Al-Qadisiya Journal For Engineering Sciences Vol. 2 No. ٢ Year 2009 ٢٨٨ A STUDY OF THE EFFECT OF IMPACT LOADING ON DSIFs AND CRACK PROPAGATION IN PLATE Dr. FATHI A. ALSHAMMA ALI FAHAD FAHEM University of Baghdad, College of Engineering University of AL-Qadisiya, College of Engineering Mechanical Engineering Department Mechanical Engineering Department Abstract In this work, a study of the effect of impact loading on dynamic crack propagation in thin and isotropic thick plates for two types material, stainless steel and aluminum, are investigated analytically and numerically to give a complete study of 3D crack growth of simple supported plate with internal crack at the center of plate. The dynamic stress intensity factor (DSIFs), velocity of dynamic crack propagation and the deep of crack normal to the crack face for different angles from crack tip have been calculated when Contact of a cylindrical impactor on isotopic plates is considered, to support the analytical result for DSIFs with a numerical package result (Ansys-10). This based on finite element method used to investigate the stress and the values of dynamic stress intensity factors at the crack tip by full transient dynamic Analysis using 3D (element, Solid-90). The results from Airy stress function method gives good agreement with the results of Ansys for long time duration in impact loading while the energy method is agreement with the Ansys package for some time and then become limited. In addition, the DSIFs and crack propagation velocity are increasing with angel from crack tip and velocity impact increasing. Keywords: Dynamic Stress Intensity Factor, Finite Element Analysis, velocity impact, Crack angle, Crack Propagation. �������� ���� ��� ����� ��� ���� ��� ����� �� �� ����� � !� "��" . ��* �B 19Bه �B%* �15 ا�2�A@ ا�(��ع. د � ا���1�9آ�، �� :� �1Cاد � آ� � ا���1�9، �� :� ا�'�د� � ( ��( � �8�. ا�9 8�. ا���1�9 ا�( ��( � ���1 ا� ا�<=>� �'& 4 آ� ا���7ر ا�q7 �8 ا�pK�$T ا� L+�� وا�,�+�6 اo�Cو@ و1+� �-�T��4* ا 2+N@ ��اء @�'+* و درا ا! ?@ ^- ^+�و�#� ���7�- pK�$T� ثs9���1د اC8 ا� q7�آ� ��� ا 8 4� ��ء درا�)�C د�����أ و اC��#+�م �t �� وT'� وم����ا ����8 ا����دن ا�� O+,1 #�د�ا ���4.86+-�#���ة ا�Q!Cد ا�u *-��- 4,�ب ?@ ، �G,#��1 q7'� دي����ا q���ف اsا�� (- q7��7ر ا��C آ� ��ا �� � � @��س ا�(�ا�� �Yد-� �pK�$T ا�oو@ و1+�#� q7�ف ا وا�� -^ =o�ف اsوا�� q7�ا x!�� ، ة�u *-���� �+'+'���ا VK��#�ا ?�@? د (- 86+-�#��� ا�Q!Cد ا�#� 86+-�#���ة ا�Q!Cد ا�u *-��-د و�Q!Cا q+���� دة�����د �'& �t �� ا��#�Y ا���C�1 د���� VK��� �-رز �#T 82s2 ا��1Cد �-�ا�#��VK ا��8 @? ا���Tل �'+�Q آ�� -��ا��� 671* . = ف ا�q7 �1���'+* ا���#�-+86 ا����C�8 ا��6-* -,�< � t� ^+1 �� ا� ي ��Q!sد -) ا�+!�-�T'� �+#-o�آ�د ا �ة ا د�� =�ل ���1+#�� ا��(�t� ^+1 q1 �� ا�(��L وا�#��VK ا���د�� . #��VK ا�� �دة�- �GYوا �-�T�ة -^ ز-^ ا $��داد . آ�ن o� q7�ا���7ر ا ���ة ا�Q!Cد ا�� آ8 و� u *-��- ان �t4s- ?@ y�M� ���WC�1 �-�T�ا �� .��o1دة ا�oوا�� -^ = ف ا�q7 و� Al-Qadisiya Journal For Engineering Sciences Vol. 2 No. ٢ Year 2009 ٢٨٩ Nomenclature A plate width B plate length D flexural rigidity of the plate DSIF dynamic Stress intensity factor EE 21 , young's modulus of steel cylinder and the plate ic G critical strain energy M equivalent mass PO peak overload observed when zerot = P t )( total instantaneous over load R radius of the steel cylindrical S velocity of impacter Thc. plate thickness crack deep correct factor length of crack 2,1 k effect stiffness m,n integer numbers from 1 to ∞. n parameter depended on properties of material p r readies of plastic zone pC r critical plastic zone u, v the length and width of the equivalent area ys σ yield stress σ 2,1 principles stresses β property characterizing take fixing =0.23 υ1,υ2 Poisson ratio for the steel cylinder and the plate α angle of crack front o α decay factor, 2.0=oα (Mario, 1980). α p principle angle of crack front ηξ , location of load Introduction One of the fundamental requirements of any engineering structure is that it should not fail in service and much of the skill of the structural engineer lies in recognizing that there are several possible modes of failure and in guarding against them in his design (Knott, 1976). In the recent years, the important increased to know the remaining service lifetime of mechanical parts that have a crack in it before that part retires from service. This interest increased after catastrophic failures in ships and airplane structures. In order to calculate the stresses around the crack tip, three common methods are introduced for calculating the crack propagation and stress intensity factor such as a- Stress intensity factors (KI, KII, and KIII) associated with the three basic modes of fracture. b- J-integral, this may be defined as a path-independent line integral that measures the strength of the singular stresses and strains near a crack tip. a o c c c o Al-Qadisiya Journal For Engineering Sciences Vol. 2 No. ٢ Year 2009 ٢٩٠ c- Energy release rate (G), which represents the amount of work associated with a crack opening or closure. Impact Loading Model i- Contact loading When the plates impacted in their middle with steel cylindrical of Radius (R) = 10 mm, weight = 75 gm, E = 206.843 G. pa and Poisson's ratio = 0.3 at various impact velocities (8 m/s, 4 m/s, 2 m/s and 1 m/s). The static force caused from impact was calculated by the following Eq.(1) (Zukas,1982). ( ) 53 2 52 2 2 2 1 2 1 4 5 11 3 4                             − + − = M v EE R P o υυ (1) Since the force that resulted from impact is now calculated, the stresses due to impact can be found by dividing it over the area of the contact of the cylinder to the plate in order to reduce the time of the numerical calculation. (Timoshenko,1959) ii- Loading from impact velocity The total response of materials and structures to intense impulsive loading is quite complex because a effective time is very short (Zukas,1982). The empirical form used to describe the impact load as )(t p in Eq. (2). Where the major part of energy is transmitted at contact time divided to two sections, first over load (positive phase) decreases to zero and second goes below zero (negative phase) within a short time as shown in Figure (1) could be taken as (Dharni,2005) Eq. (2) represents the load on the plate from impacted at any time but not all loads in plate at this time. For calculated the total load in the plate using multiple-segment of trapezoidal rule for divided the time for equal segment in positive phase only (Steven,1990). ( ) )/exp()/1( dodt ttttpoP α−−= (2) iii- duration impact and stress in plate The maximum load ( )P t occurs at a time ( ottd 5.0= ) where ( ot ) is the impact duration as shown in Eqs. (3), and (4), (Zukas,1982). (3) (4) The load from impact for any time ( d tt ≤≤0 ) obtained from substitution Eqs. (1) and (4) into Eq. (2) gives the final form to calculated the loading with time Eq.(5). 52 2/1 4 5 94.2           = Mns t o od tt 5.0= Al-Qadisiya Journal For Engineering Sciences Vol. 2 No. ٢ Year 2009 ٢٩١ ( ) ( ) ( )                     −                     −=               5 2 5 2 5 3 5 2 4 5 47.1 2.0 exp 47.1 1 2 4 54 5 Mns tt Mns M s nP t (5) The in plane stresses that caused from impact must be calculated in order to apply it on the crack faces. The following equations Eqs. (6), (7) and (8)) are used in order to calculate the σx, σy and xy τ that caused in the plate due to impact (Bairagi,1986). ( ) ( )υ π π ππ υ ππ σ 2 2 2 2 2 2 242 2 2.0 2 2 2 2 2 242 2 2.0 ,, 1 sinsin116 sinsin116 −                                     +             ⋅      −⋅⋅ −         +             ⋅      − − ⋅−=       −       − B n A m Bu ne B n A m Au me x mnD B ny A mx Smn td t po mnD B ny A mx Smn td t po thcE td t td t tyx (6) ( ) ( )υ π π ππ υ ππ σ 2 2 2 2 2 2 242 2 2.0 2 2 2 2 2 242 2 2.0 ,, 1 sinsin116 sinsin116 −                                     +             ⋅      −⋅⋅ −         +             ⋅      − − ⋅−=       −       − B n A m Au me B n A m Bu ne y mnD B ny A mx Smn td t po mnD B ny A mx Smn td t po thcE td t td t tyx (7) ( ) ( )                   −         +       ⋅      ⋅      − − ⋅⋅−= +       − 2 2 2 2 2 42 2.0 ,, 1 coscos1 16 B n A mu ne xy DABmn B ny m A mx Smn td t po thcE td t tyx πυ ππ τ (8) Where       ⋅      ⋅      ⋅      = B nu A mu B n A m Smn 2 sin 2 sinsinsin πππηπξ Al-Qadisiya Journal For Engineering Sciences Vol. 2 No. ٢ Year 2009 ٢٩٢ Results And Discussion Since the in plane stresses due to low velocity impact are known, they are resolved into components because the crack angle sometimes is oblique. One of the components is perpendicular to the crack face while the other is parallel to it. The parallel component is equal to zero, leading the KII value due to impact is zero, while the perpendicular component causes the KI mode causing an increase in the resultant of KI mode that happens in the plate at the crack tip. Figure (2) and Figure (3) shows the impact force distribution in the center of plate and crack front respectively , for Stainless Steel plate of dimensions 100×100×4.5mm obtained numerically by ANSYS-10. The Dsifs Evaluation The (KI, KII, KIII) are a constant, which gives the magnitude of the elastic stress field; there are called the stress intensity factors. Dimensional analysis chose that KI must be linearly related to stress and directly related to the square root of a characteristic length. When a higher initial KI means a higher initial crack driving force. The SIF is divided to the two types depended on the crack growth. When the amount of unstable crack growth is very limited, the dynamic effect on stress intensity can be neglected. In this case, the dynamic stress intensity factor mast be calculated dynamic KI (Wanhill,1989). There are three types of crack modes, depending on the loading condition (David,1989), the opening mode (DKI), the shearing mode (DKII), and the tearing mode (DKIII), The first mode is the most widely mode that occurs, while the second and the third modes do not occur individually, they occur with combination with mode I. The third mode was neglected in this study because of its small value that can be neglected comparing to the values of the first and the second modes. Eq.(9) is finely form to calculates the magnitude of DSIFs on mode I . (9) Where Figure (4) and Figure (5) shows the stress distribution in the crack front and tip respectively ,plate of dimensions 100×100×4.5mm made from Stainless Steel obtained numerically by ANSYS- 10. Using Eqs.(6-9) to calculated the principle stresses and DSIFs. With this way, the DSIFs were calculated for both the Stainless Steel and Aluminum plates at variable crack angle and variable aspect ratios. Figure (6-9) shows the DSIFs (DKI) and their relation with the crack angle and duration time for a plate of 100×100×4.5mm,crack 7×2 mm in dimensions and made from Stainless Steel. Figure (10) and Figure (11) shows the behavior of DKI with the increase in deep of crack and aspect ratio when the impact velocity is 1m/s . ( ) 2 2 4 1 2 2 2 2 * 88 3 cossin co ao co ao coctKI normal c ππ αα πσ +       + = pppnnormal αασασσσ sincoscos 2 2 11 +== Al-Qadisiya Journal For Engineering Sciences Vol. 2 No. ٢ Year 2009 ٢٩٣ Crack Tip Plasticity (Dugdale Model). Dugdual's analysis assumes that all plastic deformation concentrates in a strip in front of the crack. This type of behavior does indeed occur for a number of materials, but certainly not for all. For applied Dugdale theory Eq.(10), assume of elastic-perfectly plastic behavior (Giacomin,1996) 2 8        = YS p KI r σ π (10) If KIKI C = then 2 8        = YS c pc KI r σ π ; If pcp rr ≥ the crack is propagation A care must be taken in calculating readies of plastic zone value for different impact velocity for determiner the area of crack propagation. Figure (12-15) shows the plastic zone animation relation with the aspect ratio and duration time for a plate of 100×100×4.5mm,crack 7×2 mm in dimensions and made from Stainless Steel. Model Of Crack Growth Fracture mechanics deals with the study of how a crack or flaw in a structure propagates under applied loads. It involves correlating analytical predictions of crack propagation and failure with experimental results or numerical solution. The analytical predictions are made by calculating fracture parameters such as stress intensity factors in the crack region, which was used to estimate crack growth rate. Typically, the crack length increases with each application of some cyclic load, such as cabin pressurization-depressurization in an airplane. Further, environmental conditions such as temperature, extensive exposure, or dynamic loading to irradiation can affect the fracture property of a given material (Ansys,2005). Some typical fracture growth may evolve are: i- Static growth, this state happens in equilibrium condition of crack propagation (stably, slowly, controlled). ii- Quasi static growth, this state happens without kinetic-energy production, the potential energy will gradually approach zero since the fractured pieces obviously are free of stress. iii- Dynamic growth, this state happens with kinetic-energy production. The crack driving force is large that in quasi-static. The dynamic crack growth may be considered in terms of an energy balance. After initiation of energy which increases during crack growth. By the time, the crack has reached a length i a i.e. the total excess energy has converted to kinetic energy in crack tip (Wanhill,1989). Simple expression for the stored kinetic energy is obtainable from the opening displacement of the crack flanks as shown in Eq. (11) Get               −= • • normal normal ai aoE a σ σ ρ 1 (11) Where = • normalσ ( )ppp t αασασ sincoscos 2 2 1 + ∂ ∂ In this section a comparison results obtained from Eq. (11), with the reference (Sheng,2004), which is illustrated in the appendix A. Applied eq. (A-4) for one state only where this equation Al-Qadisiya Journal For Engineering Sciences Vol. 2 No. ٢ Year 2009 ٢٩٤ depended on the energy release rate theory. When a stainless steel plate (100x100x4.5mm), is impacted by steel cylinder at velocity 2 and 8 m/s the crack velocity calculated by Eq. (A-4) see Figure (16), and compare the result with Eq. (11). We noted the good agreement and the average rate error is decreasing when the velocity impact are increasing. Figure (17-19), and Figure (20) show the crack velocity relation with the aspect ratio and duration time for a plate of 100×100×4.5mm,crack 7×2 mm in dimensions and made from Stainless Steel. ENERGY METHOD When a plate element is acted by external loading, the internal fibers of the material body absorb energy in the form of the potential energy and as a result, the body shows deformed shape externally. On removal of the loads, the stored potential energy is converted to kinetic energy so that the body wholly or partially regains its original position or shape. The absorbed potential energy of the internal forces stored within the structural body is often termed as strain energy (SE) orU , and its magnitude is equal to the work done due to the internal forces (in the opposite sense). The potential of the external force V is defined as the work done by the external force (in the opposite sense) during deformation between the initial and final position. The dynamic kinetic energy DKE add to the plate from impacter is, ( 2 2 1 stmass impacter ⋅⋅ ) where the st is velocity of impacter during the time of impact. The total energy in plate 1 Π is give by (Bairagi,1986) Eq.(12). DKEVU ++=Π 1 (12) or ( ) ( ) ( ) ( ) ( )2 .2 2222 1 2 14 8 1 tsmtTW mn tABq tTABDW impmnnmmn +−+=Π π βα (13) By impacting a flat plate when diameter of impacter is large compared to its thickness in a direction perpendicular to its surface, a plane wave can be generated. Before the arrival of release waves from the edges of the plate, the center portion is in a confined state of one-dimensional or uniaxial strain (Loya,2006). The total strain energy for uniaxial strain Eq. (14) ( )( ) ( )υ υυ σ − −+ ⋅=Π 1 211 2 1 2 2 E impacter (14) The total strain energy in Eq. (13) sum as energy in Eq. (14) i.e.( 21 Π=Π ) , solving for calculated normal stress. The energy method is very important for comparison, because depended it on summations of all energy inter the plate from impact, solving Eq. (13), and Eq.(14) for calculated the normal stress on crack faces, and compared with result from airy equation for st. steel plates (100x100x4.5mm) at difference velocity impact. Draw the relation for this tow method, these results is display in Figure (21-25) noted the energy in plates are given good agreement with the airy method but not for all time, because the energy translated from impact to plate not continues to all time, but it reach to constant value. In addition, noted when the impact velocity is increasing the result from energy method is increasing the exactitude. Al-Qadisiya Journal For Engineering Sciences Vol. 2 No. ٢ Year 2009 ٢٩٥ Conclusions The major observation and conclusion from study dynamic analysis, simply supported stainless steel and aluminum cracked plated, under various impact velocities by cylindrical steel are listed as follow: 1. The results of DSIFs and velocity of crack propagation obtained by the building of programs by FORTRAN bower station-90, for impact loading. These results have been obtained by two different ways. First by using classical method, and secondly by energy method. These two ways is gives the some results with percentage error is lowest than (15%). 2. In the case of internal crack the values of dynamic stress intensity factors (DSIFs), is depended on the depth of crack and angle of local (alpha). 3. The duration of time impact is decreasing when the velocity of impact is increasing, and when the young modulus is increasing the duration time decreasing i.e. the duration of time depended on the properties of material. 4. The crack propagation activity at location when maximum DSIFs along the crack front. Also the plastic area is large then compared with a critical plastic area (Dugdale model). 5. The velocity of crack propagation in steel is larger than in aluminum because deferent by young modulus. 6. The velocity of crack propagation in plain stress is larger than in plain strain, i.e. the crack velocity is decreasing when thickness of plates is increasing. 7. The velocity of crack propagation is decreasing when the aspect ratio is increase. In addition, increasing velocity of crack when velocity impact increase, when deep of crack increase the crack velocity increasing. This behavior as applied for path material and plane stress and plane strain. 8. The strain energy method is applied for all velocity impact and gives good agreement when velocity impact increasing more than 20 m/s. Where the percentage error for result between the airy method and energy about (13%). References • • • • • • • ANSYS Release 10.0 Documentation, 2005 A. Jeffrey Giacomin, "Dynamic Fracture Toughness of Polypropylene Reinforced with Cellulose Fiber", 1996. Broek, David, “The Practical Use of Fracture Mechanics”, Kluwer Academic Publishers, 1989. H. L. Ewalds, R. J. H. Wanhill, "Fracture Mechanics", Edward Arnold, 1989. Jonas A. Zukas, Theodore Nicholas and Hallock F. Swift, "Impact Dynamics", John Wiley & Sons, Inc., 1982. J. A. Loya et .al ," Determination of the Dynamic Stress Intensity Factor of a Specimen Under One-Point Bending From Measurement of the Load-Point Displacement", Journal de Physique IV-Proceedings, 2006. Al-Qadisiya Journal For Engineering Sciences Vol. 2 No. ٢ Year 2009 ٢٩٦ • • • • • • J. F. Knott, "Fundamentals of Fracture Mechanics", Butterworth Group,1976. Loke Sworappa R. Dharni," Laminated Architectural Glass Subjected to Blast, Impact Loading", 2005. Mario Paz," Structural Dynamics", Van Nostrand Reinhold Company, 1980. N. K. Bairagi," A Text Book of Plate Analysis", Khanna Publishers, 1986. Sheng Liu et al, "Resistance to Cracking of Stretchable Semiconductor: Speed of Crack Propagation for Varying Energy Release Rate", Materials Research Society, 2004 Steven Cherpy and Raymond, "Numerical Method for Engineers with Applications"1990 S. Timoshenko, S. Woinowsky Krieger, "Theory of Plates and Shells", McGraw-Hill Company, 1959. Appendix A: Speed of crack propagation for varying energy release rate. From reference (Sheng,2004), a simulation equation to calculate the propagation velocity of successive crack as a function of energy release rate. The energy release G averaged over the advancing front of a semi-infinite crack is ( ) ( )βα συπ ⋅⋅ ⋅⋅− = g E h G 22 1 2 (A-1) Where h is the plate thickness, σ is the stress in the plate normal to the crack line, and E and υ are the young modulus and Poisson ratio of the plate, respectively. The function ( ) h L g ⋅ ⋅ = π βα 2 , (A-2) Where S EHh L µ ⋅⋅ = Where H is the substrate thickness, and s µ is the shear modulus of plate. Putting the function (A-2) into (A-1), we obtain as energy release rate G. 2 2 1 σ µ υ ⋅ ⋅⋅− = s EHh E G (A-3) Crack propagation velocity V as a function of energy release rate G was given, Al-Qadisiya Journal For Engineering Sciences Vol. 2 No. ٢ Year 2009 ٢٩٧             −⋅= 1sinh ic G G VoV β (A-4) Figure (1) Behavior of impact (Dharni, 2005) Figure (2) Impact load in Ansys Figure (3) Crack opening in Ansys Figure (4) Stress distributed crack Al-Qadisiya Journal For Engineering Sciences Vol. 2 No. ٢ Year 2009 ٢٩٨ 0.0E+00 5.0E+07 1.0E+08 1.5E+08 2.0E+08 2.5E+08 3.0E+08 3.5E+08 4.0E+08 0.E+00 2.E-06 4.E-06 6.E-06 Duration of time Sec. D y n a m ic s tr e e s i n tu n s e ty f a c to r s D S IF ( p a .m ^ -. 5 ) Alpha=0(Calculated) Alpha=0(Ansys) Alpha=45(Calculated) Alpha=90(Calculated) Figure (5) DSIF’s in Ansys Figure (6) Impact Velocity 1 m/s 0.0E+00 5.0E+07 1.0E+08 1.5E+08 2.0E+08 2.5E+08 3.0E+08 3.5E+08 0.E+00 1.E-06 2.E-06 3.E-06 Duration of time Sec. D y n a m ic s tr e s s i n te n s it y f a c to r s D K I (p a . m ^ -. 5 ) Alpha=0(Calculated) Alpha=0(Ansys) Alpha=45(Calculated) Alpha=90(Calculated) 0.0E+00 5.0E+07 1.0E+08 1.5E+08 2.0E+08 2.5E+08 3.0E+08 3.5E+08 4.0E+08 0.0E+00 5.0E-07 1.0E-06 1.5E-06 Duration of time Sec. D y n a m ic s tr e s s i n te n s e ty f a c to r s D K I (p a . m ^ -. 5 ) Alpha=0(Calculated) Alpha=0(Ansys) Alpha=45(Calculated) Alpha=90(Calculated) Figure (7) Impact Velocity 2 m/s Figure (8) Impact Velocity 4 m/s Al-Qadisiya Journal For Engineering Sciences Vol. 2 No. ٢ Year 2009 ٢٩٩ 0.0E+00 5.0E+07 1.0E+08 1.5E+08 2.0E+08 2.5E+08 3.0E+08 3.5E+08 4.0E+08 0.E+00 2.E-07 4.E-07 6.E-07 8.E-07 Duration of time Sec. D y n a k ic s tr e s s i n te n s e ty f a c to r s D K I (p a . m ^ -. 5 ) Alpha=0(Calculated) Alpha=0(Ansys) Alpha=45(Calculated) Alpha=90(Calculated) 0.0E+00 5.0E+07 1.0E+08 1.5E+08 2.0E+08 2.5E+08 3.0E+08 3.5E+08 0.E+00 2.E-06 4.E-06 6.E-06 8.E-06 Duration of time Sec. D y n a m ic s t r e e s i n t u n s it y f a c t o r s D S IF ( p a . m ^ - .5 ) Deeb=1mm Deeb=2mm Deeb=3mm Deeb=4mm Deeb=5mm Figure (9) Impact Velocity 8 m/s Figure (10) Impact Velocity is 1m/s 0.0E+00 5.0E+07 1.0E+08 1.5E+08 2.0E+08 2.5E+08 3.0E+08 3.5E+08 0.E+00 2.E-06 4.E-06 6.E-06 8.E-06 Duration of time Sec. D y n a m ic s tr e s s i n te n s e ty f a c to r s D K I (p a . m ^ -. 5 ) AR=1(Calculated) AR=1(Ansys) AR=1.5(Calculated) AR=2(Calculated) AR=2.5(Calculated) AR=3(Calculated) AR=3(Ansys) 0.0E+00 5.0E-03 1.0E-02 1.5E-02 2.0E-02 2.5E-02 3.0E-02 3.5E-02 4.0E-02 4.5E-02 0.0E+ 00 2.0E- 06 4.0E- 06 6.0E- 06 8.0E- 06 1.0E- 05 1.2E- 05 Duration of time Sec. P la s ti c z o n e m AR=1 AR=1.5 AR=2 AR=2.5 AR=3 Dugdale Theory Figure (11) Impact Velocity is 1m/s Figure (12) Impact Velocity is 1m/s Al-Qadisiya Journal For Engineering Sciences Vol. 2 No. ٢ Year 2009 ٣٠٠ 0.0E+00 5.0E-03 1.0E-02 1.5E-02 2.0E-02 2.5E-02 3.0E-02 3.5E-02 4.0E-02 4.5E-02 0.0E+00 1.0E-06 2.0E-06 3.0E-06 4.0E-06 Duration of time Sec. P la s ti c z o n e m AR=1 AR=1.5 AR=2 AR=2.5 AR=3 Dugdale Theory 0.0E+00 1.0E-02 2.0E-02 3.0E-02 4.0E-02 5.0E-02 6.0E-02 7.0E-02 0.0E+00 4.0E-07 8.0E-07 1.2E-06 Time Sec. P la s ti c a re a ( m ) AR=1 AR=1.5 AR=2 AR=2.5 AR=3 Dugdale model Figure (13) Impact Velocity is 2m/s Figure (14) Impact Velocity is 4m/s 0.0E+00 2.0E-02 4.0E-02 6.0E-02 8.0E-02 1.0E-01 1.2E-01 0.E+00 1.E-07 2.E-07 3.E-07 4.E-07 5.E-07 Time Sec. P la s ti c a re a ( m ) AR=1 AR=1.5 AR=2 AR=2.5 AR=3 Dugdale model Figure (15) Impact Velocity is 8m/s Figure (16) Velocity Compared Al-Qadisiya Journal For Engineering Sciences Vol. 2 No. ٢ Year 2009 ٣٠١ 0.0E+00 2.0E-01 4.0E-01 6.0E-01 8.0E-01 1.0E+00 1.2E+00 1.4E+00 1.6E+00 0.0E+00 2.0E-06 4.0E-06 6.0E-06 Time Sec. C r a c k v e lo c it y ( m /s ) AR=1 AR=1.5 AR=2 AR=2.5 AR=3 0.0E+00 2.0E-01 4.0E-01 6.0E-01 8.0E-01 1.0E+00 1.2E+00 0.0E+00 5.0E-07 1.0E-06 1.5E-06 2.0E-06 Time Sec. C ra c k v e lo c it y ( m /s ) AR=1 AR=1.5 AR=2 AR=2.5 AR=3 Figure (17) Velocity impact is 1 m/s Figure ( 18) Velocity impact is 2 m/s 0.0E+00 2.0E-01 4.0E-01 6.0E-01 8.0E-01 1.0E+00 1.2E+00 0.0E+00 5.0E-07 1.0E-06 1.5E-06 2.0E-06 Time Sec. C ra c k v e lo c it y ( m /s ) AR=1 AR=1.5 AR=2 AR=2.5 AR=3 0.0E+00 1.0E-01 2.0E-01 3.0E-01 4.0E-01 5.0E-01 6.0E-01 7.0E-01 8.0E-01 9.0E-01 1.0E+00 0.0E+00 5.0E-07 1.0E-06 1.5E-06 Time Sec. C ra c k v e lo c it y ( m /s ) AR=1 AR=1.5 AR=2 AR=2.5 AR=3 Figure (19) Velocity impact is 4 m/s Figure (20) Velocity impact is 8 m/s Al-Qadisiya Journal For Engineering Sciences Vol. 2 No. ٢ Year 2009 ٣٠٢ 0.0E+00 5.0E+10 1.0E+11 1.5E+11 2.0E+11 2.5E+11 3.0E+11 3.5E+11 4.0E+11 4.5E+11 0.E+00 2.E-05 4.E-05 6.E-05 Duration of time Sec. S 1 n (D y n a m ic n o r m a l s tr e s s ) p a . energy method airy method 0.0E+00 1.0E+11 2.0E+11 3.0E+11 4.0E+11 5.0E+11 6.0E+11 7.0E+11 8.0E+11 9.0E+11 1.0E+12 0.E+00 1.E-05 2.E-05 3.E-05 4.E-05 5.E-05 6.E-05 Duration of time Sec. S 1 n ( D y n a m ic n o r m a l s tr e s s ) p a . Energy method Airy method Figure (21) Velocity of impact =1 m/S Figure (22) Velocity of impact =2 m/s 0.0E+00 5.0E+11 1.0E+12 1.5E+12 2.0E+12 2.5E+12 0.E+00 1.E-05 2.E-05 3.E-05 4.E-05 5.E-05 6.E-05 Duration of time Sec. S 1 n ( d y n a m ic n o r m a l s t r e s s ) p a . Energy method Airy method 0.E+00 1.E+12 2.E+12 3.E+12 4.E+12 5.E+12 6.E+12 0.E+00 1.E-05 2.E-05 3.E-05 4.E-05 5.E-05 6.E-05 Duration of time Sec. S 1 n ( D y n a m ic n o r m a l s tr e s s ) p a . Energy method Airy method Figure (23) Velocity of impact =4 m/s Figure (24) Velocity of impact =8 m/s Al-Qadisiya Journal For Engineering Sciences Vol. 2 No. ٢ Year 2009 ٣٠٣ 0.0E+00 2.0E+12 4.0E+12 6.0E+12 8.0E+12 1.0E+13 1.2E+13 0.E+00 1.E-05 2.E-05 3.E-05 Duration of time Sec. S 1 n ( D y n a m ic n o r m a l s tr e s s ) p a . Energy method Ansys Figure (25) Velocity of impact =20 m/s