Characteristics of the Elastic Response of Isotropic Linear Micropolar Solids under Imposed Loads

World Journal of Mechanics
Vol. 3  No. 5 (2013) , Article ID: 36033 , 11 pages DOI:10.4236/wjm.2013.35025

Juan Alejandro Vazquez-Feijoo1, Pedro Montes-García1, Pator Teodoro Matadamas-Ortiz1, Nicolas Juárez-Rodríguez2, Luicita Lagunez-Rivera1

1Centro Interdisciplinario de Investigación para el Desarrollo Integral Regional, CIIDIR-Oaxaca, IPN, Engineering Group, Oaxaca, Mexico

2Instituto Tecnológico de Tuxtla Gutiérrez, Tuxtla Gutiérrez, Mexico

Email: javazquezfeijoo@yahoo.xom.mx

Copyright © 2013 Juan Alejandro Vazquez-Feijoo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Received April 12, 2013; revised May 29, 2013; accepted June 24, 2013

Keywords: Strength of Materials; Elasticity; Microcontinuum; Micropolar

This article proposes a simplified way to solve solid mechanic problems in micropolar elasticity using as a starting point the solution found in the classic theory of elasticity. In this study, an analysis of the linear isotropic micropolar elasticity is conducted based on the properties imposed on the micropolar medium by the constitutive and equilibrium equations. To ascertain how the micropolar medium responses deviate from Hooke’s theory of elasticity, different loading conditions were classified. Three cases have been found so far: the rotational couple response, the quasi-classic equilibrium of momentum response and the general case. This study is the first in a series planned to explore the use of commercial packages of finite element in order to solve micropolar elasticity problems.

1. Introduction

There are a number of elasticity theories which are based on the assumption that a solid is a continuous medium [1]. One such theory is the classic elasticity theory based on Hooke’s law and is the one on which most of the commonly used materials in engineering are based. This theory assumes that each point on the continuum possesses three degrees of freedom or displacements in three orthogonal directions. Some special materials with no symmetry in their microstructure cannot be properly modelled using the classic theory, because they present additional mechanisms which oppose to deformation. That is the case in some cellular structures such as bones, wood and special materials like liquid crystal elastomers. In the case of human bones, several factors affect the mechanical response among which are age, gender and density. Because of these factors there is a wide range in the variation among the mechanical properties of this solid [2] and alternative elasticity theories should be used to model them properly. Microcontinuum theories are one such alternative.

These theories posit that each point of a continuous medium exhibits more than three degrees of freedom. On a prefatory note these theories consider that additional degrees of freedom have to be considered from the directors which are understood to be the orientations of the medium points. The microcontinuum theories can be classified as: 1) micropolar elasticity that assumes rigid medium points that can rotate; 2) microstretch elasticity that assumes breathing directors; and 3) micromorphic elasticity that assumes fully deformable directors [3].

The theory of micromorphic elasticity states that all the points in a solid have three displacements uk. It also states that solids have nine other independent degreesof-freedom that are accounted in a 3 × 3 matrix containing the so-called director’s components xkK. Considering the simplest case in microcontinuum elasticity, the micropolar elasticity theory there are only three additional degrees of freedom namely the microrotations of the directors fk.

To date, most of the microcontinuous studies have been based on the theory of micropolar elasticity, which has been applied to granular composites [4] or to cellular and porous materials [5]. Alternatively, the theories of microstretch and micromorphic elasticity have been applied to liquid crystal and liquid crystal elastomers [6,7]. The liquid crystal elastomers have attracted a great deal of attention, because of their remarkable damping properties [8].

There have been a number of studies done on micropolar elasticity [8-10]; however, most of them are focused on mathematical solutions to the equilibrium equations in particular cases. The work developed to date is sufficient for purposes of identification, but when the objective of the work is for purposes of prediction, particularly for engineering design, the studies may not be sufficient mainly because they are not clear about the conditions under which the assumptions are still valid. There are doubts about some of the static results obtained which assume that for a particular case, the displacements obtained from the classic elasticity theory are valid for micropolar linear isotropic solids. The assumption is that these displacements obey an equilibrium of momentum equation similar to those obtained from the theory of elasticity. It is necessary, however, to investigate under what conditions this assumption is valid. There are other assumptions which may not be valid as well, as in the case of pure bending, the macrorotation and microrotation are considered to be of the same magnitude (rotational coupled case).

This paper analyses the conditions which the micropolar constitutive and equilibrium equations impose on the theoretical micropolar solid and it is intended, from an engineering point of view, to enhance the predictability of the behaviour of the micropolar solid, so that a design can be feasible.

A micropolar solid is understood to be a solid in which the microstucture is regarded as rigid non-symmetric particles in the elastic medium. Only Cartesian coordinates are used here. In the first section, the equations that conform to the linear isotropic micropolar elastic theory are presented. In the second section, the conditions under which a micropolar solid behaves as a case of rotational coupled are considered. In the third section, the focus is on the determination of the condition that does not conform to the rotational coupled behaviour. This latter is still a classical-type equilibrium of momentum equation and, therefore, a displacement field similar to the classical elasticity, can be assumed. The final section presents a methodology to solve iteratively all other loading conditions that do not fall under the cases considered in the previous sections; however it is still an exact solution.

2. Micropolar Elasticity Theory Review

This section concentrates on the Linear Isotropic Micropolar Elasticity Theory and presents the equations that are used in the remaining part of the paper. Fundamentals of this theory can be found in [3]. Micropolar theory is based on two constitutive equations, two equilibrium equations and two compatibility equations. Each of the medium points possesses six degrees-of-freedom: three displacements ux, uy, uz and three microrotations fx, fy and fz.

The constitutive equations are,

in which the comma indicates the usual partial derivative and the repeated sub-indices imply a summation. These equations define six constitutive parameters λ, μ, κ α, β and γ that are needed to guarantee positive internal energy. This restriction imposes the following conditions,

The microstrain tensor εkl that appears in Equations (1) and (2) is defined as,

As can be observed from Equation (4), the microstrain tensor possesses an antisymmetric component due to the microrotations of the medium points.

The local equilibrium equations are:

The equilibrium of momentum,

and the equilibrium of the moment of momentum,

this case, V is the volume of the micropolar medium and S is the boundary of the micropolar medium.

Microrotation and displacements can also be prescribed. In Equation (6) lk represents induced microcouples per unity of mass and j is the microinertia, which is a constant for the linear theory. The compatibility equations are,

3. Rotational Coupled Case in a Micropolar Elastic Solid

Assuming that the displacement field in the medium can be expressed by polynomial functions, the macrorotations (Rj) depend on the displacement field described by the following equation:

and therefore the macrorotations can also be expressed through polynomial functions. Using this equation in Equation (4), the microstrain tensor can be expressed as a function of the rotations,

For simplicity, consider that there are no microcouples lj. Substituting Equations (1) and (2) into (5) and (6), respectively, and using Equation (9) the equation of moment of momentum for the static case, can be rewritten as,

In this section the rotational coupled case is revised. The revision considers the convenient condition, from a mathematical point of view, that the macrorotations and microrotations are of the same magnitude, i.e.,

Once this has been established it is possible to deduce that the conditions under Equation (12) are true. According to the Equation (11), the macrorotations are a function of the microrotations and their second order derivatives, as for example fk,kj. If according to the boundary conditions, the displacement field can be expressed as a polynomial function of less than a second degree of the coordinates, all the second order derivatives are zero; therefore, Equation (11) reduces to Equation (12). Several static cases already reported indicate that simple tension, torsion and bending [11] fell into this case, and only the Equation (5) is needed to solve the displacement field.

As a consequence of Equation (12), Equation (10) is now, simply,

The symmetric strain tensor is obtained from the Green stretch tensor. The antisymmetric part of Equation (10) disappears.

An extreme example of the rotational coupled case is when a micropolar beam is under pure bending [10], which undertakes a macrorotation given by,

Substitution of Equation (14) in Equation (11) gives,

It should be clarified that there is no intention, at this point, to give a particular example of particular conditions. Rather, the intention is to consider the general results obtained from the constitutive equations. The examples given through references are provided only to demonstrate that there are particular cases that fall into the polynomial case considered in this work.

The antisymmetric part of the microstrain from Equation (10) is not considered; therefore, Equation (5) reads,

Equation (17) is similar to the classical elasticity equilibrium equation,

and therefore it can be assumed that there are identical fields of deformation for both classic elasticity and micropolar (for the case of the rotational coupled) theories. As the anti-symmetric part of the strains is zero (equation 12), the macrorotation does not contribute to the resistance of stresses produced by external forces (Equation (1)). The microrotations on the other hand do produce resistance against external moments or microcouples applied to the body (Equation (2)).

It can be concluded that a necessary condition of a micropolar solid, to respond as a rotational coupled case, is that the macro-rotation has to be a first-degree polynomial function of one of the coordinates xj or a constant. This implies that the static cases of simple tension (macro-rotation is zero) and pure beam bending (a first order function) all fall into this category.

Quasi-Classic Equilibrium of Momentum

The rotational coupled case has an advantageous circumstance because in this situation the same deformation assumptions used in the classical elasticity analysis can also be assumed in the analogue cases for the micropolar elasticity. This is possible because the equation of equilibrium of momentum resembles that of the classical elasticity theory; however, this is not the unique case in which it may happen.

Equation (5) can be expressed as function of the symmetric and antisymmetric strain tensors as follows,

For clarity, concentrated forces are eliminated in Equation (19). Note that the classical equilibrium equations appears once more if,

By the use of Equation (11), Equation (21) can be obtained,

which can be simplified as,

This equation holds for any field of displacements that produces a macrorotation with a polynomial function of a third degree or lower because third and higher order derivatives become zero. An example of this is a Timoshenko beam with two simple supports bearing concentrated force. The displacement field in the classical elasticity theory (Hooke’s Law) consists of polynomials of a maximum degree of three. A similar field can be assumed for the micropolar medium. With the Timoshenko beam the flexural moment is considered to be a function of the first order of the axe 1 which coincides with the largest dimension of the beam along the neutral axe. It can be written,

The stress can be obtained from the well known equation,

Here the axe 2 is oriented in the direction of the loads. The displacement in the x1 direction is,

which implies a macrorotation of,

which gives a macrorotation expressed by a second degree polinomial function. In a general approach, if the macrorotation can be expressed as a second degree polynomial,

the moment of momentum Equation (11) is,

where, the microrotation is assumed to be,

Because there is no other possible second degree term in Equation (25). Here, C is a constant that must balance the second term of the left hand side of Equation (25),

The substitution of Equation (26) in Equation (19) gives,

then, by the use of Equation (17),

which is analogous to the classical elasticity equation of equilibrium. Therefore, the same assumptions can be considered for the micropolar medium.

Because of the restriction on space, more details on this beam are not included in this paper; however, if further information is needed it can be found elsewhere.

If the microrotation is a function of time, as in the case of wave propagation, the microinertia has to be considered (Equation (6)). The condition for holding Equation (20) is restricted to,

Another case where Equation (27) can also be applied is when there is a constant microcouple per unit of volume, as observed by analysing Equation (11). A constant microcouple does not produce strains. No excitation of the micropolar body appears either in the equilibrium equation of momentum (Equation (27)) or in the boundary conditions (5b) and (6b). The microcouple is equilibrated only by the microrotation Analogous to the classic elasticity theory, Eringen [9] considers the case of micropolar elasticity for plane strain, in which the displacements possess the following restrictions,

The condition stated by Equation (22) is reduced to,

This implies that both sides of (21) are zero and, if so represents the rotational coupled case.

If the quasi-classic equilibrium equation is valid, then the deformations are analogous to those obtained by the classical elasticity theory; however, when the macro and microrotations are not coupled, then the microrotations are opposed to the external loads by producing microcouples and/or shear stress, as the antisymmetric part of the strains is no longer zero, see Equation (4).

4. The General Linear Isotropic Micropolar Elastic Solid in Static Conditions

The advantage of the cases treated in the previous sections is that the proposed solution for a particular case can be extrapolated directly from the deformation field, which itself can be obtained from the classical elasticity theory. In the general case, the following considerations can be taken to solve the problems which are not considered in the previous cases. Referring back to Equation (11) and bringing back into this equation the induced microcouples per unit of mass, let’s split both the macrorotation and microrotation into two terms. Then Equation (11) is now,

Because this is a linear differential equation, by the use of the superposition principle, Equation (31) can be split into two equations as follows,

Considering Equation (32), the equilibrium of momentum given by Equation (19) is now,

A displacement vector is obtained. Observe that if there are any microcouples per unit of mass, this equation becomes the quasi-static equilibrium (Equation (18)) because the microcouples participate as field forces just as gravity does. From Equation (11), the macrorotation can be obtained. Then, is obtained with Equation (32).

Thereafter, if the microrotations possess a nonzero second order derivative, the Equation (33) can be substituted in Equation (19) thus obtaining,

Equation (33) produces other sets of displacements from which it is possible to calculate the microrotations. The term is now obtained using Equation (31).

If has a non-zero second derivative in respect to the other two coordinates, then the process is repeated to obtain a new set of rotations, e.g. and, up to the point in which the corresponding second order derivative of the microrotation term is zero. The total displacement macrorotation and microrotation are,

Now that the proposed methodology has been illustrated the discussion of some examples on the general linear isotropic micropolar elastic solids are presented.

5. Examples or the General Linear Isotropic Micropolar Elastic Solids

5.1. Clamped Beam with a Free End and a Microcouple per Unit of Volume

Suppose that a micropolar solid of dimensions u0, v0 and w0, (, v0) is clamped at the end at x = 0 and that there is a free end at x = u0, as shown in Figure 1. Then, because of the magnetic or electric field, a non uniform microcouple per unit of volume is induced in accordance with,

No other external forces are applied different from the reaction at x = 0. By the use of Equation (34) one may start considering,

Under this first approximation, Equation (19) is now,

The unique equation with non trivial solution is (38b). As the microcouple is only a function of x, it is assummed that all the displacements are a function only of x. This implies that a non-normal stress is present in this equation and it can be reduced to,

Figure 1. The micropolar solid is prevented from rigid displacement and then a distributed couple per unit of mass is applied.

Considering that any displacement depends only on the direction x, the previous equation is reduced to,

The unique nonzero displacement depends on x,

The displacement field in this first iteration is,

The macrorotation is now,

Using Equation (32)

Equation (33) then gives,

Equation (19) is now,

The displacement is now,

The macrorotation is,

The total macrorotation is (adding Equations (41) and (46)),

Equations (40) and (45) give the displacement,

Equations (43) and (47) give,

Adding Equations (42) and (49),

As derivatives in Equation (32) become zero after the third iteration, no further iterations are needed. In this analysis ekk = 0, so there is no normal stress as there is when external flexural moments are applied (Figure 2). As the displacement uy is a function only of x, only shear stresses are present and each produces a sliding cross section (Figures 3 and 4). Observe that this microcouple per unit of volume does not produce any reaction force in the clamped end, as no displacement is produced at x = 0. This can be seen using Equation (4) together with Equation (1), from which is obtained,

However there is a moment, Mz, because of the presence of the microcorotation fzx. The use of Equation (50) gives,

The micromoment mzx is obtained from Equation (2),

Figure 2. The moment Mxz is applied through the distribution of σxx on the boundary s-s'.

Figure 3. Concentrated microcouple per unit of volume produce only shear strain.

Figure 4. Deformation of the micropolar body under a cubic distributed couple per unit of volume.

that when evaluated in x=0, the following is obtained,

From the previous equation the total moment Mz can be obtained by integrating,

finally obtaining,

5.2. Double Clamped Beam with a Concentrated Force

The case of a doubled clamped beam (see Figure 5) is analyzed using the same assumptions considered in [10]. As noted before, the starting point of the analysis is the traditional elasticity theory, then, if the Euler-Bernoulli equation is used in the analysis, the shear force, the moment, the angular deflexion and the vertical displacement are as follows,

where Em is the equivalent to the Young modulus,

Figure 5. Double clamped beam with symmetric transverse area.

This equation, similar to that of the classic elasticity theory, can be obtained from the matrix Equation (1) when inverted for deformations, see references [8,9].

Given a specific cut around any cross section in x, there is a moment M(x) that has to be equilibrated by the moment produced by the normal forces around the z-axis and the moment produced by the microcouples mxz distributed all around the face x. Therefore,

As in the Euler Bernoulli beam, the normal stress is a linear function of y,

In order to consider the application of Equation (2), and assuming that, which is valid as a first approach,

The integration of this equation gives,

The macrorotation is given by,

because the unique displacement is in the x direction u(x,y), then

and the integrals end to be,

Considered in the classic treatment of bending (transverse areas Ax remains without deformation). The displacement u(y,x) can be obtained from the bending angle θ(x) as shown in Figure 6.

Taking into account the geometrical parameters shown in Figure 1, it can be established that

The displacement u(x,y) is then,

Figure 6. The displacement u is function of the bending angle.

The stress is then obtained as,

Using Equation (56), the stresses are then obtained as,

From Equation (52),

and with the use of Equations (57) and (60), Equation (54) is now,

where kz is the well known radii of gyration.

The radii of curvature is,

where φ is,

Using Equations (52) and (60),

and the integration of Equation (61) gives,

From Equation (55),

The substitution of (65) in (64) gives,

The displacement u(x,y) is then in accordance with Equation (57),

The macrorotation can be rewritten as,

Now, equation (19) is valid, when ; therefore,

Substituting Equation (67) in Equation (11),

Considering Equation (67),

Using the Equation (66) and the basic concepts of mechanics of materials,

Equation (68) can then be written as,

For the particular case of concentrated forces, as in the present example, all the derivatives of V(x) are zero; therefore there is no possibility for any other derivatives. The microrotation is then,

The microcouple (moment per unit of volume exerted by the microstructure) is given by Equation (2),

when Equation (70) is used,

That in turn produces Equation (54).

Equation (51) is now modified, to include φ,

These are the equations that solve the case of the hyperstatic micropolar beam. The last two Equations (71) can now be assembled into a matrix equation, which when inverted,

gives the usual response to the classic elasticity theory. Observe that φ is not involved.

If a is the half of the beam,

The maximum vertical displacement is found when θ(x) = 0, from Equation (71),

That gives x = L/2 when a = L/2. The micropolar effect does not alter the localization of the maximum deformation. It affects only the value of the maximum displacement. For the case a = L/2 one has,

6. Conclusions

The objective of this paper is to present a simple way to solve the micropolar isotropic material strength for engineering applications. The proposed solution, which is related to the response of this kind of medium to external loads, can be obtained by an iterative process that starts from a deformation field similar to the classic elasticity theory.

The two cases which were presented here demonstrate that this iterative process leads to a consistent solution of the constitutive micropolar isotropic equations; however, some experimental work is required to verify the results obtained.

In the clamped beam with a free end and a microcouple per unit of volume case, the mathematical solution is a strain field which is not observed in the classic elasticity theory; this strain field is a particular case of flexion where only shear strains exist. This occurs because the beam cross sectional planes do not rotate but slide over one another. The displacement causes macrorotations to equilibrate the microrotations which are in opposition to the distributed microcople per unit of volume.

The solution of the double clamped beam with a concentrated force case represents a double clamped beam with a concentrated force. It can be considered an extention of the Euler-Bernoulli beam to the micropolar elasticity. The beam shows to be stiffer than this clasic elasticity counterpart. This is because of the oppositon of the microrotations to the bending moment at each transversal section.

Departing from a solution well known in the equivalent case in the classical elasticity, it is quite easier to obtain the solution to the response of the micropolar medium. In addition, the physical interpretation is much simpler.

This approach may open the possibility of a strengthening of material science for microcontinuum media, through an iterative process to start solving classic, then micropolar elasticity, then solve microstretch cases and finally obtain the micromorphic elasticity solutions. Future articles are focused on experimented data on one hand and on the other hand to expand this kind of analysis to microstrech elasticity

  1. R. Lakes, “Experimental Methods for Study of Cosserat Elastic Solids and Other Generalized Elastic Continua,” In: H. Muhlhaus, Ed., Continuum Models for Materials with Micro-Structure, Wiley, New York, 1995, pp. 1-22.
  2. J. A. Beltran-Fernández, L. H. Hernández-Gómez, G. Urriolagoitia-Calderón, A. González-Rebatú and G. Urriolagoitia-Sosa, “Biomechanics and Numerical Evaluation of Cervical Porcine Models Considering Compressive Loads Using 2-D Classic Computer Tomography CT, 3-D Scanner and 3-D Computed Tomography,” Applied Mechanics and Materials, Vol. 24-25, 2010, pp. 287-295. doi:10.4028/www.scientific.net/AMM.24-25.287
  3. C. Eringen, “Polar and Nonlocal Field Theories,” Continuum Physics, Academic Press, London, New York, 1976.
  4. R. Lakes, “Elastic and Viscoelastic Behaviour of Chiral Materials,” International Journal of Mechanical Sciences, Vol. 43, No. 7, 2001, pp. 1579-1589.
  5. J. Rosenberg, R. Cimrman and V. Novacek, “Bone Tissue Modelling Based on the Orthotropic Micropolar Continuum,” Engineering Mechanics, Svratka, 2002.
  6. C. Eringen, “A Unified Continuum Theory of Elctrodynamics of Liquid Crystals,” International Journal of Engineering Science, Vol. 35, No. 12-13, 1997, pp. 1137- 1157. doi:10.1016/S0020-7225(97)00012-8
  7. C. Eringen, “A Unified Continuum Theory for Electrodynamics of Polymeric Liquid Crystals,” International Journal of Engineering Science, Vol. 38, No. 9-10, 2000, pp. 959-987. doi:10.1016/S0020-7225(99)00089-0
  8. S. M. Clarke, A. R. Tajbakhsh, E. M. Terentjev, C. Remillant, G. R. Tomlinson and J. R. House, “Soft Elasticity and Mechanical Damping in Liquid Crystalline Elastomers”, Journal of Applied Physics, Vol. 89, No. 11, 2001, pp. 6530-6535, doi:10.1063/1.1368177
  9. R. D. Gauthier and W. E. Jahsman, “Bending of a Curved Bar of Micropolar Elastic Material,” Journal of Applied Mechanics, Vol. 42, No. 2, 1975, pp. 369-374. doi:10.1115/1.3423583
  10. A. C. Eringen, “Fracture, and Advanced Treatise,” Academic press, New York, 1968.
  11. S. Ramezani, R. Naghdabadi and S. Sohrabpour, “Analysis of Micropolar Elastic Beams,” European Journal of Mechanics-A/Solids, Vol. 28, No. 2, 2009, pp. 202-208. doi:10.1016/j.euromechsol.2008.06.006

M = bending moment V = shear force xi = coordinates in the i direction Em = micropolar Young modulus

σo= proportional constant fi = microrotation q = bending angle Ii = second moment of inertia around i dij =Dirac Delta tRL = stress l, m, k, a, br = constitutive parameter erl = symmetric strain tensor erl = symmetric strain tensor mrl = micromoments pro unit of area ui = displacement in i direction r = density fk = force per unit of mass ni = unitary vector in the i direction Ri = macrorotation lk = microcouple per unit of mass j = microinertia


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Chainlink's DeFi coin is LINK, which launched in March 2020 at $2.23. It went up over time, breaking the $3 barrier in April 2020, the $4 mark in May 2020 and $5 in July 2020. The coin's value then rocketed, reaching $19.03 on 17 August, before crashing back down to earth, landing at $7.86 on 24 September.

(See our current list of Best DeFi Rates .) Liquidity mining: This allows users to earn yield for providing liquidity (capital) to a liquidity pool on a DEX (Decentralized Exchange). Users may also be paid rewards in the DEX's in-house token, which may increase in value, a bit like owning "stock" in the company.

So, what's the top Decentralized Finance (DeFi) cryptocurrency to buy for April 2022? We've compiled a list of 7 options for you to consider and explain why they're the best you can get your hands on. 1. Uniswap Total value locked (TVL): $7.5 Billion Uniswap is a decentralized exchange (DEX) that is built on the Ethereum blockchain.

We are providing you with a list of the best Defi Coins to make sure that you are choosing the right coins. Their market cap and circulating supply values are also given. You can compare these values to decide which one is the right DeFi token or coin you want to choose for investing. 1. Uniswap

For people interested in DeFi, its applications, its potential or even creating DeFi services, these are the best online courses to help you get started. 1. Decentralized Finance (DeFi): The Future of Finance from Duke University Why take this certification?

7th Best DeFi Apps: Alpha Homora Conclusion Decentralized Finance Decentralized finance is rapidly becoming one of the popular concepts in the cryptocurrency world. Although the DeFi concept is relatively new, DeFi users utilize it to create a cheaper and faster way of making financial transactions.

Although the decentralized finance marketplace is now home to hundreds of projects - we found that the overall 10 best Defi coins are those listed below: DeFi Coin - Overall Best DeFi Coin to Buy...

Top 14 DeFi tools 1. MetaMask 2. DeFi Pulse 3. DefiLlama 4. DappRadar 5. Coinbase Wallet 6. DeFi Saver 7. Zerion 8. LiquidityFolio 9. Orion Protocol 10. Dune Analytics 11. Revert Finance 12. APY.Vision 13. DEFIYIELD 14. 1inch Honorable mention What are the best DeFi tools to use? Frequently asked questions The basics of DeFi

In light of current prices, $150,000 USD is the minimum stake amount. Lido eliminates this barrier. 10. Abracadabra (SPELL)-Defi 2.0. Abracadabra Money utilizes interest-bearing tokens (ibTKNs) like yvWETH, yvUSDC, yvYFI, yvUSDT, etc. as collateral to borrow MIM (Magic Internet Money) that is USD-pegged. MIM works like any other stablecoin.

Here are some of the best DeFi games that you can check out: 1. Decentraland With the DAO and SAB, Decentraland is one of the oldest blockchain projects. Soon after the game's launch, the DAO (Decentralized Autonomous Organization) was set up and the creators took a backseat. © Decentraland You, as a player, can buy LAND and start developing it.

DeFi Coin (DEFC) is the native digital token of DeFi Swap - a decentralized exchange that supports instant conversions, yield farming, staking, and other interest-based services. DeFi Coin operates on top of the Binance Smart Chain (BSC), which means that transactions are facilitated in a fast and extremely cost-effective manner. Community Driven

1. DeFi exchanges. Zoom 1m 3m 6m YTD 1y All Mar 15, 2022 → Jun 15, 2022 21. Mar 28. Mar 4.

DeFi Swap - Overall Best DeFi LendingPlatform for 2022. Aqru - Lend Your Idle Bitcoin and Ethereum Tokens to Earn 7% per Year. - Various DeFi Lending Accounts to Choose From. Nexo ...

Here is a list of the 10 best DeFi coins that have contributed to the rise of this decentralized marketplace. 1. Lucky Block (LBLOCK) Our number one pick when it comes to the best DeFi coins is Lucky Block. Lucky Block is an innovative crypto-lottery platform that has made waves in the market since the conclusion of its pre-sale in January 2022.

DeFi Swap: Top DEX for DeFi users . DeFi Swap is one of the best DEXs in the market. DEXs stands for decentralized exchanges. These cryptocurrency exchanges allow you to directly transfer cryptocurrencies with other users online, without relying on a centralized platform or an intermediary. With the rise of DeFi, they have become incredibly ...

Top 3 Upcoming DEFI PROJECTS 2022 With Best Potential. One of the emerging sectors in the blockchain space defi projects. The top defi projects, allow you to...

Highly optimized for mobile use, Yieldwatch is one of the best DeFi apps for smartphone usage. 7) Uniswap A decentralized exchange with a sleek interface, Uniswap enables users to exchange a number of ERC20 tokens. Moreover, you can also contribute to the liquidity pool, and make some money on Uniswap.

DeFi is a term given to financial services that have no central authority and there is no one in charge. These financial apps and protocols are governed by smart contracts*. Developers who create a DeFi app transfer the ownership to users. DeFi then becomes an asset controlled by the community coin or token.

Potentially saving the best for last, Polygon is the ultimate scaling platform for Ethereum's dApps - likely one of the biggest current drivers behind deFi as an ecosystem. MATIC's price point fluctuates within $1.20-2 at the moment, making it affordable for various strategies and positions, whether that be short- or long-term.

It's worth reading
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December 3, 2022

Game Nab

Technology Information

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Book Review: The Beast Is An Animal

The Beast is an Animal

Obviously, as an author of fantasy fiction, I love fairy tales. The Beast Is An Animal is such a story; think Brothers’ Grimm. Immersing in the story was like a cool swim on a hot day. The fictional setting felt like seventeenth century New England with small changes in the spelling of names and places to suggest a landscape that was different; forest became fforest and Alice became Alys, the story’s heroine.

The book touts the story of a girl coming of age while learning to deal with good and evil, beauty and ugliness. One cannot exist without the other, but she finds it all exists, within both her own character and her immediate environment.

I didn’t work hard to study the thematic treatments Peternelle van Arsdale infused in the work, but one especially relevant statement for our times rose to the surface, at least for me. Specifically, Beast demonstrates how extremism gives birth to evil, which continues to feed on itself until everything around it becomes tainted and ripe for destruction. In reflection, I’m not quite certain what role the Beast plays in the narrative as Alys must deal with creatures who morphed from children to something sub-human. The Beast itself did not appear to have much in the way of malevolence, and in fact, it seemed to be at risk right along with the humans of Alys’ world.

If there was one disappointment, it came at the end. Alys serves more as a spectator to the defeat of evil than an active defender of good, and the heroine who saves the world for the people she loves.

Overall, van Arsdale is a competent storyteller and should be on a short-list of authors to look for again.

World Building: An exercise in creativity…

Perhaps the most fun I’ve had in terms of writing have been those sessions where I worked to create a different world or a world within a world for a novel. For Flashback, a story where an historian tries to prevent an assassination in the past, I needed a way to deal with time-travel in a convincing way. When you write science fiction, if you don’t base your premise on plausible science, you’ll lose your audience before you start and receive countless notes on how you got it all wrong. The challenge, as an accountant by training and profession, was to learn several theories of quantum physics relative to the creation of an artificial wormhole. Physics anticipates their existence, but how do you make a credible one up? This is where the opportunity to stand on the shoulders of genius comes by. Men and women with an amperage of brainpower I will never possess have actually discussed how to get this done. I adapted the methodology into the framework of my story, created a few rules that the plot would have to live by and voilà; I had a secret world known to only a few dozen people and a science-based platform from which to launch my characters. In essence, the science (as science fiction requires) became something of a character in itself. As went the wormhole, so did the stakes for the characters. One of the lasting joys of that exercise – which took a month of reading and re-reading texts on quantum mechanics just to warp my brain around the concepts – was that I now have the foundation for an unlimited number of sequels.

My second novel, Alfheim, a story of a boy who is actually an elf (think more in terms of Legolas and not one who works for Santa) was a whole different challenge. Like a Harry Potter or Twilight, I needed to create a completely coexistent world to our own. I spent hours and hours in the library, reading and soaking up elements of mythology that suited the ethereal images that existed in my mind. Like an architect who chooses from a diverse spectrum of building materials to create the physical embodiment of his creation, I found threads of folklore that inspired me to flesh out the characters that were slowly coming to life in my head. Each discovery of some fascinating morsel forced new questions as to how or why that would work in the world that was emerging. This is the crucial point where it’s imperative to start writing down the rules, the parameters that will govern your world, for once you begin to write, everything depends on those boundaries. If you violate these rules, you will anger your reader and unwittingly introduce a deus ex machina along the way, something to avoid at all costs. In the exercise, I borrowed from Tolkien, from sources on Fairy and Elfin lore (Celtic and Scandinavian), Medieval shipbuilding techniques, Medieval clothing styles, castle structures, ancient Celtic names and words and their meanings and pronunciations, Irish history, American history, driving through the White Mountains of New Hampshire, but never once reading a book of similar genre until the book was completely done. Often, the advice is to read heavily in the genre you are interested in writing. Counter to this, I avoided it. I wanted the mythology and world building to be completely free of any other writers’ notions of what a fairy world is like (and there are many writers in the field).

The important things to remember in doing world-building are these: Create the rules for your world and stick by them, violate them only with a really good reason, make them as plausible within the expectations for the suspension of disbelief as possible, don’t be afraid to let your imagination run totally wild, and for heaven’s sake – HAVE FUN!


Published On: September 27, 2022

Small Business Capital

The term “capital” is broad and can describe anything of value to its owner. Without capital, a business cannot function, whether it is a small family business, a major corporation, or a country’s economy. Think of capital as a way of measuring of wealth. Accountants look at capital in budgeting as cash flow.

Some examples of capital are:

Financial assets – money, stocks and bonds, and bank accounts.

Intellectual assetspatents, software, and brand names.

Physical assets – machinery, equipment, cars, printers, office chairs, computers.

Capital for individuals and capital for business?

Companies use structures that include capital as debt, equity, and daily operations and expenditures. In contrast, an individual holds capital to contribute to their net worth.

Money and capital

Capital is another way of saying “money.” Remember, capital comes with a cost. If the capital is debt, then interest is the cost. If the capital is equity, the cost is distributions the shareholders receive. In the financial world, capital is viewed through the lens of current operations and future investments.

Types of capital:

Working capital

Working capital is answered with this question: Can a company’s liquid assets cover daily obligations?

To calculate this, try two assessments.

-Subtract the current liabilities from the current assets

-Subtract the accounts payable from the accounts receivable and inventory.

Liquidity is another way to view capital. Working capital is a good barometer for a company’s short-term liquidity.

Borrowing is one way a business can gain capital, and the most common way of borrowing is through bank, bonds, or other financial institution. For small businesses, loans from friends, family, online lenders, or federal loan programs are often the source of debt capital.

There are many differences regarding capital for individuals and businesses. However, in both cases, a healthy, active credit history is required. As with any typical loan, the interest rates vary depending on the borrower’s credit history and health.

In everyday life, we think of debt as a burden. However, it can be an opportunity for a business, provided it is within reason. Monitor the debt to capital ratio to prevent going under.

Bonds are a great way to raise capital when interest rates are low, which means it is cheaper to borrow. Think of the bonds as an I.O.U. between a borrower and lender. A business may issue bonds to raise capital to fund its operations and projects. However, bonds are not limited to just businesses. States, sovereign governments, and municipalities may issue bonds to finance operations and projects. As a bond owner, you are the debt holder of the issuer.

There are several forms of equity capital. They are private equity, public equity, and real estate equity.

Comparing public equity and private equity

In public equity, the stock market raises equity by listing the company’s shares on the stock exchange. As investors buy stock, a company generates more equity. The single best way to raise equity capital is with I.P.O.s. (Initial public offerings) In private equity, a group of investors raise private equity, which is not the case with public equity.

Trading capital

Trading capital is what it says, and financial institutions and brokerage firms trade capital. There are several methods investors use to add to their trading capital, which are based on calculations determining the allocation of funds to each trade.

Cash reserves

One cannot have a successful investing strategy without optimal cash reserves. Debt liability offsets debt capital on the business’s balance sheet.

Capital can define a business’ structure

A company’s balance sheet is split among assets, liabilities, and equity, and the mix of these assets defines the structure. Some metrics for analyzing business capital are based on the cost of capital, equity return, capital debt, and equity debt.

Examples of capital

Capital comes down to any financial asset an individual or business has. Even one’s bank account is capital.

Tips for securing small business capital: Build a business plan

Lenders will want to see a well-thought-out business plan that outlines marketing, management, and money. As a borrower, one must explain how the money will be used. Like any loan, the single biggest criteria a lender will look for is the ability of the borrower to repay it. In short, a lender will want to know that you know what you are getting into. Having a well-thought-out business plan will help lenders feel more confident in the borrower, as it shows the borrower did their homework.

Business history

Borrowers should have at least two years of business history. Most lenders prefer this.

Plan for the worst

Provide the lender with a business projection 12 months out, broken out month-to-month. Show the lender case scenarios of what happens if the business drops.

Borrowers must check their credit for any errors, negative marks, or other issues that may affect their ability to borrow. Get credit in order, as one would when buying a house.

Capital F.A.Q.s

Capital assets is a term that applies to both individuals and businesses, which is anything of value, such as cars, real estate, and investments. Many capital assets are not liquid, which means they cannot be easily turned into cash.

Differences of capital in business and economics

An economist may define capital as liquid assets or cash in hand that can be used for spending. All the money in circulation is how capital is defined on a global scale. Business capital is the money available to run the business on a day-to-day basis and for the future. Typically, the revenue generated by the business is the only source of its capital.

In closing

Overall capital assets are determined by calculating any possession of cash value, such as cars, real estate, equipment, and machinery. There are different types of capital and the context of the use of the word capital can determine its meaning in each situation. View capital in the same way any other financial situation is viewed: Maintain a healthy debt to income ratio, maintain a healthy and active credit score, and monitor the cash flow and balance sheets.

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