Extremum principles for certain hyperelastic materials

Int. 1. Non-Linear

Mechnics.

Vol. 8, pp. 143-153.

Pergamon

EXTREMUM

Press 1973.

Printed in Great Britain

PRINCIPLES

HYPERELASTIC

FOR CERTAIN

MATERIALS

N. C. LIND Solid Mechanics Division, University of Waterloo, Waterloo, Ontario, Canada

Abstract-A hyperelastic material is here said to be of class H,,, if the elastic potential is a homogeneous function of order m + 1 in the components of the Lagrangian displacement gradient. It is shown that a single solution to a boundary value problem generates an infmite family of solutions to a family of related boundary value problems. Assuming that a solution to a boundary value problem exists, it is shown that it is unique provided that the material is stable in the sense of Hill in a deleted neighbourhood of the stress-free state. A minimum theorem concerning the strain energy and the virtual work of the prescribed forces is established for the equilibrium configurations, and a maximum theorem concerning the virtual work of the prescribed surface displacements and the complementary stress energy is established for compatible stress fields. As an application, upper and lower bounds are found for the torsional stilfness of a cylindrical bar of square cross section under infmitesimal deformation. INTRODUCTION

and lower bound theorems for the complementary energy and the energy respectively, associated with boundary value problems of the linearized equations for an elastic continuum were established by Trefftz [l]. Similar theorems have been established for linear elastic lumped-parameter systems [2] and extended (without rigorous proof) to non-linear systems [3]. These theorems have found numerous applications in establishing upper and lower bounds for the overall properties of structural members or systems. It is not possible to extend these theorems to cases of finite deformations of general non-linear elastic bodies, as evidenced by the phenomenon of elastic instability. Instead, this work explores certain ideal constitutive equations for which related theorems can be established. As when the theorems of Trefftz are involved, it becomes a problem of practical engineering analysis to establish whether or not a particular problem admits of analysis by being within the range of validity of the theorems to be applied. Considering uniaxial deformation, the simplest generalization of Hooke’s law is the power law relating stress and deformation This paper is concerned with certain classes of hyperelastic materials to be defined in the following manner; for uniaxial deformations such materials obey a power law, and they can therefore be considered as the first generalization of the Hookean continuum. Many common engineering materials can be represented in this way when the deformations are moderate. UPPER

CLASS H,

OF HYPERELASTIC MATERIALS

Under the assumption that the motion of a continuum is adiabatic, the energy theorem can be stated from the Lagrangian viewpoint in the form [4] D&$,

r) = qj@9 rl DOIDiu@, 143

t)],

(1)

N. C. LIND

144

in which, as in the sequel, all latin subscripts range over the values 1, 2, 3 and indicate summation when repeated; Ui are the components of the instantaneous displacement vector u of a generic particle referred to a fixed rectangular Cartesian co-ordinate system Xi; a is the initial radius vector of the particle and has the components ai; Di indicates partial differentiation with respect to a, and D, the corresponding partial derivative with respect to time t; @a, r) is the instantaneous internal energy associated with a unit volume of the initial state; and T&x, t) is the Lagrangian stress tensor. For a hyperelastic material in adiabatic motion the internal energy is, by definition, an analytic function of the six independent components of Green’s strain tensor U, = (DiX$jXk - 6,)/2, from which it is clear that it may also be written as a function of the Lagrangian displacement gradient E = E(DpJ,

(2)

and it follows from equation (1) that the expression in equation (2), without loss of generality, may be written such that Tii = aE/a(Di~j).

(3)

This is equivalent to the USU~ form Tii = aE/~(DiXj), [4]. A hyperelastic material of class H,,, is defined as a material for which E(Diuj) is a homogeneous function of order m + 1: E(~iuj)

= 12m’’ E(Diuj),

(4)

where L is a positive number different from zero. It is easily verified that the n-th order partial derivatives of any such function, where they exist, are homogeneous of order (m - ?I + 1): d” E/a(~iUj) a(~Dk~3. . . = Ilm-“” a” E/a(Diuj) ~(D,u,). . . By equations 4 and 5, the Lagrangian stress tensor is homogeneous

(5)

of order m. As a neces-

sary condition for stability 0 < m. UNIQUENESS

AND STABILITY

Consider first the problem of determining the body forces and the surface tractions that must act on such a hyperelastic body if it is to be in equilibrium in the deformed state specified by r+(a). In the assumed homogeneous stress-free initial state Ui = 0 the body has the initial density p(u) and occupies the regular domain V with the surface S of external unit normal v. By the definition of the Lagrangian stress tensor the surface tractions q(v) dS and the body forces pK, dV ate uniquely determined from equilibrium considerations by Tj(V)

ds

=

ViT,i

dS =

Vi

dS aE/a(Diuj),

pKj dV = - DiT,i dV = - dvDi[aE/a(Diuj)].

(6) (7)

Let up, cj and K; in V and I;(v)0 in S represent an equilibrium field so, and let Iz be any positive non-zero number. By equations (47) inclusive it is then seen that another equilibrium held s is generated by A from so as ui = r2u,o,

(8)

Extremwn principles for certain hyperelastic materials

145

IT;i = 2’ T;,

(9)

Ki = PK;,

(10)

‘I;(y) = rz”Tj(v)o.

(11)

The totality of these states is called the family of states of state so.

Next, consider the boundary value problem 6B to determine the infinitesimal displacements &,(a) from a given equilibrium state s, caused by given infinitesimal changes 6Ki of the specific body force field in V and by given infinitesimal changes 61;(v) and 6Ui of certain components of the tractions and the complementary displacements in the surface S. It is assumed that the region V is singly connected and that the functions assigned on the surface have piecewise continuous derivatives. It is also assumed that a solution exists; here it may be sufficient to point out that it is possible to recast the incremental boundary value problem 6B into Eulerian form completely analogous to the general boundary value problem of classical elastostatics, for which the existence of a solution is assured [4, 51. To examine the question of uniqueness of solutions to the present problem, let as* and 6s denote two distinct solution fields for the problem and let b denote the field as* 6s. Since W and 6s have identical fields of specific body force and surface traction, equation 7 yields Di(ATj) = 0,

(12)

and at any surface point equation (6) and the specified boundary conditions give Jv~A~~Au~cIS

Transformation

=

0.

(13)

of this equation by Gauss’s Theorem and use of equation (12) furnishes S A~jDXAUj)dV = 0.

(14)

In accordance with equation (3) a series expansion gives aNE

1

Aqj = c

(N - l)! a(D,uj a(D,uJ . . . @Dpu,) Dk(Aul) * * ’ Dp(Aur).

(1%

N=2

Let N = n be the value of N for the first non-zero term in this expansion. Then, for sufficiently small au,* and 6ui by equation (15) equation (14) becomes

s s

PE

a(D+,> a(D,uJ neea(Dp~r)

D,(L\uj) Dk(Aur) . . . D,(Au,) dV = 0.

(16)

Equations (5) and (8) may then be applied to get equation (16) in the form a”E

a(DiUj”) a(D&)

-

. * * a(Dp@)

DXAu;) Dk(Au;) . . . Dp(Au;) dV = 0

(17)

which shows that the solution is unique if and only if the solution for the reference state so is unique. The assumption that the material is stable in a deleted neighbourhood 0~

146

N. C. LIND

uiui < a2 of the stress-free state implies that the form

is always positive definite. Accordingly, equation (17) contradicts the assumption that 6s ,and 6s* are distinct It may be concluded that the incremental boundary value problem 6B has a unique solution for a hyperelastic material of class H,,,. The discussion in connection with equations (6) and (7) show that it is always possible to devise a boundary value problem B” of any desired type, (i.e. with tractions prescribed over any desired part of the surface and displacements prescribed on the complement) in the large for which there exists at least one solution, S”. The existence of a solution S to any given boundary value problem B of finite deformation of a singly connected body of a stable, simple hyperelastic material is therefore ensured, since it is always possible to first select a problem B’ of the same type as B and then to devise a programme B” + B of continuous change which transforms the prescribed value of B” into those of B. As the programme B” + B is reversible, there is a one-to-one correspondence between the solutions s” and S. In particular, if S” is unique, so is S. In general, if B” admits of n solutions Sy, SO,,. . . , q there are n corresponding solutions S,, S2,. . . , S, to B. Thus, there may be n families C,, C2,. . . , C, of solutions to any boundary valuk problem. Any programme B” + B transforms a solution s” to a solution S of the same family, and there is no programme transforming from a solution in one family to a solution in a different one. Now, it is observed that the stress-free state is a solution S, to any homogeneous boundary value problem B; let it belong to family Cr. Hence, all solutions to any boundary value problem that can be realized by deformation from the stress free state are unique. While it is conceivable that the homogeneous boundary value problem admits of a number of solutions different from the stress free state (the “snapped” states belonging to family C1,. *. , C,,), it is clear that these cannot be realized by any continuous deformation from the stress free state. Uniqueness in the normal sense is therefore assumed. The proof of uniqueness can be extended immediately to a multiply connected body by imagining a system of cuts which renders the body singly connected; when the tractions and displacements of boundary value problem B are realized, a unique field of relative displacements are induced on the cut surfaces. This defines uniquely a supplementary boundary value problem that restores continuity at all points of the cuts and has a unique solution. Thus for the multiply connected bodies solutions exist and are unique. Finally, it may be stated that all equilibrium states s of a body of a stable, class H, hyperelastic material are stable in the sense that in the course of any motion that starts from the state with infinitesimal velocities, and during which the specific body forces and tractions are constant and the kinematical boundary conditions are satisfied, each particle remains within an infinitesimal neighbourhood of its initial position. The proof of this statement is omitted, since it follows the pattern of similar proofs in the literature [4,6]. It is also noted that the stress free state is stable if the material is stable in the sense of Hill in a deleted neighbourhood of the stress free state for a particle. GEOMETRICAL

REPRESENTATION

It is useful to consider the internal energy of an elementary volume of material as a scalar point function, the strain energy q = E(Diuj) in a 9-dimensional space with a rect-

Extremum principles for certain kyperelastic materials

147

angular Cartesian reference frame of co-ordinates Diu i As evidenced by equation (3), the Lagrangian stress tensor Kj is the gradient of the strain energy: Kj am therefore the components of a vector T which is normal to the level surfaces rp = E = constant. Similarly, the system a’E/a(Diuj) i?(D,uJ is the curvature tensor for the. surface p = ‘pO= constant. If the quadratic form of equation (18) with n = 2 is positive definite at a point, the surface ~0= cpois elliptic at that point. In general, if at a point the first n - 1 terms in the expansion equation (15) vanish and the form of equation (18) is positive definite, the surface cp = q0 in the neighbourhood of a point lies entirely in the half space formed by the tangent plane at the point and containing the origin. Thus, for a stable material the level surfaces of the strain energy are convex; they are, moreover, closed surfaces that enclose the origin. The convexity implies for a material element that if two states of deformation Diaj’ and Dig both have the same strain energy cp = (po,and if0 G a < 1, then cp(D,[rr% + (1 - ol)u;]) < 4p,, By integration over the volume, it is seen that a similar relation holds for the strain energy of the entire body; this strain energy may therefore be said to be convex in the same sense. In the following it will be assumed that the quadratic form of equation (18) for n = 2 is positive definite, so that the Hessian determinant of cp = E(Diuj) is different from zero. The Legendre transform of cp is the complementary stress energy - cp = 5

$ = lgiuj

‘I~iUj = mE

with the property that DiUj

-‘

rn~~/~~~

(20)

The gradient Dfuj of the displa~ment vector for a deformed element is therefore normal to the surface # = const. in Lagrangian stress space. STRAIN ENERGY MINIMUM

PRINCIPLE

Let s* and s now be any two neighbouring fields of stress and displacements that are not identical, and let E* and E be the corresponding values of the strain energy. Then the strain energy density difference is for any particle

It follows from the positive definite character of the expression in equation (18) that the last term in equation (21) is positive. Moreover, by eouation (20) the second term on the second right hand side equals mAE/(m + l), and equation (21) therefore gives xi% to the inequality E* - E > ~~DiUj*

-

Dill).

(22)

Now, let s denote the solution fiekl to the boundary value problem B and let UT denote a neighbouring displacement field that satisfies the kinematical boundary conditions. Since the stress field qj satisfies the equilibrium equation, equation (7) S P~j(U~

-

Uj)

d’ir + I

(Us -

Us) Cti~j

dV = 0.

(23)

148

N. C. LIND

Rewriting the last integral, using Gauss’s Theorem and equation (22) yields SE*dV-SpKjui*dV-S?;(v)ui*dS’7SEdV-SpK,scjdV-S7;(v)ujdS’, where the last integral extends that the difference between the forces and the surface tractions any neighbouring displacement

(24)

over all static boundary conditions. Equation (24) implies strain energy and the virtual work of the prescribed body is smaller for the displacement field of the solution than for field. By addition of the constant j pK,iai dV + S Uj~;(v)dS

(25)

to both sides of equation (24) this minimum principle may be worded alternatively as: The sum of the strain energy and the potential energy of the prescribed forces and surface tractions is smaller for the displacement field of the solution than for any other neighbouring displacement field that satisfies the kinematical boundary conditions.

COMPLEMENTARYSTRESS ENERGYMAXIMUMPRINCIPLE By equation (3), equation (21) with omission of its last, positive definite term and with a change in notation for the sake of distinction, can be rewritten as the inequality E** - E >

1 (T;* m

-

T,i)Diuj

where E is the (Lagrangian) stress energy. Let s denote the solution field to the boundary value problem B and let T$* denote a neighbouring equilibrium stress field that satisfies the static boundary conditions. Expressing that T$ and Tj are both equilibrium fields, using Gauss’s Theorem and the static boundary conditions, and using equation (26) yields J ViTjUj dS” - m J E dV 7 J v~T$*u~ dS’ - m J E** dV,

(27)

where the surface integrals extend over all kinematical boundary conditions. This inequality expresses the following maximum principle: The difference between the virtual work of the surface tractions on the prescribed displacement and the complementary stress energy is greater for the stress field of the solution to the boundary value problem than for any neighbouring equilibrium stress held that satisfies the statical boundary conditions.

UPPER AND LOWER BOUNDSON LUMPED BODY PARAMETERS For the solution to the boundary value problem B,

s

EdV

= &

T&ujdV; s

(28)

use of equation (7) and rewriting gives (m + 1) s E dV = j q(v) uj dS’ + f viQj

dS” + s pKpj dV.

(29)

Extretnum principles for certain hyperehstic materials

149

With this, equation (27) gives the inequality viT~*ujdS” - m s

E** dV -Z s

1

m+l

s

v&,aj dS” - -

m

m+l

7;(V) UI

dS’ - &

pK,Uj dV, s

s

(30)

and equation (24) yields 1 m+l

m vi~,~j dS’ - m+l s

REV)

Uj

dS’ - ~

PKpj dV < s

I

JE*dv-SpX,u:dv-~l;(v)u:dS’.

(31)

EXAMPLE

Consider an isotropic material with the constitutive equation T.'I = Ad.EJ (Dkuk)[“’+ B(D,uj + D J.uJ[~

(32)

where, as in the following, a bracketed exponent indicates the operation (X)&J = 1x1” sgn x

(33)

The material is evidently hy~relast~ of class I&, Equation (32) restricts the Lagrangian stress tensor to be symmetric, and the analysis in the following is therefore restricted to suitably small deformations. The restriction can be removed if additional terms are included in equation (32); for the present purpose this is an unnecessary complication. The internal energy, written as strain energy, is E=

(D,Uj + Do.)“’ (D.u. + D.u.) J 1 1 J J 1

1

The quadratic form of equation (18) becomes

in which the summation convention does not include the term “iuj + D++ Assuming that 0 < A, 0 K B this form is positive defmite, and the material is therefore stable. Let it be desired to determine the relationship between torque M and specific angle of twist 8 for a prismatic bar of length I and square cross-section with sides h, made of this material. A rectangular Cartesian reference frame x1 is located with the x,-axis parallel to the generators of the bar and with the terminal cross-sections of the bar initially at xj = 0 and xJ = 1. Surface tractions are prescribed to vanish on the sides of the bar, and Ts3 is to vanish on the terminal cross-sections; at a3 = 0 there is to he no displacement component in the xi, X, plane, and at a3 = 1 the xX- and x,-components of displacement uI and t12 are to have the values -N-z, and BIu, respectively, where 8 is an infinitesimal constant. Body forces are prescribed to vanish everywhere in the bar.

150

N.

c. LlND

The inequalities (30) and (31) simplify in the present problem to

s

T**u 3a

=

dS” - m

1

f

E** dl/C ___

T3.u, dS’:

m+l

(36)

s

(37) where, as in the following, greek indices range over the values 1,2 and indicate summation when repeated in a term. A stress field Tt* that satisfies the statical boundary conditions can be derived from a stress function Ip**(ar, az) that vanishes on the boundary of the cross section [4]:

By equation (32), D,u, = D,u, = D,u, = 0, and equation (32) is easily inverted to give, in particular, [I/ml

(DiUj + D,~i)** =

(39)

The stress energy is, accordingly [l/ml E**

+ (DzyI**(U+‘/m)

pIy**((‘+‘/N

=

1(l/W I( > B

The surface integrals in equation (36) may be rewritten as follows: j T3,pa dS” = Mel

(41)

~T;,*z& dS” = 281 j !P* dA

(42)

By equations (36,40,41 and 42) 28

s

YP**dA--

m

m+l

1

0 B

(l/m)

s

(IDJJ**l(l+lI”‘)

+ IDzlff**I(‘+‘/“‘))dA<

---&

Me. (43)

Let !P** be written as a$ where ti = $(a,, a,) and CLis a scalar parameter to be given the value which maximizes the left hand side of equation (43). With the notation j$dA

= C,

(49

~(~&~l(‘+l’m) + ID,t,bI(‘+‘~m))dA=

F,

(45)

the left hand side of equation (43) becomes 2tlaC -

5

B(l/rn) a(1 + l/m)

F.

(46)

Excremumprinciples for certain hyperelastic materials

151

This function of a has the maximum value 2#El+1 _-em+’

BC”+l F-“;

m+f

(47)

hence, M > Zrn+l BP+'

8”.

(48)

(1 - 4a;/P),

(49)

F-"

Using the stress function lj = (1 - 4&P) equations (44) and (45) give respectively C = 4h2/9, F

=

4(1+11m)J(n)-

m

(50)

s(5/2 + l/m)

2m -t 1 T(5m +

h(l-l/m)

(51)

2/2m)

So that M , 2m+1 2m + 1 m3-2m-2s-(I,Zj ( m >

r(5/2 + l/m) ( r(2 + l/m) >

m-l Bh”

+ 38”

(52)

A field u: that satisfies the kinematical boundary conditions is %* =

-&z,a,,u;

=

@a,a,,tij

=

@*(al,4

(53)

The only non-zero components of DiUj + DJUi are D,u,

+ D,u,

= D,@*

-

@a,;D,u, + D3uz = D2@* + @a,

(54)

and equation (34) gives the strain energy

E*

=

&(lDl@*

- f&I”+1

+ jD#*

+ Ba,I”+l)

(55)

A simple, but somewhat crude upper bound is obtained by neglecting warping of the cross-section (~3 = @* 3 0). Integration of equation (54) through the volwne and use of equation (37) then gives M (. s

Bhm+38”

(56)

The numerical coefficients of equations (51) and (55) are denoted by K- and kl+ respectively and are given in Table 1 for some representative values of m.

152

N. C. LIND TABLE~.TOR~IONAL STIFF= m

+O

0.25 0.50 0.75 1

co-JJFFICIFJNTS FOR ASQUARE

K-

K+

@2222 0.2 579 0.2191 0.1764 0.1389

05000 03736 0.2828 0.2162 0.1667

BAR

CONCLUSIONS

A body ma& of a continuous stable hyperelastic material with a strain energy that is a homogeneous function of the displacement gradient has a unique stable configuration that satisfies the requirements of continuity and equilibrium when it is subjected to an arbitrary continuous field of body forces and piecewise differentiable tractions and complementary displacements on the surface. Equations (30) and (31) can be used to establish lower and upper bounds respectively for the overall stiffness of such bodies under prescribed proportional loading. Acknowledgements-The material in this paper forms part ofa study sponsored by theNational Research Council of Canada. The author is grateful for the assistance of Dr. A. Kulkarni and for the helpful comments of Dr. R. N. Dubey.

REFERENCES E. TREFFTZ,Em Gegensttick zum Ritzschen Verfahren, Proc. Second Znt. Congr. Appl. Mech., Zurich, pp. 131-137 (1927). S. H. CRANDALL, Engineering Analysis, McGraw-Hill, New York (1956). J. H. ARGYRISand S. Karsav, Energy Theorems and,Structurul Analysis, Butterworth, London (1960) W. PRAGER, Introduction to Mechanics of Continua, Ginn and Co. (1961). I.S. SOKOLNIKOFF, Mathematical Theory of Elasticity, 2nd Edn., McGraw-Hill, New York (1956). R. HILL, On uniqueness and stability in the theory of finite elastic strain, J. Mech. Phys. Solidr 5,229 (1957). (Received 6 December

1969)

Zusammenfassung-Ein hyperelastisches Material wird hier als der Klasse H,,, zugehiirig betrachtet, wenn das elastische Potential von der Ordung m + 1 in den Komponenten des Lagrange’ schen Versetzungsgradienten ist. Es wird gezeigt, dass eine einzige Losung eines Randbedingungsproblems eine Familie unendlich vieler Liisungen zu einer Familie verwandter Randbedingungsprobleme erzeugt. Unter der Annehme, dass eine Losung zu einem Randbedingungsproblem existiert, wird gezeigt, dass diese eindeutig ist, vorasusgesetzt dass das Material im Sinne Hills stabil ist in einer ausgelassenen Nachbarschaft des spannungsfreien Zustandes. Fiir die Gleichgewichtsanordnung wird ein Minimumstheorem beziiglich der Dehnungsenergie und der virtuellen Arbeit der vorgegebenen Krafte aufgestellt; ein Maximumstheorem beziiglich der virtuellen Arbeit der vorgegebenen Oberflsichenversetzungen und der komplementaren Spannungsenergie wird fur vereinbare Spannungsfelder aufgestellt. Als Anwendung werden die untere und overe Grenze der Verdrehungssteitigkeit eines zylindrischen Stabes mit quadratischem Querschnitt unter infinitesimalen Verformungen bestimmt. ArsrsoTaqwa-Prinepynpyrnti eCJlEIeI'0 YllpJWii

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Extremum principles for certain hyperelastic &III

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