Mechanical behavior of two kinds of rubber materials

\ PERGAMON

International Journal of Solids and Structures 25 "0888# 4434Ð4447 www[elsevier[com:locate:ijsolstr

Mechanical behavior of two kinds of rubber materials Y[C[ Gaoa\\ Tianjie Gaob a

b

Northern Jiaoton` University\ 099933\ Beijin`\ P[R[ China Department of Physics\ University of Pennsylvania\ PA 08093!5285\ U[S[A[ Received 5 December 0886^ in revised form 01 August 0887

Abstract Two kinds of elastic laws have been adopted by Gao "0889 and 0886# to analyze the deformation _elds near a crack tip[ One of them contains the response to volume change and shape change^ the other contains the response to extension and compression[ In this paper the two kinds of constitutive relations are examined by typical large deformation\ and the restrictions on constitutive parameters are discussed[ Þ 0888 Elsevier Science Ltd[ All rights reserved[

0[ Introduction An important problem in nonlinear elastic theory is to give a reasonable and applicable elastic law[ Many attempts have been made to develop a theoretical stressÐstrain relation that can _t experimental results for highly elastic materials[ However\ in general\ the strain energy function is complicated[ Ogden "0861a\ b# proposed a form of strain energy which is a linear combination of strain invariants[ An excellent agreement between Ogden|s formula and Treloar|s "0847# exper! imental data up to 6 times extension is obtained[ When we consider a problem with singular point "such as crack tip\ concentrated force#\ the situation is di}erent from the ordinary _nite deformation case[ Actually\ near a singular point in rubber like material\ the strain has a tendency to go to in_nity\ that renders the problem compli! cated[ Theoretical analysis requires the elastic law to be expressed as simple as possible[ However\ the simplicity often violates rationality[ In order to re~ect the material behavior near a singular point\ i[e[ strain tends to in_nity\ two kinds of constitutive relations were introduced by Gao "0889 and 0886#^ the earlier one represents the response to shape change and to volume change^ the latter represents the response to tension and compression[ These two constitutive relations were successfully used in the analysis of a singular point by Gao and Gao "0883#\ Gao and Shi "0884#\ Gao and Liu "0884\ 0885#\ Gao and Gao "0885#\ Gao "0886#[ Furthermore\ these two kinds of

 Corresponding author[ Tel[] 99 75 09 52139316^ fax] 99 75 09 51144560^ e!mail] ycgaoÝcenter[njtu[edu[cn 9919!6572:88:, ! see front matter Þ 0888 Elsevier Science Ltd[ All rights reserved PII] S 9 9 1 9 ! 6 5 7 2 " 8 7 # 9 9 1 4 8 ! 4

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Y[C[ Gao\ T[J[ Gao : International Journal of Solids and Structures 25 "0888# 4434Ð4447

strain energy were directly used in _nite element calculation and were shown to be stable by Zhou and Gao "0887# in the Total Lagrangian "L[T[# method\ the maximum tension of a line element reached 49 times[ It is well known that every constitutive relation contains some parameters that are restrained in certain value reach to ensure the material behavior to be reasonable[ In the present paper the two elastic laws will be examined by typical load[ The constitutive parameters will be discussed in detail[ Finally\ these two elastic laws are compared with Ogden|s "0861a\ b# formulae and Knowles et al[ "0862# formulae[

1[ Basic formulae Let P and Q denote the position vectors of a material point before and after deformation\ respectively\ xi "i  0\ 1\ 2# is the Lagrangian coordinate[ Two sets of local triads are de_ned as follows\ Pi 

1P

1Q \ Q  i 1xi 1xi

"0#

The displacement gradient tensor is F  Qi & P i

"1#

Where P i is the conjugate of Pi\ & the dyadic symbol\ and the summation rule is implied[ The right and left CauchyÐGreen strain tensors are D  F T = F\ d  F = F T

"2#

where superscript T indicates transposition[ D and d possess the same invariants such as I0  D ] U  d ] U\ I1  D ] D  d ] d\ I2  D 1 ] D  d 1 ] d\ [ [ [

"3#

I−0  D −0 ] U  d −0 ] U\ I−1  D −0 ] D −0  d −0 ] d −0 \ [ [ [

"4#

in which U denotes unit tensor\ ] denotes duel multiplication[ Among these invariants\ there are only three independent[ Besides\ a common used invariant is K\ K  05 "I 02 −2I0 I1 ¦1I2 #

"5#

Let li denote the values of principal strain\ then I0  l01 ¦l11 ¦l21 \ I−0  l0−1 ¦l1−1 ¦l2−1 \ K  l01 l11 l21

"6#

For isotropic material\ the strain energy per unit undeformed volume can be expressed by three independent invariants\ for example\ W  W"I0 \ I1 \ K# The Kirchho} stress is

"7#

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s1

1W 1D

"8#

then the Cauchy stress is t  K −0:1 F = s = F T

"09#

2[ Two elastic laws From physics point of view\ the necessary conditions for a reasonable constitutive relation of a solid can be stated in two ways] "0# A material element must possess sti}ness to resist both shape change and volume change[ "1# A material element must possess sti}ness to resist both extension and compression[ According to statement "0#\ a strain energy formula that only contains two terms was proposed by Gao "0889#\ Wa

n

0 1 I0

K

0:2

¦b"K−0# m K −q

"00#

where a\ b\ n\ m\ q are positive constitutive parameters m should be an even integer[ Noting eqn "6#\ it is found that the _rst term of eqn "00# only depends on the ratio of li but not on their magnitudes\ therefore\ it only re~ects pure shape change^ the second term of eqn "00# re~ects pure volume change[ Equations "8#Ð"00# can be used to give n

6 0 10

t  1K −0:1 na

I0

K 0:2

1

0

17

q d U m − ¦b"K−0# m K 0−q U − I0 2 K−0 K

"01#

Evidently\ the _rst term in the brackets is a stress deviator while the second term is a hydrostatic stress[ According to statement "1#\ another strain energy formula that is even simpler was proposed by Gao "0886#\ N # W  A"I 0N ¦I −0

"02#

Using equations "8#\ "09# and "02# we obtain N−0 −0 d # t  1NAK −0:1 "I 0N−0 d−I −0

"03#

3[ Material behavior according to eqn "01# 3[0[ Small strain case Let o  01 "d−U#\ e  o ] U Assuming o to be small\ eqn "01# is reduced to

"04#

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0

1

e t  3na2n−0 o− U ¦1m bmem−0 U 2

"05#

For m  1\ the linear elastic relation is obtained\ and it follows that n

1b−n2n−1 a 3b¦n2n−1 a

\ E

7n2n ab 3b¦n2n−1 a

"06#

where E and n are Young|s modulus and Poisson|s ratio\ respectively[ Evidently\ n ³ 9[4 is automatically satis_ed[ The condition n × 9 requires that n b × 2n−1 a 1

"07#

3[1[ Uniaxial stress Without loss of generality\ we consider a cubic material element with arrises of unit length as shown in Fig[ 0"a#[ Under the action of normal stress t00 along x0!direction\ the arris lengths become l\ m and m\ respectively\ but the arrises still remain perpendicular\ as shown in Fig[ 0"b#[ Let ei "i  0\ 1\ 2# denote the unit vectors along xi!direction\ then according to eqns "1#Ð"6#\ it follows that F  le0 & e0 ¦m"e1 & e1 ¦e2 & e2 #

"08#

d  l1 e0 & e0 ¦m1 "e1 & e1 ¦e2 & e2 #

"19#

d −0  l−1 e0 & e0 ¦m−1 "e1 & e1 ¦e2 & e2 #

"10#

I0  l1 ¦1m1 \ I−0  l−1 ¦1m−1 \ K  l1 m3

"11#

Further\ eqn "01# becomes

Fig[ 0[ "a# A material cubic element[ "b# Uniaxial load[

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t

1

6

na"l1 −m1 #

lm1 2"l1 ¦1m1 #

3:2

$0 1 l m

¦1

1:2 n

01 % m l

"1e0 & e0 −e1 & e1 −e2 & e2 #

7

¦bð"m−q#l1 m3 ¦qŁ"l1 m3 −0# m−0 "lm1 # −1q U

"12#

Considering t11  t22  9\ eqn "12# gives na"l1 −m1 # 2"l1 ¦1m1 #

3:2

$0 1 l m

¦1

1:2 n

01 % m l

−bð"m−q#l1 m3 ¦qŁ"l1 m3 −0# m−0 "lm1 # −1q  9

"13#

then eqn "12# becomes t

1na "l1 −m1 # lm1 l1 ¦1m1

3:2

$0 1 l m

¦1

1:2 n

01 % m l

e0 & e0

"14#

When l Ł 0"m ð 0#\ eqn "13# can be used to give 2 3"2s¦n#

0 1

na m 2sb

2s−1n

l− 1"2s¦n#

"15#

where "16#

s  m−q eqn "15# shows that when l :  if m : 9\ the following condition is required s × 1n:2

"17#

Substituting eqn "15# into "14#\ it follows that\ t  1"na# a "2sb# 0−a l1na e0 & e0

"18#

in which 2 1s−0 a = 1 2s¦n

"29#

In order to ensure that t :  when l : \ s × 9[4 is required[ The total load acting on the element is 2s

n

L  m1 t00  1"na# 2s¦n "2sb# 2s¦n l

2s"1n−0#−n 2s¦n

"20#

eqn "20# shows that if L increase with l\ the following condition is required\ s×

n 2"1n−0#

The work to extend the specimen is

"21#

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W

g

l

L dl c

0

2s n 5sn 2s¦n "na# 2s¦n "2sb# 2s¦n l2s¦n 2sn 

"22#

Evidently\ when l : \ W : [ When l ð 0"m Ł 0#\ eqn "13# gives m

2 1"5q¦n#

0 1 5qb n

1 na

2q−n

l− 5q¦n

"23#

eqn "23# shows that when l ð 0 if m Ł 0 the following is required\ "24#

q × n:2 Substituting eqn "23# into "14# it follows that t  −"1n na# b "5qb# 0−b l−nb e0 & e0

"25#

where b

2"1q¦0# 5q¦n

"26#

The total load acting on the element is 5q

n

L  m1 t00  −"1n na# 5q¦n "5qb# 5q¦n l−

5q"n¦0#¦n 5q¦n

"27#

eqn "27# shows that the material element is always stable since n\ q × 9[ The work to compress a specimen to become a plate with thickness l is W

g

l

L dl c

0

5q n −5qn 5q¦n n 5q¦n "5qb# 5q¦n l5q¦n "1 na# 5qn 

"28#

3[2[ Biaxial stress Consider the same cubic material element as shown in Fig[ 0"a#[ Under the action of normal stress t11 and t12 "t11#\ the arris length becomes l\ m and m\ respectively\ as shown in Fig[ 1[ Equations "08#Ð"12# are still valid but the relation of l and m must be given by t00  9\ i[e[

Fig[ 1[ Biaxial load[

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1na"l1 −m1 # 2"l1 ¦1m1 #

3:2

$0 1 l m

¦1

1:2 n

01 % m l

4440

¦bð"m−q#l1 m3 ¦qŁ"l1 m3 −0# m−0 "lm1 # −1q  9

"39#

then eqn "12# is reduced to t

1na "m1 −l1 # lm1 l1 ¦1m1

3:2

$0 1 l m

¦1

1:2 n

01 % m l

"e1 & e1 ¦e2 & e2 #

"30#

When m Ł 0"l ð 0#\ eqn "39# gives 2 1"2s¦n#

1n na l 2sb

0 1

5s−n

m− 2s¦n

"31#

t "1n na# a "2sb# 0−a m1na "e1 & e1 ¦e2 & e2 #

"32#

then eqn "30# becomes

where the a is given by eqn "29#[ The total load on one side is 2s

n

2s"1n−0#−n 2s¦n

L  lmt11  lmt22 "1n na# 2s¦n "2sb# 2s¦n m

"33#

eqn "33# shows that if L increase with m\ the condition "21# is required[ The work to extend the specimen is W1

g

m

L dm c

0

2s n 5sn 2s¦n n 2s¦n "2sb# 2s¦n m2s¦n "1 na# 2sn 

"34#

When m ð 0"l Ł 0#\ eqn "39# gives 2 1"2q¦1n#

0 1

2qb l 1na

m−

1"2q−n# 2q¦1n

"35#

eqn "35# shows that if eqn "24# is satis_ed\ l :  when m : 9[ Substituting eqn "35# into "30# it follows that t  −"1na# g "2qb# 0−g m−3ng "e1 & e1 ¦e2 & e2 #

"36#

where g

2"1q¦0# 1"2q¦1n#

"37#

The load on one side is 2q

1n

L  lmt11  −"1na# 2q¦1n "2qb# 2q¦1n m− The work to compress the specimen is

2q"3n¦0#¦1n 2q¦1n

"38#

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W1

g

m

L dm c

0

2q 1n 01qn 2q¦1n "1na# 2q¦1n "2qb# 2q¦1n m− 2q¦1n 5qn

"49#

4[ Material behavior according to eqn "03# 4[0[ Small strain case From eqn "04# it follows that d −0  U−1o\ I−0  2−1e

"40#

Substituting eqns "04# and "40# into "03# it follows that

$

%

"41#

N−0 7N 1 N 2 A\ n  1N¦0 1N¦0

"42#

t  7NA2N−0 o¦

N−0 eU 2

then E

The condition n × 9 requires N×0

"43#

4[1[ Uniaxial stress Consider the specimen shown in Fig[ 0[ Equations "08#Ð"12# are still valid\ then eqn "03# gives t

1NA 1

lm

N−0 −1 ð"I 0N−0 l1 −I −0 l #e0 & e0 ¦"I 0N−0 m1 −I 0N−0 m−1 #"e1 & e1 ¦e2 & e2 #Ł

"44#

The conditions t11  t22  9 give "l1 ¦1m1 # N−0 m1 −"l−1 ¦1m−1 # N−0 m−1  9

"45#

then eqn "44# becomes t

1NA lm1

0

I 0N−0 l1 −

m3 l1

1

e0 & e0

"46#

When l Ł 0"m ð 0#\ eqn "45# gives N−0

1

m  11"N¦0# l−0¦ N¦0

"47#

If m : 9 when l : \ the condition "43# is required[ From eqns "11# and "47# we can see\ if K increase with l the following condition is required\

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N³2

4442

"48#

Equations "46# and "47# can be combined to give 1

3

t  1N¦0 ANl1N¦0− N¦0 e0 & e0

"59#

The total load is L  m1 t00  1NAl1N−0

"50#

The work to extend the specimen is W

g

l



L dl c Al1N

0

"51#

When l ð 0"m Ł 0#\ eqn "45# gives 1

N−0

m  1− 1"N¦0# l−0¦ N¦0

"52#

then eqn "46# is reduced to 1N

3

t  −1N¦0 ANl−1N¦0− N¦0 e0 & e0

"53#

The total load is L  m1 t00  −1NAl−1N−0

"54#

The work to compress the specimen is W

g

l



0

L dl c Al−1N

"55#

4[2[ Biaxial stress Consider the cubic material element as shown in Fig[ 0"a#\ under the action of t11 and t22 "t11# it is deformed as shown in Fig[ 1[ Using the condition t00  9 and eqn "44# we obtain "l1 ¦1m1 # N−0 l1 −"l−1 ¦1m−1 # N−0 l−1  9

"56#

then eqn "44# is reduced to t

1NA lm1

0

I 0N−0 m1 −

l3 m1

1

"e1 & e1 ¦e2 & e2 #

"57#

When m Ł 0"l ð 0#\ eqn "56# gives N−0

1

l  1− 1"N¦0# m−0¦ N¦0 Then eqn "57# is reduced to

"58#

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N−0

t  1N¦ 1"N¦0# ANm1N−0− N¦0 "e1 & e1 ¦e2 & e2 #

"69#

The load on the side is L  lmt11  NA1N m1N−0

"60#

The work to extend the specimen is W1

g

m

0

L dm c 1N Am1N

"61#

When m ð 0"l Ł 0#\ eqn "56# gives 1

N−0

l  11"N−0# m−0¦ N¦0

"62#

then "57# is reduced to 1

N−0

t  −1N− 1"N¦0# ANm−1N−0− N¦0 "e1 & e1 ¦e2 & e2 #

"63#

The load on the side is L  lmt11  −1N NAm−1N−0

"64#

The work to compress the specimen is W1

g

m



0

L dm c 1N Am−1N

"65#

5[ An example We consider a spherical rubber membrane subject to internal pressure p as shown in Fig[ 2[ Let "R\ U\ F# and "r\ u\ 8# denote the spherical coordinates for the framework before and after deformation\ respectively[ The deformation is described by the mapping function

Fig[ 2[ "a# A thin spherical membrane[ "b# Spherical coordinates[

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uU

6

r  f"R#

\ 8F

4444

"66#

Let ei "i  r\ u\ 8# denote the unit vectors along the coordinate lines r\ u and 8\ then PR  er \ PU  R sin Feu \ PF  Re8 QR 

df e  f ?er \ QU  r sin Feu \ QF  re8 dR r

"67# "68#

therefore\ d  l1 er & er ¦m1 "eu & eu ¦e8 & e8 #

"79#

where l  f ?\ m 

f R

"70#

Then I0\ I−0 and K are given by eqn "11#[ It is assumed that initial thickness H is much smaller than the radius of the membrane R9\ so that the strain and stress along the thickness can be considered as constants\ i[e[ the values of l and m in eqn "70# are constants[ Further\ since H ð R9\ we assume that trr ð tuu "t88 #

"71#

then the equilibrium condition for the membrane can be written as r tuu ¦t88 − p  9 h

"72#

where h is the thickness after deformation\ h  Hf ?  lH

"73#

The analysis of deformation will be given according to two elastic laws of eqn "01# and eqn "03#\ respectively[ 5[0[ Accordin` to eqn "01# From eqns "30#\ "72#\ "73#\ and "70# it follows that 1na m1 −l1 1

1

1

lm l ¦1m

3:2

$0 1 l m

¦1

1:2 n

01 % m l



mRp 1lH

"74#

When m Ł 0\ eqn "30# becomes eqn "32# then eqn "74# is reduced to 2s n 2 Rp "1n na# 2s¦n "2sb# 2s¦n m2s¦n ðs"1n−2#−nŁ 1H

If p increase with m\ the following condition is required

"75#

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n × 2:1\ s ×

n 1n−2

"76#

The work to swell up the membrane is W  3pR 2

g

m



m1 p dm c 3pR 1 H

0

2s n 5sn 2s¦n n 2s¦n "2sb# 2s¦n m2s¦n "1 na# 2sn 

"77#

5[1[ Accordin` to eqn "03# From eqns "46#\ "72#\ "73# and "70# it follows that 1NA 1

lm

0

I 0N−0 m1 −

l3 1

m

1



mRp 1lH

"78#

When m Ł 0\ eqn "78# is reduced to Rp  1N NAm1N−2 1H

"89#

If p increase with m it is required that N × 2:1

"80#

The work to swell up the membrane is W  3pR 2

g

m

0



m1 p dm c 3p1N R 1 HAm1N

"81#

6[ The relation of eqns "00# and "02# with other forms of energy Ogden "0861a\ b# proposed a quite general but convenient form of strain energy\ using the notation of this paper\ it can be written as W  s mi f"ai #¦F"K#

"82#

i

where K is given in "6#\ mi are constants\ f"ai# are strain invariants with exponent ai that may not be integer\ f"ai # "l0ai ¦l1ai ¦l2ai −2#:ai

"83#

If F 0 9\ and only two terms in "82# are taken\ let\ a0  1\ a1  −1\ m0  1A\ m1  −1A

"84#

then "82# becomes an equivalent form of "02# for the case of N  0[ Ogden "0861a\ b# obtained a su.cient condition for satisfying Hill|s constitutive inequality\

Y[C[ Gao\ T[J[ Gao : International Journal of Solids and Structures 25 "0888# 4434Ð4447

mi ai × 9

"each i\ no summation#

4446

"85#

Evidently\ eqn "02# meets this condition for N  0[ As for the general case\ 0 ³ N ³ 2\ eqn "02# is not equivalent with "82#\ the Hill|s inequality is di.cult to discuss\ but the analysis in this paper directly revealed the reasonable reach of N[ From Ogden|s original idea\ see eqn "3# of Ogden "0861b#\ the general form of "82# can be written as W  s mi f"ai # = K gi ¦F"K#

"86#

i

If only one f"ai# is taken and a0  1\ g0  −0:2\ m0  1a\ let F  b"K−0# m K −q ¦2aK −0:2

"87#

then "86# becomes a special case of "00#\ i[e[ n  0[ Therefore\ both eqns "00# and "02# possess some common feature with the energy form given by Ogden "0861a\ b#[ Knowles and Sternberg "0862# proposed another form of energy and has been used by many authors\ W "AI0 ¦BJ¦CI0 J −1 # N

"88#

where J  K 0:1

"099#

The Hill|s inequality was not discussed by Knowles and Strenberg "0862# for "88#[ The interesting fact is that the same essential solutions were obtained according to di}erent energy forms "00#\ "02# and "88#\ see Gao and Gao "0888#[ The inherent relations between "00#\ "02#\ "86# and "88# should be further discussed in concrete application[

Acknowledgement This work is supported by the National Science Foundation of China "No[ 08461990#[

References Gao\ Y[C[\ 0889[ Elastostatic crack tip behavior for a rubber like material[ Theoretical and Appl[ Fract[ Mech[ 03\ 108Ð 120[ Gao\ Y[C[\ 0886[ Large deformation _eld near a crack tip in rubber!like material[ Theoretical and Appl[ Fract[ Mech[ 15\ 044Ð051[ Gao\ Y[C[\ Gao\ T[J[\ 0888[ Analytical solution to a notch tip _eld in rubber!like materials under tension[ Int[ J[ Solids Structures 25\ 4448Ð4460[ Gao\ T[S[\ Gao\ Y[C[\ 0883[ A rubber wedge under compression of a line load[ Int[ J[ Solids Structures 20 "06# 1282Ð 1395[

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Y[C[ Gao\ T[J[ Gao : International Journal of Solids and Structures 25 "0888# 4434Ð4447

Gao\ Y[C[\ Gao\ T[S[\ 0885[ Notch!tip _eld in rubber!like materials under tension and shear mixed load[ Int[ J[ Fracture 67\ 172Ð187[ Gao\ Y[C[\ Liu\ B[\ 0885[ Stress singularity near the notch tip of a rubber like specimen under tension[ Eur[ J[ Mech[\ A:Solids 04 "1#\ 088Ð100[ Gao\ Y[C[\ Liu\ B[\ 0884[ A rubber cone under the tension of a concentrated force[ Int[ J[ Solids Structures 21 "00#\ 0374Ð0382[ Gao\ Y[C[\ Shi\ Z[F[\ 0884[ Large strain _eld near an interface crack tip[ Int[ J[ Fracture 58\ 158Ð168[ Knowles\ J[K[\ Sternberg\ E[\ 0862[ An asymptotic _nite deformation analysis of the elastostatic _eld near the tip of a crack[ J[ of Elasticity 2\ 56Ð096[ Ogden\ R[W[\ 0861a[ Large deformation isotropic elasticity*on the correlation of theory and experiment for incom! pressible rubber!like solids[ Proc[ R[ Soc[ Lond[ A[ 215\ 454Ð473[ Ogden\ R[W[\ 0861b[ Large deformation isotropic elasticity*on the correlation of theory and experiment for com! pressible rubber!like solids[ Proc[ R[ Soc[ Lond[ A[ 217\ 456Ð472[ Treloar\ L[R[G[\ 0847[ The Physics of Rubber Elasticity\ 1nd ed[ Oxford University Press[ Zhou\ Z[\ Gao\ Y[C[\ 0887[ Analysis and _nite element calculation of crack tip _eld in rubber like material under tension and shear mixed load\ to be published[