser processing, and computer embroidery [15].
\nThe screen printing is to stretch and fix synthetic fibers, silk fabrics, or mental wire meshes on the frame, using the method of making the hand-painted film or photochemical plate to make the screen printing plate, and the metal ink is squeezed from the mesh of the pattern portion, which is a process for extruding onto a fabric to form a sample. Figure 34 shows a ring-shaped frequency selective surface of a complementary structure prepared by the screen printing method.
\nThe ways of screen printing processing. (a) The patch type and (b) The aperture type.
The complementary cross-type frequency selective surface prepared by the laser processing method is directly produced by laser processing on the flexible medium to which metal materials are pasted. The patch type (slit type) sample torn off the excess metal (patch metal) at the gap to obtain the final sample, thus ensuring the process precision of the patch type frequency selective surface (Figure 35).
\nThe laser engraving. (a) The patch type and (b) The aperture type.
The frequency selective surface with ring patch type is prepared by computer embroidery technology. According to the unit pattern and size of frequency selective surface period designed, the processing personnel uses the sample programming system to make the programming sample, and the needle position data of the design pattern is designed. Using these needle position data to control the computer embroidery machine, the silver-plated yarns are embroidered onto the fabric to produce the fabric material frequency selective surface (Figure 36).
\nThe screen printing technology is more adaptable and can be applied to the flexible medium surface in printing; besides, the process is simple, the cost is low, and the quality is relatively stable. However, the screen printing processing has low production efficiency and is only suitable for small batch production, and the image accuracy produced is not high, which has a certain influence on the frequency response characteristics of the product.
\nThe computer embroidery processing.
The laser processing technology is characterized by high quality, high efficiency, and low cost. The laser processing is a kind of non-contact processing, and the frequency selection surface patterns at the sharp corners such as precise polygons can be obtained, and the products have high precision. Because the excess metal needs to be removed during the sample preparation processing to obtain the desired sample, the production efficiency of the product is affected.
\nThe precision of computer embroidery technology when preparing flexible frequency selective surface is affected by the fineness of the needle, but the process is simple and the production efficiency is the highest. It is suitable for mass production and exhibits better band resistance characteristics at the resonance frequency. The fabric-based frequency selective surface structure can be directly integrated into various textiles such as tents, clothing, and decorative products, and has the advantages of portability, maintenance-free, and low cost.
\nThe single-performance frequency selective surface can no longer meet complex electromagnetic wave environments. Incident angle stability, multi-band, wide passband, miniaturization, flexibility, and active frequency selective surfaces are the research hotspots of frequency selective surface in recent years. The use of various processing techniques to convert a two-dimensional FSF into a three-dimensional structure can bring more performance to the frequency selective surface [16].
\nThe 3D FSF consists of the structural unit (white part in Figure 37), the dielectric unit (black part in Figure 37), and the base medium (gray part in Figure 37). In the z-axis direction, the conical stereoscopic periodic structure composed of the structural unit and the dielectric unit and a composite structure of the dipole plane periodically loaded with a base medium structure. At the same time, the 3D FSF has a multi-scale structure and easily deformable feature, and the electromagnetic parameters can be adjusted at multiple scales or by deformation.
\nThe multi-scale structure of 3D FSF.
Compared with metal periodic structural materials, 3D FSF is characterized by flexibility and can achieve deformation control. During use, the base medium will be the main bearer for external force, especially the elastic base medium with large deformation. The deformation of the substrate medium will cause changes in the size and other dimensions of the various scale structures fixed thereon, further leading to changes in electromagnetic response characteristics [17]. It can be seen that the electromagnetic response of the 3D FSF can be regulated by deformation.
\nThe authors disclosed receipt of the following financial support for the research, authorship/or publication of this article: the authors acknowledge support from the National Natural Science Foundation of China (grant number 51673211).
\nThis chapter will mainly discuss about 3D slope stability analysis using Limit Equilibrium Method (LEM) and Finite Element Method (FEM). These two methods are widely developed by academics and have been applied by many mining geotechnical practitioners in slope stability analysis. In recent time most of the analysis are performed in 2D method because of its simplicity and lower operational cost [1]. However, 3D analysis is more justifiable to represent the actual geometry condition. Thus, to obtain 3D slope stability analysis has its own importance. In this regard, 2D and 3D analysis can both be performed for the same slope stability analysis problem to obtain a more convincing and realistic results. These two methods can be used to validate one to another [2]. The analysis of the 3D approach is carried out by [3, 4, 5] in the 1970’s. Ref. [5] modifying the slice model used in 2D analysis to become a column in 3D analysis. The study of 3D slope stability analysis model is further developed and evaluated by [2, 6, 7] states that 3D FoS value is always greater than 2D analysis, therefore 2D results are considered more conservative [8]. Most existing three-dimensional 3D slope stability analysis methods are based on simple extensions of corresponding two-dimensional 2D methods of analysis and a plane of symmetry or direction of slide is implicitly assumed. 3D asymmetric slope stability models based on extensions of Bishop’s simplified, Janbu’s simplified, and Morgenstern–Price’s methods are developed by [8].
The stability of a slope can be determined by 2 criteria of considerations, which is the value of the safety factor (FoS) and also the value of the probability of failure (PoF), these two criteria are used to determine the optimal geometry of the pit opening. In order to obtain accurate analysis results, the information data regarding the geotechnical conditions of the model must be repetitive to the actual conditions. In general, the principal of the value of safety factor concept is the ratio between the shear strength along the slip surface required to maintain the slope at a stable condition, and the available shear strength [9]. The above definition can be mathematically described as:
The slope is assumed to be a model of an inclined plane [9]. By determining the resultant overall forces and moments acting in equilibrium state, the slope factor of safety value can be determined by comparing the amount of the resisting force to the driving force, or the resisting moment to overturning moment can see in Figure 1.
Equilibrium force and moment in inclined plane [9].
Based on the Mohr-Coulomb failure criteria, the definition of the value of the safety factor can be determined as follows:
Where c and ϕ are effective cohesion and internal friction angle.
The probabilistic failure is an approach that consider various input parameters that generates different Factor of Safety (FoS) values [10]. This information is based on the fact that every random input parameter has the same probability to yield a certain value of FoS. Regarding the difficulty and high expense of field and laboratory data collecting, this method is more attractive because of its representativeness. Figure 2 presents the concept of failure probability and the amount of uncertainty. Slope PoF is determined from the ratio between the area under the curve of the distribution of FoS <1 value to the distribution of FoS ≥1 value. The greater the range of distribution of FoS values, the higher the uncertainty of FoS values with the same PoF values.
Concept of probabilistic failure.
By definition there is a linear relationship between the PoF value and the likelihood of failure, while this does not apply to the FoS relationship with the chance of failure [10]. A large FoS does not represent a more stable slope, because the implicit uncertainty is not captured by the FoS value. Slopes with FoS of 3 do not mean that they are 2 times more stable than FoS of 1.5, while slopes with a PoF value of 5% are 2 times more stable than slopes with a PoF value of 10%. Slope stability in general performed in two-dimensional analysis. But in modeling complex geometries, 2D analysis cannot simulate them properly. Therefore, the 3D analysis is considered to be able to describe the conditions in the field better than the 2D analysis. Analysis of slope stability with 3D limit equilibrium method starts by assuming the geometry of the sliding mass (Figure 3).
Comparison between 3D and 2D single slope analysis [11].
The results of the calculation of slope stability can be expressed in safety factor. In this method, safety factor is not only influenced by the direction sliding, but also by the slip surface that safety factor is sensitive to critical slip surface locations. Therefore, the determination of a critical slip surface is very important. Safety factor can be obtained correctly if the determination of critical slip surface is accurate.
Optimal slope geometry is obtained from a step-by-step assessment process [12] state there are 5 stages of the process, which is models, domains, design, analysis and implementation. The initial stage of the geotechnical model is determined by 4 parameters, namely the geological model, structural conditions, rock mass and hydrogeological model. At the domains stage, the failure modes are determined by 2 parameters, namely the strength of the material and the condition of the structure. A single slope design arrangement is determined by the regulations or standards used by the company and the capabilities of the equipment used. Determination of the haulage road width and also the overall slope angle is based on mine planning related to the economic aspects of the opening geometry made. Furthermore, the stability analysis of the slope geometry that has been designed refers to the parameters (structure, strength, groundwater, and in-situ stress). After the final design is obtained, a risk assessment is carried out to mitigate the potential for landslides that may occur. In the implementation stage, the functions of dewatering, blasting and monitoring of the progress of the design model and the movement of rock masses (Figure 4).
Slope design process [12].
These days the needs and pressure to analyze a slope 3 dimensionally is more sounds. This is because 2D analysis assumes that the width of slope is infinitely wide so then it neglects 3D effect [11]. In most of the cases the width to height ratio is not sufficiently long and varies perpendicular to the slide movement. Therefore, 3D analysis is considered important to be done to produce the representative FoS. Moreover, in 3D analysis the volume of failure can also be estimated while 2D analysis cannot. If the volume can be determinate, it can be useful as one of the considerations in giving failure prevention recommendation.
The 3-dimensional model is a refined version of the 2-dimensional by projecting the skid plane into a column and determining the resultant force, as well as the moment based on the x, y, and z directions. The equilibrium force and moments acting on the overall column mass are used to determine the following 3 possible direction of the slip plane:
The column moves in the same direction
The column moves towards one another
The column moves in the opposite direction
For the 3-dimensional analysis, the mass potential of the slip plane is divided into several columns. Ref. [8] give the equation of the Simplified Janbu method deduced from the Morgenstern-Price method to obtain a safety factor value of 3-dimensional analysis (Figure 5).
3 dimensional column [8].
ai is space angle for sliding direction with respect to the projected x – y plane, ax, ay are base inclination along x and y directions measure at the center of each column, Exi, Eyi are inter-column normal forces in x and y directions, respectively, Hxi, Hyi are lateral inter-column shear forces in x and y directions, N’i, Ui are effective normal and base pore watery force, Pvi, Si is vertical external force, and base mobilized shear force, and Xxi, Xyi are vertical inter-column shear force in plane perpendicular to x and y directions.
With the mohr-coulomb collapse criteria, the safety factor is determined using the following equation:
where Sfi is ultimate resultant shear force available at the base of column i, N’i is the effective base normal force, Ci is (c. Ai) and c and Ai are effective cohesive strength and the base area of the column. The base shear force Si and normal base force Ni are expressed as the components of forces with respect to x, y, and z directions for column i.
where f1, f2, f3 and g1, g2, g3 = unit vectors in the direction of Si and Ni. The projected shear angles a’ = same for all columns in the x – y plane in the present formulation, and by using this angle, the space shear angle ai found for each column.
An arbitrary intercolumn shear force function f (x, y) is assumed in the present analysis, and the relationships between the intercolumn shear and normal forces in the x- and y-directions are given as:
Where λx and λy are intercolumn shear force mobilization factors in x and y directions, respectively, and λxy and λyx are intercolumn shear force mobilization factors in xz and yz planes. Considering the vertical and horizontal force equilibrium for the ith column in the z, x, and y directions produces the following equations (Figure 6):
Force equilibrium in columns [8].
Solving Equation, the base normal and shear forces can be expressed as
Considering the overall force equilibrium in x-direction internal force E cancels out.
Considering overall moment equilibrium in the x-direction
Considering overall force equilibrium in the y-direction
Considering overall moment equilibrium in the y-direction
The directional safety factor Fx and Fy is determined as follows:
Formulation 3D Bishop’s methods by considering the overall moment equilibrium equations in x or y direction and neglecting the inter-column vertical and horizontal shear forces.
Considering overall moment equilibrium about an axis passing through (x0, y0, z0) and parallel to the z axis gives:
For the 3D asymmetric Bishop’s method, at moment equilibrium point, the directional factors of safety, Fmx, Fmy, and Fmz are equal to each other. Under this condition, the global factor of safety Fm based on moment can be determined as
Formulation 3D simplified Janbu’s methods by considering the overall force equilibrium equations and neglecting the inter-column vertical and horizontal shear forces.
For 3D asymmetric Janbu’s method, at the force equilibrium point, the directional factors of safety, Fsx, and Fsy are equal to each other. Under this condition, the global factor of safety Ff based on force is determined as follows:
The safety factor is also used in vertical and 3D force equilibrium to achieve the simplified Janbu’s method.
One of the methods that can be used to determine the critical slip surface is the grid search method [11]. In the grid search method, the first thing to do is to determine the size of the grid box with dimensions x, y, and z. After the grid box is available, the user determines the number of grid points that you want to use in the x, y, and z directions. This point serves as the center of rotation. Each center of rotation can produce a number of circles that are used as slip surfaces. The number of circles produced at each center of rotation from the minimum radius to the maximum radius is called the radius increment. Illustration of the number of grid points and radius increment can be seen in Figures 7 and 8.
Illustration of grid point [11].
Illustration of the radius increment in the grid search [11].
After assuming the field geomaterial failure, the next step is mass discretization of the sliding mass into a number of columns. Square nets are applied to the sliding mass so that the sliding mass is divided into columns. There are two kinds of columns; the active column where the column is inside the sliding mass boundary line, and the inactive column where these columns are outside the sliding mass boundary line. In the calculation, the inactive columns are ignored so that the discrete sliding mass is determined only from the sum of the active columns. Figure 9 shows the illustration of the discretization of the sliding mass using a square grid.
Discretization of the sliding mass using a square grid [11].
After discretizing the sliding mass, internal and external forces in each column can be calculated based on moment equilibrium, force equilibrium, or both depending on what method of calculation is used (Figure 10).
3D model for slope stability analysis [11].
The grid search method is used to find critical slip surface. The grid search method starts by specifying the grid box dimension. The location and dimensions of the grid box must cover the entire study area so that the search for critical slip surface can be performed optimally, for the influence of number of grid point and radius increment in determining safety factor result can be seen in Table 1. The result of 3D slope stability analysis using grid search can see in Figure 11.
Radius increment | Number of grid points | Factor of safety | Volume (m3) | Location | Direction of sliding | Center of rotation | ||
---|---|---|---|---|---|---|---|---|
X | Y | Z | ||||||
20 × 20 × 10 | 0.868 | 2,245,470 | North-HW | 246.7 | 371,070 | 9,586,190 | 470 | |
10 | 30 × 30 × 15 | 0.874 | 1,998,950 | North-HW | 246.6 | 371,069 | 9,586,190 | 446 |
40 × 40 × 20 | 0.873 | 2,009,660 | North-HW | 246.5 | 371,068 | 9,586,190 | 443 | |
20 × 20 × 10 | 0.868 | 2,245,470 | North-HW | 246.7 | 371,070 | 9,586,190 | 470 | |
20 | 30 × 30 × 15 | 0.874 | 2,028,600 | North-HW | 246.5 | 371,069 | 9,586,200 | 429 |
40 × 40 × 20 | 0.872 | 1,934,080 | North-HW | 246.4 | 371,069 | 9,586,200 | 428 | |
20 × 20 × 10 | 0.868 | 2,245,470 | North-HW | 246.7 | 371,070 | 9,586,190 | 470 | |
30 | 30 × 30 × 15 | 0.906 | 1,628,780 | North-HW | 246.2 | 371,092 | 9,586,210 | 352 |
40 × 40 × 20 | 0.867 | 1,937,390 | North-HW | 246.1 | 371,076 | 9,586,170 | 459 | |
20 × 20 × 10 | 0.868 | 2,245,470 | North-HW | 246.7 | 371,070 | 9,586,190 | 470 | |
40 | 30 × 30 × 15 | 0.876 | 1,756,790 | North-HW | 246.2 | 371,076 | 9,586,200 | 404 |
40 × 40 × 20 | 0.872 | 1,934,080 | North-HW | 246.4 | 371,069 | 9,586,200 | 428 | |
20 × 20 × 10 | 0.868 | 2,245,470 | North-HW | 246.7 | 371,070 | 9,586,190 | 470 | |
50 | 30 × 30 × 15 | 0.876 | 2,068,950 | North-HW | 246.3 | 371,077 | 9,586,180 | 459 |
40 × 40 × 20 | 0.879 | 1,968,990 | North-HW | 246.3 | 371,078 | 9,586,190 | 421 |
The influence of number of grid point and radius increment in determining safety factor.
Grid search LEM 3D analysis result [11].
Grid Search is commonly used as slip surface searching method because the principle is simple and easy to understand [11]. However, this method can only calculate the circular slip surfaces, so it cannot represent the stability of slope in real condition. To make it more representative, non-circular slip surface option is also available in slope stability simulation software, and one of the searching methods in non-circular option is CS method.
CS is an algorithm which is used for solving the optimization problems. CS has been used in engineering field, such as welded beam and spring design optimization but there is still a few that use it for slope stability issue. Nowadays, the necessity to analyze 3D slope stability is more essential. The reason why slope stability problem should not be assumed 2 dimensionally that should be taken into account is the importance of determining volume of failure for risk management volume of failure is counted as one of the consequences, and it can be obtained by analyzing slope stability in 3D, indeed. Furthermore, there is a relationship between failure probability and volume of failure. Another issue exists when 3D analysis is performed using Grid Search on a vast area. The use of 3D analysis on a vast area can be complicated, and in real cases more advanced optimization methods are required. Therefore, CS is tried to be applied in order to determine the slip surface with minimum SF in 3D analysis. Thus, it can be suggested to be used as an alternative or other better option in 3D analysis.
CS means a metaheuristic optimization method that was developed by [13]. This method was inspired from cuckoo’s breeding behavior. In this research, CS is used as a slip surface search tool that has the lowest SF. CS is coupled with Lévy Flights random walk. There are few rules to use this algorithm as follows:
Each cuckoo lay one egg at a time, and dumps it in a randomly chosen nest;
The best nests with high quality of eggs (solutions) will carry over to the next generations;
The number of available host nests is fixed, and a host can discover an alien egg with a probability pa ∈ [0, 1]. In this case, the host bird can either throw the egg away or abandon the nest so as to build a completely new nest in a new location.
The last rule can be approached using a fraction pa to determine the worst solutions of n nests that will be replaced with a new nest randomly. In order to solve the problem, it can be simply illustrated that every egg in a nest represents one new solution. The purpose is to use the new and potentially better solution to replace the current solution in the nest. In a certain condition, the nest may have 2 eggs (2 solutions) but this problem is simplified so one nest has only 1 solution.
The random walk as determinated by Lévy Flights can be described in the following formula:
where α > 0 is the step size and Lévy(λ) is the position function from Lévy Flights (Figure 12).
Cuckoo search LEM 3D analysis result [11].
CS has been applied in many optimizations and computer intelligence with promising efficiency, this has been proved from design application in engineering field, scheduling problems, thermodynamic calculations, etc. Few examples of CS application in engineering field are designing spring, welded beam, and steel frame. The CS’s performance has also been compared with some metaheuristic algorithms such as PSO and GA, and the result shows that CS has higher success rate than. The result of the influence of max columns in x or y and max iteration in determining safety factor can be seen in Table 2.
Max iteration | Max columns in X or Y | FoS | Volume (m3) |
---|---|---|---|
40 | 50 | 2.06 | 159,816 |
60 | 2.02 | 179,523 | |
70 | 2.10 | 229,704 | |
80 | 2.10 | 303,186 | |
90 | 2.01 | 183,631 | |
100 | 2.01 | 187,898 | |
80 | 50 | 2.03 | 179,192 |
60 | 2.02 | 192,951 | |
70 | 2.03 | 167,961 | |
80 | 2.02 | 178,858 | |
90 | 2.02 | 171,821 | |
100 | 2.01 | 183,469 | |
120 | 50 | 2.02 | 175,113 |
60 | 2.02 | 180,573 | |
70 | 2.01 | 178,671 | |
80 | 2.01 | 193,606 | |
90 | 2.01 | 180,490 | |
100 | 2.02 | 190,336 | |
160 | 50 | 2.00 | 194,125 |
60 | 2.01 | 174,180 | |
70 | 2.03 | 170,430 | |
80 | 2.01 | 179,233 | |
90 | 1.00 | 19,008 | |
100 | 1.99 | 199,321 | |
200 | 50 | 2.01 | 180,821 |
60 | 2.01 | 179,535 | |
70 | 2.10 | 258,302 | |
80 | 2.19 | 270,897 | |
90 | 2.04 | 172,299 | |
100 | 1.99 | 192,171 | |
400 | 50 | 2.00 | 195,539 |
60 | 2.05 | 174,848 | |
70 | 2.02 | 199,719 | |
80 | 2.00 | 195,231 | |
90 | 2.00 | 194,572 | |
100 | 2.00 | 186,047 | |
600 | 50 | 2.05 | 157,244 |
60 | 2.04 | 171,805 | |
70 | 2.02 | 178,812 | |
80 | 2.01 | 184,185 | |
90 | 2.00 | 200,779 | |
100 | 2.00 | 195,602 | |
800 | 50 | 0.82 | 38,239 |
60 | 2.01 | 192,786 | |
70 | 2.01 | 204,299 | |
80 | 2.01 | 196,060 | |
90 | 2.01 | 187,613 | |
100 | 2.01 | 187,709 | |
1000 | 50 | 2.02 | 185,006 |
60 | 2.04 | 173,577 | |
70 | 2.03 | 175,154 | |
80 | 2.01 | 187,845 | |
90 | 2.01 | 189,787 | |
100 | 2.01 | 177,988 |
The influence of max columns in X or Y and max iteration in determining safety factor.
3D limit equilibrium analysis method, data regarding 3D slope geometry model, material properties and 3D geological models are required. The 3D slope geometry model required for this method can be obtain from the reconstruction of pit surface model from either terrestrial or photogrammetric measurement methods. The next step is to create an external volume from the pit surface model which will then be analyzed to determine the position and shape of the slip surface, example of 3D slope geometry model can be seen in Figure 13.
3D slope geometry model.
Limit equilibrium analysis method uses the properties values of materials obtained from laboratory tests results to calculate the value of the safety factor. An example of the input parameter data used in the analysis can be seen in Table 3.
Input parameter | Limonite | Saprolite | Bedrock | Unit |
---|---|---|---|---|
Unit Weight | 17.19 | 15.77 | 29 | γn (kN/m3) |
Cohesion | 41.14 | 38.06 | 3270 | C (KPa) |
Internal Friction Angle | 15.67 | 12.58 | 41.68 | ϕ (°) |
Modulus Young | 20,000 | 20,000 | 150,000 | ε (Kpa) |
Poisson Ratio | 0.30 | 0.40 | 0.20 | ν |
Material properties.
Geological modeling is the process of creating visual description geometry of rock lithology into software that represents the actual conditions. However, there are limitations in the modeling process, that related to the limited information on the data held and errors in data interpretation carried out. One of the methods that can be used to create the 3D geological model is interpolation the lithology information data from geotechnical drilling results. The 3D geological model will be used in the 3-dimensional slope stability analysis to determine the distribution characteristics of the rock lithology. In this analysis, the 3D geological floor model of limonite, saprolite and bedrock lithology is used and can be seen in Figure 14.
3D geological floor model.
The results of the analysis using the Bishop Simplified method for the slip surface search method, both grid search and cuckoo search can be seen in Figures 15 and 16. The analysis results with the grid search show the safety factor value is 1.104, while the cuckoo search is 1.089. The position of the slip surface with the grid search and cuckoo search is the same position it indicates the accuracy of the cuckoo search, while the grid requires a grid box which must represent the actual slip surface position.
Result 3D analysis using grid search.
Result 3D analysis using cuckoo search.
The finite element method has been widely applied by mining geotechnical practitioners in slope stability analysis, with the advantage that the stress–strain analysis in the material allows to determine the displacement and strain values acting on the model elements, but this method has weaknesses in the process. This analysis uses a large number of calculation matrices so that it requires a long computation time, especially if the analysis is carried out in 3D, the number of elements and nodes used will also be high so that the computation time can run for days depending on the number of nodes and types on mesh used. The computation time is closely related to the maximum number of iterations in the calculation of the force error or load imbalance (solid tolerance) in determining the convergence of model, the higher the maximum number of iterations, the calculation in determining the convergence level in the analysis result model will be more accurate, but unfortunately it will take a long time in the analysis.
Slope stability analysis using the finite element method takes into account the stress–strain analysis that works on each element of the model and focuses more on the analysis of the value of the deformation that occurs rather than the level of slope stability [2]. The advantages of analysis using the finite element method when compared to the limit equilibrium method are as follows:
No need to assume the position of the slip surface, failure occur in zones where the shear strength of the material cannot maintain stability, due to the shear stress that works due to gravity.
It does not require the concept of slice or columns, so it does not require an approach of forces acting on global equilibrium.
Analyze the stress–strain so that it can see the deformation and the effective stress.
The finite element method can monitor the movement of rock masses towards failure.
The value of the safety factor (SF) in the finite element method is defined as the ratio of the actual material shear strength to the shear strength of the material when the model failure. This concept is similar to the limit equilibrium method, where the shear strength ratio of a material to its driving force [1]. The strength of the material at failure can be written in the following equation:
With strength reduction factor (SRF) is the value of the factor for the decrease in the strength of the material, cf is the cohesion value when the model failure and ϕf is the value of the internal friction angle when the model failure. In determining the strength of the material at failure, the technique of shear strength reduction (SSR) is used, where the actual material strength parameter is decreased step by step until the model become failure condition (non-convergent) [14]. The value of the material strength reduction factor for the Mohr-Coulomb criteria can be determined as follows:
The systematic stage in the model analysis to determine the critical value of the material shear strength reduction factor (Critical SRF) is as follows:
The first stage is to prepare the model for analysis using the finite element method, determine characteristics of strength and deformation properties of the material.
The second stage is to increase the value of the material strength reduction factor (SRF) and calculate the material parameters using the Mohr-Coulomb criteria, then input the new material properties data into the model and recalculate and record the maximum total deformation value.
Repeat the second stage using a systematic increase in the value of the material strength reduction factor, until the model becomes a non-convergent condition (failure). The critical SRF is determined at the highest srf value in the model to achieve the convergent condition.
3D model analysis is an extension of 2D analysis, but in modeling complex geometries 2D analysis cannot simulate them properly. The 2D analysis assumes that the slope width is infinitely wide [11]. However, in many cases the 2D analysis is considered more conservative because it results in a lower safety factor value compared to the 3-dimensional analysis. The weakness of the analysis which is carried out in 3 dimensions is that it takes a lot of time and money, which makes practitioners not want to switch. The results of the 3D analysis can be conservative if they are validated with a 2D analysis in the most critical areas [2]. For conservative results, the 2D and 3D analyzes should not be significantly different, however in many cases the 3D analysis provides a higher safety factor value. The advantage given if the analysis is carried out in 3D is that it can represent the actual slope conditions in real terms and can determine the critical position. The value of the safety factor in the finite element method analysis with the 3-dimensional model is determined by the use of the material shear strength reduction factor (SRF) technique. in order to obtain the true safety factor, the srf is gradually increased until the model become failure (non-convergent). When this critical value is found, the safety factor of the slope model is equal to the reduction in material strength (SF) ≈ (SRF).
The determination of the critical value of srf is determined by increasing and decreasing the SRF value step by step until the highest SRF value is obtained in the model to be able to achieve the convergence criteria, see Table 4, the slope model enters a non-convergent (failure) at srf 1.05, so that the critical value of srf is 1.04 and highlighted in bold color.
Step | SRF | Solid Tolerance | Convergence |
---|---|---|---|
1 | 1 | 0.0005 | Yes |
2 | 1.3 | 0.2813 | No |
3 | 1.14 | 0.0691 | No |
4 | 1.06 | 0.0152 | No |
5 | 1.02 | 0.0008 | Yes |
6 | 1.04 | 0.0009 | Yes |
7 | 1.05 | 0.0078 | No |
Critical SRF determination.
Finite element analysis method employs the utilization of elements and nodes to perform stress–strain analysis acting on each element. The process of establishing these elements is called meshing. Mesh type will affect the analysis result. Greater number of elements will lead to a more accurate analysis result. However, this will result a more time-consuming computation. There are various mesh types in 3-dimensional analysis, there are 4-nodes and 10-nodes with uniform and graded shape available to this method. The example of elements and nodes utilization in FEM are given in Figure 17.
4-node graded and uniform.
To recapitulate the results of the analysis can be seen in Figure 18. From this figure provides information that there is an effect of using the mesh type on the srf value of the analysis results obtained, this is because the number of elements and nodes used is also different.
SRF VS mesh type.
In model analysis using the finite element method, the determination of the convergence criteria is limited by the analysis calculation (maximum iteration), the more iterations allowed, the more accurate the analysis results will be, but it also needs to be considered in terms of the slope model being analyzed, in the use of the maximum number. Optimal iteration related to the level of efficiency in the analysis so as not to waste too long.
At first, the model will be analyzed the stresses that work on each element due to the applied load, it will get the principal major stress and minor for each element, then enter the material properties for each material and start entering the material strength reduction factor (SRF) value for each element, at this stage the material properties value can be increased or decreased depending on the error value (solid tolerance) obtained. Furthermore, an Elasto-plastic analysis was performed using the Mohr-Coulomb failure criterion to obtain a new error value (solid tolerance). If the error value is still above the maximum value within the allowable iteration calculation limit, the SRF value will be lowered until the error value is below the maximum limit. The recapitulate the analysis results can be seen in Table 5.
Max iteration | SRF | Max displacement (m) | Computing time (s) |
---|---|---|---|
10,000 | 1.04 | 8.60 | 110,459 |
5000 | 1.04 | 8.74 | 57,990 |
2500 | 1.04 | 8.48 | 26,214 |
1000 | 1.04 | 24.29 | 14,622 |
500 | 1.04 | 14.34 | 6021 |
450 | 1.04 | 12.90 | 5422 |
400 | 1.04 | 12.72 | 4788 |
350 | 1.03 | 5.25 | 3650 |
300 | 1.03 | 5.42 | 3341 |
250 | 1.03 | 6.51 | 3159 |
200 | 1.01 | 1.70 | 2300 |
150 | 1.01 | 1.70 | 1969 |
100 | 1.00 | 1.67 | 1704 |
50 | 0.98 | 1.20 | 1541 |
25 | 0.68 | 1.09 | 1501 |
Influence of variable max iteration.
The analysis shows that the higher the maximum number of iterations, the SRF value also increases Table 4. As the SRF value increases, in the maximum number of iterations of 400 the SRF value constant at a value of 1.04, so this provides information that at a maximum of 400 iterations is the maximum optimal number of iterations. If seen from the effect of the number of iterations on the total displacement, the results will fluctuate, this is closely related to the srf value because the higher the srf value, the total displacement will also increase, but at 1.04 the srf value does not change/is consistent but the total displacement fluctuates because it is solid tolerance, the resulting solid tolerance also varies due to the use of different maximum iterations, this is because the slope conditions remain non-convergent (energy equilibrium is not achieved) above srf 1.04 in the number of iterations that have been set, however, solid tolerance will get closer to the maximum value, while for the computation time it is very clear that the higher the maximum number of iterations, the computation time will also increase, because it will take more time to search for convergence with the maximum iteration limit given, although the results will remain the same, that non-converging above 1.04 and the last converging value is 1.04.
Analysis using the finite element method uses the same data as the analysis carried out with boundary equilibrium, namely rock characteristics, 3D slope geometry and 3D geological models. Young and poisson ratio are input parameters in this analysis. If the boundary equilibrium analysis requires the position and shape of the slip plane, the analysis does not require these assumptions but uses the elements and nodes that are attached to the model. For an example of a 3D mesh model, see Figure 19. For this example, case a 4-node element type with a graded gradient is used.
3D meshing model.
After the model is successfully meshed, it is necessary to determine the limit of the resistance that works in the field of analysis to determine the internal force that works on the restraint installation which plays a role in determining the deformation limit of the model. For an example of the restraint model can be seen in Figure 20.
3D restraint model.
Analysis with the finite element method can see information about stress acting on the model and the total displacement (Figure 21).
3D FEM analysis result.
Slope stability using LEM shows that the cuckoo algorithm is reliable in obtaining position and shape of slip surface. In finite element analysis method, the optimum iteration number needs to be considered to improve analysis efficiency and preserving accuracy.
The authors would like to thank PT X who has facilitated the collection of field data, as well as the Mining Computation Laboratory of Mining Engineering Department FTKE of University Trisakti which has facilitated the author in using the software used in modeling research data.
.
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