30120130406015

International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –

6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 6, November - December (2013) © IAEME

122

THE EFFECT OF DYNAMIC IMPACT LOADING WITH COMBINED

BUCKLING STRESSES ON THE DYNAMIC SURFACE CRACK

PROPAGATION IN PLATES SUBJECTED TO THERMAL STRESSES

Dr. Fathi Al-Shammaa1, Khawla A .AL-Zubaidy

2

1Asst. Prof., Department of mechanical Eng. Baghdad University / Iraq 2Lecturer, Department of mechanical Eng. Baghdad University / Iraq

ABSTRACT

When plates subjected to the application of large in-plane loads either compressive or shear

they buckle in a non-linear behavior which is characterized by increase of the displacements

associated with the small increment of the loads. In this work a theoretical study of the dynamic

growth of a crack in plates with mixed complex boundary conditions under in plane loading causes

shear, compression and combined shear and compression buckling subjected to low velocity impact

at the edge of crack in the middle of simply supported plate under various thermal condition.

Two methods of approximate analytical solution using in the first one Airy stress function,

equilibrium equation and large deflection plate theory to find the expression of the deflection and

the second method using energy equation which modified for including the impact loading with

thermal stresses to find dynamic crack propagation. The dynamic stress intensity factors (SIF),

velocity of dynamic crack propagation with deep of crack normal to the crack face have been

calculated using numerical package (Ansys-10) to investigate the stress and the values of dynamic

stress intensity factor at the crack tip by full transient dynamic analysis in three dimensional

elements.

INTRODUCTION

Unlike columns, the plate failure does not occur when the critical buckling load is reached.

Plates continue to resist the in-plane load for in excess to the critical load before failure, thus the post

buckling behavior of plates plays an important role in determining the ultimate carrying capacity.

For thin plates (large values of length to thickness ratio) made from a typical strain hardening

material with yield stress by, instability occurs at an average stress �� that is much less than the

yield stress, now the complicated of this case is the pressence of the Impact loading acting on the tip

of a crack in the middle of a thin plate with different thermal stress which are commonly found in

INTERNATIONAL JOURNAL OF MECHANICAL ENGINEERING

AND TECHNOLOGY (IJMET)

ISSN 0976 – 6340 (Print)

ISSN 0976 – 6359 (Online)

Volume 4, Issue 6, November - December (2013), pp. 122-137

© IAEME: www.iaeme.com/ijmet.asp Journal Impact Factor (2013): 5.7731 (Calculated by GISI) www.jifactor.com

IJMET

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123

aero engine components such as vanes. However existing solution are based on constant thermal

stresses through the plate structure.

Extensive work has been carried out to determine expressions for critical buckling loads with

bending and shear C.A. Farther stone et al (1) and (2) ,the experimental results were used to examine

whether or not finite element analysis can be used as on alternative to determine collapse load and

post buckling behavior.

In C.A. Farther stone (3) two approaches are used ,a linear bifurcation buckling analysis were

carried out to determine the bifurcation load of the structure and the second method is a fully non-

linear analysis have been performed with deflections geometric imperfections and plasticity properly

modeled Z.y. zhang et al (4) investigated low velocity impact induced non - penetration damage in

pultruded glass fiber reinforced polyester (GRP) composite material using an instrumented falling

weight impact test machine with a chisel shaped imparter while Yung - Tze chen (5) studied the

crack propagation of linear elastic cracked plates using analytical solution to uniform static loading

with simply supported boundary condition to predict the path and life for crack growth in a plate

under mired mode I- Mode II with complex loading , L.F . Martha et al (6) have investigated the

velocity of crack growth numerically and experimentally

In this work the dynamic crack growth of a crack in plate under buckling shear and bending

with Impact loading and thermal stresses have been studied analytically and numerically to compare

the results achieved.

FUNDAMENTAL SOLUTION

In the first, it must be considered that two kind of deflection have been calculated in the

theoretical analysis:-

1 - w0 initial deflection that induced in the plate from the environmental to and boundary conditions

( like the difference in the temperature ,the constraint that prevented the plate from extension, etc).

2 - w1 additional deflection that causes from the in plane and lateral loads applied at the plate ( as

bending and combined longitudinal and shear buckling ) then the total deflection is:

�� ……… . . �1�

The general governing equation for deflection of plates subjected to lateral load (p) is :

�� ……… . �2�

where p is the lateral force due to impact

�� …… .. �3�

Substituting eq.1 in eq.3 for the initially curved plate the governing equation will be of the form :-

�� 1∆�� 2�� !… . �4�



124

Knowing that, the influence of the initial curvature on the total deflection of the plate is

equivalent to the influence of some fictitious lateral load of intensity pf expressed as:-

�# � �� 2�� … . . �5�

As mentioned before, the thermal effect will causes combined three kind of a) Bending and

buckling produced by direct compression form the constraint against thermal expansion and can be

obtained from the values of buckling parameter for all edges simply supported with different aspect

ratio which given in table (1) and is given by [7]:-

��% � &'(�)12�1 * +�� ,-./� * * * �6�

Where:-

Kb=Bending buckling stress parameter

Table (1) bending buckling parameter with aspect ratio of all edges simply supported

a/b 0.5 0.6 0.667 0.75 0.8 0.9 1 1.5 2

Kb 25.6 24.1 23.9 24.1 24.4 25.6 25.6 24.1 23.9

Let the initial deflection represented by

�� 12344567∞

38(�9 5677(�.

∞

2 * * * *�7�

Where: ;23=The initial deflection at the center of the plate for thermal condition 123 � <∆=.>� if we

assume the initial curve is very small where α = thermal expansion of the plate for the condition of

applying only Nx and putting Nxy , P and Ny equal to zero gives

�� 1231 * ?44567∞

38(�9 567 7(�.

∞

2 * * * *�8�

Where

? � ��(�A9� B1 � C9.D�E�



125

And due to shear only then Nxy have a value and put Nx , p and Ny equal to zero gives:

�F � 9F44567∞

38(�9 567 7(�.

∞

2 * * * *�9�

� 12344H1� 2�� A(��9� � .��9I.I JK78(�9 JK7 7(�. * 2�� A L∞

3∞

2

. M567 8(�9 567 7(�. N * * * �10�

For the case of combined shear and direct compression by adding eq. 8 and eq.10 gives

�F� � 12344H 11 * ? � P1 2�� A(��9� � .��9I.I JK78(�9 JK7 7(�. * 2�� A QL∞

3∞

2

. M5678(�9 567 7(�. N * * * �11�

Now the pressure distribution due to Impact can be derived from the Impact low velocity [8] as

J� � 2.94 � 548�7�RS2�.T!�T * * * �12�

Where to is the total impact duration and the pressure distribution at the contact region is:-

��U�, R�, J� � ��J� �1 * U��W�� * R��W��! * * * �13�

Where

��J� � 3758(XY8� ,1567 (JR2.941/ �Z * * * �14�

The final equation of representing the deflection in case of study the impact compression with

shear will be

�S�F � �� 16[�J�(\A . 445678(]K 567 7(7^ 5678(U�9 567 7(R�. 5678(�9 567 7(�.87 _,8�9� � 7�.�/� * ��A C89(D� * 2 �� A(� C879.D`

∞

3∞

2 * *�15�



126

Substitute the value of W0 in eq. 15 gives:-

�S�F � a123 � 16[�J�(\A b. b44 5678(cK 567 7(7^ 5678(U�9 567 7(R�.87 _,8�9� � 7�.�/� * ��A C89(D� * 2 �� A(� C879.D`

∞

3∞

2 deeef C5678(�9 567 7(�. D * *�16�

b) Shear buckling :-For the case of shear force only then the critical shear buckling stress will be:-

;�� &F(�)12�1 * +�� ,-./� * * * �17�

Where

Ks=shear buckling stress parameter (from table 2)

Table(2) values of shear buckling parameter(Ks)for simply supported plate

A/B 1 1.2 1.4 1.5 1.6 1.8 2 2.5 3 4

Ks 9.34 8 7.3 7.1 7 6.8 6.6 6.1 5.9 5.7

c) combined bending and shear buckling:-

This can be done when there is lateral loud with shear on the edge of the plate then the critical

buckling stress will be: �� g��2'(�. )12�1 * +�� ,-./� * * * *�18�

Where

Kcomb= buckling parameter for combined shear and direct compression (shown in table 3)

depending on the ratio(σ/τ)

Table(3) Buckling parameter of combined shear and direct compression

σ/τ 0 0.5 1.0 1.5 2.0

kcomb 14.71 7.09 4.5 3.24 2.51

The stresses due to Impact, shear and direct compression with thermal effect can be found by

substituting the value of W and P(t) and taking the maximum stress induced at the surface �X � h�� in the following equations:-



127

�� )i�1 * +�� aj� kj�� + j�kj�� l � � mn�opq� Mrq sr q � + rqsr�q N……………….. (19 )

;� � mn�tp� Mrq sr�r N

Gives:

�� (� )u2�1 * +��44v�89�� +�7.��w v123 � 67�F(x[8�W �1sin |1|R �⁄2.94 ~ � a FS3�� FS3�� FS3�� FS3��23v��q�qt�q�q�qo�� qo��q � �� lw � 5672�� 567 3� � �……….. (20 )

� � (� )u2�1 * +��44v�7.�� +�89��w v123 � 6 7�F(x[8�W �1sin |1|R �⁄2.94 ~(\. A �

H 5678(wK 567 7( ^ 5678(U�9 567 7(R�.87v�8�9� � 7�.�� * ��A � 8(9 �� * 2��^(� C 879. DwLw C 5678(�9 567 7(�. D��21�

;� � (� )u2�1 � +�44�879.� v 123 �67�F(x[8�W �1sin (;R2.94 ~ � �Z(\. A

H 5678(wK 567 7(̂ 5678(U�9 567 7(R�.87v�8�9� � 7�.�� * ��A � 8(9 �� * 2��^(� C 879. DwLw C ��58(�9 ��5 7(�. D��22�

The principal stresses can be calculated for various times:-

� � �� 2 � �� * ��2 �� ;� �

�� 2 � �� * ��2 �� ;� �

�Y � 12 KW�JK7 �� * ��2 �

Where �� = angle of principal stress.



128

Now if we take the condition in fig(1) considering that ơ = ơ1, ʎ ơ = ơ2 , and θp=α , then the

stresses and dynamic intensity factor will be:-

�� 5 �� M1 * 567 �� 567 I�� N * �� 567 �� M2 � ��5 �� 5 I�� N+��1 * 1��52�….(23 )

� � g �2�� 5 �2 B1 � 567 �2 567 3�2 E � g �2�� 567 �2 B��5 �2 ��5 3�2 E ;� � �� M567 �� 5 �� 5 I�� N � �� 5 �� M1 * 567 �� 567 ��N……..( 24 )

�3 � ��5�1 , ; � �567 1 cos 1 , g � �3 √(K, g � ;√(K

Knowing that when the crack has elliptical shape specially in think plates correction factor

(¤�)should be used [8] this factor has a magnitude depend on the geometry of the crack, the value of

this factor can be shown in table from [9] so that kІ and kІІ will be:-

& � ¤��(K� ¥ � ,K�¥ / , & � ¤�;�(K� ¥ � ,K�¥ /

The dynamic crack growth may be considered in terms of energy balance, and the excess

energy can be expressed as

U� � ¦ �§ * ¨��K � *¨�K6 * K�� ¦ �©qªm«�opqªSª�ªSª� �K ………… (25 )

¬ U� � (��2)′ �K6 * K��K6 � K� * 2�………�26 �

Where )′ � )1�1 * +��

From the opening displacement of the crack flank as[9]

� 2�) ��KS� * �� ¤ �KS)′ 6¥ k® JKg® � � ¤K K7� ¤ � 2��1 * ¤�� ¬ �R�J � ′ � ¤) ��˙KS � S��……… . . �27 �

The kinetic energy | � � A ¦ ¦¯˙�°�� A ¦ ¦ ��qm�q ��K��……… . . �28 �

Equating the strain energy with the kinetic energy:-

(��2m′ �KS * K��KS � K� * 2� � 12 A ¤�)′� ±±��˙K * ��… . . �29�



129

Since and a are not function of x and y and both ��, �� ., � .� will be order �567� 2�� 567� 3� � � then :-

4(��)′A¤� � KS * K�� KS � K� * 2� � 2��K� * 2K�K� � K��² …… . �30�

K. � K� �.� � 2³ ()′A¤2 �KS * K��KS � K� * 2�…… �31� K� � K� ©�©� � 2� �m′�´�� KS * K��………(32)

Where , � , represent the velocity of crack propagation from J� � ,

Fig (1): represent the principal stresses � � � , �� µ� for a plate with crack 2a

NUMERICAL ANALYSIS

In the complicated case of buckling, direct and thermal stresses, approximate numerical

methods are the approaches that can be employed for the solution of practically important plate

problems which could be used to compare with the exact theoretical analyses and provide an

understanding of physical plate behave under an applied loading.

Finite element method is based on concept that one can replace any continuum by an

assemblage of simple shaped element with well defined force displacement and material

relationships



130

Form Fig(2 ),one can note that there is affine mesh of the element at the crack tip and near it

because this is the region of study at which the stresses distribution is very useful , from the nodes

drawn generate the elements (solid 95 element) can be used have Fig(3) shows the generation of

elements around the crack and all the plate.

Fig (2): applying loads on plate under thermal condition using (ANSYS10)

Fig (3): PHOTO from solution movie using ANSYS10



131

RESULTS AND DISCUSSION

Fig. (4) shows the behavior of variation of the critical buckling load with respect ratio for flat

plates under the action of combined shear and bending forces to verification the analytical method

and to find the percentage of error between the practical experiments by Featherstone 1998 and the

analytical values achieved by this study which is between (1.83 to 3.5%) this percentage of error

decreases as the aspect ratio tend to 1.5.

Fig. (4) Comparison of analytical and experimental results of AL flat plate

Figure (5) shows the variation of time duration of contact ,with velocity of impact for two

materials (AL and St. steel) the time duration decreases as the velocity of Impact increase and the AL

plate takes more time duration of impact because its hardness is less than that of St. steel so the

impact go deeper in AL plate for the same impact velocity.

Fig. (5) Variation of time duration with velocity of AL and St. steel with aspect ratio=1

0

100

200

300

400

500

600

700

800

0 1 2 3

crit

ica

l b

uck

lin

g l

oa

d N

Aspect ratio

al flat analytical

results

al flat featherston test

0

0.000005

0.00001

0.000015

0.00002

0.000025

0.00003

0.000035

0.00004

0.000045

0.00005

0 5 10 15 20 25 30 35

tim

e d

ura

tio

n (

se

c)

velocity (m/s)

steel time duration

AL time duration



132

In fig(6) it can be shown that from the relation between KI and impact velocity, the graduate

increases with impact velocity for a range of temperature while when failure takes place a sudden

increase in values of KI appears.

Fig. (6) Effect of impact velocity on KI for different thermal condition of St. steel plate

The same effect has been shown in fig (7) for the relation between KII with the impact

velocity but the increase in KI is more effective than KII for the same temperature so that mode 1 is

more effect in the crack propagation than mode 2.

Fig. (7) Effect of impact velocity on KII for different thermal condition of St. steel plate

0

0.5

1

1.5

2

2.5

3

3.5

0 5 10 15 20 25

KI(

Mp

a√

m)

impact velocity (m/s)

300°c

400°c

500°c

600°c

0

0.5

1

1.5

2

2.5

3

0 5 10 15 20 25

KII

(Mp

a√

m)

impact velocity (m/s)

300°c

400°c

500°c

600°c



133

In fig (8) and (9) it can be shown that in AL plate the same effect of temperature on KI and

KII are the same which denotes that when the material become more ductile then the effect of KI and

KII on the crack propagation are the same which different than in steel.

Fig. (8) Effect of impact velocity KI for different thermal condition of AL plate

Fig. (9) Effect of impact velocity KII for different thermal condition of AL plate

Fig (10) and (11) shows the effect of velocity of impact on (� , ��) for AL and steel with

high temperature. The variation of (�) has been effected more than (��) which means that in high

temperature the crack will grow in the mode 1 crack propagation. Also Fig (12) shows the variation

of the dynamic crack growth with time duration of impact for AL and steel with the same thickness

for low and high temperature, the dynamic crack growth in the steel is greater than that in AL this is



134

because the modulus of elasticity is effected by two factors, the rate of loading which effected the

rate of stress applied and second the rate of temperature distribution which effected the rate of

thermal stress induced in the plate.

Fig.(10) Effect of velocity of impact on the principal stresses variation with time of AL plate using

Ansys 10

Fig.(11) Effect of velocity of impact on the principal stresses variation with time of st plate using

Ansys 10

0

10

20

30

40

50

60

70

80

90

0 5 10 15 20 25 30 35

d σ

/dt

(Mp

a/s

ec)

velocity of impact m/s

0

10

20

30

40

50

60

70

80

0 5 10 15 20 25 30 35

dσ/

dt(

mp

a/s

ec)

velocity of impact m/sec

σ1

σ2



135

Fig.(12): Comparison of Dynamic Crack Growth of AL and Steel Under High Temperature

CONCLUSION

In this study a new analytical solution has been derived to predict the dynamic crack growth

in the steel and Aluminum plate under the effect of temperature and combined bending and shear

buckling with direct impact loading. A numerical method also has been obtained using (Ansy 10)

with the same steel and Aluminum plates to compare the results with analytical one.

The results show that the value of KI has been affected with temperature than KII and

dynamic crack growth for steel is higher than for Aluminum at higher temperature.

REFERENCES

1. C.A. Farther stone and C. Ruiz “Buckling of flat plates under bending and shear “journal of

mechanical Eng. Science. Vol.212,P.249.

2. C.A. Farther stone and C. Ruiz “Buckling of curved plates under combined shear and

compression “journal of mechanical Eng. Science. Vol.218, -.183.

3. C.A. Farther stone, “the use of finite element analysis in the examination of instability in flat

plants and curved panels under compression and shear journal of non-linear mechanics,

(2000), uk.

4. Z.Y. zhang and M.O.W. Richardson “Low velocity Impact induced damage evaluation

and its effect on the residual flexural properties of pultvuded GRP Composites” paper

University of Portsmouth,( Hampshire Po1 3D ), 2006, uk.

5. Yung-Tze chen” crack propagation of cracked plates paper, sinotech Engineering consultants

Inc.2003. Taiwan.

0

5

10

15

20

25

30

35

40

45

50

0 0.1 0.2 0.3 0.4 0.5

d a

/dt

(m/s

ec)

time duration fraction

steel

AL



136

6. L.F. Martha , A.C.O Miranda, M.A. Meggiolaro and J.T.P. Castro ”path and life predictions

under mixed mode І Mode ІІ complex loading ”Brazilian Sinotech of mechanical. Science

Engineering”,2007

7. Stephen P. Timoshenko, “Theory of elastic stability”, Book, Mc Graw-Hill book company,

1936 , u.S.A.

8. Jonas A. Zukas, Theodore Nicholas, Hallock. F. Swift, Longin B. Ggreszczuk and Donald R.

Curran” Impact Dynamics ”,Book, Wicly interscience Publication, 1982ز

9. H.L.Ewalds and R.J.H. Wanhill “Fracture Mechanics”, Book, Edward Inc. publishers, 1984,

U.S.A.

10. Akash.D.A, Anand.A, G.V.Gnanendra Reddy and Sudev.L.J, “Determination of Stress

Intensity Factor for a Crack Emanating from a Hole in a Pressurized Cylinder using

Displacement Extrapolation Method”, International Journal of Mechanical Engineering &

Technology (IJMET), Volume 4, Issue 2, 2013, pp. 373 - 382, ISSN Print: 0976 – 6340,

ISSN Online: 0976 – 6359.

11. Manjeet Singh and Dr. Satyendra Singh, “Estimation of Stress Intensity Factor of a Central

Cracked Plate”, International Journal of Mechanical Engineering & Technology (IJMET),

Volume 3, Issue 2, 2012, pp. 310 - 316, ISSN Print: 0976 – 6340, ISSN Online: 0976 – 6359.

12. Dr. Yadavalli Basavaraj and Pavan Kumar B K, “Modeling and Analysis of Base Plate for

Brake Spider Fixture by Fem using ANSYS Software”, International Journal of Mechanical

Engineering & Technology (IJMET), Volume 4, Issue 5, 2013, pp. 26 - 30, ISSN Print:

0976 – 6340, ISSN Online: 0976 – 6359.

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