Interaction due to a mechanical source in transversely isotropic micropolar media

Abstract

The present investigation is concerned with interaction due to a mechanical source in transversely isotropic micropolar elastic media, determined using the finite element method. A particular type of normal force has been taken to illustrate the utility of the approach. The components of displacement, stress and microrotation are obtained and depicted graphically for a specific model. A special case of interest is also deduced from the present investigation.

Keywords

Transversely isotropic micropolar media finite element method

1. Introduction

Classical mechanics deals with the basic assumption that the effect of the microstructure of a material is not essential for describing mechanical behavior. Such an approximation has been shown in many well-known cases. Often, however, discrepancies between the classical theory and experiments are observed, indicating that the microstructure might be important. For example, discrepancies have been found in the stress concentrations in the areas of holes, notches and cracks; elastic vibrations characterized by high-frequency and small wavelengths, particularly in granular composites consisting of stiff inclusions embedded in a weaker matrix, fibers or grains; and the mechanical behavior of complex fluids, such as liquid crystals, polymeric suspensions and animal blood. In general, granular composites, for example porous materials, are widely used in the area of passive noise control as sound absorbers, and the effect of acoustical waves characterized by high frequencies and small wavelengths become significant.

To explain the fundamental departure of microcontinuum theories from the classical continuum theories, a continuum model embedded with microstructures to describe the microscopic motion or a nonlocal model to describe the long range material interaction is developed. This theory extends the application of the continuum model to microscopic space and short time scales. Micromorphic theory (Suhubi and Eringen, 1964; Eringen, 1999) treats a material body as a continuous collection of a large number of deformable particles, with each particle possessing finite size and inner structure. Using assumptions such as infinitesimal deformation and slow motion, micromorphic theory can be reduced to Mindlin's microstructure theory (Mindlin, 1964). When the microstructure of the material is considered rigid, it becomes the micropolar theory (Eringen, 1966).

Eringen's micropolar theory is more appropriate for geological materials such as rocks and soils, since this theory takes into account the intrinsic rotation and predicts the behavior of material with an inner structure. Different researchers have discussed different types of problems in transversely isotropic elastic material. Abubakar (1962) discussed free vibrations of a transversely isotropic plate. Suvalov et al. (2005) described the long-wavelength onset of the fundamental branches for a free anisotropic plate with arbitrary through-plate variation of material properties. Payton (1992) has studied wave propagation in a restricted transversely isotropic elastic solid whose slowness surface contains conical points. Tomar (2005) and Kumar and Deswal (2006) studied some problems of wave propagation in micropolar elastic media with voids. However, no attempt has been made to study the deformation in micropolar transversely isotropic material using the finite element method.

The exact solution of the governing equations of the micropolar generalized thermoelastic theory for a coupled and nonlinear/linear system exists only for very special and simple initial and boundary problems. A numerical solution technique is used to calculate the solution of general problems. For this reason the finite element method is chosen.

The finite element method is a powerful technique originally developed for the numerical solution of complex problems in structural mechanics, and it remains the method of choice for complex systems. A further benefit of this method is that it allows physical effects to be visualized and quantified regardless of experimental limitations. Abbas and Abd-alla (2008), Abbas and Palani (2010), Abbas and Othman (2011, 2012), Othman and Abbas (2011, 2012) and Abbas (2012a, 2012b) have successfully applied the finite element method to various problems in generalized thermoelastic materials.

The aim of the present study is to enhance our knowledge about the application of the finite element method in a micropolar transversely isotopic media. This study has many applications in various fields of science and technology, namely, atomic physics, industrial engineering, thermal power plants, submarine structures, pressure vessels, aerospace, chemical pipes and metallurgy.

The present investigation determines the components of displacements, microrotation and stress due to a mechanical source in transversely isotropic micropolar media. Such mechanical normal loading may produce a severe deformation in a thin zone near the half-space surface and thereby cause excessive wear and even cracking near the contact zone. It is therefore useful to analyze this class of problems by using a formulation that is as exact as possible and to provide results for the surface and/or near-surface field quantities (displacements, microrotation, stresses) that may be required for design purposes. The solution to the problem investigated here has practical applications in the fields of geomechanics, engineering, fiber-wound composites and laminated composite materials.

2. Basic equations

Following Eringen (1999), the constitutive relations and balance laws in general micropolar anisotropic medium possessing a center of symmetry, in the absence of body forces and body couples, are given by

Constitutive relations:

σ_{ij} = A_{ijkl} E_{kl} + G_{ijkl} Ψ_{kl},

m_{ij} = G_{ijkl} E_{kl} + B_{ijkl} Ψ_{kl},

(1)

The deformation and wryness tensor are defined by

E_{ji} = u_{i, j} + ɛ_{ijkl} φ_{k}, Ψ_{ij} = φ_{i, j} .

Balance laws:

σ_{ji, j} = ρ {\overset{··}{u}}_{i},

m_{ik, i} + ɛ_{ijk} t_{ij} = ρ j {\overset{··}{φ}}_{k},

(2)

where

σ_{ji}

and

m_{ij}

are, respectively, the stress tensor and couple stress tensor, ρ is bulk mass density and u_i and

φ_{i}

are, respectively, the components of the displacement vector and microrotation vector.

A_{ijkl}, G_{ijkl}, B_{ijkl}

are characteristic constants of material following the symmetry properties given by Eringen (1999).

3. Formulation of the problem

We have used appropriate transformations, following Slaughter (2002), on the set of Equations (1) to derive equations for the micropolar transversely isotropic medium. In the present paper, we consider a homogeneous, transversely isotropic micropolar elastic half-space $x \geq 0 .$ A continuous normal force is assumed to be acting on the micropolar elastic half-space, as shown in Figure 1. All the considered functions will depend on the time t and the coordinates x and z. Thus we assume that the displacement vector and microrotation vector are, respectively, of the form

u = (u (x, z), 0, w (x, z)), φ = (0, φ_{y} (x, z), 0) .

(3)

Figure 1.

Geometry of the problem.

Making use of (3) in Equations (1) and (2) yields

A_{11} \frac{\partial 2 u}{\partial x 2} + A_{55} \frac{\partial 2 u}{\partial z 2} + (A_{13} + A_{56}) \frac{\partial 2 w}{\partial x \partial z} + K_{1} \frac{\partial φ_{y}}{\partial z} = ρ \frac{\partial 2 u}{\partial t 2},

(4)

A_{66} \frac{\partial 2 w}{\partial x 2} + A_{33} \frac{\partial 2 w}{\partial z 2} + (A_{13} + A_{56}) \frac{\partial 2 u}{\partial x \partial z} + K_{2} \frac{\partial φ_{y}}{\partial x} = ρ \frac{\partial 2 w}{\partial t 2},

(5)

B_{77} \frac{\partial 2 φ_{y}}{\partial x 2} + B_{66} \frac{\partial 2 φ_{y}}{\partial z 2} - X φ_{y} - K_{1} \frac{\partial u}{\partial z} - K_{2} \frac{\partial w}{\partial z} = ρ j \frac{\partial 2 φ_{y}}{\partial t 2},

(6)

where

K_{1} = A_{56} - A_{55}, K_{2} = A_{66} - A_{56}, X = K_{2} - K_{1} .

For further considerations, it is convenient to introduce the dimensionless variables defined by

\begin{matrix} (x', z') = \frac{ω *}{c_{1}} (x, z), (u', w') = \frac{ω *}{c_{1}} (u, w), t_{ij}' = \frac{t_{ij}}{A_{11}}, \\ m_{ij}' = \frac{m_{ij} c_{1}}{B_{66} ω *} φ_{y}' = \frac{φ_{y} A_{55}}{K_{1}}, t' = ω * t, ω * 2 = \frac{X}{ρ j}, \\ c_{1}^{2} = \frac{A_{11}}{ρ} \end{matrix} .

(7)

Using Equation (7) in Equations (2)–(6), we obtain the equations in nondimensional form after suppressing the primes as

\frac{\partial 2 u}{\partial x 2} + d_{1} \frac{\partial 2 u}{\partial z 2} + d_{2} \frac{\partial 2 w}{\partial x \partial z} + d_{3} \frac{\partial φ_{y}}{\partial z} = \frac{\partial 2 u}{\partial t 2},

(8)

d_{4} \frac{\partial 2 w}{\partial x 2} + d_{5} \frac{\partial 2 w}{\partial z 2} + d_{2} \frac{\partial 2 u}{\partial x \partial z} + d_{6} \frac{\partial φ_{y}}{\partial x} = \frac{\partial 2 w}{\partial t 2},

(9)

d_{7} \frac{\partial 2 φ_{y}}{\partial x 2} + d_{8} \frac{\partial 2 φ_{y}}{\partial z 2} - d_{9} φ_{y} - d_{10} \frac{\partial u}{\partial z} - d_{11} \frac{\partial w}{\partial x} = \frac{\partial 2 φ_{y}}{\partial t 2},

(10)

σ_{zz} = d_{12} \frac{\partial u}{\partial x} + d_{5} \frac{\partial w}{\partial z},

(11)

σ_{xx} = \frac{\partial u}{\partial x} + d_{12} \frac{\partial w}{\partial z},

(12)

σ_{xz} = d_{13} \frac{\partial w}{\partial x} + d_{4} \frac{\partial u}{\partial z} + d_{14} φ_{y},

(13)

m_{xy} = d_{15} \frac{\partial ϕ_{y}}{\partial x},

(14)

where

\begin{matrix} (d_{1}, d_{2}, d_{3}, d_{4}, d_{5}, d_{6}) = \\ \frac{1}{A_{11}} (A_{55}, A_{13} + A_{56}, \frac{K_{1}^{2}}{A_{55}}, A_{66}, A_{33}, \frac{K_{1} K_{2}}{A_{55}}), \end{matrix}

\begin{matrix} (d_{7}, d_{8}) = \frac{1}{ρ {jc}_{1}^{2}} (B_{77}, B_{66}), (d_{9}, d_{10}) \\ = \frac{1}{ρ j ω * 2} (X, A_{55}), d_{11} = \frac{K_{2}}{K_{1}} d_{10}, \\ (d_{12}, d_{13}, d_{14}) = \frac{1}{A_{11}} (A_{13}, A_{56}, K_{1}), d_{15} = \frac{B_{77} k_{1}}{B_{66} A_{55}} \end{matrix}

4. Initial and boundary conditions

The above equations, solved subject to initial conditions and boundary conditions, are

u = w = φ_{y} = 0, \overset{\cdot}{u} = \overset{\cdot}{w} = {\overset{\cdot}{φ}}_{y}, t = 0, x \geq 0, - \infty < z < \infty,

(15)

\begin{matrix} t_{xx} = - f_{0} H (t) (2 l - | z |), t_{xz} = 0, m_{xy} = 0, \\ x = 0, t > 0, - \infty < z < \infty, \end{matrix}

(16)

where H is the Heaviside unit step and f₀ is the magnitude of the force.

5. Finite element formulation

In this section, the governing equations of micropolar transversely isotropic half-space are summarized, followed by the corresponding finite element equations. In the finite element method, the displacement components u, w and microrotation component $φ_{y}$ are related to the corresponding nodal values by

u = \sum_{i = 1}^{m} N_{i} u_{i} (t), w = \sum_{i = 1}^{m} N_{i} w_{i} (t), φ_{y} = \sum_{i = 1}^{m} N_{i} φ_{yi} (t),

(17)

where m denotes the number of nodes per element, and N_i are the shape functions. The eight-node isoparametric, quadrilateral element is used for displacement components, microrotation and conductive temperature calculations. The weighting functions and the shape functions coincide. Thus

δ u = \sum_{i = 1}^{m} N_{i} δ u_{i} (t), δ w = \sum_{i = 1}^{m} N_{i} δ w_{i} (t), δ φ_{y} = \sum_{i = 1}^{m} N_{i} δ φ_{yi} (t),

(18)

It should be noted that appropriate boundary conditions associated with the governing equations (8)–(10) must be adopted in order to properly formulate a problem. Boundary conditions are either essential (or geometric) or natural (or traction) types. Essential conditions are prescribed displacements $u, w$ and microrotation $φ_{y}$ , while the natural boundary conditions are prescribed tractions and couple stress, which are expressed as

σ_{xx} n_{x} + σ_{xz} n_{z} = {\bar{σ}}_{x}, σ_{xz} n_{x} + σ_{zz} n_{z} = {\bar{σ}}_{z}, m_{xy} n_{y} = {\bar{m}}_{y},

(19)

where

n_{x}, n_{y}

and n_z are direction cosines of the outward unit normal vector at the boundary,

{\bar{σ}}_{x}, {\bar{σ}}_{z}

are the given tractions values and

{\bar{m}}_{y}

is the given couple traction component. In the absence of body force and body couples, the governing equations are multiplied by weighting functions and then are integrated over the spatial domain .Ω. with the boundary

Γ

. Applying integration by parts and making use of the divergence theorem reduce the order of the spatial derivatives and allow for the application of the boundary conditions. Thus, the finite element equations corresponding to Equations (8)–(10) can be obtained as

\begin{matrix} \sum_{e = 1}^{m} ([\begin{matrix} M_{11}^{e} 0 0 \\ 0 M_{22}^{e} 0 \\ 0 0 M_{33}^{e} \end{matrix}] {\begin{matrix} \overset{··}{u} e \\ \overset{··}{w} e \\ {\overset{··}{ϕ}}_{y}^{e} \end{matrix}} \\ + [\begin{matrix} K_{11}^{e} K_{12}^{e} K_{13}^{e} \\ K_{21}^{e} K_{22}^{e} K_{23}^{e} \\ K_{31}^{e} K_{32}^{e} K_{33}^{e} \end{matrix}] {\begin{matrix} u e \\ w e \\ ϕ_{y} \end{matrix}} = {\begin{matrix} F_{1}^{e} \\ F_{2}^{e} \\ F_{3}^{e} \end{matrix}}), \end{matrix}

(20)

where the coefficients in Equation (20) are given below:

\begin{matrix} M_{11}^{e} = \int_{Ω} [N] T [N] d Ω, M_{22}^{e} = \int_{Ω} [N] T [N] d Ω, \\ M_{33}^{e} = \int_{Ω} [N] T [N] d Ω, \end{matrix}

\begin{matrix} K_{11}^{e} = \int_{Ω} ([\frac{\partial N}{\partial x}] T [\frac{\partial N}{\partial x}] + d_{1} [\frac{\partial N}{\partial z}] T [\frac{\partial N}{\partial z}]) d Ω, \\ K_{12}^{e} = \int_{Ω} d_{2} [\frac{\partial N}{\partial z}] T [\frac{\partial N}{\partial x}] d Ω, \end{matrix}

K_{13}^{e} = \int_{Ω} d_{3} [\frac{\partial N}{\partial z}] T [N] d Ω, K_{21}^{e} = \int_{Ω} d_{2} [\frac{\partial N}{\partial x}] T [\frac{\partial N}{\partial z}] d Ω,

\begin{matrix} K_{22}^{e} = \int_{Ω} (d_{4} [\frac{\partial N}{\partial x}] T [\frac{\partial N}{\partial x}] + d_{5} [\frac{\partial N}{\partial z}] T [\frac{\partial N}{\partial z}]) d Ω, \\ K_{23}^{e} = \int_{Ω} d_{6} [\frac{\partial N}{\partial x}] T [N] d Ω, \end{matrix}

K_{31}^{e} = \int_{Ω} d_{10} [N] T [\frac{\partial N}{\partial z}] d Ω, K_{32}^{e} = \int_{Ω} d_{11} [N] T [\frac{\partial N}{\partial x}] d Ω,

K_{33}^{e} = (\int_{Ω} d_{7} [\frac{\partial N}{\partial x}] T [\frac{\partial N}{\partial x}] + d_{8} [\frac{\partial N}{\partial z}] T [\frac{\partial N}{\partial z}] - d_{9}) d Ω,

F_{1}^{e} = \int_{Γ} [N] T {\bar{σ}}_{x} d Γ, F_{2}^{e} = \int_{Γ} [N] T {\bar{σ}}_{z} d Γ, F_{3}^{e} = \int_{Γ} [N] T \bar{m} d Γ .

(21)

Symbolically, the discretized equations of Equations (20) can be written as

M \overset{··}{d} + Kd = F ext,

(22)

where M, K and

F ext

represent the mass, stiffness matrices and external force vectors, respectively.

d = [u, w, φ_{y}] .

On the other hand, the time derivatives of the unknown variables have to be determined by the Newmark time integration method or other methods (see Wriggers, 2008; Vikas, 2011; Ellahi and Zeeshan, 2012; Venkatachalam et al., 2012).

6. Special case

If we take $A_{11} = A_{33} = λ + 2 μ + K, A_{55} = μ + K, A_{13} = λ, A_{56} = μ, B_{77} = B_{66} = γ,$ then the above results reduce to the micropolar isotropic elastic material.

7. Numerical results and discussion

For the purpose of numerical computation, the following values of the relevant parameter are taken as A₁₁ = 15.974 × 10¹⁰Nm^–2, A₃₃ = 13.843 × 10¹⁰Nm^–2, A₅₅ = 5.357 × 10¹⁰ Nm^–2, A₆₆ = 5.42 × 10¹⁰ Nm^–2, A₁₃ = 9.59 × 10¹⁰ Nm^–2, A56 = 5.89 × 10¹⁰ Nm^–2, B₇₇ = 1.779 × 10⁹ N, B₆₆ = 2.779 × 10⁹ N, ρ = 1.74 Kg/m³, j = 0.2 m² $, l = 0.25 .$

Figures 2–7 exhibit the variation of displacement, stress components and couple stress components with $x .$ The solid lines corresponds to $t = 0.1$ , small dashed lines correspond to $t = 0.2$ and dotted lines correspond to $t = 0.3 .$

Figure 2.

Variation of displacement component u with distance x.

Figure 7.

Variation of couple stress component m_xy with distance x.

Figure 2 depicts the variations of normal displacement u with x. The values of u decrease for smaller values of x, whereas for higher values of x, it becomes constant (for all values of t). It is noticed that the values of u remain more for the case of (in comparison with $t = 0.1$ , $t = 0.2$ ) for smaller values of x, although for higher values of x, the values of u become approximately equal for all values of $t .$ Figure 3 depicts the variation of horizontal displacement w with x. It is evident that the behavior and variation of w is similar to u, but the values are different.

Figure 3.

Variation of displacement component w with distance x.

Figure 4 depicts the variations of microrotation component $φ_{y}$ with x. The values of $φ_{y}$ decrease for smaller values of x, whereas for higher values of x it becomes constant (for all values of t). It is noticed that the values of $φ_{y}$ remain higher for the case of $t = 0.3$ (in comparison with $t = 0.1$ , $t = 0.2$ ) for smaller values of x, although for higher values of x, the values of u become approximately equal for all values of $t .$ Figure 5 depicts the variation of normal stress component $σ_{xx}$ with x. It is evident that the values of $σ_{xx}$ increase initially, but for higher values of x they become constant. It is noticed that the values of $σ_{xx}$ in the case of $t = 0.1$ remain higher in comparison with $t = 0.1$ , $t = 0.2 .$

Figure 4.

Variation of microrotation component φ_y with distance x.

Figure 5.

Variation of stress component σ_xx with distance x.

Figure 6 exhibits the variation of tangential stress component $σ_{zx}$ with x and it indicates that the values of $σ_{zx}$ remain oscillatory for smaller values of x, although for higher values of x it become dispersionless. It is noticed that for smaller values of x, the values of $σ_{zx}$ in the case of $t = 0.3$ remain higher in comparison with others, and finally the values of $σ_{zx}$ become approximately equal at the boundary surface. Figure 7 depicts the variation of couple stress component $m_{xy}$ with x. It is evident that the values of $m_{xy}$ remain oscillatory for smaller values of x, although for higher values of x they become constant.

Figure 6.

Variation of stress component σ_xz with distance x.

8. Conclusion

The two-dimensional problem of micropolar transverse material is solved numerically by the finite element method due to normal force. It is observed that as t increases, the values of the displacement and microrotation components decrease; near the application of the source, the values are higher; away from the source, the values decrease. Also, the stress components have oscillatory behavior; near the application of the source, the stress components have higher values; away from the application of the source, they oscillate and then converge to the boundary surface.

Footnotes

Funding

This research received no specific grant from any funding agency in the public, commercial or not-for-profit sectors.

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