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viscoplastic solution for a thick-walled spherical container subjected to impulsive loading
Int. 1. ImpactEngngVol. 6, No. 1, pp. 23-34, 1987 Printedin Great Britain
0"/34-743X/87$3.00+0.00 PergamonJournalsLtd.
A DYNAMIC ELASTO/VISCOPLASTIC SOLUTION FOR A T H I C K - W A L L E D S P H E R I C A L C O N T A I N E R SUBJECTED TO IMPULSIVE LOADING TOSHIAKI HATA Faculty of Education, Shizuoka University, Shizuoka City, 422, Japan
(Received 30 November 1985; and in revised form 5 December 1986) Smnnmry--A general method for the solution of dynamic elasto/viscoplastic solid problems is presented. This method is an extension of ray theory to cater for dynamic elasto/viscoplastic deformations. The method reduces the elasto/viscoplastic problem to a sequence of elastic problems with initial strains. The solution of this problem is determined by using four displacement functions. Using the foregoing method, the solution is derived for the dynamic elasto/viscoplastic behaviour of a thick-walled spherical shell subjected to internal impact load. The numerical results show how the dynamic stresses in a sphere with viscoplastic properties vary with time.
NOTATION n r S Tn T U I a, b CL kp e
•/2 P r s# t .~j % ~y
k,t~ v p
surface normal position vector strain dyadic surface traction stress dyadic displacement vector idemfactor inner and outer radii of a sphere, respectively velocity of longitudinal wave yield stress strain second invariant of the deviatoric stress tensor pressure radial co-ordinate deviatoric stress time poles viscoplastic constants stress Lame's constants Poisson's ratio density
BP,~ p,
×P ~s,~s
plastic-displacement functions elastic-stress functions
subscripts i,j refers to i(i= 1,2,3) and j(j= 1,2,3) directions r,0 refer to radial and tangential directions, respectively i,o refer to components of internal and external surfaces, respectively superscripts ' refers to elastic components " refers to plastic components s,p refer to elastic and plastic components, respectively differentiation with respect to time
INTRODUCTION
In several fields of engineering and science, there are requirements for structures, such as blast chambers and nuclear reactors, which must be able to withstand the effects of 23
24
TOSHIAK1 HATA
internally applied high transient pressures. Since these structures may have to withstand static, as well as transient pressures, they are usually constructed in the form of thickwalled pressure vessels, such as spheres or cylinders. In designing such structures, one must not solely rely on static pressure design criteria, but must also take into consideration the dynamic response to the high transient pressure [1-3]. The simplest analytical problem relating to the design of these structures is that of the response of a thick-walled spherical shell subjected to a spherically symmetric transient pressure pulse which acts on the inner and external surfaces. One important conclusion arising from the static analysis of a thick-walled elastic, spherical shell is that the radial stress in the shell is always compressive, and less in magnitude than the applied internal pressure. However, this may not be the case when the shell is subjected to dynamic loadings. It is generally understood that when a spherical wave, which is generated by a dynamic pressure at the inner surface, is reflected at a traction free outer boundary the radial stress may change from compression to tension as a result of the reflection. An elasto/viscoplastic material [4] is assumed to be one with distinct viscous properties only in plastic regions, and which is ideally elastic. In this case, the theoretical assumption that a solid undergoing dynamic loading can be rate-dependent under plastic deformations, but rate-independent in the elastic range appears to be fairly generally accepted for many metals [4]. Materials which exhibit this type of behaviour are usually classified as elasto/viscoplastic. In the following analysis the sphere is assumed to be an elasto/viscoplastic one. Here it is very important to obtain an accurate elastic solution because the accuracy of the elasto/viscoplatic analysis depends on the accuracy of the elastic predictions. There are a few types of dynamic solution for thick elastic spherical shells in the literature: one group is a numerical analysis, which contains FEM, the finite-differential analysis and the boundary-element method, and the other group contains theoretical analyses, which includes normal mode analysis, the method of characteristics and the ray method. The numerous wave problems concerning elasto/viscoplastic media are solved [4] by using the method of characteristics. The accuracy of the solution of characteristics depends on the mesh size which is used for the numerical integration. But in the analyses of dynamic response of an elastic thick-walled spherical shell the method of ray theory [5] represents the wave motion exactly, because the ray integral represents the wave motions exactly. This method is quite adequate for the analysis of initial dynamic response of the solid. In this paper the method of ray theory will be extended to the dynamic elasto/plastic deformation of a sphere subjected to a dynamic loading at the inner and external surfaces. In the calculation of plastic strains the approximate method of solution is employed, in which the plastic-strain distribution is developed in small increments of time. During a time interval, the stresses and strains vary continuously owing to the presence of plastic properties due to the surface forces. The incremental stresses of the first step are added to the stresses which exist at the beginning of the time interval. This result represents the first approximation to the stress distribution which exists at the end of this interval. The stress distribution in the next time increment is determined as before. The solution of the problem may be determined successively by the foregoing procedure, which is fully illustrated in Ref. [6]. In this case, the stress-strain relation is assumed to be elastic during the unloading process. In the procedure, the incremental stress problems are reduced to the boundary-value problems similar to the welt-known thermoelastic analysis. Then the incremental plastic stresses are determined by using four displacement functions similar to the NeuberPapkovitch stress functions [7]. We derive the solution for a dynamic elasto/viscoplastic problem of a thick-walled spherical shell subjected to an internal impact load. The numerical results show how the dynamic stresses in a sphere having viscoplastic properties vary with time.
A dynamicelasto/viscoplasticsolution
25
FORMULATION OF PROBLEM The constitutive equations and field equations for a solid with plastic deformation are presented in this section. The solution of problem requires the determination of the dynamic stress tensor T and strain tensor S in the region 3~ with boundary F. The stress-strain relations may be expressed as follows:
T=2~S'+XlS'II,
(1)
where S' represents the elastic-strain tensor and I, h and ~ are idemfactor and Lame's constants, respectively, and the trace of a dyadic S" is represented by Is'q. At any time, the total strains consist of a sum of elastic and plastic strains, S=S'+S",
(2)
T=XISII+20.S-[XIS"II+2~S"].
(3)
where S" is the plastic tensor. Substitution of S' into (1) yields
The equation of motion without body forces may be written as V.T=pi~ in Z.
(4)
Tn=n.T on F.
(5)
The surface traction Tn is given by
Substituting equations (3) into (4) and (5), we obtain hV[SI + 21xV'S-[~vls"l + 21xv.s"] =pii
(6)
Tn= hnIS[ + 2p,n.S-[hn[S"[ + 2 p,n.S"].
(7)
The strain-displacement relations are
S=(Vu+uV)/2.
(8)
Substitution of equations (8) into (6) yields the equation of motions as follows: V.Vu+
1 1-2v
VV.u=pi~+2V.S~+2
l+v 1-2v
V.S"s,
(9)
where
(10) and
sg-s-1/31s"l 1, ss=v31s"lI.
(11)
If we assume the rate of plastic deformation is constant during the time interval, total displacement u consists of the plastic displacement u p and the elastic displacement u ~.From equation(9) we have
26
TOSHIAK! HATA
V.VuP+
1
VV.uP=2V.S~+2
1-2v 1
V.VuS+
1-2v
l+v 1-2v
V.S"
VV.uS=pii s.
(12) (13)
The solution of equation (12) may be simply derived by using four displacement functions. Here, we consider the three scalar components of vector up in terms of four functions, the scalar function ~P and the three scalar components of vector Bp. That is oP=4(1 -v)BP-2gr~ p.
(14)
Substituting equation (14) into (12), we know the system of equation (12) is satisfied if V.VB p=
1
V.S~
2(1-v)
(15)
V'V~p=zV'Bp- 1 (l+v__._~)Is~l. 3 (l-v)
(16)
We introduce the displacement scalar function ×P in the forms ~P=xP+r'B p,
(17)
where r is a position vector. Using equation (17), equation (16) takes the following form: V.V×p=
1
1 l+v
2(1-v)
3 1-v
--Is71.
(18)
The corresponding stresses may be written as follows: l+v T p= 21~[2vV.BPI + 2( 1 - v) (VBp+ BPV)- VV~ p] -- 21~[S~+ - S:]. 1-v
(19)
The hypotheses of imcompressibility is denoted in the form
Is"l = Is:l =0.
(20)
Next, we express the components of the displacement vector # in terms of derivatives of potentials. It is well known that the displacement vector can be written in the form US=V+S+Vx, s.
(21)
Equation (21) satisfies the equation of motion (13) if
v.v,l,s=,bs/c v.vqjs=i
s/c2 '
(22)
where C2= (h +2~)/p, C2= 0dp
(23)
and the additional constraint condition is V.+S=0.
(24)
A dynamicelasto/viscoplasticsolution
27
The corresponding elastic stresses are obtained from TS=XlSSlI+2~SS
(25)
The elastic strain-displacement relation is given by
SS=(VuS + #V)/2.
(26)
Finally total stresses may be given by T = T P + T s.
(27)
A DYNAMIC ELASTIC SOLUTION The motion of an isotropic, homogeneous, elastic shell subject to radially symmetric loads on both surfaces shown in Fig. 1 may be determined from a displacement potential d~S(r,t) of equation (22) with t~s=0 which satisfies the following equation, 02dOs 2 O+s ~s 0---7-- q r -Or - - -C~ -"
(28)
The displacement u s and the radial and tangential components of the stress ~ and trg are given from equations (21), (25) and (26) by uS=~bS,r
(29)
#,lpC2=$s/c2-(4~r),~,r, tr~/pC2=(1--2K)$s/c2 +(2~r)d~S,r,
(30)
where VL
1-2v
K= X+2-----~ -- 2(1--V-----~ "
(31)
For a shell subjected to internal and external pressures, the boundary conditions are
O~r(a,t) = -- Px (t) , or~(b,t) = - P2(t).
(32
The shell is at rest prior to time t=0 and the initial conditions are uS(r,o)=0, OuS/Ot=Oat t=0.
FIG. 1. Co-ordinatesand surface forces.
(33)
28
TOSHIAKI HATA
As an explicit example, we consider the case of a suddenly applied uniform pressure at the inside surface. Thus Pl(t)=PoH(t) and P2(t)=0 in equation (32). In Ref. [8], Pao and Ceranoglu presented an analysis of the solution of this problem by the ray theory as follows:
uS(r,t)/a = ~
~bn(r,t)
n=0 oo
~(r,t)/Po=(pC2/Po) Z
[¢bn(r,t),r+(m+ 2)tbn(r,t)(a/r) l
(34)
n=O
a
(m+2)
trg(r,t)/Po=(pC2/po) Z [ q b . ( r , t ) , r + - - d ~ n ( r , t ) ] - n=o vr 2
where the sum of the first n terms means the sum of the first n-th rays and
m=-4(1-2v)/{2(1-v)}=-4k,
C=CL.
(35)
The function ¢,, is given by
2 a(ogr+ia)H(tC_r+a) e -ieq(tC-r+a)/a dpo(r,t) =i Z j=o r2(3~+8kegi-4k) +l(r,t) = 4
a(cgr -ia)(otyZb 2+ 4kabegi- 4kaZ)M(tC+ a - 2b + r) y=0 r2{5bZotj4-16 kab(1-b)otj3i-12kotjZ(a2-4kab+bZ)+32ot]k2a(b-a)i+16k2a2}
(36)
X e -ieq(tC+a-2b+r)/a
oo
d~zn(r,t)= I [8(C'r+2a-2b)+qc(a")ld~2n-2(r,t-'r)d'r --o¢
d~2n+l(r,t)= f [~(Ca"+2a-2b)+qc(x)]tb2n-l(r,t-a") dT and function q¢(-r) is written as qc('r) = ~( Ca"+ 2 a - 2b) + q(a-),
(37)
where
4 i4kog(a- b) (ba2+ 4ka)H(C'r + 2a- 2b) e-ieq(Cx+2a- 2b)/a q(a")= Z j=l {2b2o93+6kb(b-a)egzi+4(4k2ab-kb2-ka2)°9+8k2(b-a) i (38) where ~(t) is the delta function and the pole 09 is given by ~a ~*a Oto=O' O t ' l = ~ * ' O t 2 = - - ~ ' O t 3 = Z ' Or4= b "
(39)
where
~=2k(~v/- l +k-l +i), 8*=2k(X/- l + k - l - i ) .
(40)
A DYNAMICVISCOPLASTICSOLUTION The theoretical assumption utilized here is that a solid undergoing dynamic loading can be rate-dependent under plastic deformation. A particular form of this constitutive equation, proposed by Perzyna [9], is written as
A dynamicelasto/viscoplasticsolution
r)>sd;
29 (41)
where the symbol is defined as follows: < ¢ ( F ) > = O : F_-<0, < ¢ ( F ) > = ¢ ( F ) : F > 0 .
(42)
Perzyna [4] also showed that a power form best suited the biaxial configuration and proposed the form (43)
~p(F) = F ~, F= V'~2/k p - 1,
where ~ is a material constant and k p is the yield stress in pure shear and F=0 denotes the Mises yield condition. If we extend the resulting relations of equation (41) into the polar symmetric spherical co-ordinates and integrate them during the time increment, we obtain he"=~/0<¢(F)> 2(cr'-Gr°) At
3v7;
Aeg=~/o<¢(F)> (O'0--O'r) At
(44)
J2=(Crr-O'o)2/3.
(45)
3VY7
where
If elasto-dynamic stresses in equations (34) are allowed to redistribute for an increment of time at the plastic rate of equations (44), the stresses caused by plastic strains can be obtained from equations (14)-(20). Equation (14) is represented in the polar symmetric spherical co-ordinates as UP= 4 ( 1 - v)BP-(xP + rBP), r.
Equations (15) and (18) with
Is l=0 are
(46)
represented as
2 1 de~ 2 V2Bp- r--~Bp= 2(1-v) [ dr +--(eT-e~)] r Pt
r der & 2 (e"-e[)], 2(1--2v) [ dr r
V2×p=
(47)
where V2= dE/dr2+ (2/r) (d/dr).
(48)
The solutions of equations (47) may be obtained as follows:
p
C1
1
1 r
(1-2v) I! d~
a =Cor+ ~- + 2(1-2v----~7Ia'q2 [ "O~-) ×P=- ~
ir
1
2(1--'1))
a
n2(1 -
"q. de"
2
d~ + ~ (e"-e~)}d~]d'q 2 - - (e'-eg)]dn,
7)[--~-~ "q- 'Y]
where Co and C 1 a r e unknown constants. The plastic stresses are found from equation (19) as 2 cr~=2~[2v(BP,r + - BP) +4(1-v)BPr- (×P+rBP),rr]-2txe" r
(49)
30
TOSH1AKIHA:A
2 Bp trPo=o-P=Zp,[Zv(BP+ - BP)+4(1-v) r r
(xP+rBP)'r]_ztxe,~.
(50)
r
Substituting equations (49) into (50), we obtain CrPr=C0
2C: r3
4~ r3 I~ "q2f(xl)d'q+(k+Ev°)f(r)-2txd~ a
2Ix r
(51)
' 2 (1-2v) fr [dg'+--(#'-eg)]d"q p r)= ( l - v ) Ja a~ n
(52)
C:
7 + '-~-I a "q2f(~)d'q+kf(r)-21xe'd'
°r~=°'P=C°'+ where /,,f
\
The undetermined constants Co and C1 in equations (51) are determined in order to satisfy the boundary conditions of
crP~(r,t)=Oat r=a,b.
(53)
The constants are determined as follows:
C°= (a~-b3) [-2I i ~q2f(rl)d~+h + 2~ b3f(b)-b3 g~lr=b a3b 3 2 b CI=I'L (a3----~31 " [ - ~ - g l a "q2f('q)d~+
~
,
+ ,
(54)
f(b)--g'lr=b grit=a].
Finally the elasto/viscoplastic stresses are obtained as follows: L
~ilt=tL-cr ilt=tL Z _
s
+
or Plat=at,
(i=r,0).
(55 /
1=1
NUMERICAL RESULTS AND DISCUSSION
A numerical example was evaluated for a hollow sphere with b/a= 2. The plastic data were fitted to the viscoplastic law of equation (43) with the following parameters:
~1o a/c= l , P0/kP=500
(56)
and Poisson's ratio v=0.3 and Po/pC2= 1 are employed. The numerical integrations were carried out by using Simpson's formula. The computations described in the foregoing sections were carried out with the aid of a HITAC M240H electronic computer of Tokyo University. The results of numerical evaluation of stress variation are illustrated from Figs 2-5. In these figures we used the non-dimensional time variable t* = Ct/a and took the increments of time as At*=0.05. The time history of tangential stresses for the spherical shell at various locations is shown in the case of 8=1 in Fig. 2 and 8=3 in Fig. 3. In Fig. 2 we observe that the maximum dynamic stress in a sphere without plastic strains, which occurs at t*=3 on r=a, is about 1.65; loc. cit. the static tangential stress in a sphere with the constant internal pressure is about 0.71 on r=a. The maximum dynamic stress at the internal surface is about two times higher than the corresponding static value.
A
~o
dynamic elasto/viscoplastic solution
31
05
I0
-05
I I I/
-I0 FIG. 2.
Variation of tangential stress w i t h r in the case of 8 = 1 . redistributed stress through viscoplasticity.
:
elastic stress,
- . . . .
:
The plastic strains occur when the stresses satisfy the yield conditions F=0. In this problem the elastic stresses satisfy the yield condition at t*=0.5. Then the values of tangential stresses after t* =0.5 are lower than those of the elastic ones, because plastic strains occur on the internal surface. Comparing the tangential stresses in an elastic sphere with those in elasto/viscoplastic one at t*=5, the elastic stress on the internal surface is about 0.5 of tension, but the elasto/viscoplastic one is about 0.18 of compression. We also observe that the stress distributions at r/a=l.5 and 2.0 in an elastic sphere are slightly higher than those in the elasto/viscoplastic one. In this figure we can observe that the travelling time of wave in an elastic sphere is the same as that in the elasto/viscoplastic one. The same conclusion has been shown in Ref. [4]. In Fig. 3 we show the time history of tangential stresses in the case of 8=3. This stress distribution has the same tendency as before. In order to clarify the influence of plastic f
15
~
/ ~/~/rla ~ I
I,/\
r i o = I. 5
II
,o
ff
,
I0
I i
t
./
I I
-I,0 -l~o.
3.
~I
Variation of tangential stress with r in the case of 8 = 3 . redistributed stress through viscoplasticity.
:
elastic stress,
- . . . .
:
32
TOSHIAKI HATA
strains on the tangential plastic stresses, we show the time history of plastic tangential stresses for an elasto/viscoplastic sphere with various locations in Fig. 4. We observe that these stresses occur at t*=0.5 and increase with time. In particular, remarkable changes are observed on the internal surface where the maximum elastic tangential stresses Occur.
In Fig. 5 we show the time history of radial stress distributions at r/a= 1.2 and 1.5 in the case of 8=3. From this figure, we observe that the difference of the radial stresses between elastic and elasto/viscoplastic spheres is small. The same tendency may be observed in the case of 8= 1. To examine the accuracy, we preferred another time increment of At*=0.1. The difference may be negligible. Then the calculations may be convergent. Hereafter we comment on the numerical example. We use the non-dimensional parameters of equation (56), because it is easier to generalize the results. The example
02~--
.rig
=2
__\':
,o
-o2-~ b
04
~rla
1.5
!~,,~r/a =I
; V , -08--
FIG. 4. V a r i a t i o n o f t a n g e n t i a l plastic stress w i t h v a r i o u s locations. ..... : in the case of G=3.
: in t h e case of 8 = 1 ,
05[ ID--
0 i1 o
-iD
--
rio = I
-I 5
--
Fro. 5. V a r i a t i o n of r a d i a l stress w i t h r in the case of 8 = 3 . r e d i s t r i b u t e d stress t h r o u g h viscoplasticity.
: elastic stress, - . . . .
:
A dynamic elasto/viscoplastic solution
33
corresponds to the 6061T6 aluminium sphere in Ref. [9]. The data, which satisfies equations (56), are: E=713 kbars, kP=l.7 bars, 0=2.7 g CC- 1 , "y0=5000 S- 1 and a = l . 2 m. Next, we may discuss the advantage of the method of the ray theory over any of the existing methods. One of the existing numerical methods is FEM. Reference [10] uses the method of finite elements and obtains the dynamic response of a shallow elasto/viscoplastic spherical cap subjected to a uniformly distributed external step pressure. In Ref. [10] the plastic strain is calculated under the same assumption as in this paper, under which the plastic properties do not affect the travelling time of wave. However, it is simply concluded that the arrival times of the initial peaks are delayed relative to the elastic cases. Another numerical methods [10] cannot obtain the discontinuity of the stress wave and the travelling time of wave. Moreover it is impossible to obtain the stress distribution inside the sphere because the stress distribution is discontinuous. However, the method of characteristics is very good for the analysis of an elasto/viscoplastic problem. Reference [4] shows numerous solutions which were obtained with the method of characteristics. A defect of the method is that the accuracy of the solution depends on the mesh size which is used for the numerical integration. Thus, it may be difficult to estimate values of the dynamic stresses with high accuracy in a thick-walled elasto/viscoplastic sphere. CONCLUSIONS
This paper develops a method for calculating the dynamic stress redistribution accurately in a viscoplastic medium. The author extends the ray theory to predict the dynamic elasto/viscoplastic deformation of a thick-walled sphere subjected to dynamic loading on its surfaces. Adopting the procedure of time increments, the solution of plastic deformation has been determined by using four displacement functions similar to the Neuber-Papkovitch stress functions. A numerical example of this method has been presented for a thick-walled spherical shell subjected to an internal impact load. The figures show that the elastic tangential stresses at the internal surface of a sphere are moved to the compression side by the viscoplastic properties, but those for the locations inside the sphere are slightly moved to the tension side. They also show that the difference in the radial stresses between elastic and elasto/viscoplastic sphere is negligible. Acknowledgments--The author wishes to acknowledge the encouragement and the direction of Emeritus Professor A. Atsumi, Tohoku University, in the preparation of this paper. Thanks are also expressed to Professor Yih-Hsing. Pao, Cornell University, for his advice.
REFERENCES 1. J. H. Hua'n and J. D. COLE, Elastic-stress waves produced by pressure loads on a spherical shell. J appl. Mech. 22, 473-478 (1955). 2. J. L. RosE, S. C. Cnou and P. C. CrIou, Vibration analysis of thick-walled spheres and cylinders. J. acoast. Soc. Am. 53, 771-776 (1973). 3. R. R. KARrp et al., Response of containment vessels to explosive blast loading. J. Pressure Vessd Technol. 105, 23-27 (1983). 4. W. K. NowActo, Stress Waves in Non-Elastic Solids, Pergamon Press, Oxford (1978). 5. Y.-H. PAO and A. N. CEV.ANOaLU,Determination of transient responses of a thick-walled spherical shell by the ray theory. J. appl. Mech. 45, 114-122 (1978). 6. S. S. MANSON, Thermal Stress and Low-Cycle Fatigue. McGraw-Hill, New York (1966). 7. H. NEtYaER, Kerbspannungslehre. Springer, Berlin (1937).
34
TOSHIAKIHATA
8. P. PERZYNA,Fundamental problems in viscoplasticity. Adv. appl. Mech. 9, 243-377 (1966). 9. I. M. FYFE, The applicability of elastic/viscoplastic theory in stress wave propagation. J. appl. Mech. 42, 141-146 (1975). 10. S. NAGARAJANand E. P. PoPov, Non-linear dynamic analysis of axisymmetric shells. Int. J. Numer. Meth. Engng. 9, 535-550 (1975).
0"/34-743X/87$3.00+0.00 PergamonJournalsLtd.
A DYNAMIC ELASTO/VISCOPLASTIC SOLUTION FOR A T H I C K - W A L L E D S P H E R I C A L C O N T A I N E R SUBJECTED TO IMPULSIVE LOADING TOSHIAKI HATA Faculty of Education, Shizuoka University, Shizuoka City, 422, Japan
(Received 30 November 1985; and in revised form 5 December 1986) Smnnmry--A general method for the solution of dynamic elasto/viscoplastic solid problems is presented. This method is an extension of ray theory to cater for dynamic elasto/viscoplastic deformations. The method reduces the elasto/viscoplastic problem to a sequence of elastic problems with initial strains. The solution of this problem is determined by using four displacement functions. Using the foregoing method, the solution is derived for the dynamic elasto/viscoplastic behaviour of a thick-walled spherical shell subjected to internal impact load. The numerical results show how the dynamic stresses in a sphere with viscoplastic properties vary with time.
NOTATION n r S Tn T U I a, b CL kp e
•/2 P r s# t .~j % ~y
k,t~ v p
surface normal position vector strain dyadic surface traction stress dyadic displacement vector idemfactor inner and outer radii of a sphere, respectively velocity of longitudinal wave yield stress strain second invariant of the deviatoric stress tensor pressure radial co-ordinate deviatoric stress time poles viscoplastic constants stress Lame's constants Poisson's ratio density
BP,~ p,
×P ~s,~s
plastic-displacement functions elastic-stress functions
subscripts i,j refers to i(i= 1,2,3) and j(j= 1,2,3) directions r,0 refer to radial and tangential directions, respectively i,o refer to components of internal and external surfaces, respectively superscripts ' refers to elastic components " refers to plastic components s,p refer to elastic and plastic components, respectively differentiation with respect to time
INTRODUCTION
In several fields of engineering and science, there are requirements for structures, such as blast chambers and nuclear reactors, which must be able to withstand the effects of 23
24
TOSHIAK1 HATA
internally applied high transient pressures. Since these structures may have to withstand static, as well as transient pressures, they are usually constructed in the form of thickwalled pressure vessels, such as spheres or cylinders. In designing such structures, one must not solely rely on static pressure design criteria, but must also take into consideration the dynamic response to the high transient pressure [1-3]. The simplest analytical problem relating to the design of these structures is that of the response of a thick-walled spherical shell subjected to a spherically symmetric transient pressure pulse which acts on the inner and external surfaces. One important conclusion arising from the static analysis of a thick-walled elastic, spherical shell is that the radial stress in the shell is always compressive, and less in magnitude than the applied internal pressure. However, this may not be the case when the shell is subjected to dynamic loadings. It is generally understood that when a spherical wave, which is generated by a dynamic pressure at the inner surface, is reflected at a traction free outer boundary the radial stress may change from compression to tension as a result of the reflection. An elasto/viscoplastic material [4] is assumed to be one with distinct viscous properties only in plastic regions, and which is ideally elastic. In this case, the theoretical assumption that a solid undergoing dynamic loading can be rate-dependent under plastic deformations, but rate-independent in the elastic range appears to be fairly generally accepted for many metals [4]. Materials which exhibit this type of behaviour are usually classified as elasto/viscoplastic. In the following analysis the sphere is assumed to be an elasto/viscoplastic one. Here it is very important to obtain an accurate elastic solution because the accuracy of the elasto/viscoplatic analysis depends on the accuracy of the elastic predictions. There are a few types of dynamic solution for thick elastic spherical shells in the literature: one group is a numerical analysis, which contains FEM, the finite-differential analysis and the boundary-element method, and the other group contains theoretical analyses, which includes normal mode analysis, the method of characteristics and the ray method. The numerous wave problems concerning elasto/viscoplastic media are solved [4] by using the method of characteristics. The accuracy of the solution of characteristics depends on the mesh size which is used for the numerical integration. But in the analyses of dynamic response of an elastic thick-walled spherical shell the method of ray theory [5] represents the wave motion exactly, because the ray integral represents the wave motions exactly. This method is quite adequate for the analysis of initial dynamic response of the solid. In this paper the method of ray theory will be extended to the dynamic elasto/plastic deformation of a sphere subjected to a dynamic loading at the inner and external surfaces. In the calculation of plastic strains the approximate method of solution is employed, in which the plastic-strain distribution is developed in small increments of time. During a time interval, the stresses and strains vary continuously owing to the presence of plastic properties due to the surface forces. The incremental stresses of the first step are added to the stresses which exist at the beginning of the time interval. This result represents the first approximation to the stress distribution which exists at the end of this interval. The stress distribution in the next time increment is determined as before. The solution of the problem may be determined successively by the foregoing procedure, which is fully illustrated in Ref. [6]. In this case, the stress-strain relation is assumed to be elastic during the unloading process. In the procedure, the incremental stress problems are reduced to the boundary-value problems similar to the welt-known thermoelastic analysis. Then the incremental plastic stresses are determined by using four displacement functions similar to the NeuberPapkovitch stress functions [7]. We derive the solution for a dynamic elasto/viscoplastic problem of a thick-walled spherical shell subjected to an internal impact load. The numerical results show how the dynamic stresses in a sphere having viscoplastic properties vary with time.
A dynamicelasto/viscoplasticsolution
25
FORMULATION OF PROBLEM The constitutive equations and field equations for a solid with plastic deformation are presented in this section. The solution of problem requires the determination of the dynamic stress tensor T and strain tensor S in the region 3~ with boundary F. The stress-strain relations may be expressed as follows:
T=2~S'+XlS'II,
(1)
where S' represents the elastic-strain tensor and I, h and ~ are idemfactor and Lame's constants, respectively, and the trace of a dyadic S" is represented by Is'q. At any time, the total strains consist of a sum of elastic and plastic strains, S=S'+S",
(2)
T=XISII+20.S-[XIS"II+2~S"].
(3)
where S" is the plastic tensor. Substitution of S' into (1) yields
The equation of motion without body forces may be written as V.T=pi~ in Z.
(4)
Tn=n.T on F.
(5)
The surface traction Tn is given by
Substituting equations (3) into (4) and (5), we obtain hV[SI + 21xV'S-[~vls"l + 21xv.s"] =pii
(6)
Tn= hnIS[ + 2p,n.S-[hn[S"[ + 2 p,n.S"].
(7)
The strain-displacement relations are
S=(Vu+uV)/2.
(8)
Substitution of equations (8) into (6) yields the equation of motions as follows: V.Vu+
1 1-2v
VV.u=pi~+2V.S~+2
l+v 1-2v
V.S"s,
(9)
where
(10) and
sg-s-1/31s"l 1, ss=v31s"lI.
(11)
If we assume the rate of plastic deformation is constant during the time interval, total displacement u consists of the plastic displacement u p and the elastic displacement u ~.From equation(9) we have
26
TOSHIAK! HATA
V.VuP+
1
VV.uP=2V.S~+2
1-2v 1
V.VuS+
1-2v
l+v 1-2v
V.S"
VV.uS=pii s.
(12) (13)
The solution of equation (12) may be simply derived by using four displacement functions. Here, we consider the three scalar components of vector up in terms of four functions, the scalar function ~P and the three scalar components of vector Bp. That is oP=4(1 -v)BP-2gr~ p.
(14)
Substituting equation (14) into (12), we know the system of equation (12) is satisfied if V.VB p=
1
V.S~
2(1-v)
(15)
V'V~p=zV'Bp- 1 (l+v__._~)Is~l. 3 (l-v)
(16)
We introduce the displacement scalar function ×P in the forms ~P=xP+r'B p,
(17)
where r is a position vector. Using equation (17), equation (16) takes the following form: V.V×p=
1
1 l+v
2(1-v)
3 1-v
--Is71.
(18)
The corresponding stresses may be written as follows: l+v T p= 21~[2vV.BPI + 2( 1 - v) (VBp+ BPV)- VV~ p] -- 21~[S~+ - S:]. 1-v
(19)
The hypotheses of imcompressibility is denoted in the form
Is"l = Is:l =0.
(20)
Next, we express the components of the displacement vector # in terms of derivatives of potentials. It is well known that the displacement vector can be written in the form US=V+S+Vx, s.
(21)
Equation (21) satisfies the equation of motion (13) if
v.v,l,s=,bs/c v.vqjs=i
s/c2 '
(22)
where C2= (h +2~)/p, C2= 0dp
(23)
and the additional constraint condition is V.+S=0.
(24)
A dynamicelasto/viscoplasticsolution
27
The corresponding elastic stresses are obtained from TS=XlSSlI+2~SS
(25)
The elastic strain-displacement relation is given by
SS=(VuS + #V)/2.
(26)
Finally total stresses may be given by T = T P + T s.
(27)
A DYNAMIC ELASTIC SOLUTION The motion of an isotropic, homogeneous, elastic shell subject to radially symmetric loads on both surfaces shown in Fig. 1 may be determined from a displacement potential d~S(r,t) of equation (22) with t~s=0 which satisfies the following equation, 02dOs 2 O+s ~s 0---7-- q r -Or - - -C~ -"
(28)
The displacement u s and the radial and tangential components of the stress ~ and trg are given from equations (21), (25) and (26) by uS=~bS,r
(29)
#,lpC2=$s/c2-(4~r),~,r, tr~/pC2=(1--2K)$s/c2 +(2~r)d~S,r,
(30)
where VL
1-2v
K= X+2-----~ -- 2(1--V-----~ "
(31)
For a shell subjected to internal and external pressures, the boundary conditions are
O~r(a,t) = -- Px (t) , or~(b,t) = - P2(t).
(32
The shell is at rest prior to time t=0 and the initial conditions are uS(r,o)=0, OuS/Ot=Oat t=0.
FIG. 1. Co-ordinatesand surface forces.
(33)
28
TOSHIAKI HATA
As an explicit example, we consider the case of a suddenly applied uniform pressure at the inside surface. Thus Pl(t)=PoH(t) and P2(t)=0 in equation (32). In Ref. [8], Pao and Ceranoglu presented an analysis of the solution of this problem by the ray theory as follows:
uS(r,t)/a = ~
~bn(r,t)
n=0 oo
~(r,t)/Po=(pC2/Po) Z
[¢bn(r,t),r+(m+ 2)tbn(r,t)(a/r) l
(34)
n=O
a
(m+2)
trg(r,t)/Po=(pC2/po) Z [ q b . ( r , t ) , r + - - d ~ n ( r , t ) ] - n=o vr 2
where the sum of the first n terms means the sum of the first n-th rays and
m=-4(1-2v)/{2(1-v)}=-4k,
C=CL.
(35)
The function ¢,, is given by
2 a(ogr+ia)H(tC_r+a) e -ieq(tC-r+a)/a dpo(r,t) =i Z j=o r2(3~+8kegi-4k) +l(r,t) = 4
a(cgr -ia)(otyZb 2+ 4kabegi- 4kaZ)M(tC+ a - 2b + r) y=0 r2{5bZotj4-16 kab(1-b)otj3i-12kotjZ(a2-4kab+bZ)+32ot]k2a(b-a)i+16k2a2}
(36)
X e -ieq(tC+a-2b+r)/a
oo
d~zn(r,t)= I [8(C'r+2a-2b)+qc(a")ld~2n-2(r,t-'r)d'r --o¢
d~2n+l(r,t)= f [~(Ca"+2a-2b)+qc(x)]tb2n-l(r,t-a") dT and function q¢(-r) is written as qc('r) = ~( Ca"+ 2 a - 2b) + q(a-),
(37)
where
4 i4kog(a- b) (ba2+ 4ka)H(C'r + 2a- 2b) e-ieq(Cx+2a- 2b)/a q(a")= Z j=l {2b2o93+6kb(b-a)egzi+4(4k2ab-kb2-ka2)°9+8k2(b-a) i (38) where ~(t) is the delta function and the pole 09 is given by ~a ~*a Oto=O' O t ' l = ~ * ' O t 2 = - - ~ ' O t 3 = Z ' Or4= b "
(39)
where
~=2k(~v/- l +k-l +i), 8*=2k(X/- l + k - l - i ) .
(40)
A DYNAMICVISCOPLASTICSOLUTION The theoretical assumption utilized here is that a solid undergoing dynamic loading can be rate-dependent under plastic deformation. A particular form of this constitutive equation, proposed by Perzyna [9], is written as
A dynamicelasto/viscoplasticsolution
r)>sd;
29 (41)
where the symbol
(42)
Perzyna [4] also showed that a power form best suited the biaxial configuration and proposed the form (43)
~p(F) = F ~, F= V'~2/k p - 1,
where ~ is a material constant and k p is the yield stress in pure shear and F=0 denotes the Mises yield condition. If we extend the resulting relations of equation (41) into the polar symmetric spherical co-ordinates and integrate them during the time increment, we obtain he"=~/0<¢(F)> 2(cr'-Gr°) At
3v7;
Aeg=~/o<¢(F)> (O'0--O'r) At
(44)
J2=(Crr-O'o)2/3.
(45)
3VY7
where
If elasto-dynamic stresses in equations (34) are allowed to redistribute for an increment of time at the plastic rate of equations (44), the stresses caused by plastic strains can be obtained from equations (14)-(20). Equation (14) is represented in the polar symmetric spherical co-ordinates as UP= 4 ( 1 - v)BP-(xP + rBP), r.
Equations (15) and (18) with
Is l=0 are
(46)
represented as
2 1 de~ 2 V2Bp- r--~Bp= 2(1-v) [ dr +--(eT-e~)] r Pt
r der & 2 (e"-e[)], 2(1--2v) [ dr r
V2×p=
(47)
where V2= dE/dr2+ (2/r) (d/dr).
(48)
The solutions of equations (47) may be obtained as follows:
p
C1
1
1 r
(1-2v) I! d~
a =Cor+ ~- + 2(1-2v----~7Ia'q2 [ "O~-) ×P=- ~
ir
1
2(1--'1))
a
n2(1 -
"q. de"
2
d~ + ~ (e"-e~)}d~]d'q 2 - - (e'-eg)]dn,
7)[--~-~ "q- 'Y]
where Co and C 1 a r e unknown constants. The plastic stresses are found from equation (19) as 2 cr~=2~[2v(BP,r + - BP) +4(1-v)BPr- (×P+rBP),rr]-2txe" r
(49)
30
TOSH1AKIHA:A
2 Bp trPo=o-P=Zp,[Zv(BP+ - BP)+4(1-v) r r
(xP+rBP)'r]_ztxe,~.
(50)
r
Substituting equations (49) into (50), we obtain CrPr=C0
2C: r3
4~ r3 I~ "q2f(xl)d'q+(k+Ev°)f(r)-2txd~ a
2Ix r
(51)
' 2 (1-2v) fr [dg'+--(#'-eg)]d"q p r)= ( l - v ) Ja a~ n
(52)
C:
7 + '-~-I a "q2f(~)d'q+kf(r)-21xe'd'
°r~=°'P=C°'+ where /,,f
\
The undetermined constants Co and C1 in equations (51) are determined in order to satisfy the boundary conditions of
crP~(r,t)=Oat r=a,b.
(53)
The constants are determined as follows:
C°= (a~-b3) [-2I i ~q2f(rl)d~+h + 2~ b3f(b)-b3 g~lr=b a3b 3 2 b CI=I'L (a3----~31 " [ - ~ - g l a "q2f('q)d~+
~
,
+ ,
(54)
f(b)--g'lr=b grit=a].
Finally the elasto/viscoplastic stresses are obtained as follows: L
~ilt=tL-cr ilt=tL Z _
s
+
or Plat=at,
(i=r,0).
(55 /
1=1
NUMERICAL RESULTS AND DISCUSSION
A numerical example was evaluated for a hollow sphere with b/a= 2. The plastic data were fitted to the viscoplastic law of equation (43) with the following parameters:
~1o a/c= l , P0/kP=500
(56)
and Poisson's ratio v=0.3 and Po/pC2= 1 are employed. The numerical integrations were carried out by using Simpson's formula. The computations described in the foregoing sections were carried out with the aid of a HITAC M240H electronic computer of Tokyo University. The results of numerical evaluation of stress variation are illustrated from Figs 2-5. In these figures we used the non-dimensional time variable t* = Ct/a and took the increments of time as At*=0.05. The time history of tangential stresses for the spherical shell at various locations is shown in the case of 8=1 in Fig. 2 and 8=3 in Fig. 3. In Fig. 2 we observe that the maximum dynamic stress in a sphere without plastic strains, which occurs at t*=3 on r=a, is about 1.65; loc. cit. the static tangential stress in a sphere with the constant internal pressure is about 0.71 on r=a. The maximum dynamic stress at the internal surface is about two times higher than the corresponding static value.
A
~o
dynamic elasto/viscoplastic solution
31
05
I0
-05
I I I/
-I0 FIG. 2.
Variation of tangential stress w i t h r in the case of 8 = 1 . redistributed stress through viscoplasticity.
:
elastic stress,
- . . . .
:
The plastic strains occur when the stresses satisfy the yield conditions F=0. In this problem the elastic stresses satisfy the yield condition at t*=0.5. Then the values of tangential stresses after t* =0.5 are lower than those of the elastic ones, because plastic strains occur on the internal surface. Comparing the tangential stresses in an elastic sphere with those in elasto/viscoplastic one at t*=5, the elastic stress on the internal surface is about 0.5 of tension, but the elasto/viscoplastic one is about 0.18 of compression. We also observe that the stress distributions at r/a=l.5 and 2.0 in an elastic sphere are slightly higher than those in the elasto/viscoplastic one. In this figure we can observe that the travelling time of wave in an elastic sphere is the same as that in the elasto/viscoplastic one. The same conclusion has been shown in Ref. [4]. In Fig. 3 we show the time history of tangential stresses in the case of 8=3. This stress distribution has the same tendency as before. In order to clarify the influence of plastic f
15
~
/ ~/~/rla ~ I
I,/\
r i o = I. 5
II
,o
ff
,
I0
I i
t
./
I I
-I,0 -l~o.
3.
~I
Variation of tangential stress with r in the case of 8 = 3 . redistributed stress through viscoplasticity.
:
elastic stress,
- . . . .
:
32
TOSHIAKI HATA
strains on the tangential plastic stresses, we show the time history of plastic tangential stresses for an elasto/viscoplastic sphere with various locations in Fig. 4. We observe that these stresses occur at t*=0.5 and increase with time. In particular, remarkable changes are observed on the internal surface where the maximum elastic tangential stresses Occur.
In Fig. 5 we show the time history of radial stress distributions at r/a= 1.2 and 1.5 in the case of 8=3. From this figure, we observe that the difference of the radial stresses between elastic and elasto/viscoplastic spheres is small. The same tendency may be observed in the case of 8= 1. To examine the accuracy, we preferred another time increment of At*=0.1. The difference may be negligible. Then the calculations may be convergent. Hereafter we comment on the numerical example. We use the non-dimensional parameters of equation (56), because it is easier to generalize the results. The example
02~--
.rig
=2
__\':
,o
-o2-~ b
04
~rla
1.5
!~,,~r/a =I
; V , -08--
FIG. 4. V a r i a t i o n o f t a n g e n t i a l plastic stress w i t h v a r i o u s locations. ..... : in the case of G=3.
: in t h e case of 8 = 1 ,
05[ ID--
0 i1 o
-iD
--
rio = I
-I 5
--
Fro. 5. V a r i a t i o n of r a d i a l stress w i t h r in the case of 8 = 3 . r e d i s t r i b u t e d stress t h r o u g h viscoplasticity.
: elastic stress, - . . . .
:
A dynamic elasto/viscoplastic solution
33
corresponds to the 6061T6 aluminium sphere in Ref. [9]. The data, which satisfies equations (56), are: E=713 kbars, kP=l.7 bars, 0=2.7 g CC- 1 , "y0=5000 S- 1 and a = l . 2 m. Next, we may discuss the advantage of the method of the ray theory over any of the existing methods. One of the existing numerical methods is FEM. Reference [10] uses the method of finite elements and obtains the dynamic response of a shallow elasto/viscoplastic spherical cap subjected to a uniformly distributed external step pressure. In Ref. [10] the plastic strain is calculated under the same assumption as in this paper, under which the plastic properties do not affect the travelling time of wave. However, it is simply concluded that the arrival times of the initial peaks are delayed relative to the elastic cases. Another numerical methods [10] cannot obtain the discontinuity of the stress wave and the travelling time of wave. Moreover it is impossible to obtain the stress distribution inside the sphere because the stress distribution is discontinuous. However, the method of characteristics is very good for the analysis of an elasto/viscoplastic problem. Reference [4] shows numerous solutions which were obtained with the method of characteristics. A defect of the method is that the accuracy of the solution depends on the mesh size which is used for the numerical integration. Thus, it may be difficult to estimate values of the dynamic stresses with high accuracy in a thick-walled elasto/viscoplastic sphere. CONCLUSIONS
This paper develops a method for calculating the dynamic stress redistribution accurately in a viscoplastic medium. The author extends the ray theory to predict the dynamic elasto/viscoplastic deformation of a thick-walled sphere subjected to dynamic loading on its surfaces. Adopting the procedure of time increments, the solution of plastic deformation has been determined by using four displacement functions similar to the Neuber-Papkovitch stress functions. A numerical example of this method has been presented for a thick-walled spherical shell subjected to an internal impact load. The figures show that the elastic tangential stresses at the internal surface of a sphere are moved to the compression side by the viscoplastic properties, but those for the locations inside the sphere are slightly moved to the tension side. They also show that the difference in the radial stresses between elastic and elasto/viscoplastic sphere is negligible. Acknowledgments--The author wishes to acknowledge the encouragement and the direction of Emeritus Professor A. Atsumi, Tohoku University, in the preparation of this paper. Thanks are also expressed to Professor Yih-Hsing. Pao, Cornell University, for his advice.
REFERENCES 1. J. H. Hua'n and J. D. COLE, Elastic-stress waves produced by pressure loads on a spherical shell. J appl. Mech. 22, 473-478 (1955). 2. J. L. RosE, S. C. Cnou and P. C. CrIou, Vibration analysis of thick-walled spheres and cylinders. J. acoast. Soc. Am. 53, 771-776 (1973). 3. R. R. KARrp et al., Response of containment vessels to explosive blast loading. J. Pressure Vessd Technol. 105, 23-27 (1983). 4. W. K. NowActo, Stress Waves in Non-Elastic Solids, Pergamon Press, Oxford (1978). 5. Y.-H. PAO and A. N. CEV.ANOaLU,Determination of transient responses of a thick-walled spherical shell by the ray theory. J. appl. Mech. 45, 114-122 (1978). 6. S. S. MANSON, Thermal Stress and Low-Cycle Fatigue. McGraw-Hill, New York (1966). 7. H. NEtYaER, Kerbspannungslehre. Springer, Berlin (1937).
34
TOSHIAKIHATA
8. P. PERZYNA,Fundamental problems in viscoplasticity. Adv. appl. Mech. 9, 243-377 (1966). 9. I. M. FYFE, The applicability of elastic/viscoplastic theory in stress wave propagation. J. appl. Mech. 42, 141-146 (1975). 10. S. NAGARAJANand E. P. PoPov, Non-linear dynamic analysis of axisymmetric shells. Int. J. Numer. Meth. Engng. 9, 535-550 (1975).