Home Calculation and 3D analyses of ERR in the band crack front contained in a rectangular plate made of multilayered material
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Calculation and 3D analyses of ERR in the band crack front contained in a rectangular plate made of multilayered material

  • Arzu Turan Dincel EMAIL logo and Surkay D. Akbarov
Published/Copyright: May 31, 2018
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Open Chemistry
From the journal Open Chemistry Volume 16 Issue 1

Abstract

An investigation into the values of the Energy Release Rate (ERR) at the band crack’s front in the rectangular plate made of multilayered composite material is carried out for the opening mode. The corresponding boundary-value problem is modelled by using threedimensional linear theory and solved numerically by using 3D FEM (Three Dimensional Finite Element Method). The main purpose of the current investigation is to study the influence of mechanical and geometrical parameters on the Energy Release Rate (ERR) at this crack front. The numerical results related to the ERR, and the effect of the mechanical and other problem parameters on the ERR are presented and discussed.

Keywords: Energy Release Rate (ERR); band crack; 3D FEM (Three Dimensional Finite Element Method); rectangular plate

1 Introduction

Since material defects such as cracks have a significant role in the processes of fractures, fracture mechanics investigates the mechanical behavior of fracture parameters (Stress Intensity Factors (SIF) or Energy Release Rates (ERR)) from various aspects such as load, material property and crack length. From recent research into this area, it can be seen that many of them have focused on determining the effects of anisotropy and other problem parameters on the SIF or ERR within the framework of the two-dimensional (2D) problem formulation [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. Up to now, there have not been any investigations regarding the corresponding 3D macro-crack problems for anisotropic bodies.

In connection with this, the purpose of this current paper is to investigate the effects of anisotropic properties of the plate material and other problem parameters on the ERR in the macro-band crack contained in an anisotropic plate.

2 Problem Formulation and Solution Method

Let us consider a thick rectangular plate(Figure 1)occupying the domain Ω = {0 ≤ x1 ≤ ℓ1, 0 ≤ x2 ≤ h, 0 ≤ x3 ≤ ℓ3} in the Cartesian Coordinate system Ox1x2x3, which contains a band crack located in the region Ω′ = {(ℓ1 /2 − ℓ0/2) ≤ x1 ≤ (ℓ1 / 2 + ℓ0 / 2), x2 = hc ± 0, 0 ≤ x3 ≤ ℓ3} . Assume that on the crack’s edge planes, the opening uniformly distributed normal forces with intensity p act.

Figure 1 Geometry of a rectangular plate with a band crack.
Figure 1

Geometry of a rectangular plate with a band crack.

The field equations and boundary conditions for the case under consideration are:

∂σij∂xi=0,σii=Aijεjj,Aij=Ajj,σij=2μijεij,ati≠j,εij=12(∂ui∂xj+∂uj∂xi),u2|x1=0=u2|x1=ℓ1=0,u2|x3=0=u2|x3=ℓ3=0,σ1k|x1=0=σ1k|x1=ℓ1==0,σ3k|x3=0=σ3k|x3=ℓ3=0,σ2i|x2=0=0,σ2i|x2=h=0,σ2i|Ω′=−pδi2,i,j=1,2,3;k=1,3.(1)

To solve the foregoing problem (1), we employ the 3D FEM and for this purpose we introduce the following functional:

Π=12∫Ω∫−∫Ω′(σ11ε11+2σ12ε12+2σ13ε13+2σ23ε23+σ22ε22+σ33ε33)dΩ−∫0ℓ3∫0ℓ1pu2|x2=hdx1dx3−∫0ℓ3∫0ℓ1pu2|x2=0dx1dx3(2)

Using the Ritz technique [13], δΠ = 0, we obtain the equilibrium equation and boundary conditions in Eq. (1). In this way, the validity of the functional (2) for the FEM modelling of the considered problem is proven.

In these cases, the solution domain Ω is divided into a finite number of finite elements, which are selected as standard rectangular prisms (bricks) with eight nodes. Selection of the number of degrees of freedom (NDOF) is determined from the requirements that the boundary conditions should be satisfied with very high accuracy and that the numerical results obtained for various NDOFs should converge.

After consideration of the FEM modelling, the ERR (denoted by γ) is calculated by using the expression:

γ≈12U(Sc+ΔSc(s/ℓ1))−U(Sc)ΔSc(s/ℓ1).(3)

In Eq. (3), U is the strain energy, Sc shows the area of the crack’s edge surface, ΔSc is the increment of the crack’s edge area, and s / ℓ1 = (0.5ℓ3 − x3)/ ℓ1. In the solution procedure, the values of ΔSc are selected regarding convergence of the numerical results obtained for γ.

Ethical approval: The conducted research is not related to either human or animals use.

3 Numerical Results

We assume that the plate is made of multilayered composite material consisting of alternating layers of two isotropic homogeneous materials parallel to the Ox1x3 plane and introduce the following notation for the matrix and the reinforcing layers: E1, and E2 are the Young’s moduli; ν1 and ν2 are the Poisson ratios; and η1 and η2 are the concentrations of the constituents. For the selected composite material, the effective anisotropic material constants Aij are determined according to the expressions given in [14].

The numerical investigations are performed for the case where η2 = 1 −η1 = 0.5, ν2 = ν1 =0.3 and h / ℓ = 0.20. Since the plate is symmetric with respect to the planes x1 = ℓ/2 and x3 = ℓ3 / 2, then FEM solutions are obtained in a quarter part of the domain. This domain was divided into 30, 12, and 30 brick elements along the Ox1x2x3 axes.

Before obtaining numerical results, the PC programs composed and used by the authors, were tested with the known results given in [15].

Table 1 shows the values of KI / KI∞ and KIS/KI∞(whereKI∞=pπℓ0) for various values of the ratios ℓ0 / ℓ1, ℓ0 / h and ℓ3 / ℓ1. Note that in Table 1, the values of KI / KI∞ and KIS/KI∞ are SIF of Mode I; KI / KI∞ is relevant to the plane strain state; and KIS/KI∞ is calculated by using exact solution for an infinite plate, for the approximate series as detailed in the handbook [15]. In this case, based on mechanical considerations, with increasing of the ratio ℓ3/ ℓ1, KI(=γ/(1−ν2)πℓ0) at s / ℓ1 = 0 must approach the corresponding values obtained in [15] for the plane strain state. This prediction is confirmed by the data given in Table 1. Moreover, our results agree with the mechanical consideration according to which the values of KI / KI∞ must approach 1 with increasing of the ratios ℓ3 / ℓ1 and decreasing of the ratios ℓ0 / ℓ1 and ℓ0 / h . This comparison shows good agreement and confirms the validity of the algorithm and programs used in the present investigations.

Table 1

The values of SIF at s / ℓ1 = 0 obtained for various values of the ratio ℓ3 / ℓ1 in the case where the material of the plate is isotropic.

ℓ0/ℓ1ℓ0/hℓ3/ℓ1
1510152050KIS / KI∞ [15]
0.0800.800.97611.17711.20001.20291.20361.23561.2406
0.0750.750.97421.17201.19221.19431.19471.19481.2009
0.0600.600.92781.11291.13041.13291.13351.13411.1444
0.0500.500.89061.06581.08711.09091.09201.09311.0931

Now, we analyze the numerical results which illustrate the influence of the problem parameters on the values of the ERR. Table 2 shows the influence of the ratios s /ℓ1 and E2 / E1 on the ERR obtained in the case where ℓ0 / 2ℓ1 = 0.15 and hu / ℓ1 = 0.1 (i.e. hu = h / 2). It follows from Table 2, as can be predicted, that the absolute maximum value of the ERR occurs at s / ℓ1 = 0 and the values of the ERR monotonically decrease with s / ℓ1. Moreover, it follows from Table 2, that the magnitude of the influence of the material anisotropy on the ERR increases with a decrease in the values of the ratio E2 / E1.

Table 2

The influence of the ratio E2/E1 on the values of the ERR in the various s / ℓ1 in the case where ℓ0/2ℓ1 = 0.15 and ℓ3/ ℓ1 = 1.

E2 / E1s / ℓ1
00.10.1330.1660.20.30.4
0.29.12669.08279.03558.95638.82667.86994.9238
0.54.98654.98294.96774.93904.85834.20812.3733
13.32333.32713.32123.30113.25642.81871.5627

Table 3 illustrates the influence of the ratios ℓ3 / ℓ1 on the ERR for various values of E2 / E1 obtained in the case where ℓ0/2ℓ1 = 0.15, s / ℓ1 = 0 and hu / ℓ1 = 0.1 (i.e. hu = h / 2). As can be seen, the values of the ERR monotonically increase with ℓ3 / ℓ1. This effect becomes more pronounced with decreasing of the ratio E2 / E1.

Table 3

The influence of the ratio E2/E1 on the values of the ERR in the various ℓ3 /ℓ1 in the case where ℓ0/2ℓ1 = 0.15 and s/ℓ1 = 0.

E2 / E1ℓ3 / ℓ1
0.20.511.52
0.20.93916.45169.12669.820410.3322
0.50.30873.29384.98655.45055.8451
10.18832.19243.32333.64873.9346

Table 4 shows the effect of the crack’s location and crack length on the ERR for various values of E2 / E1 in the case where s / ℓ1 = 0 . It can be concluded that the values of the ERR increase as the crack location approaches the upper face plane of the plate i.e. as the values of hu / ℓ1 decrease. This effect becomes significant with crack length, i.e. ℓ0 / 2ℓ1.

Table 4

The influence of the ratio E2/E1 and hu / ℓ1 on the values of the ERR in the various ℓ0 / ℓ1 in the case where s / ℓ1 = 0.

E2 / E1hu / ℓ1ℓ0 / 2ℓ1
0.040.100.150.25
0.50.100.34841.81294.986520.9955
0.08330.35471.90375.312822.9299
0.06660.37742.22726.506430.2514
0.050.43383.04279.546049.4419
10.100.21791.18783.323314.4063
0.08330.22201.24923.543915.7873
0.06660.23711.47144.352920.9374
0.050.27502.02416.462034.5191
50.100.10720.60281.59806.2118
0.08330.10990.63071.69356.7353
0.06660.11920.73072.04118.6579
0.050.14040.97672.922213.7326

4 Conclusion

In the current paper, the ERR in the band crack’s front which is contained in the thick rectangular plate made of multilayered composites is investigated by employing 3D FEM. The opening mode case which is considered is based on the numerical results of the influence of the plate material anisotropy and crack’s location geometry on the ERR, which are presented and discussed. At the same time, the approach used here can be applied to the related materials considered in works [16,17].

  1. Conflict of interest: Authors state no conflict of interest

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Received: 2018-01-22
Accepted: 2018-03-05
Published Online: 2018-05-31

© 2018 Arzu Turan Dincel, Surkay D. Akbarov, published by De Gruyter

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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  131. Comparative study of in vitro antioxidant, acetylcholinesterase and butyrylcholinesterase activity of alfalfa (Medicago sativa L.) collected during different growth stages
  132. Isothermal Oxidation Behavior of Gadolinium Zirconate (Gd2Zr2O7) Thermal Barrier Coatings (TBCs) produced by Electron Beam Physical Vapor Deposition (EB-PVD) technique
  133. Optimization of Adsorption Parameters for Ultra-Fine Calcite Using a Box-Behnken Experimental Design
  134. The Microstructural Investigation of Vermiculite-Infiltrated Electron Beam Physical Vapor Deposition Thermal Barrier Coatings
  135. Modelling Porosity Permeability of Ceramic Tiles using Fuzzy Taguchi Method
  136. Experimental and theoretical study of a novel naphthoquinone Schiff base
  137. Physicochemical properties of heat treated sille stone for ceramic industry
  138. Sand Dune Characterization for Preparing Metallurgical Grade Silicon
  139. Catalytic Applications of Large Pore Sulfonic Acid-Functionalized SBA-15 Mesoporous Silica for Esterification
  140. One-photon Absorption Characterizations, Dipole Polarizabilities and Second Hyperpolarizabilities of Chlorophyll a and Crocin
  141. The Optical and Crystallite Characterization of Bilayer TiO2 Films Coated on Different ITO layers
  142. Topical Issue on Bond Activation
  143. Metal-mediated reactions towards the synthesis of a novel deaminolysed bisurea, dicarbamolyamine
  144. The structure of ortho-(trifluoromethyl)phenol in comparison to its homologues – A combined experimental and theoretical study
  145. Heterogeneous catalysis with encapsulated haem and other synthetic porphyrins: Harnessing the power of porphyrins for oxidation reactions
  146. Recent Advances on Mechanistic Studies on C–H Activation Catalyzed by Base Metals
  147. Reactions of the organoplatinum complex [Pt(cod) (neoSi)Cl] (neoSi = trimethylsilylmethyl) with the non-coordinating anions SbF6– and BPh4–
  148. Erratum
  149. Investigation on Two Compounds of O, O’-dithiophosphate Derivatives as Corrosion Inhibitors for Q235 Steel in Hydrochloric Acid Solution
https://doi.org/10.1515/chem-2018-0057
Keywords for this article
Energy Release Rate (ERR); band crack; 3D FEM (Three Dimensional Finite Element Method); rectangular plate
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