Nonlinear effects in a plane problem of the pure bending of an elastic rectangular panel

International Journal of Engineering Science xxx (2014) xxx–xxx

Contents lists available at ScienceDirect

International Journal of Engineering Science

journal homepage: www.elsevier .com/locate / i jengsci

Nonlinear effects in a plane problem of the pure bendingof an elastic rectangular panel

http://dx.doi.org/10.1016/j.ijengsci.2014.02.0230020-7225/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Tel.: +7 9043402211.E-mail addresses: [email protected] (M. Karyakin), [email protected] (N. Shubchinskaya).

Please cite this article in press as: Karyakin, M., et al. Nonlinear effects in a plane problem of the pure bending of an elastic rectpanel. International Journal of Engineering Science (2014), http://dx.doi.org/10.1016/j.ijengsci.2014.02.023

Mikhail Karyakin ⇑, Vitaliy Kalashnikov, Nataliya ShubchinskayaSouthern Federal University, Rostov-on-Don, Russia

a r t i c l e i n f o

Article history:Received 3 January 2014Accepted 13 February 2014Available online xxxx

Dedicated to Prof. Leonid M. Zubov on theoccasion of his 70th birthday.

Keywords:Pure bendingLarge strainsStabilityBifurcations

a b s t r a c t

The paper presents a modification of the semi-inverse representation of the pure bendingdeformation of a prismatic panel with rectangular cross-section that is suitable for themethod of successive approximations (or Signorinis expansion method). This modificationwas used to investigate with an accuracy to second order terms the plane problem of purebending for three different models of nonlinearly elastic behavior: harmonic material, Blatzand Ko material, Murnaghan material. It was found that the typical diagram of bending hasmaximum point followed by falling region. To study the stability of the bent panel thebifurcation approach was used. Some results about the position of bifurcation points atthe loading diagram depending on the material and geometrical parameters are presented.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

The bending as well as tension and torsion is one of the main type of a deformation of construction elements so manyproblems on different types of bending were thoroughly investigated. Within the framework of the linear elasticity the prob-lem of bending of a prismatic body was solved by Saint-Venant (1856). Later the problem was generalized to different bodyshapes, loading types and material properties in the case of small strains.

As to the finite strains the title problem has been studied extensively for incompressible materials for many years, datingback at least to the contributions of Rivlin (1949). For incompressible materials the transformation describing bending of arectangular bar into the cylindrical panel is universal, see e.g. Saccomandi (2001), that means it satisfies the equilibriumequations for every type of isotropic constitutive relation. Different aspects of the stability of the incompressible bent panelwas studied by Triantafyllidis (1980); Haughton (1999), Coman and Destrade (2008) and Destrade, Gilchrist, and Murphy(2010).

The corresponding compressible problem has received less attention, although there is a clear analysis in the book byOgden (1984), which includes an analytical solution for a particular class of materials. Exact nonlinear solution of a planeproblem on pure bending of a bar in the case of large strains based upon the harmonic type material model was publishedin the classical treatise of Lurie (1970). Analytical expression describing bar bending for the Blatz and Ko compressible mate-rial model was presented by Carroll and Horgan (1990). There are also nice discussions in the papers (Aron & Wang, 1995;Bruhns, Xiao, & Meyers, 2002).

angular

http://dx.doi.org/10.1016/j.ijengsci.2014.02.023

mailto:[email protected]

mailto:[email protected]


http://www.sciencedirect.com/science/journal/00207225

http://www.elsevier.com/locate/ijengsci


2 M. Karyakin et al. / International Journal of Engineering Science xxx (2014) xxx–xxx

The spatial nonlinear theory of pure bending of a prismatic bar was developed by Zelenina and Zubov (2000). In theirworks three-dimensional problem of bending was reduced to the two-dimensional nonlinear boundary value problem forthe plane region having the shape of the bar cross-section. Linearized problem on the bending of a pre-stressed prismaticbar within the framework of the superimposing the small strains on the large one was solved by Zubov (1985).

As to an investigation of the effects of the second order during bending the authors usually analyze the case of initiallybent bodies (see e.g. Batra, Dell’Isola, & Ruta, 2005). Such ‘‘ignoring’’ the case of a straight rod may be connected with somespecific feature of traditional semi-inverse relations describing the bending deformation. These relations are valid for everynon-zero value of bending parameter (e.g. the panel curvature) but don’t admit direct transition to the undeformed state ofthe panel. This makes the application of the successive approximations method difficult because of the absence of the firstterm in the expansion.

Present paper first of all clearly defines the problem. On the basis of analysis of the exact solution of the bending problemfor harmonic material a modification of the semi-inverse representation of the bending deformation is proposed. The feasi-bility of this modification is confirmed by investigation second order effects for three different models of nonlinearly elasticbehavior: harmonic material, Blatz and Ko material, Murnaghan material. Analytical expression for the relative change of thepanel thickness is obtained. It can be useful for experimental determination of the elastic modulus of the second order.

The numerical analysis of this boundary value problem for a function that describes the radius of a point in the deformedstate shows that the diagram of the bending (the graph of the bending moment with respect to angle of bending) has a max-imum point followed by a falling segment. So the question about the stability of the obtained solutions arises. To study thestability the bifurcation approach based upon the linearization of the equilibrium equations was used. The bifurcation pointis assumed to be identical with the point of existence of the non-trivial solutions of linearized uniform boundary value prob-lem. The influence of geometrical parameters of the panel upon the bifurcation points distribution along the loading diagramwas studied numerically.

2. Rectangular block bending. General relations

Let us consider nonlinearly elastic panel with rectangular cross-section �a=2 6 x 6 a=2;�h=2 6 y 6 h=2 in the unde-formed configuration (Fig. 1 (a)). We denote Cartesian coordinates in the reference configuration as x; y; z and choose cylin-drical coordinates r;u; z as coordinates in the actual (deformed) state. Correspondent unit vectors ex, ey; ez and er ; eu; ez arealso shown in Fig. 1. We assume no displacement in the z-direction though the stresses in that direction are not zero thatcorresponds to the case of the plane strain (Lurie, 1970). The bending of the panel is described by following semi-inverserepresentation (see Fig. 1)

Pleasepanel.

r ¼ rðxÞ; u ¼ By; ð1Þ

where B ¼ c=h ¼ const. Transformation (1) turns the rectangular cross-section into the sector of the circular ring so itdescribes the bending of the panel. At this bending the cross-sections y ¼ const stays plane suffering the distortion alongthe direction er that is characterized by rðxÞ and rotation around vector eZ through an angle of By. The value q ¼ 1=B is adistance between the origin and the position of the rectangle’s center of gravity in the deformed state.

The equilibrium equations in the volume in the absence of body forces have the form

Div P ¼ 0: ð2Þ

Here Div is the divergence operator at the reference configuration, P – non-symmetric Piola–Kirchhoff stress tensor, consti-tutive relation for which can be written as follows

P ¼ @WðFÞ@F

: ð3Þ

(a) (b)

Fig. 1. Plane strain of the rectangular block at the pure bending: (a) reference configuration; (b) deformed state.

cite this article in press as: Karyakin, M., et al. Nonlinear effects in a plane problem of the pure bending of an elastic rectangularInternational Journal of Engineering Science (2014), http://dx.doi.org/10.1016/j.ijengsci.2014.02.023


M. Karyakin et al. / International Journal of Engineering Science xxx (2014) xxx–xxx 3

In (3) WðFÞ is the specific (per unit volume in the reference configuration) potential energy, F – deformation gradient.Assuming that the lateral surface of the panel is free from loads we get following the boundary conditions there

Pleasepanel.

P � exjx¼�a2¼ 0: ð4Þ

Conditions (4) should hold exactly, while the boundary conditions at the ends ðy ¼ �h=2Þ could be satisfied in the integralsense meaning absence of the axial force and the equality of the resulting momentum of the stresses distributed at the end tothe given bending torque.

Calculating basic geometrical characteristics of the strain describing by (1) we find following expressions for deformationgradient F ¼ Grad r

F ¼ r0ðxÞer � ex þ rðxÞB e/ � ey þ ez � ez ð5Þ

and Cauchy strain measure C ¼ FT � F

C ¼ ðr0ðxÞÞ2ex � ex þ ðrðxÞBÞ2ey � ey þ ez � ez: ð6Þ

For isotropic material potential the strain energy W is a function of the principal invariants of the strain measure C

W ¼WðI1; I2; I3Þ

and constitutive relation (3) can be written (Lebedev, Cloud, & Eremeyev, 2010) as follows

P ¼ 2F � @W@I1

Iþ @W@I2ðI1I� CÞ þ @W

@I3I3C�1

� �; ð7Þ

where I – identity tensor. Using expressions (5) and (6) one can easily see that Piola tensor (7) has only three non-zero com-ponents Prx; Pyu; Pzz and equilibrium equations (2) are reduced to the single relation

dPrx

dx� BPyu ¼ 0: ð8Þ

Also we get only one non-trivial equation from boundary conditions (4), exactly

Prxjx¼�a2¼ 0: ð9Þ

The first problem with using transformation (1) in the case of small strains, i.e. at the formal transition B! 0, is the limitvalue of the angular coordinate: it seems from (1) that in this case u! 0. The latter means that the rectangle degeneratesinto the interval r 2 � h

2 ;h2

� �;u ¼ 0. What actually happens is that in this limit the function rðxÞ, which depends upon the

parameter B, increases without bounds so that

limB!0

rðxÞB – 0: ð10Þ

This removes the problem of the region degeneration (the limiting state is a rectangle of finite sizes while the origin of thecoordinate system moves to the infinity), but invokes the other: according to (10) the dependence of the radius r in thedeformed configuration upon the parameter B is singular at the point B ¼ 0. This fact makes impossible the direct usageof the successive approximations method to determine the radius.

3. Solution for the harmonic type material and the modification of the semi-inverse representation

The problem on the plane bending of the nonlinearly elastic bar from harmonic material was solved by Lurie (1970) bymeans of complex-value transformation of coordinates in the reference configuration using harmonic functions of complexvariables. In this paper to solve this problem we use approach based on the semi-inverse the representation (1).

The specific potential energy of harmonic type material introduced by John (1960) is written as a function of the invari-ants of the left tensor of distortion U ¼ C1=2,

W ¼ 12

k tr2ðU� IÞ þ l tr½ðU� IÞ2�: ð11Þ

Piola stress tensor in this case can be written as follows

P ¼ F � ½ðks1 � 2lÞU�1 þ 2lI�; ð12Þ

where s1 ¼ tr U� 3; k;l– elastic modulus (Lame parameters).Expressing the parameter k through shear modulus l and a Poisson’s ratio m we can rewrite expression (12) in a form

P ¼ 2lF � m1� 2m

s1 � 1� �

U�1 þ Ih i

: ð13Þ

Taking into account relationships (5), (6) and (13) the nonzero components of Piola stress tensor become




Pleasepanel.

Prx ¼2l

1� 2mr0ðxÞð1� mÞ þ mrðxÞ B� 1ð Þ;

Pyu ¼ �2l

1� 2mrðxÞBðm� 1Þ � mr0ðxÞ þ 1ð Þ;

Pzz ¼2lm

1� 2mðr0ðxÞ þ rðxÞB� 2Þ:

ð14Þ

The boundary problem (8) and (9) now has the form

r00ðxÞ ¼ rðxÞB2 � 11�m B;

ðm� 1Þr0ðxÞ � mrðxÞBþ 1jx¼�a2¼ 0:

(ð15Þ

It is linear, that allows defining function rðxÞ in an explicit form

rðxÞ ¼ e�BxC1 þ eBxC2 �1

Bðm� 1Þ ; ð16Þ

where

C1 ¼ eBa2m� 1

Be

B2a24 þ 1

� ��1; C2 ¼ ð2m� 1ÞC1:

It is easy to see that expansion of the solution (16) into the series upon the B powers begins with a singular term

rðxÞ ¼ 1Bþ xþ mð4x2 � a2Þ

8ðm� 1Þ Bþ ð4x2 � 3a2Þx24

B2 þ OðB3Þ: ð17Þ

This term 1B ¼ q defines a position of the line which is passing through the center of gravity of a rectangle after deformation.

In other words, it expresses displacement of origin of co-ordinates while the bending angle increases. To separate the sin-gularity in (17) we modify semi-inverse representation (1) as follows

rðxÞ ¼ 1Bþ AðxÞ: ð18Þ

From the mathematical point of view, replacement (18) is standard procedure of allocation of the singularity. Geometri-cally we represent the distance from the point in the deformed state to the origin as a sum of distance between origin and thecenter of the cross-section and a term that describes the change in the thickness.

The modified semi-inverse representation will look like

r ¼ 1Bþ AðxÞ; u ¼ By: ð19Þ

The sense of the initial relation (1) thus remains. Moreover, function AðxÞ in (19) gains physical sense and describes themodification of the thickness at bending. It is visible from (19) that if AðxÞ is bounded function of parameter B in the neigh-borhood of the point B ¼ 0 then at passage to undeformed state of a body (B! 0) the expression ru does not approach tozero, but to the finite limit coinciding with ordinate of a point of the underformed cross-section of the panel.

Boundary value problem (8) and (9) can be solved starting with a relation (19), or by supposing rðxÞ ¼ 1Bþ AðxÞ in the

resultant boundary value problem (15). Strictly speaking, the analytic property (and even boundedness) of the AðxÞ shouldbe proved for each concrete model of a nonlinearly elastic material. The asymptotic scheme constructed below is verified bycomparison with an analytical solution for Blatz and Ko material and with numerical calculations for Murnaghan material.This makes it suitable for determination of the second order effects for the materials described by elastic potentials of rathergeneral kind.

4. Solution by means a method of successive approximations

4.1. A harmonic type material

The method of successive approximations, also known as Signorinis expansion method, is applied, naturally, for studyingthe nonlinear boundary value problems whose solution could not be obtained analytically. In the given section the scheme ofthis method for a problem of pure bending will be completed on an example of the problem (15) which is linear and hasanalytical solution (16) with known expansion (17). It is made intentionally to specify the scheme, to discover possible prob-lems and to plan the methods of their solution. The received technique will be applied further to essentially nonlinearboundary value problems.

With replacement (18) in a boundary value problem (15) we shall search the function of distortion of a cross-sectionshape AðxÞ in the form

AðxÞ ¼ A0ðxÞ þ A1ðxÞBþ A2ðxÞB2 þ � � � ð20Þ




As a result we will obtain a set of following linear boundary value problems

Pleasepanel.

A00nðxÞ ¼ FðA0n;An; xÞ;PðnÞrx ða=2Þ ¼ 0; n ¼ 0;1;2; . . .

PðnÞrx ð�a=2Þ ¼ 0;

8><>:

After solving each of these problems one could discover unknown functions AnðxÞ. Being interested in so-called second ordereffects we would like to limit this series with the n ¼ 2 but further considerations will show that to find solutions forn ¼ 0;1;2 one should analyze at least third or even fourth problem.

Let’s consider a boundary value problem corresponding to zero degree of expansion

A000ðxÞ ¼ 0;

A00ðxÞ � 1jx¼�a2¼ 0:

ð21Þ

The obvious solution of this linear problem is written as follows

A0ðxÞ ¼ xþ C0; ð22Þ

where C0 is a constant of integration which remains indefinite. It happens because the system of boundary conditions in (15)after a linearization (21) becomes linearly dependent. Note that such obstacle does not arise when solving original, generallyspeaking nonlinear, problem and both constants can be determined completely. Some additional condition is required todetermine C0. We will leave it indefinite for a while and consider a problem corresponding to the first degree of expansion

A001ðxÞ ¼m

ðm� 1Þ ;

ðm� 1Þ A01ðxÞ � 1�

� m xþ C0ð Þjx¼�a2¼ 0:

ð23Þ

The solution of a problem (23) can be written as

A1ðxÞ ¼m

m� 1x2

2þ C0x

� �þ C1: ð24Þ

Boundary conditions coincide again, and we get new indefinite constant C1 as a result.Let us consider a problem corresponding to the second degree of B

A002ðxÞ ¼ xþ C0 þ 1; ð25Þ

ðm� 1ÞA02ðxÞ �m2

m� 1x2

2þ C0x

� �� mC1jx¼�a

2¼ 0: ð26Þ

The general solution of Eq. (25) in turn has two arbitrary constants, say, Cð1Þ2 and Cð2Þ2 . Substituting this solution in the systemof boundary conditions (26) we obtain following equations system for the determination of Cð1Þ2 ; Cð2Þ2

ðm� 1ÞCð1Þ2 ¼m2

m� 1� mþ 1

� �a2

8þ mC1 þ

m2

m� 1� mþ 1

� �a2

C0;

ðm� 1ÞCð1Þ2 ¼m2

m� 1� mþ 1

� �a2

8þ mC1 �

m2

m� 1� mþ 1

� �a2

C0;

ð27Þ

that is, in general, incompatible. The condition of its resolvability is the coincidence of right hand parts of Eq. (27) that gives

m2

m� 1� mþ 1

� �a2

C0 ¼ 0:

From here we find that the constant C0 which has appeared at the solution of a problem in a zero approximation (21) is equalto zero. Taking into account this fact we find the solution of (25) and (26) in a form

A2ðxÞ ¼x3

6þ a2ð2m� 1Þ þ 8mðm� 1ÞC1

8ðm� 1Þ2xþ C2: ð28Þ

Similarly, to determine the constant C1 it is necessary to examine a problem for the third approximation, and a consistencyrelation for boundary conditions in this case implies that

C1 ¼ �ma2

8ðm� 1Þ ð29Þ

and the solution has a form




Pleasepanel.

A3ðxÞ ¼m

m� 1x4

24� a2x2

16þ C2x

� �þ C3: ð30Þ

Thus the uncertain constant of the ith approximation can be found from a requirement of resolvability of a boundary-valueproblem for the ðiþ 2Þth approximation.

Such scheme of determination the unknowns is not very convenient as for some materials bulky enough expressions arisewhen solving such boundary value problems. The method of determination of indefinite constants which is based on the factthat the axial force

Q ¼Z a=2

�a=2Pyudx ð31Þ

corresponding to an initial semi-inverse relation is equal to zero seems to be more effective. This equality strictly followsfrom the equilibrium equations: having integrated Eq. (8) through a thickness of a panel
Z a=2
�a=2

dPrx

dxdx� B

Z a=2

�a=2Pyudx ¼ 0;

we see that the first integral is equal to zero due to boundary conditions (9) so the second one, representing axial force, isequal to zero too.

The indefinite constant of the ith approximation can be determined from a condition of absence of a longitudinal force atthe account in it’s expressions terms of ðiþ 1Þth order, i.e. from the requirement that in expansion of axial force on B powersthe expansion factor at Biþ1 is equal to zero. We will illustrate the specified technique with an example of the definition of aconstant C1 and then we will find the constant C2.

For this purpose we write

rðxÞ ¼ 1Bþ A0ðxÞ þ A1ðxÞBþ A2ðxÞB2;

where functions A0ðxÞ; A1ðxÞ; A2ðxÞ are defined above by expressions (22), (24) and (29) respectively, and a constant C2 in(31) is considered indefinite, while the value of the constant C0 ¼ 0 is discovered earlier. By the representation for rðxÞwe build Piola stress tensor components (14) and the expression for axial force Q (31) which is then expanded into the powerseries in B. In this expansion the terms with a power one unit above the expansion degree which constant is required to bedefined are kept and equated to zero. In particular, for definition of a constant C1 of the first approximation we receive theequation

d2Q

dB2 jB¼0 ¼ 0; ð32Þ

or

�8C1ð2m2 � 3mþ 1Þ þ ma2ð2m� 1Þ2ð2m� 1Þðm� 1Þ2

a ¼ 0;

that leads to the expression for the C1 coinciding with (29).Similarly, we will determine a constant C2. We write

rðxÞ ¼ 1Bþ A0ðxÞ þ A1ðxÞBþ A2ðxÞB2 þ A3ðxÞB3;

where functions A0ðxÞ; A1ðxÞ; A2ðxÞ; A3ðxÞ are defined above by expressions (22), (24), (28) and (30) respectively in whichconstants C0; C1 are defined earlier, and constants C2; C3 are unknown. By representation for rðxÞwe again build Piola stresstensor components and expression for longitudinal force at panel bending under the formula (31), keeping items not abovethe third power

Q ¼ �2laC2

m� 1B3 þ OðB4Þ:

The expression d3QdB3 jB¼0 ¼ 0 results in C2 ¼ 0. Thus, we have constructed a solution of a boundary value problem by a meth-

od of successive approximations

AðxÞ ¼ xþ mð4x2 � a2Þ8ðm� 1Þ Bþ ð4x2 � 3a2Þx

24B2 þ OðB3Þ

coinciding under the form with an expansion in a series of the analytical solution (17).As to the general scheme of construction the solution accounting for the second order effects it can be formulated as fol-

lows. At first it is necessary to perform the solution process up to the third approximation with one indefinite constant in it.Constants C0;C1 can be discovered from compatibility conditions on a course of this process. As to the constant C2, it is more




convenient to find it from a condition of absence of longitudinal force without a solution (and even without the formulation)of a problem for the fourth order approximation.

4.2. Blatz and Ko material

Using the procedure featured above, we will construct an asymptotic solution of a problem of bending in the case ofmaterial model presented by Blatz and Ko (1962), expression of a specific strain potential energy for which is given bythe formula

Pleasepanel.

W ¼ l2

I2

I3þ 2

ffiffiffiffiI3

p� 5

� �:

The constitutive relation for the Piola stress tensor has the form

P ¼ lI3

F � I1I� Cþ I323 � I2

� �C�1

� �:

Tensor P components presenting in the equilibrium equation (8) with account for (5), (6) and (18) become

Prx ¼ l AðxÞB� 1A0ðxÞ

þ 1� �

;

Pyu ¼ lA0ðxÞðAðxÞ3B3 þ AðxÞ2B2 þ AðxÞBþ 1Þ � 1

ð1þ AðxÞBÞ3

and boundary problem (8) and (9) has the form

A00ðxÞ ¼ �13

A0ðxÞ4B

AðxÞ3B3 þ 3AðxÞ2B2 þ 3AðxÞBþ 1;

AðxÞB� 1A0ðxÞ

þ 1 jx¼�a2¼ 0:

ð33Þ

It should be noticed here that use of the semi-inverse representation without singularity allocation in this case leads to morecompact boundary value problem, the analytical solution (more precisely speaking, analytical expression of x from r, con-taining two unknown constants subjected to numerical definition) for which was given by Carroll and Horgan (1990). Theasymptotic solution constructed below will be compared to a solution constructed on this analytical basis.

Seeking for the solution of boundary value problem (32) in the form

AðxÞ ¼ A0ðxÞ þ A1ðxÞBþ A2ðxÞB2 þ � � �

and using the procedure developed for the case of harmonic material one could sequentially find

A0ðxÞ ¼ xþ C0;

A1ðxÞ ¼ �x2

6� 1

3C0xþ C1;

A2ðxÞ ¼1354

x3 � 13

a2

3þ C1

� �xþ C2

and then determine constants

C0 ¼ 0; C1 ¼29

a2; C2 ¼ 0:

Thus, the solution of a boundary value problem (32) constructed by method of successive approximations looks like

AðxÞ ¼ x� x2

6� 2

9a2

� �Bþ 13

54x3 � 5

27xa2

� �B2 þ O B3

� �: ð34Þ

Fig. 2 shows good correlation of the asymptotic solution (33) (circles) with the analytical one (solid line). The calculationswere performed for the cross-section h=a ¼ 5 when bending parameter B is equal to 0:1=a.

4.3. Murnaghan model

The solution of the problem for the material model introduced by Murnaghan (1967) is of the greatest interest from thepoint of view of the theory of the second order effects. The elastic potential for this case looks like



−0,5 0,5 1

0,5

−0,5

−1

1

Fig. 2. Function of the cross-section shape distortion, Blatz and Ko material.


Pleasepanel.

W ¼ 14�3k� 2lþ 9

2lþ n

2

� �I1 þ

12

kþ 2l� 3l� 2mð ÞI21 þ 3m� 2l� n

2

� �I2 �mI1I2 þ

16ðlþ 2mÞI3

1 þn2ðI3 � 1Þ

� �ð35Þ

where k, l – Lame parameters of linear elasticity, l;m;n – elastic moduli of second order.Murnaghan model allows complete considering the effects of the second order, the necessity to account them at working

out of modern high-precision equipment is described, e. g. by Stepanenko (1997). Besides, to within the square terms, it ispossible to establish connection between Murnaghan constants and material parameters of the isotropic nonlinearly-elasticmaterials which models are described by functions of principal invariants of a Cauchy strain measure using formulas (Lurie,1990)

n ¼ 8@W0

@I3; l ¼ 4

@

@I1þ 2

@

@I2þ @

@I3

� �2

F

" #0

;

m ¼ n2� 4

@2W@I1@I2

þ @W@I3þ @2W@I1@I3

þ 3@2W@I2@I3

þ @2W

@I23

þ 2@2W

@I22

!0

; ð36Þ

F ¼ @W@I1þ ðI1 � 1Þ @W

@I2þ I3

@W@I3

:

Upper index ‘‘0’’ means that a value relates to the reference configuration, i.e. after differentiation it is assumed thatI1 ¼ I2 ¼ 3; I3 ¼ 1. In particular for the Blatz and Ko model studied above we obtain

m ¼ �5l; n ¼ �8l; l ¼ �l2: ð37Þ

Similar relations can be also received for the harmonic material. The expressions of the derivatives of the term trðU� IÞentering into expression of energy of the harmonic material at the point I1 ¼ I2 ¼ 3; I3 ¼ 1 could be determined by twoways. One of them is differentiation of the explicit formulas given in Zubov and Karyakin (2006) expressing principal invari-ants of a square root from a positively definite tensor through principal invariants of the tensor itself. Another way is thesolution of the linear systems of equations received after consecutive differentiation with respect to I1; I2; I3 up to the thirdderivative the following relations, connecting principal invariants J1; J2; J3 of the left tensor of distortion U with principalinvariants of a Cauchy strain measure

J21 � 2J2 ¼ I1; J2

2 � 2J1J3 ¼ I2; J23 ¼ I3:

Both methods lead to the following expression of Murnaghan parameters for harmonic material

m ¼ �12ðkþ 3lÞ; n ¼ �3l; l ¼ � k

2: ð38Þ

On the basis of semi-inverse representation (18) the equilibrium equation in a bending problem (6) for a material (35) iswritten in a form




Pleasepanel.

A00ðxÞ ¼ Að1Þ

Að2Þ; ð39Þ

where

Að1Þ ¼ A0ðxÞ43lð2m� 1ÞB½BAðxÞ þ 1� þ þ2A0ðxÞ2BðBAðxÞ þ 1Þ½BlAðxÞð2m� 1ÞðBAðxÞ þ 2Þ � lð2m� 1Þ � 2lm� � AðxÞ5B6ð2m

� 1Þð2mþ lÞ � 5AðxÞ4B5ð2m� 1Þð2mþ lÞ � AðxÞ3B4½ð2m� 1Þð16mþ 6lÞ þ 4lðm� 1Þ�

� 2AðxÞ2B3 ð2m� 1Þð4m� lÞ þ 6lðm� 1Þ½ � þ AðxÞB2½3lð2m� 1Þ � 4lð3m� 2Þ� � ½lð2m� 1Þ þ 4ml�B;

Að2Þ ¼ 5A0ðxÞ4ð2m� 1Þð2m� lÞ � A0ðxÞ2½ð2m� 1Þð6lAðxÞ2B2 þ 12lAðxÞB� 6ð2mþ lÞÞ þ 12lðm� 1Þ� � AðxÞ4B4lð2m� 1Þ

� AðxÞ3B34lð2m� 1Þ � 2AðxÞ2B2ðlð2m� 1Þ � 2mlÞ þ 4AðxÞBðlð2m� 1Þ þ 2mlÞ � ð2m� 1Þð2mþ lÞ þ 4lðm� 1Þ

and boundary condition (8) becomes

Prxjx¼�a2¼ A0ðxÞ5ð2m� 1Þð2mþ lÞ þ A0ðxÞ3½ð2m� 1Þð2lAðxÞ2B2 þ 4lAðxÞB� 2l� 4mÞ þ 4lðm� 1Þ�

þ A0ðxÞ½ð2m� 1ÞðlAðxÞ4B4 þ 4lAðxÞ3B3 þ 2mþ lÞ � 4lðm� 1Þ�þ 2AðxÞ2B2ðlð2m� 1Þ � 2mlÞ � 4AðxÞBðlð2m� 1Þ þ 2mlÞjx¼�a

2¼ 0: ð40Þ

It is not possible to obtain the solution of a problem (38) and (39) in an analytical form. The solution on the basis of the suc-cessive approximations method will be compared to outcomes of direct numerical calculations.

As well as above, discovering the solution in an form

AðxÞ ¼ A0ðxÞ þ A1ðxÞBþ A2ðxÞB2 þ � � �

we sequentially obtain

A0ðxÞ ¼ xþ C0;

A1ðxÞ ¼m

ðm� 1Þx2

2þ C0x

� �þ C1;

A2ðxÞ ¼mðm� 1ÞC1 þ 1

8 a2ð2m� 1Þðm� 1Þ2

x� x3½ð4mþ 8l� lÞm3 þ ð4l� 2m� 12lÞm2 þ ð6l� 4lÞmþ l� l�6lðm� 1Þ3

þ C2

and then determine that

C0 ¼ 0;

C1 ¼ð4mþ 8lÞm3 � ð12lþ 6mþ 6lÞm2 þ ð6lþ 9lþ 6mÞm� 3l� 2m� l

24lðm� 1Þ2a2: ð41Þ

Let us notice, that for Murnaghan model the system which is obtained at the third step of a method is extremely bulky. Itssolution necessary for definition of a constant C2 looks like

A3ðxÞ ¼ �1

24l2ðm� 1Þ5Að1Þ3 x4 þ Að2Þ3 x2a2� �

þ mm� 1

C2xþ C3

where

Að1Þ3 ¼ ð8ml� l2 � 144lm� 48m2 þ 16ll� 96l2Þm5 þ ð48m2 � 36ml� l2 þ 240l2 � 8llþ 264lmÞm4 þ ðl2 � 12m2

� 240l2 � 180lmþ 40mlþ 136llÞm3 þ ð54lm� 8ml� 104llþ 8l2 þ 120l2Þm2 þ ð37ll� 10l2 � 6lm� 30l2

� 6mlÞmþ 3l2 þ 3l2 � 5llþ 2ml;

Að2Þ3 ¼ð32lm�2mlþ32l2þ8m2�4llÞm6þð3l2�2ll�80lm�ml�16m2�96l2Þm5

þ 41lm�212

l2þ18m2þ96lmþ13mlþ120l2� �

m4þ 332

l2�72lm�10m2�1472

ll�80l2�12ml� �

m3

þð2m2þ34lm�15l2þ30l2�mlþ55llÞm2þ 152

l2�19ll�6l2þ4ml�9lm� �

mþ52

llþ lmþ l2

2�3

2l2�ml:

Nevertheless constant C2 is equal to zero as in both previous cases.




Finally, the solution of a boundary value problem (38) and (39) by the method of successive approximations to withineffects of the second order becomes

Fig. 3.circles

Pleasepanel.

AðxÞ ¼ xþ A1ðxÞBþ A2ðxÞB2; ð42Þ

where

A1ðxÞ ¼m x2

2ðm� 1Þ þ C1;

A2ðxÞ ¼C1mðm� 1Þ þ 1

8 a2ð2m� 1Þðm� 1Þ2

x� 4C1

a2ðm� 1Þ þ1

6lðm� 1Þ3ð�lm3 þ 10lm2 þ 4mm2 � 13lm� 6mmþ 2mþ 4mÞ

!x3

and constant C1 is determined by (41).At the account of relations between constants of Murnaghan and Blatz and Ko materials (37), from the formula (42) we

come to expression (33), and using (38) – to expression for a harmonic material (17). Thus, having received a solution (42)for Murnaghan material, we have really received a set of solutions for arbitrary isotropic nonlinearly elastic materials.

It is necessary to notice, that material parameters of the second order l and m enter into the factor at the B in the firstpower. In this connection, the solution AðxÞ ¼ xþ A1ðxÞB that depends linearly from B is not really a solution of the lineartheory of elasticity.

To validate the formula (42), we consider a numerical solution of a problem (39) and (40), received by a shooting method.For numerical calculations we use values of Murnaghan constants for steel, copper, plexiglass and quartz from Lurie (1990).Fig. 3 presents the typical graphs of dependence of the function of distortion of the cross-section shape AðxÞ constructednumerically (solid line) and on the basis of (42) (circles) when the panel width a=h ¼ 1=5. One can see that at comprehen-sible values of curvature the asymptotic solution (42) is close enough to the numerical solution of the corresponding bound-ary value problem. The displacement of a neutral line of a panel cross-section can be also noticed. It is impossible to predictin advance in what direction the neutral (not deformed) line will move: this direction depends upon material parameters asFig. 4 demonstrates.

5. Study of the second order effects

5.1. The bending moment

Magnitude of the specific (per unit length of of the panel in z-direction) bending moment is defined by the formula

M ¼Z rþ

r�

Tuurdr; ð43Þ

where Tuu ¼ eu � T � eu is the component of Cauchy stress tensor T, r� ¼ rð�a=2Þ – inner and outer radii of the bent panelcross-section. Taking into account the relation between Piola and Cauchy stress tensors

−1 −0,5 0,5 1

0,5

−0,5

−1

1

−1 −0,5 0,5 1

0,5

−0,5

−1

1

(a) (b)Function of distortion of the cross-section shape for the copper panel. (a): Ba ¼ 0:06, (b): Ba ¼ 0:1. Solid line corresponds to the numerical solution,– to the asymptotic solution (42).



−0,02 0,02

0,05

−0,05

−0,1

0,1

V I

I I I

I I

I

Fig. 4. Influence of material parameters on the neutral line displacement: I – copper, II – steel , III – plexiglass, IV– quartz. Ba ¼ 0:1.


Pleasepanel.

T ¼ ðdet FÞ�1P � FT

and Eq. (5) relation (43) can be re-written in the reference configuration as follows

M ¼Z a=2

�a=2PyurðxÞdx: ð44Þ

With the account in (44) second order terms for the material models considered above we receive following relations:

- harmonic material

Mla2 ¼

16ð1� mÞBa� 1

30ð1� mÞ ðBaÞ3; ð45Þ

- Blatz and Ko material

Mla2 ¼

29

Ba� 1431215

ðBaÞ3; ð46Þ

- Murnaghan material

Mla2 ¼

16ð1� mÞBa� 1

360l2ð1� mÞ5Mð3ÞðBaÞ3; ð47Þ

where

Mð3Þ ¼ 40 4l2 þm2 þ 4lm� �

m6 � 12 13m2 þ 8llþ 52lmþ 4mlþ 52l2� �

m5

þ 12 19m2 þ 4l2 þ 80l2 þ 78lmþ 28llþ 14ml� �

m4 � 4 55m2 þ 190l2 þ 187lmþ 60mlþ 114llþ 42l2� �

m3

þ 2ð174lmþ 90mlþ 150llþ 165l2 þ 75m2 þ 111l2Þm2 � 3 20m2 þ 44l2 þ 25l2 þ 30lmþ 32ll� 24ml� �

m

þ ð10mþ 12lÞðmþ lÞ þ 7l2 þ 30l2:

It is visible from the formulas above that nonlinearity in expression of the bending moment is exhibited only in terms of thethird order. The linear item in expressions (45)–(47) coincides with well-known expression of the linear theory of elasticity(see, e.g. Lurie (1970)) if to realize the passage from a plane strain to a plane stress state, i.e. to substitute m by m=ð1þ mÞ. Theabsence of square terms in expression for the bending moment on the one hand confirms high level applicability of classicalformula of an engineering mechanics for rather large curvatures, and on the other hand means unfitness of the dependenceof the moment from curvature as a source of the experimental information on elastic constants of the second order.

The moment–angle relation at bending for the cross-section with a=h ¼ 1=5 is presented in Fig. 5. As the nonlinearityexhibits not in the second but in the third order terms the linear theory typically gives very good approximation for thebending moment value for large values of bending angle (up to p=2 for the Blatz and Ko model).

5.2. The panel thickness change

Determination of analytical expression of the relative change of the panel thickness at pure bending could be of greatimportance for definition of the material parameters, in particular, for determination of values of Murnaghan constants.




This thickness change is expressed by the relation

Fig. 5.theory,

Pleasepanel.

d ¼ Aða=2Þ � Að�a=2Þ � aa

: ð48Þ

If we restrict the consideration to the two first terms in the expansion thickness function AðxÞ, i.e. assume thatAðxÞ ¼ xþ A1ðxÞB where as stated above function A1ðxÞ is even, then d ¼ 0 that, generally speaking, corresponds to the lineartheory. At the account of square terms in this expansion we get that d – 0 that means that the change of the panel thickness(as well as effect of Poynting (1909)) is effect of the second order.

For harmonic material value of d can be obtained explicitly using expression (16)

d ¼ �aþ 2kþ eakða� 2kÞ

a eak þ 1

� ;

or up to the second order terms

d ¼ � a2

12k2 ; ð49Þ

where k ¼ h=a;a ¼ Bh – bending angle.For Blatz and Ko material

d ¼ � a2

8k2 : ð50Þ

For Murnaghan material

d ¼ a2

k2

ð4mþ 8lÞm3 � ð12lþ 6mþ 5lÞm2 þ ð6lþ 6lþ 2mÞm� 2l� l

24lðm� 1Þ2: ð51Þ

Taking into account (37) the formula (51) passes into (50), and taking into account (38) the formula (51) passes into (49). Itmeans that value of d is completely defined by constants of the second order, i.e. the formula (51) describing the relativechange of the panel thickness at pure bending can be used for determination or correcting the Murnaghan parameters.

To do this we can add the relation (51) to the two well-known relations of the second order. The first of them is connectedwith tensile test of the rod of length L by distributed load. In this case the elongation of the rod up to of the second orderterms is determined by the relation given in Lurie (1990)

e ¼ e3 �32

e23 �

e23

2lð1þ mÞ ½lð1� 2mÞ3 þ 2mð1þ mÞð1� m� 2m2Þ þ 3nm2�; ð52Þ

where e3 ¼ r3= 2lð1þ mÞð Þ – elongation of the rod given by linear elasticity theory.The second relationship is associated with the torsion test on the circular rod of length L and radius R. The arising elon-

gation (the Poynting effect) is expressed in terms of the Murnaghan constants by following relations (see, e.g. Kalashnikov &Karyakin (2006))

D ¼ w2R2

41

ð1þ mÞ 2m� nm4l� ð1� 2mÞ m

2l

� �� 1

� �; ð53Þ

(a) (b)The dependence between applied moment and angle of bending: (a) Blatz and Ko material, (b) Murnaghan material (quartz). Dashed line – linearsolid line – numerical solution of nonlinear equations, circles – account for second order terms.




where w – twisting angle per unit length.The relation (51) for the bending test for the panel with rectangle cross-section of depth a and width h ¼ ka closes the

system for the determination of the Murnaghan constants. In the case m – 1=2 this system (51)–(53) has unique solution thatusing notations

Fig

Pleasepanel.

Ce ¼ ee2

3� 1

e3þ 3

2 ;

Ct ¼ 4Dw2R2 ;

Cb ¼ d k2

a2 ;

can be written as

m ¼ 12ð2m� 1Þ ½2lðmþ 1ÞCe � 12ml mþ 1ð ÞCt � ðm� 1Þ2Cb þ lð7m2 � 6m� 2Þ�;

n ¼ 1m½2lðmþ 1ÞCe � 4l 3m2 þ 4mþ 1

� Ct � ðm� 1Þ2Cb þ lð7m2 � 2m� 6Þ�;

l ¼ 1

ð2m� 1Þ3½2mlð1� m2ÞCe � 12lm2ð1� m2ÞCt þ ðm� 1Þ2ðm2 � mþ 1ÞCb � lð7m4 � 13m3 � m2 þ 8m� 2Þ�:

6. Stability of a bending

The characteristic feature of bending diagram is the existence of point of maximum followed by the falling region. Onecan see it in Fig. 5 for the solution of the second-order theory. More important fact is the existence of such regions at thediagrams corresponding to numerical solutions for the Murnaghan model as well as the analytical solution for the harmonicmaterial (see Fig. 6). Such falling region usually means some instabilities of the loading process. The stability analysis is madehere using the bifurcation approach based on the linearization of the equilibrium equations in the vicinity of the constructedsolution.

Let disturb the known position of equilibrium of the bent panel by adding small displacement to all its points. This meanschanging the semi-inverse representation (1) by following relations:

r ¼ rðxÞ þ eu x; yð Þ;u ¼ Byþ evðx; yÞ;

ð54Þ

where e – small parameter. Corresponding to (54) deformation gradient can be rewrite

F ¼ F0 þ e _F; ð55Þ

where F0 – deformation gradient that corresponds to the basic solution the stability of which is testing, tensor _F linearlydepends on functions U x; yð Þ, V x; yð Þ. The linearization of other geometric deformation characteristics is carried out similarly,the general formula of this process can be represented as (Lurie, 1990):

_G ¼ dde

GðF0 þ e _FÞje¼0:

The equations of neutral equilibrium – linearized equation (2) – are written as follows

V I

I I I

I I

I

. 6. Bending diagrams: dashed line – harmonic material, solid lines – Murnaghan models (I – copper, II – steel , III – quartz, IV – plexiglass).




Pleasepanel.

Div _P ¼ 0;

or in component form

@ _Prx

@xþ @

_PyR

@y� B _Puy ¼ 0;

@ _Pxu

@xþ @

_Puy

@yþ B _PyR ¼ 0:

ð56Þ

Non-zero components of linearized Piola stress tensor _P for harmonic type material are following

_Prx ¼ ðkþ 2lÞðu;x þ krv;y þ kBuÞ;

_Pxu ¼ðkþ 2lÞðr2v ;xBþ r02v þ rr0v ;xÞ þ kðr2vB2 � u;yBr � u;yr0Þ

rBþ r0þ 2ðkþ lÞðr0vBr � Bvr � rv;x � vr0Þ

rBþ r0;

_PyR ¼kðBr2v ;x þ rr0v ;x þ r2vÞ � ðkþ 2lÞðB2vr2 � rBu;y � r0u;yÞ

rBþ r0þ 2ðkþ lÞðrv;x � rvBþ rr0Bv þ u;y � r0vÞ

rBþ r0;

_Puy ¼ ðkþ 2lÞðPv ;y þ BuÞ þ ku;x;_Pzz ¼ kðPv ;y þ uBþ u;xÞ

and for Blatz and Ko material they are

_Prx ¼ l2B3uAA04 � j2

1ð3u;xBþ uB2A04 þ v ;yA04ÞBA04j2

1

;

_Pxu ¼ lBj3

1ðv þ u;yA03Þ � j21ðBu;y þ v;xA0Þ þ vBA03j4

1 � u;yA02B

BA03j31

;

_PyR ¼ lv;xA03j4

1 � j1ðv ;xA02 þ u;yA0BÞ þ j31 A04vB� v ;x� �

þ vBA03

BA03j31

;

_Puy ¼ l3v ;yA02 þ j3

1BA02u;xBA02j1

;

_Pzz ¼ lj1 Bu;xA02 þ v ;yA03� �

þ uA03B2

BA02:

where j1 ¼ 1þ AðxÞB; f ;x � @f=@x, f ;y � @[email protected] of the components of linearized Piola stress tensor for the Murnaghan model are too lengthy and will not be

presented here.The boundary conditions now have the form

_Prx �a2; y

� �¼ 0; _Pxu �

a2; y

� �¼ 0; ð57Þ

_PyR x;� h2

� �¼ 0; v x;� h

2

� �¼ 0: ð58Þ

The relationships (58) mean sliding attachment at the panel borders.The system (56) is a system of differential equations for the unknown functions U x; yð Þ and V x; yð Þ. The replacement

vðx; yÞ ¼ VðxÞ sinðanyÞ;uðx; yÞ ¼ UðxÞ cosðanyÞ;

ð59Þ

where an ¼ pn=ð2hÞ;n 2 Z reduces it to the system of ordinary differential equations and identically satisfies to the boundaryconditions (58).

The study of the possible existence of non-trivial solutions of the system (56) and (57) was carried out according to thefollowing scheme. The general solution was sought in the form:

U ¼ C1U1 þ C2U2; V ¼ C1V1 þ C2V2; ð60Þ

where ðUk; VkÞ; k ¼ 1;2 are linearly independent sets of functions. These functions are obtained as a result of the solution ofthe Cauchy problem for (56). The first set of the initial conditions for this problems is choosing in the form

U1ð�a=2Þ ¼ 1; V1ð�a=2Þ ¼ 0;U2 �a=2ð Þ ¼ 0; V2ð�a=2Þ ¼ 1

ð61Þ



Fig. 7. Typical distribution of bifurcations points along the bending diagram.


and the second one – for derivatives U0k �a=2ð Þ;V 0k �a=2ð Þ – is obtained from Eq. (57) at the point x ¼ �a=2 taking into account(61). Satisfaction of the boundary conditions (57) at the point x ¼ a=2 leads to linear system of equations for the C1 and C2.The critical values of the parameter B, or bifurcation points, are found from the characteristic equation – the condition ofvanishing of the determinant of this system meaning the existence of non-trivial solutions for C1;C2.

For all material models considered and for all types of considered buckling modes (i.e. for all considered values of theparameter n, greater than zero) the bifurcation points were found. A typical distribution of these points, common to differentmodels of nonlinear elastic behavior of the material, in the case of a thin panel is shown on the bending diagram in Fig. 7. Thefirst bifurcation point P1 – the first root of the characteristic equation for the first mode – is situated considerably to the leftof the bending diagram maximum point. The second point of bifurcation Q1 for the first mode, i. e. the second root of thesame equation, is very close to the maximum point and also does not exceed it. The pairs of first and second bifurcationpoints for the other modes ðPk;QkÞ are located inside the segment ½P1;Q 1�. It should be noted that for the function rðxÞ in(54) we used either analytical solution or numerical solution of the nonlinear boundary value problem thus allowing forarbitrary level of initial strains.

7. Conclusion and results

A modification of semi-inverse representation of a deformation of plane pure bending of a straight nonlinearly elasticpanel suitable for application of a method of successive approximations is offered. With the use of the offered representationto within the second order effects the bending problem is solved for three models of nonlinearly elastic behavior: a harmonicmaterial, Blatz and Ko material, Murnaghan material. The results of the comparison of a second order solution with analyticaland numerical solutions of corresponding boundary value problems have shown high degree of its applicability even in theregion of rather large bending. The absence of square terms in the dependence of the bending moment upon the curvature ofa bar makes this dependence unsuitable for deriving experimental data for definition of elastic constants of the second order.At the same time, the relative change of the panel thickness at bending is the effect of the second order, and received explicitformula of the thickness dependence upon material and deformation parameters allows to recommend it as a mean of deriv-ing or checking of the information on magnitude of Murnaghan constants.

All considered bending diagrams exhibit the similar behavior: all of them have the point of maximum followed by thefalling region. Typically this region is located in the range of extra-high bending strains corresponding to turning the rect-angle cross-section into the circular ring and seems to be of no interest for real applications. However the bifurcation anal-ysis shows that the stability loss at bending can occur significantly earlier than the maximum point and the falling region isachieved.

Acknowledgment

This work was supported by Grant 213.01-24/2013-78 of Southern Federal University.

References

Aron, M., & Wang, Y. (1995). Remarks concerning the flexure of a compressible nonlinearly elastic rectangular block. Journal of Elasticity, 40, 99–106.Batra, R. S., Dell’Isola, F., & Ruta, G. C. (2005). Second-order solution of Saint-Venant’s problem for an elastic bar predeformed in flexture. International

Journal of Non-Linear Mechanics, 40, 411–422.Blatz, P. J., & Ko, W. L. (1962). Application of finite elasticity to the deformation of rubbery materials. Transactions of the Society of Rheology, 6, 223–251.Bruhns, O. T., Xiao, H., & Meyers, A. (2002). Finite bending of a rectangular block of an elastic Hencky material. Journal of Elasticity, 66, 237–256.Carroll, M. M., & Horgan, O. (1990). Finite strain solutions for compressible elastic solid. Quarterly of Applied Mathematics, XLVIII(4), 767780.

Please cite this article in press as: Karyakin, M., et al. Nonlinear effects in a plane problem of the pure bending of an elastic rectangularpanel. International Journal of Engineering Science (2014), http://dx.doi.org/10.1016/j.ijengsci.2014.02.023

http://refhub.elsevier.com/S0020-7225(14)00044-5/h0010








Coman, C., & Destrade, M. (2008). Asymptotic results for bifurcations in pure bending of rubber blocks. Quarterly Journal of Mechanics and AppliedMathematics, 61, 395–414.

Destrade, M., Gilchrist, M. D., & Murphy, J. G. (2010). Onset of non-linearity in the elastic bending of blocks. ASME Journal of Applied Mechanics, 77, 061015.Haughton, D. H. (1999). Flexure and compression of incompressible elastic plates. International Journal of Engineering Science, 37, 1693–1708.John, F. (1960). Plane strain problems for a perfectly elastic material of harmonic type. Communications on Pure and Applied Mathematics, 13, 239–296.Kalashnikov, V. V., & Karyakin, M. I. (2006). Second-order effects and Saint Venant’s principle in the torsion problem of a nonlinear elastic rod. Journal of

Applied Mechanics and Technical Physics, 0021-8944, 47(6), 879–885. http://dx.doi.org/10.1007/s10808-006-0127-8.Lebedev, L. P., Cloud, M. J., & Eremeyev, V. A. (2010). Tensor analysis with applications in mechanics. New Jersey: World Scientific.Lurie, A. I. (1970). Elasticity theory. Moscow: Nauka (in Russian).Lurie, A. (1990). Nonlinear theory of elasticity. Amsterdam: North-Holland.Murnaghan, F. (1967). Finite deformation of an elastic solid. New York, New York: Dover.Ogden, R. W. (1984). Nonlinear elastic deformations. Chichester: Ellis Horwood.Poynting, J. H. (1909). On pressure perpendicular to the shear planes in finite pure shears, and on the lengthening of loaded wires when twisted. Proceedings

of the Royal Society of London Series A, 82, 546–559.Rivlin, R. S. (1949). Large elastic deformations of isotropic materials. V. The problem of flexture. Proceedings of the Royal Society A, 195, 463–473.Saccomandi, G. (2001). Universal results in finite elasticity. In Y. B. Fu & R. W. Ogden (Eds.), Nonlinear elasticity: Theory and applications (pp. 97–134).

Cambridge: Cambridge University Press (chap. 3).Saint-Venant, A.-J.-C. (1856). Mémoire sur la flexion des prismes. Journal de Math’ematiques de Liouville, Series II, 1, 89.Stepanenko, Y. P. (1997). Poyntyngs effect: Calculation schemes and experiment. In Advanced problems of continuum mechanics: Proceedings of III

international conference, Rostov-on-Don, Vol. 1 (pp. 64–68) (in Russian).Triantafyllidis, N. (1980). Bifurcation phenomena in pure bending. Journal of the Mechanics and Physics of Solids, 28, 221–245.Zelenina, A. A., & Zubov, L. M. (2000). The non-linear theory of the pure bending of prismatic elastic solids. Journal of Applied Mathematics and Mechan, 64,

399–406.Zubov, L. M. (1985). Linearized bending problem and Saint-Venant principle. Proceedings of North Caucasus Scientific Center of Higher School (4), 34–38 (in

Russian).Zubov, L. M., & Karyakin, M. I. (2006). Tensor calculus: Foundations of the theory. Moscow: Vuzovskaya kniga (in Russian).

Please cite this article in press as: Karyakin, M., et al. Nonlinear effects in a plane problem of the pure bending of an elastic rectangularpanel. International Journal of Engineering Science (2014), http://dx.doi.org/10.1016/j.ijengsci.2014.02.023






http://dx.doi.org/10.1007/s10808-006-0127-8



















Documents

Nonlinear effects in a plane problem of the pure bending of an elastic rectangular panel