stress rate are used to develop axisymmetric finite strain, elastic, closed form ... applications of Eq. (1) in obtaining finite strain closed form solutions for an ...
Acta Mechanica 97, 1 - 2 2 (1993)
ACTA MECHANICA 9 Springer-Verlag 1993
Finite strain elastic closed form solutions for some axisymmetric problems C. S. Wu, M. Budhu, and S. EI-Kholy, Tucson, Arizona (Received June 27, 1991; revised January 8, 1992)
Summary. In this paper, rate type constitutive equations using the Jaumann and 2nd Piola-Kirchhoff stress rate are used to develop axisymmetric finite strain, elastic, closed form solutions for a variety of loading conditions. We examined several cases comprizing compression-extension loading conditions, simple shear and cavity expansion conditions. Results from small strain analyses are used to indicate strain ranges for which such analyses will provide satisfactory solutions. For all the cases examined, except simple shear, the 2rid Piola-Kirchhoff stress rate does not appear to be a suitable stress rate to describe a material which follows a rate-type constitutive equation for strains greater than about 40%. The Jaumann stress rate solution shows oscillatory shear stress for axisymmetric simple shear similar to that found earlier by many other authors for rectangular (or cuboidal) condition. Negative excess pore water pressure (suction) at the cavity wall during the expansion of a cylindrical cavity was also observed in Jaumann stress rate solution.
1 Introduction A number of constitutive equations, mainly based on the modification of the theories of hyperelasticity [8] and hypoelasticity [16], has been proposed for the analyses of large deformation. The application of hypoelasticity, called rate-type constitutive equation when extended to elasto-plasticity, has recently become popular due to the ease of implementation into finite element procedures. The other advantage of using rate-type constitutive equation is, that time t is an explicit variable such that the constitutive equation can be easily extended to visco elastic-plastic or dynamic analysis. There are many stress and strain measures which can be selected to formulate large deformation solutions. One of the key criteria when selecting stress measures is that of objectivity (or frame-indifference). Truesdell and Noll [16] examined the question of objectivity in developing constitutive equations and concluded that any objective stress rate can be used in formulating constitutive equations. Several stress rates have been proposed and examined with respect to the problem of simple shear [1], [5], [9], [15]. In engineering problems, many loading conditions are possible and the suitability of a stress rate should be examined using a wide range of loading conditions. In geotechnical engineering, elastic analyses are often used, as a first approximation, to interpret soil behavior. Indeed, settlement of soil structures is usually computed assuming elastic response. Recently, there has been an increasing interest in studying soil behavior at large strains. Both the Jaumann [10] and the 2nd Piola-Kirchhoff [3] stress rates have been employed. During construction and in rapid loading of soils, the soil is assumed to be an incompressible solid (undrained condition). In this case, the Jaumann stress rate is preferred [2]. For long term loading
2
C.S. Wu et al.
(post construction), the excess pore water pressure developed as a result of the imposed loading will dissipate with time (consolidation). In this case, the 2nd Piola-Kirchhoff stress rate is preferred [2]. In this contribution, we use two rate-type constitutive equations to develop closed form solutions for a few axisymmetric problems under a variety of loading conditions. The practical application of these closed form solutions to engineering problems, in particular geotechnical engineering problems, is highlighted. A critical examination of the suitability of these two constitutive equations is provided. The first of these constitutive equations is for a hypoelastic material as proposed by Truesdell [14] with the exception that the Jaumann stress rate [7], rather than the Truesdell stress rate [14], is employed as shown in the constitutive equation
(r~ = 2dkk6ij + 2#dlj
(1)
where the superimposed dot (-) denotes the material derivative, d~ is the Jaumann stress rate,
vEy
gy
d~j is the rate of deformation, 2 = (1 + v) (1 - 2v)' # - 2(1 + v)' Er is Young's modulus and v is Poisson's ratio. The use of the Jaumann stress rate is to fulfill the requirement of frameindifference (objectivity). According to the assumed constitutive equation (Eq. (1)), the stress rate is a function only of the current values of Cauchy stress and the rate of deformation. The applications of Eq. (1) in obtaining finite strain closed form solutions for an isotropic elastic material in a rectangular coordinate system are given in many contributions [1], [6], [9]. The second of these constitutive equations is
Sij = )~Ekk~ij + 2/~/~ij
(2)
where S~j is the 2nd Piola-Kirchhoff stress tensor and E~j is the Green-Lagrangian strain tensor. By definition, we have
[~Sij dt -= Sij
(3)
and
S Eij dt = Eli.
(4)
If 2 and ]z are constants during loading, Eq. (2) reduces to S~j = )~Ek~cSij+ 2/~Eij
(5)
such that the solutions using Eq. (2) and Eq. (5) would be the same. The inclusion of the Ei~ (or/~j) reveals that the constitutive equations (2) and (5) depend on the total deformation from the reference configuration which is expressedly excluded from hypoelasticity [9]. We shall refer to the solution using Eq. (1) as Jaumann stress rate solution and the solution using Eq. (2) (or (5)) as the 2nd Piola-Kirchhoff stress solution. Before the derivation of the closed form solutions, a brief description of necessary background of kinematics is given in the next Section.
Finite strain elastic closed form solutions
3
2 Kinematics The motion of a deformable continuum is described by the mapping Xi(x~, t) = x~ + u~(x~, t), in which X~ is the deformed position vector, x~ is the reference position vector, u~ is the displacement vector and t denotes time. For an axisymmetric deformation, this is denoted as
R(r, z, t) = r + u,(r, z, t)
(6)
Z(r, z, t) = z + u~(r, z, t)
(7)
where u. and u~ are displacements in the radial and vertical directions respectively, R and Z are coordinates after deformation while r and z are reference coordinates. The deformation gradient F in axisymmetry is
[SXl] IF] : L~xjJ
-SR 8r = 8Z
~
8R 8z 8Z a~
0
0
0 o
(8)
R -/,
where the bracket [] denotes matrix. By definition, we have the Green-Lagrangian strain tensor [11] [El : ~ ([F] r [F] - [I])
(9)
in which [I] is the unit matrix and the superscript T denotes transpose. The spatial gradient of velocity in axisymmetry is l ~e
aR 0
8R
[L]= LaXj]
8Z
OR 8Z 0
0
(10)
0
The rate of deformation is 1 [n] = [ a j = ~ ([L] ~ + [L])
(11)
and the vorticity is
1 [w] = [co,j] = 7 ([L]~ - [L]).
(12)
For the ease of interpreting the results of Jaumann stress rate solution, we adopt the strain definition eo = S dij dt, that is, err = ~ drr dt, ezz = ~ d~r dt, ~oo = ~ doo dt and ~rz = ~ dr~ dt, where err, e= and e00 denote the strains in the radial, vertical and tangential directions respectively and
4
C.S. Wu et al.
erz denotes shear strain. These strains are the generalization of the natural-strain and may not have physical significance [11], especially when the material undergoes large rotation. The Jaumann stress rate is defined as [7]
(13)
( ~ = dYij - - ffip(Dpj - - O'jpU)pi
where cqj is Cauchy stress tensor. The 2rid Piola-Kirchhoff stress is defined as [10] [S] = J[F -1] [a] [F-l] r
(14)
where J denotes Jacobian ( = det JFD. Alternatively, we have 1
[~] = 7 [F] [S]
[F]r.
(15)
The stretches, by definition, are: R 2~ = --
(16)
r
Z 2z = --.
(17)
Z
3 Case studies
Case A
Unconfined vertical compression (or extension)
Loading a cylindrical sample vertically (Fig. 1) is one of the most common laboratory tests used in engineering. Loading with zero confining pressure is called unconfined compression test. Loading with a constant confining pressure is called conventional triaxial compression test (although only two stresses are controlled independently).
1
z,Z
L
ro
--,=-
(U,)o-~
T
I
(Uz)~kot-~
E C' D (r,z)
Ur
1 Uz
D'(R ,Z)
0
A
Ro
A' .._J --i
t
-r,R
Fig. 1. Vertical compression of a cylindrical sample (cases A and E)
Finite strain elastic closed form solutions We assume that the deformation is homogeneous such that Ro
R
ro
r
Zo
Z
Z0
Z"
(18)
(19)
If the deformation at the top of a cylindrical bar (Fig. 1) is undergoing a motion
(20)
Zo = Zo + kot
where ko is a constant rate, the vertical displacement field then becomes koz Z=z+--t=
(zo + kot~ - z.
Zo
(a) J a u m a n n
\
(21)
Zo /
stress rate solution:
Substituting Eqs. (18)-(20) into Eqs. (10) and (11), we have
-SR
0
0
82
? x , ] __ 0
~
0
0
[C] = k ~ J
0
= [D].
(22)
Applying Eq. (21), we obtain
82
ko
8Z
Zo + kot
(23)
Thus, we h a v e e = =
f 882z d t = l n 2 ~ "
Substituting Eq. (22) into Eq. (1) and noting that 6-~ = 6-o (no rotation is involved in homogeneous uniaxial deformation) and that from the boundary conditions 6-r~= 6-00= 0, we obtain
OR
R
8R
R
-2
a2
2(2 + #) 8Z
-
a2
v--.
8Z
(24)
Thus, e~ = e00 = - v In 2~ and ko azz = f E~ zo + kot
dt
Ey In 2z.
This is exactly the same solution as the logarithmic strain solution.
(25)
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C.S. Wu et al.
(b) t h e 2 n d P i o l a - K i r c h h o f f
stress solution"
Applying Eqs. (18), (19), (21) and (8), we have the deformation gradient matrix
[iR
o
[F] =
2z
0
:]
(26)
2~
and the Jacobian OR J = det IFI = ~ 9o,9~.
(27)
The Cauchy stress matrix in cylindrical coordinates for vertical loading only is
[a] =
a=
(28)
.
0 Substituting Eqs. (26), (27) and (28) into Eq. (14), the 2nd Piola-Kirchhoff stress in axial deformation can be obtained as:
fo_ 8R
[S] = J [ F 1] [a] [F 1]r =
~ .
a=
(29)
.
0
Similarly, substituting Eq. (26) into Eq. (9), the Green-Lagrangian strain matrix is obtained as: m
7 \\0r/
1 [E] = ~- ([F]r [F] - [I]) :
- 1
0
0
o
1 7 (~z2 - 1)
o
o
o
1 (2,. 2 1) 7 -
(30)
By substituting Eqs. (29) and (30) into Eq. (5), it is straightforward to obtain Err + Eoo ~?R R = --2vE~z, Err = Eoo such that - 2r and Err = Eoo = - r E = . The 2nd Piola-Kirchhoff 0r r stress solution as well as the Jaumann stress rate solution are given in Table 1. The relationship, 2~e = 1 + v - v2z2 (Table 1) depicts the limits of the applicability of this closed form solution to uniaxial extension. This is because that 2~2 must remain greater than zero so that the u
~
maximum value of 2z is equal to / l + v 2, would be 2.08. X/ v
For example, if v = 0.3, the maximum value of "
Finite strain elastic closed form solutions
7
Table 1. Unconfined vertical compression Constitutive law
Strains
Stresses
(rs = ,~Sijdkk + 21~dij
e~ = In 2~ ~rr ~ ~00 ~
o-= = Ev In 2= --I)~zz
Grr ~ (700 ~ 0
2r = 2z -v S i j = }~)ijEkk 4-
21~E~j Err = Eoo = --vEz~
a~ = Croo = 0
2r2 = 1 +v(1--2= 2)
0.0
-0.2-0.4-~
2
n
d
Piola-Kirchhoff
-0.6-0.8Jaumann
-1.0-1.2. -1.4, -1.6. -1.8 1.0
6.9
d.a
6.z
d.6
d.s
6.4
o.a
o.2
z
Fig. 2. Predictions of vertical stress in unconfined vertical compression
A comparison of normalized vertical stress ratio azz obtained from the two constitutive laws, Ev together with the small strain solution in the case of unconfined vertical compression, is given in Fig. 2. There is no significant difference between the small strain solution and these large strain solutions for stretch 2z larger than 0.95. The 2nd Piola-Kirchhoff stress solution predicts a maximum stress at stretch )~zabout 0.61 and then drops towards zero when 2z approaches zero. This is inconsistent with the expectation from homogeneous elastic compression. We may term this observation "finite elastic yield". The ratio or= from the Jaumann stress rate solution tends to Ey infinity as 25 approaches zero. The Jaumann stress rate solution has more physical meaning (monotonously increasing stress) with respect to homogeneous elastic compression than the 2nd Piola-Kirchhoff stress solution as shown in Fig. 2.
Case B.
Unconfined
radial compression
(or extension)
The deformation mode for unconfined radial compression is shown in Fig. 3. In this case, loading is applied in the radial direction while the vertical direction is free (or held constant). A typical example of this kind of loading condition is the conventional triaxial extension test. The
8
C.S. Wu et al.
z,Z I
Ro ___-.N_(u,~,_.. t B' q
T
I I I
C ,>,
T
(Uz)o B
I
D'(R ,Z)
E E
I
Zo
Zo
.< u~ O(qz) I I I I
0
A' Li -
A
ro
-0.5-
Fig. 3. Radial compression of a cylindrical sample (cases B and F)
_l i
•••__••Ki•,hhoff
-1I. I..
io
-1.5-
-2-
-2.5
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
Fig. 4. Predictions of radial stress in unconfined radial compression
Table 2. Unconfined radial compression Constitutive law
Strains
Stresses
--2v
~ = 26ijdkk + 2#d u
~zz = 0
l--v 8rr
=
200
In 2r
=
(Trr = (500 =
Er In 2r 1--V
2~ = 2r ~-~ Sij = 2(~ijEkk + 2#Eij
E= -
--2v 1--v
0"= = 0
Er, 1
1
Err = Eoo = ~ (2~2 - 1 ) (SZy
\az) * E v=E~+E=+Eoo
=
l+v
1-~
+
--2v
i-
G~ = ~roo = ~ 2~~
8z
(2Ev* + 2#E,,)
Finite strain elastic closed form solutions
9
derivation of unconfined radial compression follows similar procedures as unconfined vertical compression and will not be repeated here. However, the solutions are summarized in Table 2. O'rr
A comparison of normalized radial stress - - obtained from the two constitutive laws, together Ey with small strain solution, is shown in Fig. 4. It is observed that the solution follows a similar trend to the solutions of vertical compression mentioned above, that is, the 2nd Piola-Kirchhoff solution tends to a finite value when the stretch 2~ approaches zero but the J a u m a n n stress rate solution tends to infinity.
Case C." Constrained vertical compression (1-D consolidation)
This type of deformation can be considered as uniaxial loading with zero radial deformation. The deformation mode is shown in Fig. 5. A typical example of this deformation pattern is the one-dimensional consolidation test (odometer test) used in soil mechanics. Adopting the kinematic Eqs. (18)-(21) except that R = r, u~ = 0 a n d / ~ = 0 due to the constrained radial z.Z L
to=
Ro
--I B
(Uz)o=ko
Zo
4
'
!o
B'
0
A
r.R
Fig. 5. Constrained vertical compression of a cylindrical sample (case C)
Table
3. Constrained vertical compression
Constitutive law ~
=
2~)udkk q 2#d u
Strains
Stresses
e= = In 2~
az= = (2 + 2#) In 2~ ~r~ = Croo= 2 In 2~
Err ~ ~00 ~ 0
2r= 1 Sij = 26ijEkk + 2#E u
1
E= = ~ (2z2 -- 1)
azz = G(2 + 2#) E=
Err = Eoo = 0
~rr = GO0 = ~ Ezz
2 Ar z 1
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c.S. Wu et al.
deformation, we have for the rate of deformation du,
1
[D] = ~ ([L]r + [L]) =
a2
~
.
(31)
0 By substituting Eq. (31) into Eq. (1), the Jaumann stress rate solution is determined as shown in Table 3. The deformation gradient is
[~]
.~:
=
(32)
0
and J = 2z. The Green-Lagrangian strain for constrained vertical compression is o 1
[E] = ~ ([F] r [F] - [I]) =
1
~ (A=2 - 1)
.
(33)
0 The application of Eqs. (33) and (5) leads to
[
[S]=
2E= 0
0 (2+2~)Gz
0
0
:;
.
(34)
2E=
Substituting Eqs. (32) and (34) into Eq. (15), we obtain the 2nd Piola-Kirchhoff stress solution as shown in Table 3. For a hypoelastic material, the stretch 2z after deformation, with respect to different tractions, can be determined from 2= = exp [Gz/(,;t + 2/~)] = exp [fi(1 + v) (1 - 2v)/(1 - v)]
(35)
where O-zz
/~ = - - .
Ey
(36)
Similarly, the 2nd Piola-Kirchhoff stress solution is 2=(2~2 - 1) = fi[2(1 + v) (1 -- 2v)/(1 - v)].
(37)
Equations (35) and (37) can be used to determine the consolidation settlement of an elastic soil under large loading pressure. The comparison of these two predictions, together with small strain prediction, is given in Fig. 6. The smaller the stretch 2=, the larger the settlement. As seen from Fig. 6, for a specific vertical traction ratio fi, Jaumann stress rate solution gives the least
Finite strain elastic closed form solutions
11
1.0"
~
0.9"
0.80.7" )..
0.60.5-
II
,.,//
/ , 2rid Piola-Kirchhoff
0.4-
v
0.30.20.10.0
1.0
13.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
().1
0.0
Fig. 6. Predictions of total settlement in constrained vertical compression
settlement while the 2nd Piola-Kirchhoff stress solution predicts the largest settlement. It is also observed that the 2nd Piola-Kirchhoff stress prediction is valid only when/? < 0.28. Otherwise, it predicts negative stretch 2z which is not permissible.
Case D." Constrained radial compression The deformation mode of constrained radial compression is shown in Fig. 7. The derivation of constrained radial compression follows similar procedures as constrained vertical (Case C) and will not be repeated here. The solutions are summarized in Table 4.
Case E." Undrained vertical compression (or extension) The deformation mode of undrained vertical compression is similar to unconfined vertical compression (Fig. 1) except the loading is applied very fast. The undrained triaxial compression test is usually used to determine the undrained shear strength of soils. The test is done by z,Z
~ .,
Ro
"l
-
(U,)o -----,H
g o
zo = Zo
~///////////////////,~ ro
rA"
A
"~.R
--!,
Fig. 7. Constrained radial compression of a cylindrical sample (case D)
12
C. S. Wu et al.
Table 4. Constrained radial compression Constitutive law
Strains
Stresses
(~ = 2bijdl,k + 2#dlj
e= = 0 er~ = eoo = In 2~ 2~=1
o-~ = 22 In 2~ o-~ = Ooo = 2(2 + ,u) In 2~
2#Eij
Sij = ]~g)ijEkk +
E~ = 0
22 ~z = - - E~ .~r2
1 E~ = goo = ~ (2~2 -- 1)
o-~ = aoo = 2(2 + #) E~
2~=1
applying a vertical l o a d i n g keeping the h o r i z o n t a l l o a d i n g c o n s t a n t o n a cylindrical sample. The term " u n d r a i n e d " m e a n s the l o a d i n g is applied very fast a n d the water is n o t allowed to drain out. Thus, if the soil is saturated, volumetric change is zero in u n d r a i n e d l o a d i n g a n d the c o n d i t i o n of incompressibility applies. A d o p t i n g the k i n e m a t i c Eqs. (18)-(21), we o b t a i n
[F] =
o
o
0
2z
0
0
0
2~
(3s)
8R a n d the J a e o b i a n is J = det iF] = 8 r 2r2~. The rate of d e f o r m a t i o n is
T~ [D] =
0
0
o2
0
--
0
0
(39)
0
8Z
R
The c o n d i t i o n of incompressibility d e m a n d s that J = 1 such that 2zR dR = r dr.
(40)
Solving the last e q u a t i o n yields e 2
~r
(41)
= ~-1.
C o m b i n i n g Eqs. (41) a n d (23), we o b t a i n R
8#,
R
OR
- 1 OZ 2
8Z
-- 1
ko
2 zo+kot
(42)
The J a u m a n n stress rate solution can be o b t a i n e d by a p p l y i n g Eqs. (39), (42) a n d (1) a n d is s u m m a r i z e d in Table 5. It is observed, from Table 5, that ]ez~J = 2 [err[ = 2 [eoHa[ a n d [cr2z[ = 2 [Crrrl = 2 Io-0ol. If we apply Terzaghi's principle, that is, a~j = aij + 8ijuw, where cr/tjis the total stress, crij is the effective stress a n d uw is pore water pressure, the excess pore water pressure,
Finite strain elastic closed form solutions
13
Table 5. Undrained vertical compression Constitutive law d/j = .~g)ijdkk q-
2#du
Strains
Stresses
G~ = In 2z
azz = 2# In 22
1 err = e00 = - - In 2~ 2
Gr = aoo = - # In 2~
2 r = )~z2
Ab/w =
E= = ~ (2z2 -- 1)
o-~ = 2~2(.a,Ev + 2,uE=)
-1
Sij = 2g;ijEkk q- 2#Eij
E , = Eoo = ~
)
-- 1
# In
2=
1
G~ = aoo = 2~ (2E~ + 2#E,)
-1
--i
;~ = ;~z~
AUw = ~ - (xE~ + 2ue,3
0.2 J
0.0 r -0.2-0.4-0.6-0.8-1.0-1.2-
-1.4
0.9
6.8
0.7
d.6
0.5
Fig. 8. Predictions of vertical stress in undrained vertical compression Au.,,, generated is # In 2Z ( = - G r ) - However, it s h o u l d be n o t e d that the stress state is d e t e r m i n e d Gzz from the stresses applied at the b o u n d a r y . The predictions of the n o r m a l i z e d effective stress - # are s h o w n in Fig. 8. Again, the 2nd P i o l a - K i r c h h o f f stress s o l u t i o n becomes tensile (positive) d u r i n g c o m p r e s s i o n w h e n the stretch 2z is less t h a n 0.64. This is physically impossible.
Case F: Undrained radial compression (or extension) The d e f o r m a t i o n m o d e of u n d r a i n e d radial compression, similar to u n c o n f i n e d radial c o m p r e s s i o n (Case B), is s h o w n in Fig. 3. F o r a m a t e r i a l subjected to h o m o g e n e o u s l y radial compression, the k i n e m a t i c e q u a t i o n s c a n be a s s u m e d as Ro = ro + kot
(43)
kor (ro + kot~ R = r + u, = r + - - t = r ro \ ro /
(44)
Z = z + uz.
(45)
Rz The c o n d i t i o n of incompressibility also gives the relationship ~5- = 25 -1 (2z = 2r-2).
C. S. Wu et al.
14 Table 6. Undrained horizontal compression Constitutive law
Strains
Stresses
d~ = 26ijd~k + 2#dij
e~ = - 2 In 2~ e,.r = e00 = In 2r
a~_, = - 4 # in ;t~ a~, = aoo = 2# in 2r Au~ = 4# In 2,
Z~ = Z~ -2 1Z 4 -1) E==5(~-
Sq = ZbzjEkk + 2#Eij
1)2
/~r,- = E00 = ~ ( ~
a= = 2~ ~(2E~ + 2#E~) -
1)
a ~ = a00 = Zr2(ZE~ + 2#Err)
Auw = --Zr-4(ZE~ + 2#E~z)
The derivations of strains and stresses are similar to the procedures described in Case E and will not be repeated here. The solutions are given in Table 6. Comparing the Jaumann stress solutions in undrained radial compression (Case F) with undrained vertical compression (Case E), it is observed that (i) the vertical strain is also twice the radial strain and tangential strain as in the case of vertical undrained compression, that is, lez~l = 2 [errl = 2 I~001; (ii) the effective stresses obtained are similar to undrained vertical compression, that is, lazzl = 2 I~r,r[ = 2 ICr00]but the excess pore water pressure is equal to lazzl, not l~r,.]. Case G : A x i s y m m e t r i c f i n i t e s i m p l e shear
Simple shear strain is defined as a plane strain state with zero volumetric change and homogeneous lateral displacement in one direction. The major difference of simple shear from other loading conditions is the axes of principal stresses and strains rotate freely throughout the shearing process. An example of the deformation mode for simple shear in axisymmetry is shown in Fig. 9, where 7 represents the rotation angle of the element and kt = tan 7 = K. Many practical problems in engineering can be simulated as problems of axisymmetric finite simple shear. For examples, in geomechanics, the deformation of soil elements adjacent to the shaft of a frictional pile or the wall of a sample tube [3], [14], or in mechanical engineering, the deformation of a bushing due to a shaft movement can be best simulated by simple shear. For simple shear condition, the solution using rate-type constitutive laws in a rectangular coordinate system was found by many investigators [1], [6]. However, the solution for the above problems which represent simple shear under axial symmetry is not available, based on a literature survey by the authors. Assuming the deformation is homogeneous, the displacement field of axisymmetric simple shear in a cylindrical coordinate system can be described as (Fig. 9) R = r
(46)
Z = z + K r = z + k(r - to) t
(47)
O = 0
(48)
where ro is the reference radius (the radius at point A (or D) in Fig. 9); O and 0 are tangential angles in the cylindrical coordinate system, which are equal due to axial symmetry. The deformation gradient matrix becomes
[F] =
1
=
t
0 and the Jacobian is J
1 0
=
det IF] = 1 implying no volumetric change.
(49)
Finite strain elastic closed form solutions
15
Pile
z,Z
___,q D
f
c
kerr i ~z
o'
E
R,r Fig. 9. TEe deformation mode of axisymmetric finite simple shear (case G)
It is straightforward to d e t e r m i n e that
[L] =
o
(50)
0 and
[!k] !kl] o
[D] = ~ ([L] r + [L]) =
)-
o
0
(51)
0
o ~ o
[W] =
1
([L] T + [L]) =
--
0
0
.
(52)
16
C.S. Wu et al.
Table 7. Simple shear strain
Constitutive law
Strains
Stresses
d~ = 26~jdkk + 2/,dia
&~ = 0 err = 0
a= =/,(1 - cos kt) G~ = - # ( 1 - cos kt)
8oo = 0
0"oo = 0
kt
S~s = 20~;Ekk + 2#E~
~ = -2
a,.~ = # sin kt
E~=0
o= = (2 + 2#) ~ - +
--K4 t2 + 2 # ) K 2
K 2
K 2
E,~ = T
a,.,. =
(2
Eoo = 0
Goo =
--
+
2#) T
2
2
K 2
K
K3
E,.~ = 2
G~ = (2 + 2/,) ~ - + #K
A p p l y i n g Eqs. (51), (52) a n d (1) a n d following a procedure similar to that described by Dienes [6] for a r e c t a n g u l a r c o o r d i n a t e system, we o b t a i n the J a u m a n n stress rate s o l u t i o n for a material with zero in situ stress as shown in Table 7. If the m a t e r i a l were subjected to n o n - z e r o initial stresses, the closed form solution is Orr ~
- - O roz
sin kt + #(cos kt
0 1) + G~
(53)
0 o ~ = 0-,9 sin kt - #(cos kt - 1) + ~=
(54)
r
(55)
= r176
0 G~ = G~ cos kt + # sin kt.
(56)
The G r e e n - L a g r a n g i a n strain t e n s o r Eij is
i
1
K2
[E] = ~ (IF] T [F] - [I1) = 7
0
.
0
(57)
By substituting Eq. (57) into (5), we have
[s] = ~
~K
;~K ~
0
Oo1
(58)
.
)~K2
T h e C a u c h y stresses can be d e t e r m i n e d from Eqs. (49), (58) a n d (15) as
[a]=
I
SK, KS,~ + S~z Sr~ + Sr~ KZSrr + 2KS, z + S= 0
0 1 0
(59)
Soo
where K = kt. The explicit s o l u t i o n is given in Table 7. It is observed, from Table 7, that the J a u m a n n stress rate solution in a x i s y m m e t r y predicts oscillatory shearing stresses similar to the predictions in the r e c t a n g u l a r c o o r d i n a t e system [6].
Finite strain elastic closed form solutions
17
3.o-
2.503 09
2.0-
2nd Piola-Kkchhoff~
rJ) 1.5. Q3 t-. r~ 1.0. 0.5. 0.00.0 l~
o.1
ola
0'.3
0'.4
0'.5
o'.6
o'.r
0'.8
0'.9
1.0
kt
Fig. 10. Predictions of shear stress in axisymmetric finite simple shear
The comparison of normalized shear stress ( ~ ) i s
given in Fig. 10 where one observes that the
prediction of the 2nd Piola-Kirchhoff stress gives much larger shearing stresses compared with the other two solutions. The Jaumann stress rate solution yields a very close solution (within 8%) to the small strain solution up to a rotation angle equal
4
(kt = 1). When kt is larger than ~, the
Jaumann stress rate solution oscillates but the 2rid Piola-Kirchhoff stress solution keeps increasing. Dienes [6] suggested that the Jaumann stress rate solution should be applied only when the shear strain is tess than 40%. It appears that, for simple shear, the 2nd Piola-Kirchhoff stress solution may yield a better agreement with observed material behavior than the Jaumann stress rate.
Case H .
Undrained cylindrical cavity expansion
Assuming that the wall of a cylindrical cavity is expanding with a constant velocity k0 such that (Fig. 11) Ro = ro + kot,
(60)
the displacement field at an arbitrary radius can be described as R = r + ur(r, t)
(61)
z = z
(62)
where u~ is the displacement in the radial direction, R and Z are deformed coordinates while r and z are reference coordinates.
18
C.S. Wu et al.
~,
R~
..,----Ro .~-- ro - , . ~
U,o,. 4
Fixed
bounda~
I I I
1 I
