STATIC AND DYNAMIC ANALYSIS OF CENTER CRACKED FINITE PLATE SUBJECTED TO UNIFORM TENSILE STRESS USING FINITE ELEMENT METHOD

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1 INTERNATIONAL JOURNAL OF MECHANICAL ENGINEERING AND TECHNOLOGY (IJMET) International Journal of Mechanical Engineering and Technology (IJMET), ISSN (Print), ISSN (Print) ISSN (Online) Volume 6, Issue 1, January (2015), pp IAEME: Journal Impact Factor (2014): (Calculated by GISI) IJMET I A E M E STATIC AND DYNAMIC ANALYSIS OF CENTER CRACKED FINITE PLATE SUBJECTED TO UNIFORM TENSILE STRESS USING FINITE ELEMENT METHOD Najah R.Mohsin Southern Technical University, Technical Institute-Nasiriya, Mechanical Technics Department ABSTRACT The study of crack behavior in a plate is a considerable importance in the design to avoid the failure. This paper deals with investigation of stress intensity factor, Von-Misesstress (Ϭ Von-mises ), natural frequency, mode shape and the effect of excitation frequency on the finite center cracked plate subjected to uniform tensile loading depends on the assumptions of Linear Elastic Fracture Mechanics (LEFM) and plane strain problem. The stress intensity factors mode I (K I ) are numerically calculated by finite element solution using ANSYS (ver. 15) software and theoretically using standard equations for different crack lengths and plate dimensions. Generally, the results show that there are no major differences between the two methods. However, the difference between the two methods occur if we take the plate length parameter in considerate. Furthermore, Ϭ Von-mises at crack tip region, 10th natural frequencies and the effect of excitation frequency on the crack tip stresses are studied for three different materials. Keywords: Crack Tip, Stress Intensity Factor, Natural Frequency, Finite Element Method (FEM), Harmonic Analysis, Linear Elastic Fracture Mechanics (LEFM). 1- INTRODUCTION In general, a fracture is defined as the local separation of an object or material into two or more pieces under the action of stress. Usually, the fracture of a plate occurs due to the development of certain displacement discontinuity surfaces within the plate. Recent development in engineering structures shows that fracture can be caused by small cracks in the body of structures despite the authenticity of elasticity theory and strength of materials. 56

2 As a result, fracture mechanics filed which is concerned with the propagation of cracks in materials has developed to study more about this subject, Ali and et al. [1]. According to the types of load, there are three linearly independent cracking modes are used in fracture mechanics (Fig.1) as follow: Fig.(1): Mode I, Mode II Mode I : a tensile stress normal to the plane of the crack (tensile mode), Mode II: a shear stress acting parallel to the plane of the crack and perpendicular to the crack front (in-plane shear mode). Mode III : a shear stress acting parallel to the plane of the crack and parallel to the crack front (out- of- plane shear mode) Where Mode I is the most common load type encountered in engineering design. The problem of determining the stress intensity factors of cracks in a plate is of considerable importance in the design of safe structures because of stress intensity factors the main key value defining the stresses around the crack tip arising from that crack. An approach based on the continuous dislocation technique was formulated by Huang and Kardomateas[2]to obtain the Mode I and II stress intensity factors K I and K II in a fully anisotropic infinite strip with a central crack. The elastic solution was applied to calculate K I for a center crack in an anisotropic strip with the effects of crack length and material anisotropy. The problem of a crack in a general anisotropic material under conditions of LEFM was examined by Banks-Sills and et at.[3]. General material anisotropy was considered in which the material and crack coordinates at arbitrary angles. A three-dimensional treatment was required for this situation in which there may be two or three modes present. Azevedo[4] was study the stress intensity factors K I and K II for an inclined central crack on a plate subjected to uniform tensile loading were calculated for different crack orientations (angles) using FEM analysis, which was carried out in ABAQUS software. The stress intensity factors were obtained using the J integral method and the modified Virtual Crack Closure Technique (VCCT). Both methods produced results for K I and K II which were close to the analytical solution. The effects of the boundary conditions were discussed. Ergun and et at.[5]were used the FEM to analyze the behavior of repaired cracks in 2024-T3 aluminum with bonded patches made of unidirectional composite plates. The K I was calculated by FE Musing displacement correlation technique. Jweeg and et at.[6] were studied natural frequencies of composite plates with 57

3 the effect of crack orientation, crack position, crack size and based on the shape of the fibers. They made a comparison between the analytical results and the results get by finite element solution using ANSYS (ver.14) software. Bhagat and et at.[7] were studied a finite rectangular plate of unit thickness with two inclined cracks (parallel and non-parallel) under biaxial mixed mode condition were modeled using FEM. The FEM was used for determination of stress intensity factor by ANSYS software. Effects of crack inclination angle on stress intensity factor for two parallel and non-parallel cracks are investigated. The significant effects of different crack inclination parameters on stress intensity factor were seen for lower and upper crack in two inclined crack. Al-Ansari [8] was a comparison between six models, calculating K I for central cracked plate with uniform tensile stress, was made in order to select the suitable model. These models were three theoretical models and three numerical models. The three numerical models are half ANSYS model, quarter ANSYS model and weight function model. Crack geometry, crack length, plate length and the applied stresses are the parameters that used to compare between the models. He concludes that the three theoretical models can recognize the effect of the width of the plate, the crack length and applied stress but they failed to recognize the effect of the length of the plate. Ali and et at.[1] were attempts to analyze the stress intensity factor in various edge cracks along the length of a finite plate which was under a uniform tension. FEM was utilized for the analysis. In addition, Neural Network Method (NNM) was used to predict the correlation of stress intensity factor and the position of edge crack along the length of a finite plate. For certain cracked configurations subjected to external forces, it is possible to derive closedform expressions for the stresses in the body, assuming isotropic linear elastic material behavior. If we define a polar coordinate axis with the origin at the crack tip Fig.(2), it can be shown that the stress field in any linear elastic cracked body is given by Anderson[9] σ = f θ + A. r.g θ,.(1) where Ϭ ij stress tensor, r and θ are as defined in Fig.(2), kconstant, f ij dimensionless function of θ in the leading term, A m is the amplitude for the higher-order terms and g ij (m) is a dimensionless function of θ for the m th term. Fig.(2): Definition of the coordinate Mode III crack loading axis ahead of a crack tip 58

4 The higher-order terms depend on geometry, but the solution for any given configuration contains a leading term that is proportional to 1/ r.as r 0, the leading term approaches infinity, but the other terms remain finite or approach zero. Thus, stress near the crack tip varies with 1/ r, regardless of the configuration of the cracked body. It can also be shown that displacement near the crack tip varies with r. Eq.(1) describes a stress singularity, since stress is asymptotic to r =0. Most cracks are long and sharp tips. These can be of atomic dimensions in brittle materials. In 1938, Westergaard solved the stress field for an infinitely sharp crack in an infinite plate. The elastic stresses were given by the equations, Rae [10]. σ = π cos θ 1 sin θ sin θ..(2) σ = π cos θ 1+sin θ sin θ..(3) σ = π cos θ sin θ cos θ....(4) Similar expressions for displacements u u = µ π cos θ k 1+2sin θ...(5) u = µ π sin θ k+1 2cos θ,..(6) Where µ denotes the shear modulus, k the small difference in formulas for plane stress and plane strain which is equal to. Plane Stress k=. (7) 3 4v. Plane Strain Based on the above equations, we show that, the K I defines the amplitude of the crack-tip singularity. That is, stresses near the crack tip increase in proportion to K I. Moreover, the stress intensity factor completely defines the crack tip conditions, if K I is known, it is possible to solve for all components of stress, strain, and displacement as a function of r and θ. This single-parameter description of crack tip conditions turns out to be one of the most important concepts in fracture mechanics, Anderson [9]. Table (1): Mechanical properties used for selected material, Kulkani [12] Material type Modulus of elasticity density Poisons ratio (Mpa) (Kg/m 3 ) Aluminum(alloy) 0.71e Carbon Steel 2.02 e Nickel Silver e

5 2- MATERIALS AND METHODS 2.1- Based on the assumptions of LEFM and plane strain problem, K I to a center cracked plate under static load were numerically calculated using FEM for three materials Carbon Steel, Aluminum(alloy) and Nickel Silver materials are shown in Table (1)and theoretically using the two standard equations as follows a) From Ergun and et at.[5] and Rae[10] K =σ π a Sec π (8) b) From Al-Ansari[8] K = σ π a (9) 2.2- For practical considerations, any kinds of stress, strains and deformations in single parts or assemblies can be better approximated using FEM. FEM is a numerical technique in which the governing equations are represented in matrix form, which is to be solved by computer software. The solution region is represented as an assemblage of small sub-regions called finite element. The element is the basic building unit with a predetermined number of degrees of freedom (d.o.f) and can take various forms, e.g. beam, plate, shell or solid elements. The selection of the best element depends on the type of problem, geometry of boundaries, boundary conditions, accuracy required, size of the available computer and the maximum allowable computing cost. The main purpose of this paper is to observe the behave or of finite plate with a central crack under the effect of some parameters such as a ratio of crack length to width plate, ratio of width to length plate, applied stress for different materials by calculating the stress intensity factors K I and Ϭ Von-mises to the crack tip. Natural frequencies, mode shapes and also the effect of excitation frequencies are studied in this paper. Quarter model is selected to represent the finite plate with center crack in the FEM software (ANSYS ver.15) because the geometry and load applied for specimen is symmetry (Fig.3). To compute the required results in a faster and accuracy way, programs are written with APDL (Ansys Parameter Design Language).The first step of these softwares is to discretize the structure into finite elements connected at nodes. It is necessary to discretize the plate structure into a sufficient number of elements in order to obtain a reasonable accuracy, on the other side, the more elements that are used, the more costly it will be. In this paper, PLANE183 is used as a discretization element. 60

6 Fig. (3): Center cracked plate specimen with dimensions Fig. (4): Geometry, node locations and the co-ordinate ordinate system for PLANE183 Element, ANSYS help [11] 3- PLANE183 ELEMENT DESCRIPTION PLANE183 is an isoperimetric eight or six nodes(i,j,k,l,m,n,o,p) quadratic or triangle displacement behavior element which is better suited to modeling irregular meshes.the element may be used as a plane element (plane stress, plane strain and generalized plane strain) or as an axisymmetric element. It has two degree of freedom (translation in X and Y directions) at each node. Fig.(4)shows the geometry, node locations and the co-ordinate system for PLANE183 element. The model used in this paper with elements, ements, nodes, boundary conditions and mesh generation are shown in Fig.(5). 61

7 Fig. (5): Mesh generation for quarter model with 556 elements, 1709 nodes and boundary conditions 4- APDL PROGRAM In the written APDL program, there are four important processors are used 4.1- Preprocessor (/PREP7):This command contains what you need to use to build a model such as define element types, real constant,material properties,create model geometry and mesh the object created Solution Processor (/SOLU):This command allows to apply boundary conditions, loads and create the concentration key point (crack tip) using the(kscon)command.this processor contents different analysis such as: Static analysis (ANTYPE, STATIC): To determine the structure analysis under static loads Mode Analysis (ANTYPE, MODAL): It is used to determine the magnitude of natural frequencies and mode shapes for the structure Harmonic Analysis (ANTYPE, HARMIC): Harmonic response analysis is a technique used to determine the steady-state response of a linear structure to loads that vary sinusoidally (harmonically) with time Postprocessor (/Post1): This command used to display the results of stress intensity factor in lists and display Ϭ Von-mises in lists, plots or curves The Time-History Postprocessor (/POST26): This command is used to evaluate solution results at specific points in the model as a function of time, frequency, or some other change in the analysis parameters that can be related to time. Fig.(6) shows the APDL flow chart. Carbon Steel, Aluminum (alloy) and Nickel Silver materials (Table 1) are studied in this paper to calculate K I, Ϭ Von-mises with range of a/b at b=0.3 m, range of b/hat b=0.3m, range of b/h at h=0.3m, range of applied stress, first 10 th natural frequencies, mode shapes and range of excitation frequency, where a, b and h are defined in Fig.(3). 62

8 Start Define element type with plane strain option Define material properties Define keypoints and lines for model geometry with center crack Discretize lines into suitable division number Great crack point using KSCON command Great the model area Apply boundary conditions (displacement & applied stresses) Mesh the model Solution Static analysis Model analysis Harmonic analysis Solve General Postproc Define crack face path Define local crack tip CS Change active CS to specified CS Calculate K I &Ϭ Von-mises using KCALC& PRNSOL command Select no. of mode to extract Solve General Postproc Calculate natural frequency and plot mode shape Exit Select freq. range and no. of substeps Solve Time History Variable Plot ϬVon-Mises with the reang of Fig.(6): Flow chart of APDL program 63

9 5- RESULTS AND DISCUSSIONS 5.1- Stress Intensity Factors K I K I are numerically calculated using ANSYS ver. (15) (STATIC Analysis) for three different materials (Nickel Silver, Aluminum (alloy) and Carbon Steel) and theoretically by Eq.(8) and Eq.(9). Fig. (7) to Fig.(10) show that the variation of stress intensity factor with different values of a/b, tensile stresses Ϭ t, b/h at specified h=0.3m and b/h at specified b=0.3m. From these figures, it can be seen that increasing the ratio of a/b and applied stresses leads to increasing the value of K I in a high level for all the selected values. Fig.(9) shows that small effect for b/h ratio at h=0.3mon the K I magnitude. From Fig.(10), it is clear that at a specific value b=0.3m in the numerical solution, the K I increases with ratio of b/h increases but remain unchanged in theoretical solution because the theoretical equations don t take the parameter h in a consideration. In all mentioned figures, it is found that the material type variation is not important to calculate K I and also we show that K I calculated from theoretical solution either equal or less (with small ratio) than that of numerical solution except when b/h ratio increasing at specific bas mention above. Fig.(7): Variation of stress intensity factor with (a/b) ratio numerically for different materials and theoretically using Eq.(8) and Eq.(9) Fig.(8): Variation of stress intensity factor with tensile stress numerically for different materials and theoretically using Eq.(8) and Eq.(9) Fig.(9): Variation of stress intensity factor with (b/h) Fig.(10): Variation of stress intensity factor with ratio for h=0.3m numerically for different materials (b/h) ratio at b=0.3m numerically for different and theoretically using Eq.(8) and Eq.(9) materials and theoretically using Eq. (8) and Eq.(9) 64

10 5.2- Ϭ Von-mises (SEQV) (STATIC Analysis) Fig.(11) to Fig.(14) show that the variation of Ϭ Von-mises (in Mpa) at crack tip with different values of a/b ratio, tensile stress, b/h ratio at specified h=0.3m and b/h ratio at specified b=0.3m. From these figures, it can be seen that increasing a/b ratio and magnitude of tensile stress leading to increase Ϭ Von-mises in a large value and in small value in case of increasing b/h ratio at fixed b =0.3m. Increasing b/h ratio at fixed h=0.3m from 0.5 to 2.5m with step 0.25m doesn t change Ϭ Von-mises in a considerable value. Ϭ Von-mises in these figures are numerically approached only due to difficulty of the theoretical approach. It can be observed that there is a small difference between the Von-Misses stress when we use a different types of materials. Generally, Ϭ Von-mises in Carbon Steel is greater than of Nickel Silver and both of them greater than of Aluminum (alloy) due to the difference in mechanical properties (modulus of elasticity and poisons ratio).fig. (15) to Fig. (18) show a plot results for different states three for quarter model and one for half model. Fig.(11): Variation of Ϭ Von-mises with (a/b) ratio numerically for different materials Fig.(12): Variation of Ϭ Von-mises with tensile stress numerically for different materials Fig.(13): Variation of Ϭ Von-mises with (b/h) ratio at h=0.3m numerically for different materials Fig.(14): Variation of ϬVon-mises with (b/h) ratio at b=0.3m numerically for different materials 65

11 Fig.(15): ϬVon-mises distribution for quarter model with b = 0.3 m and Ϭt =70 Mpa Fig.(16): ϬVon-mises distribution for quarter model with b = 0.15 m and Ϭt =100 Mpa Fig. (17): ϬVon-mises distribution for quarter model with b = 0.3 m and Ϭt =250 Mpa Fig.(18): ϬVon-mises distribution for half model with b = 0.3 m and Ϭt =250 Mpa 5.3- Free Vibration Analysis (MODAL Analysis) This analysis consists of studying the vibration characteristics such as natural frequencies and mode shapes for the central cracked plate. The first 10 th natural frequencies were numerically calculated and reported in Table (2) for three different materials. We can see that the magnitudes of natural frequencies (ω) are different from one material to another because the natural frequencies of the structure depend on the material stiffness and density. Furthermore, four mode shapes are shown in Fig.(19) to Fig.(22) to the quarter model to explain the plate deformation with respect to class of mode shape. 66

12 No. of natural frequency Table (2): First 10 th natural frequencies for central cracked plate Natural frequency (Hz) Nickel silver Aluminum(alloy) Carbon steel h = 0.3 m b = 0.3 m a = 0.09 m Fig.(19): 4th mode shape for quarter model to Nikel-Silver material with b=h=0.3m and a=0.09 m Fig.(20): 7th mode shape for quarter model to Aluminum (alloy) material with b=h=0.3m and a=0.09 m Fig.(21): 9th mode shape for quarter model to Nikel-Silver material with b=h=0.3m and a=0.09 m Fig.(22): 10th mode shape for quarter model to Carbon Steel material with b=h=0.3m and a=0.09 m 67

13 5.4- Harmonic Analysis (HARMIC Analysis) In this analysis, an external tensile stresses and wide range of excitation frequency was applied in order to illustrate the behavior of crack tip stresses under deferent values of excitation frequencies. Fig.(23) to Fig.(25) illustrate the variation of crack tip Ϭ Von-mises with wide range of excitation frequencies 0 to 10Hz to cover more than the first 5 th natural frequencies with the parameters a=0.09m, h=0.3m,b=0.3m, Ϭ t =130Mpa for three different materials Nickel Silver, Aluminum(alloy) and Carbon Steel. From these figures, it is clear that the magnitude of Ϭ Von-Mises take a huge value in crack tip when the excitation frequencies equal to any frequency from plate natural frequencies (resonance phenomenon) especially at fundamental frequency in Carbon Steel plate (Ϭ Von-mises = Mpa) and Aluminum (alloy) plate (Ϭ Von-mises = Mpa) and at the second natural frequency in Nickel Silver plate (Ϭ Von-mises = Mpa). Fig.(23): Variation of ϬVon-mises with excitation frequencies for Carbon steel material When b = 0.3 m, h = 0.3 m and a = 0.09 m and Ϭt = 130 Mpa Fig.(24): Variation of ϬVon-mises with excitation frequencies for Aluminum (alloy) material When b = 0.3 m, h = 0.3 m and a = 0.09 m and Ϭt = 130 Mpa 68

14 Fig.(25): Variation of Ϭ Von-mises with excitation frequencies for Nickle-Silver material When b = 0.3 m, h = 0.3 m and a = 0.09 m and Ϭt = 130 Mpa 6- CONCLUSIONS The main conclusions of this work are reported below 6.1- Increasing the crack length and applied stresses lead to increasing the value of K I. In the other hand, K I value change with the change of plate length in the numerical solution but remains constant in theoretical solution as the theoretical equations don t take this parameter in a consideration There is no sensitive effect of the material type on the value of K I The first 10 th natural frequencies for three material types are shown to be different for the same plate dimensions and boundary conditions because the natural frequency depends on the stiffness and density of the material Ϭ Von-mises value takes a huge value at crack tip region when the excitation frequency equals to any frequency from plate natural frequencies (resonance phenomenon) especially at fundamental frequency in Carbon Steel and Aluminum (alloy) plate) and at the second natural frequency in Nickel Silver plate. 7- REFERENCES [1] Z. Ali, K. Esfahan, S. Meysam, A. Iman, B. Aydin and B. Yashar, FEM Analysis of Stress Intensity Factor in Different Edge Crack Positions, and Predicting their Correlation using Neural Network Method, Research Journal of Recent Sciences, Vol.3(2), p.p , [2] H. Huang and G.A. Kardomateas, Stress intensity factors for a mixed mode center crack in an anisotropic strip, International Journal of Fracture, Vol. 108, p.p , [3] L. Banks-Sills, P.A. Wawrzynek, B. Carter, A.R. Ingraffea and I. Hershkovitz, Methods for calculating stress intensity factors in anisotropic materials: Part II Arbitrary geometry, Engineering Fracture Mechanics, Vol.74, p.p ,

15 [4] P.C.M. Azevedo, Stress intensity factors determination for an inclined central crack on a plate subjected to uniform tensile loading using FE analysis, [5] E. Ergun, S. Tasgetiren and M. Topcu, Stress intensity factor estimation of repaired aluminum plate with bonded composite patch by combined genetic algorithms and FEM under temperature effects, Indian Journal of Engineering & Materials Sciences, Vol. 19, P.P , [6] M. J. Jweeg, A, S. Hammood and M. Al-Waily, Analytical Solution to Oblique Crack Effect for Difference Composite Material Plates, ARPN Journal of Science and Technology, Vol. 2, NO 8, p.p , [7] R. K. Bhagat, V. K. Singh, P. C. Gope and A.K. Chaudhary, Evaluation of stress intensity factor of multiple inclined cracks under biaxial loading, Fratturaed Integrità Strutturale, Vol. 22, p.p. 5-11, [8] Dr. L. S. Al-Ansari, Calculating Stress Intensity Factor (Mode I) for Plate with Central Crack: Review and Comparison between Several Techniques of Calculations, Asian Transactions on Engineering, Vol. 2, p.p ,2012. [9] T.L.Anderson, Fracture Mechanics Fundamentals and Applications, Third Edition, Taylor &Francis Group, CRC Press, [10] Dr. C.Rae, Fracture and Fatigue, Natural Sciences Tripos Part II, Material Science, Easter Term , Department of Materials Science and Metallurgy, C15, University of Cambridge, [11] ANSYS help. [12] S.G.Kulkani, Machine Design, Sixth reprint, McGraw-Hill companies,