A Stationary Edge Crack in a Finite Body: A Finite Element Approach

 

Sourav Kr. Panja*, Subhas Ch. Mandal

Department of Mathematics, Jadavpur University, Kolkata 700032, West Bengal, India.

 

Abstract:

In the present paper, a stationary edge crack is considered in a finite elastic body under the normal loading condition. A compact analysis of finite element approach of an edge crack is studied and the numerical procedure is implemented in the MATLAB software. The displacements along the cracked surface and the stress components are presented graphically. The convergency of the solution is exhibited by means of graphs.

 

 

INTRODUCTION:

Inclusion and crack problems are normal phenomena in most of the fabricated materials. Edge cracks are occured due to overload at the edge of a body, shear and normal failure or erosion in the edge. The edge crack initiation and its impact due to thermal stress in a functionally graded plate is studied by Noda¹⁵. Das et al.^5,6,7,8 studied edge crack in orthotropic materials under normal loading. P-wave interaction with an edge crack in an infinite strip is examined by Munshi and Mandal¹³. Some recent papers on the wave propagation in thermoelastic bodies are discussed by many authors^20-29. Theory of elastic-plastic shells is studied by Verma²². Some authors^24,25 investigated vibration analysis in the presence of fracture.

 

The numerical solution of the boundary value problem in the presence of static crack in a finite body is determined by the finite element method (FEM). This method contains some conventional steps. The first step is to derive the elemental equation from variational statement and assemble the elemental equation into global form according to the discretization of the whole domain. The second step is to impose the boundary condition i.e. Dirichlet and Neumann boundary conditions and solve the assembled equations. Finally, the postprocessing of the obtained result is taken into consideration. Elasto-static crack problem is analyzed by Liebowitz and Moyer¹¹ through finite element method. Alshoaibi and Fageehi⁴ investigated quasi static crack problem by finite element method. Sukumar and Prevost¹⁸ studied quasi-static crack growth with the help of extended finite element method. dynamic crack analysis by finite element techniques is examined by enderlein et al.^9,10.

 

In last few decades, the development of analytical, semi-numerical and numerical models under different loading conditions for several crack geometries have been studied. A finite element formulation for an edge crack due to anti-plane shear wave is discussed by Alex³. Yi et al.¹⁹ studied analytical analysis of finite width cracked plate by crack line analysis method. Two mixed boundary value problems through displacement potential approach for plane stress and plane strain conditions is examined by Debnath¹⁴. Recently Panja and Mandal¹⁷ exhibited the effect of magnetic field for a Griffith crack in an infinite medium for normal and in- plane shear loading. Cracked surface displacement and stress distributions is analyzed for a penny shaped crack in infinite thermoelastic medium by Abbas^1,2. effect of rotation on Plane waves in an infinite fibre-reinforced half-space is studied by Othman and Abbas¹⁶. Mohsin¹² examined static and dynamic analysis of a cracked finite plane under uniform tensile stress through ANSYS workbench software.

Cracks or defects occurs, either as natural or as a consequence of fabrication process. In case of finite body, the analytical solution become more complicated due to the presence of boundaries. Many problems are studied containing one or more cracks in an infinite medium or semi-infinite elastic medium. The mixed boundary value problem associated with a finite crack in a finite body have enormous number of applications in construction engineering. Although a plenty number of papers related with finite cracked body have been solved numerically through finite element software like ANSYS Workbench, ABAQUS, COMSOL Multiphysics, FreeCAD, etc., the problem a stationary edge crack in a finite region under normal load is still not attempted semi-numerically.

 

The current article goals to show the mathematical model of a static edge crack present in a finite body under normal loading condition. A brief analysis of finite element model is constituted using equilibrium equations and mixed boundary conditions. The finite domain is discretized into some sub domains and obtained the displacement components after assembling the elemental equations into global form. The convergency of the solution is checked and presented by means of graphs after refining the mesh. The cracked surface displacement and stress state of the body are illustrated graphically.

 

PROBLEM OUTLINE:

Let us consider a cracked body with an edge crack occupying in the  plane and locating at  along the thickness  in the  direction. Also, we assume the dimension of the material is  . The cracked body is subjected to normal load along the crack surface, causes the displacements are in the  and  direction. The complete analysis of the problem is taken in the  plane and it is sufficient to consider a rectangular cross section geometry as shown in Fig. 1. Moreover, the displacement is symmetric with respect to the  plane and therefore we consider only the upper half of the geometry.

 

Figure 1. Cross section of the cracked geometry.

 

 

FINITE ELEMENT FORMULATION:

 

 

 

 

Figure 2. Transformation from global coordinate system to local coordinate system.

 

 

 

i.e., the local coordinate and hence the non-zero shape functions in (18) are

,.(23)

 

 

 

NUMERICAL RESULTS:

 

 

 

 

Figure 3. Physical element with refined mesh.

 

 

 

 

Figure 4. Crack surface displacement versus x.

 

 

Figure 5. Normal stress versus x.

 

 

In Figure 5 the normal stress  is plotted against  for different materials. It is observed that  is very high just outside the crack and decreases as the distance from the crack tip increases. Moreover, the nature of the  curve is like a wave as it starts from positive value and become negative through zero level when it goes away from the crack.

 

Figure 6. Convergency of the solution

 

In Figure 6, the convergence of the present solution is rendered by taking more elements in the refinement of discretization. Initially we have  elements and further we discretize the whole domain into  elements and so on. In this current problem the refinement is not uniform i.e. not global refinement as the solution is required along the crack edge and outside the crack line i.e. at . Therefore, the discretization is taken care about the crack line and it reduces the number of elements and computational work load than global refinement. It is clear

from Figure 6 that the solution is convergent in each of the materials Aluminum, Stainless Steel and Iron.

 

CONCLUSION:

In this paper a semi-numerical finite element approach is implemented to study a static edge crack problem in a finite region. The main focus of this present study is to analyze crack surface displacement and normal stress analysis outside the crack. The following bullet points have been highlighted from this study:

·         Crack surface displacement is varied with the material constants as the current problem disclosed that crack surface opened more in case of Aluminum than Iron and Steel.

·         Normal stress outside the crack behaves wave like nature and impact of materials is not so much on normal stress.

·         The convergency of the solution obtained through semi numerical finite element method is depicted through different material and represented a steady solution by remeshing the domain.

·         The research outcome may be very significant in construction sector as pre-existing cracks are present in any finite body like wall, beam, pillar, any wooden structure, etc. Some extended models, such as two or more cracks, cracks in an infinite domain, cracks in the presence of magnetic field and effect of temperature on crack surface can be investigated with the help of current semi-numerical finite element approach.

 

CONFLICT OF INTEREST:

The authors have no conflicts of interest regarding this investigation.

 

ACKNOWLEDGMENTS:

The authors would like to thank UGC, New Delhi, India for financial support during this research.

 

REFERENCES:

1.        Abbas, Ibrahim A. Finite element analysis of the thermoelastic interactions in an unbounded body with a cavity. Forschung im Ingenieurwesen. 2007; 71(3): 215-222.

2.        Abbas, Ibrahim A. Finite element analysis of internal penny-shaped crack problem in an unbounded thermoelastic medium. Journal of Thermal Stresses. 2016; 39(10): 1171-1181.

3.        Alex, Rajan. An analytical and finite element analysis of fracture in nonhomogeneous viscoelastic material. 1994.

4.        Alshoaibi, Abdulnaser M., and Yahya Ali Fageehi. Simulation of quasi-static crack propagation by adaptive finite element method. Metals. 2021; 11(1): 98.

5.        Das, S., et al. Symmetric edge cracks in an orthotropic strip under normal loading. International Journal of Fracture. 2008; 153: 77-84.

6.        Das, S., S. Mukhopadhyay, and R. Prasad. Stress intensity factor of an edge crack in bonded orthotropic materials. International Journal of Fracture. 2011; 168: 117-123.

7.        Das, S., R. Prasad, and S. Mukhopadhyay. Stress intensity factor of an edge crack in composite media. International Journal of Fracture. 2011; 172: 201-207.

8.        Das, Subir. Weight function for an edge crack in an infinite orthotropic strip under normal point loading. ZAMM‐Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik: Applied Mathematics and Mechanics. 2010; 90(4): 271-277.

9.        Enderlein, M., A. Ricoeur, and M. Kuna. Comparison of finite element techniques for 2D and 3D crack analysis under impact loading. International Journal of Solids and Structures. 2003; 40(13-14): 3425-3437.

10.      Enderlein, Marco, A. Ricoeur, and M. Kuna. Finite element techniques for dynamic crack analysis in piezoelectrics. International Journal of Fracture. 2005; 134: 191-208.

11.      Liebowitz, H., and E. T. Moyer Jr. Finite element methods in fracture mechanics. Computers and Structures. 1989; 31(1): 1-9.

12.      Mohsin, Najah R. Static and dynamic analysis of center cracked finite plate subjected to uniform tensile stress using finite element method. Int. J. Mech. Eng. Technol. 2015; 6(1): 56-70.

13.      Munshi, Nasima, and Subhas Chandra Mandal. Diffraction of p-waves by edge crack in an infinitely long elastic strip. JSME International Journal Series A Solid Mechanics and Material Engineering. 2006; 49(1): 116-122.

14.      Nath, SK Deb. Analytical solution of mixed boundary value problems using the displacement potential approach for the case of plane stress and plane strain conditions. Int J Appl Mech Eng. 2017; 22(2): 269-291.

15.      Noda, Naotake. Thermal stresses in functionally graded materials. Journal of Thermal Stresses. 1999; 22(4-5): 477- 512.

16.      Othman, Mohamed IA, and Ibrahim A. Abbas. Effect of rotation on plane waves at the free surface of a fibre- reinforced thermoelastic half-space using the finite element method. Meccanica. 2011; 46: 413-421.

17.      Panja, Sourav Kumar and S. C. Mandal. Impact response of a finite crack in the presence of magnetic field. Engineering Fracture Mechanics. 2021; 253: 107851.

18.      Sukumar, N., and JH. Prévost. Modeling quasi-static crack growth with the extended finite element method Part I: Computer implementation. International journal of solids and structures. 2003; 40(26): 7513-7537.

19.      Yi, Zhijian, et al. Analytical analysis on the Dugdale model of a finite-width cracked plate by using crack line analysis method. AIP Advances. 2020; 10(11).

20.      Singh, Baljeet, Anand Kumar Yadav, and Sachin Kaushal. Effect of impedance boundary on reflection of plane waves from free surface of a rotating thermoelastic solid half space. Research Journal of Engineering and Technology. 2017; 8(4): 405-413.

21.      Singh, Baljeet. On Rayleigh wave in a generalized porothermoelastic solid half-space. Research Journal of Engineering and Technology. 2018; 9(2): 179-188.

22.      Verma, Gaurav. Theory of Elastic-Plastic Shells. Research Journal of Science and Technology. 2016; 8(1): 41-44.

23.      Verma, K. L. On the Guided Waves in Generalized Dynamical Theories of Thermoelasticity. Research Journal of Science and Technology. 2017; 9(1): 189-194.

24.      Jain, N. K. Vibration Analysis of Fibrous Composite Plates with Central Circular Hole. International Journal of Technology. 2014: 1-3.

25.      Sahoo, Manoj Kumar, O. P. Sahu, and Manish Verma. Vibration analysis of a V-groove edged cracked cantilever beam. International Journal of Technology. 2012: 65-70.

26.      Pathania, Vijayata, and S. Pathania. Thermoelastic Waves Propagation in Layered Plates in Anisotropic Media. Research Journal of Science and Technology. 2013; 5(1): 123-129.

27.      Kumar, R., B. Kumar, and N. Sharma. Reflection and transmission coefficients at the boundary surface of micropolar viscothermoelastic solids with two temperatures. Research Journal of Science and Technology. 2017; 9(3): 323-338.

28.      Thakur, Anita D., and J. N. Sharma. Generalized Piezothermoelastic Continuum Subjected To Temperature Input. Research Journal of Science and Technology. 2013; 5(1): 207-212.

29.      Pathania, Vijayata. Circular Waves Propagation in Thermoelastic Solid-Liquid Interface. Research Journal of Science and Technology.  2017; 9(1): 179-183.

 

 

 

Received on 31.05.2024       Modified on 18.06.2024 Accepted on 15.07.2024      ©A&V Publications All right reserved Research J. Science and Tech. 2024; 16(3):211-218. DOI: 10.52711/2349-2988.2024.00031