
The primary goal of this study is to create a novel mathematical model based on the time-stepping boundary element method (BEM) scheme for solving three-dimensional coupled dynamic thermoelastic fracture issues in anisotropic functionally graded materials (FGMs). The crack tip opening displacement determines the dynamic stress intensity factor (SIF). The effects of anisotropy, graded parameters, and angle locations on the SIF were studied for three-dimensional coupled dynamic thermoelastic fracture situations. The results show that the novel method is exceptionally exact and efficient at assessing the fracture mechanics of fractured thermoelastic anisotropic FGMs. In addition, this paper provides a theoretical framework for analyzing a wide range of real engineering applications.
Citation: Mohamed Abdelsabour Fahmy, Ahmad Almutlg. A time-stepping BEM for three-dimensional thermoelastic fracture problems of anisotropic functionally graded materials[J]. AIMS Mathematics, 2025, 10(2): 4268-4285. doi: 10.3934/math.2025197
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The primary goal of this study is to create a novel mathematical model based on the time-stepping boundary element method (BEM) scheme for solving three-dimensional coupled dynamic thermoelastic fracture issues in anisotropic functionally graded materials (FGMs). The crack tip opening displacement determines the dynamic stress intensity factor (SIF). The effects of anisotropy, graded parameters, and angle locations on the SIF were studied for three-dimensional coupled dynamic thermoelastic fracture situations. The results show that the novel method is exceptionally exact and efficient at assessing the fracture mechanics of fractured thermoelastic anisotropic FGMs. In addition, this paper provides a theoretical framework for analyzing a wide range of real engineering applications.
The head groups of membrane lipids have either single charge (e.g. tetraether lipids [1], phosphatidic acid (PA), phosphatidylserine (PS), phosphatidylethanolamine (PE), and phosphatidylinositol (PI)) or electric dipole (e.g. phospholipids, such as dimyristoyl-, dipalmitoyl- and distearoylphosphatidyl choline (DMPC, DPPC and DSPC, respectively)).
Between lipids containing head groups with electric dipole there is short range interaction, i.e. where the two-body potential decays algebraically at large distances with a power equal or larger than the spatial dimension [2]. Theoretical models of lipid membranes usually focus on systems where there is short range lateral interactions between nearest neighbor lipids [3],[4] because it is enough to consider only the interactions between nearest-neighbor lipid molecules. It is much more difficult to model a lipid membrane containing single charged head groups [5]. Between lipids with single charged head groups there is long range interaction, i.e. where the two-body potential decays algebraically at large distances with a power smaller than the spatial dimension [2] and thus modeling this system one has to consider the entire system rather than the interactions between the nearest-neighbor lipids. In order to get closer to the solution of this problem recently we developed a generalized version of Newton's Shell Theorem [6],[7] to calculate the electric potential, V around a surface-charged sphere (of radius R1) surrounded by electrolyte at a distance Z from the center of the sphere (see also Eqs 9,10 in ref.7):
where
Using the Screened Poisson Equation (Eq A4) one can calculate the potential energy of an electrolyte that contains also external charges. The external charges are embedded into the electrolyte (like the charges of the surface-charged sphere) but not part of the electrolyte itself. For the solution one has to know the charge density of the external charges (see Eq 4 in ref.7 or Eq A5 in Appendix 1), i.e. distribution of the charges on the surface-charged sphere and not the distribution of the ions in the electrolyte. In our case it is assumed that the charges on the surface of the sphere are homogeneously distributed and in this case Eqs 1,2 is the exact solution of the Screened Poisson Equation.
Note that recently by using Eqs 1,2 electric energies have been calculated [10], such as the electric potential energy needed to build up a surface-charged sphere, and the field and polarization energy of the electrolyte inside and around the surface-charged sphere.
In this paper the density of electric field energy is calculated around two surface-charged spheres where the smaller sphere is located inside the larger one and the entire system is embedded in neutral electrolyte. This system is close to a charged vesicle [1] or to a cell [11] where charged lipids are located both on the outer and inner leaflet of the membrane, i.e. two concentric surface-charged spheres. It also models an eukaryote [12] where neutral phospholipids such as sphingomyelin and zwitterionic phosphatidylcholine are located primarily in the outer leaflet of the plasma membrane, and most anionic phospholipids, such as phosphatidic acid (PA), phosphatidylserine (PS), phosphatidylethanolamine (PE), and phosphatidylinositol (PI) are located in the inner leaflet of the plasma membrane (represented by the large surface-charged sphere of our model). Eukaryotes also have a single nucleus enveloped by double layer of lipid membranes which may contain charged lipids too (representing the smaller surface-charged sphere of our model). Note that these two charged spheres of an eukaryote are not necessarily concentric. Finally, our model is generalized for the case when the large surface-charged sphere contains several smaller surface-charged spheres. This system may also model osteoclast cells [12] containing many nuclei.
In this work the density of the electric field energy inside and outside of two surface-charged spheres are calculated at different locations. The density of the electric field energy at a point can be calculated by the following equation [13]:
where E is the vector of the electric field strength at the considered point, ϵ0 is the absolute vacuum permittivity and ϵr is the relative permittivity of the electrolyte.
Here by using the recently generalized Shell Theorem [7] we calculate the density of electric field energy, uF produced by two surface-charged spheres (see Figure 1) surrounded outside and inside by electrolyte where the smaller sphere is located inside the larger sphere.
Z: the distance between the centers of the spheres (dashed blue line); R1 and RL is the radius of the smaller and larger sphere, respectively; D1 and DL is the distance between point P1 and the center of the smaller and larger sphere, respectively; E1 and EL is the field strength created in point P1 by the smaller and larger surface-charged sphere, respectively.
The origin of the coordinate system (x, y) is attached to the center of the larger sphere and the coordinates of point P1 are xp and yp. The coordinates of the center of the larger and smaller sphere are
In order to calculate the density of electric field energy one has to determine the electric field strength (see Eq 3), i.e. the gradient of the electric potential. The potential produced by the smaller sphere, V1 at a distance D1 from its center can be calculated by Eqs 1,2 (or Eqs 9,10 in ref.[7]). The electric field strength created by the smaller sphere at point P1 (see Figure 1) is:
where
where λD is the Debye length and
Similarly, the electric field strength created by the large sphere at point P1 (i.e. at a distance DL from its center; see Figure 1) is:
where one can construct
Here by using
Inside a large surface-charged sphere of radius
Dotted red line:
The connection point between the large sphere and the small sphere (represented by orange circle in
and
where x1 is the x coordinate of the center of the small sphere.
Dotted red line:
Dotted red line:
Note in
It is also important to note that
In this work the solution of the screened Poisson equation ([7] and Eq A5 in Appendix 1) is used to calculate the field energy density around two surface-charged spheres where the small sphere is located inside the large sphere. This solution is not restricted to small potentials (<< 25 mV) like in the case of the Debye-Hückel approximation of the Poisson-Boltzmann equation [9] where the superposition principle is not applicable either. This is an important advantage because the measured absolute value of the Zeta potentials of the cells are usually higher than 25 mV (e.g. –57.89 ± 22.63 mV on ARO cells, –40.41 ± 5.10 mV on C32TG cells, −46.99 ± 18.71 mV on RT4 cells, –40.13 ± 9.28 mV on TK cells, and −43.03 ± 5.52 mV on UM-UC-14 cells [14].
The considered two spheres (with homogeneously charged surfaces) electrically interact. If the lateral movement of the charges on the spheres would not be restricted the interaction of the smaller sphere (located inside the larger sphere) with the larger sphere would result in inhomogeneous distribution of the surface charges on both spheres. However, the free lateral diffusion of proteins and lipids are usually restricted in biological membranes not only by direct collisions with structures where immobile proteins are crowded, but also by electrostatic deflection, hydrophobic mismatches, and other mechanisms [15].
The density of the electric field energy depends on the electric field strength (Eq 3), i.e. the gradient of the electric potential (Eqs 4,6). In the case of a single surface-charged sphere surrounded by electrolyte with low ion concentration the potential inside the sphere is close to constant (see red curve in Figure 3A in ref. [7]) and thus the absolute value of the electric field strength is close to zero. On the other hand, outside the sphere the absolute value of the potential and also the electric field strength decrease with increasing distance from the surface of the sphere (see red curve in Figure 3A in ref. [7]). At higher electrolyte ion concentration, because of the increased screening effect, the absolute value of the potential and also the electric field strength decrease faster with increasing distance from the surface of the sphere. In this case inside the sphere toward its center the absolute value of the potential and the electric field strength also decrease (see curves in Figure 3A,B in ref. [7]).
In this work two surface-charged spheres (with the same surface charge density) are considered where the smaller sphere is located inside the larger sphere. The above mentioned electric properties of a single surface-charged sphere remain the same for the smaller sphere (located inside a larger sphere) if the surfaces of the spheres are far enough from each other (farther than 4 λD), i.e. the absolute value of the potential decreases close to zero between the surfaces of the two spheres. However, when part of the surfaces of the two spheres are close enough to each other one sphere contributes to the potential and electric field strength around the other sphere. The electric field energy density is particularly high at the place where the surfaces of the two spheres touch each other. This maximal electric field energy density is very close to the outer surface of the larger sphere. Thus one can detect at the outer surface of the erythrocyte when the nucleus is getting close.
The electric field energy density has maximum when the horizontal line crosses the circles in
This maximal electric field energy density is very close to the outer surface of the larger sphere (
In general the first maximum of uF (see the left maximum in Figures 3–5) is getting smaller when the center of the small sphere approaches the center of the large sphere. This is the case because the interaction between the spheres is reducing when the average distance between the surfaces of the two spheres is increasing.
In the case of horizontal lines where
When the location of the center of the small and large sphere is identical (i.e.
When
In the case of
Cross # | xP [m] | uF [J] | E1x [V/m] | ELx [V/m] | E1y [V/m] | ELy [V/m] |
Z = 0.8 RL | ||||||
1 | −9.9·10−7 | 7.4·107 | 1.8·108 | 2.47·108 | −1.42·108 | −3.74·107 |
2 | −9.4·10−7 | 2.2·107 | 2.29·108 | −1.16·108 | −2.45·108 | 1.86·107 |
3 | −6.6·10−7 | 4.65·107 | −2.29·108 | −5.45·107 | −2.45·108 | 1.24·107 |
4 | 9.9·10−7 | 2.15·107 | −9.7·104 | −2.47·108 | −8.2·103 | −3.74·107 |
Z = 0.5 RL | ||||||
1 | −9.9·10−7 | 2.73·107 | 3·107 | 2.47·108 | −9.22·106 | −3.74·107 |
2 | −6.4·10−7 | 2.97·107 | 2.29·108 | −5.16·107 | −2.45·108 | 1.2·107 |
3 | −3.6·10−7 | 4.1·107 | −2.29·108 | −2.2·107 | −2.45·108 | 9.16·106 |
4 | 9.9·10−7 | 2.15·107 | −3.2·105 | −2.47·108 | −3.22·104 | −3.74·107 |
Z = 0.0 RL | ||||||
1 | −9.9·10−7 | 2.2·107 | 2.6·106 | 2.47·108 | −3.97·105 | −3.74·107 |
2 | −1.4·10−7 | 3.63·107 | 2.29·108 | −7.6·106 | −2.45·108 | 8.17·106 |
3 | 1.4·10−7 | 3.63·107 | −2.29·108 | 7.6·106 | −2.45·108 | 8.17·106 |
4 | 9.9·10−7 | 2.2·107 | −2.6·106 | −2.47·108 | −3.97·105 | −3.74·107 |
For example in the case of
As an other example in the case of
Finally, the analytical equation, Eq 7, for the calculation of the electric field energy density of two surface-charged spheres (the smaller sphere located inside the larger sphere), can be generalized for the case when N small surface-charged spheres are located inside the large sphere (see Appendix 2). Also when the radius of the smaller sphere approaches zero the total surface charge of the smaller sphere, Q1 approaches zero too and consequently the electric field strength of the smaller sphere, E1 approaches zero. Thus, based on Eq 7 one can calculate the field energy density around a single charged sphere by:
Based on the generalized version of Newton's Shell Theorem [7] the electric field energy density, uF around two surface-charged spheres surrounded by electrolyte where the smaller sphere is inside the larger one is analytically calculated. According to the calculations when the surfaces of the spheres are farther from each other than four times of the Debye length the field energy density around and inside the smaller sphere is basically independent from the presence of the larger sphere. The electric field energy density is maximal when the smaller sphere touches the inner surface of the larger sphere and the maximum of uF is located at the touching point on the outer surface of the larger sphere.
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Cross # | xP [m] | uF [J] | E1x [V/m] | ELx [V/m] | E1y [V/m] | ELy [V/m] |
Z = 0.8 RL | ||||||
1 | −9.9·10−7 | 7.4·107 | 1.8·108 | 2.47·108 | −1.42·108 | −3.74·107 |
2 | −9.4·10−7 | 2.2·107 | 2.29·108 | −1.16·108 | −2.45·108 | 1.86·107 |
3 | −6.6·10−7 | 4.65·107 | −2.29·108 | −5.45·107 | −2.45·108 | 1.24·107 |
4 | 9.9·10−7 | 2.15·107 | −9.7·104 | −2.47·108 | −8.2·103 | −3.74·107 |
Z = 0.5 RL | ||||||
1 | −9.9·10−7 | 2.73·107 | 3·107 | 2.47·108 | −9.22·106 | −3.74·107 |
2 | −6.4·10−7 | 2.97·107 | 2.29·108 | −5.16·107 | −2.45·108 | 1.2·107 |
3 | −3.6·10−7 | 4.1·107 | −2.29·108 | −2.2·107 | −2.45·108 | 9.16·106 |
4 | 9.9·10−7 | 2.15·107 | −3.2·105 | −2.47·108 | −3.22·104 | −3.74·107 |
Z = 0.0 RL | ||||||
1 | −9.9·10−7 | 2.2·107 | 2.6·106 | 2.47·108 | −3.97·105 | −3.74·107 |
2 | −1.4·10−7 | 3.63·107 | 2.29·108 | −7.6·106 | −2.45·108 | 8.17·106 |
3 | 1.4·10−7 | 3.63·107 | −2.29·108 | 7.6·106 | −2.45·108 | 8.17·106 |
4 | 9.9·10−7 | 2.2·107 | −2.6·106 | −2.47·108 | −3.97·105 | −3.74·107 |
Cross # | xP [m] | uF [J] | E1x [V/m] | ELx [V/m] | E1y [V/m] | ELy [V/m] |
Z = 0.8 RL | ||||||
1 | −9.9·10−7 | 7.4·107 | 1.8·108 | 2.47·108 | −1.42·108 | −3.74·107 |
2 | −9.4·10−7 | 2.2·107 | 2.29·108 | −1.16·108 | −2.45·108 | 1.86·107 |
3 | −6.6·10−7 | 4.65·107 | −2.29·108 | −5.45·107 | −2.45·108 | 1.24·107 |
4 | 9.9·10−7 | 2.15·107 | −9.7·104 | −2.47·108 | −8.2·103 | −3.74·107 |
Z = 0.5 RL | ||||||
1 | −9.9·10−7 | 2.73·107 | 3·107 | 2.47·108 | −9.22·106 | −3.74·107 |
2 | −6.4·10−7 | 2.97·107 | 2.29·108 | −5.16·107 | −2.45·108 | 1.2·107 |
3 | −3.6·10−7 | 4.1·107 | −2.29·108 | −2.2·107 | −2.45·108 | 9.16·106 |
4 | 9.9·10−7 | 2.15·107 | −3.2·105 | −2.47·108 | −3.22·104 | −3.74·107 |
Z = 0.0 RL | ||||||
1 | −9.9·10−7 | 2.2·107 | 2.6·106 | 2.47·108 | −3.97·105 | −3.74·107 |
2 | −1.4·10−7 | 3.63·107 | 2.29·108 | −7.6·106 | −2.45·108 | 8.17·106 |
3 | 1.4·10−7 | 3.63·107 | −2.29·108 | 7.6·106 | −2.45·108 | 8.17·106 |
4 | 9.9·10−7 | 2.2·107 | −2.6·106 | −2.47·108 | −3.97·105 | −3.74·107 |