Holpzaphel_-_Nonlinear-Solid-Mechanics-a-Contin

material parameters .as given above.

The analytical solutions of the six parameter material model proposed by Ogden are in -excellent .agreement with experimental data by TRELOAR [ 1944] (see also NEEDLEMAN [1977]) who solved the problem on the basis of the Ritz-Galerkin method. Ex- perimental data show a very stiff initial stage (as seen with any party balloon) in which the inflation pressure Pi rises steeply with the circumferential st.retch. After the pressure has reached a maximum the rubber balloon will suddenly 'snap through', and a

of the pressure will allow it to 'snap back' (see Figure 6.4, see also the stud-

OGDEN fl972a] and BEATTY [1987] among others). This effect is caused by the deformat.ion dependent pressure load, is dynamic in character and known as snap buckling. The pressure-stretch path clearly shows the .existence of a local maximum

.and minimum, the maximum and mini.mum pressures are at ,\ = l.38 and ,\ = 4.32, respectively.

The curve for the Mooney-Rivlin model, with c1/c2 = 7, shows the characteristic behavior of a spherical rubber balloon, but, however, the results based on Ogden's model {and Treloar's experimental data on vulcanjzed rubber) are significantly different

The neo-Hookean and Varga form of the strain -energy reproduces more or Jess the real behavior of the balloon for small strain ranges. Reasonable correlations for all material models are obtained at the low strain level. However, for finite strains the typical characteristic of the load-deflection curve cannot be reproduced with the. simplified .neo-Hookean and Varga model.

Experimental investigations show that the balloon develops a bulge on one side and becomes aspherical (compare with NEEDLEMAN [l 9771). The bifurcations of pressur- ized elastic spheric.al shells from an analytical po.int of view are studied by HAUGHTON and OGDEN [1978], HAUGHTON [1980] .and ERICKSEN [ 1998, Chapter 5]. Compress- ibility effects are considered by HAUGHTON [ 1987]. II

Yeoh, Arruda and Boyce model for i.ncompressible (rubber-like) materials. Nearly all practical engi.neer.ing elastomers .contain .reinforcing fil.lers such as carbon black (in natural rubber vulcanizate) or silica (in silicone rubber). These finely distributed fillers, whkh have typical dimensions of the order of 1..0 - 2~0 · 10- 12-rn, form physical and chemical bonds w.ith the polymer chains. The fine filler particles are added to the elastomers in order to improve their physical properties which .are mainly tensile and tear strength, or abrasion resistance. The associated stress-strain behavior is observed to be highly nonlinear (see_, for example, GENT [.1962]). Carbon-black filled rubbers have important applications in the manufacture of automotive tyres and many other engineering components.

It turns out that for carbon-black filled rubbers the strain-energy functions described hitherto in this section are not adequate to approximate the observed physical behavior. For example, consider a simple shear deformation of a filler-loaded rubber. Physical observations show· that the shear modulus Jt of the .material varies with deformation in a significant way. To be more specific, JL decreases with increasing deformation initially and then rises again at large deformations (s·ee YEOH .[1990, Figure 2]). The associated relation for the shear stress is clearly nonlinear. Now" taking, for example, the ·Mooney-Rivlin model according to strain-energy function (6. l27h, then, from the explicit expression (6.78h we may specify a shear modulus

The relation for the shear stress is, however, .linear with the constant slope 2(c1 + c2), i.e. the shear .modulus. Apparently the .Mooney-Rivlin (and its neo-Hookean spe- c.ialization) model is too simple for the characterization of the elastic properties of carbon-black tiIled rubber vulcanizates.

The phenomenological material model by YEOH [ 1990] is motivated in order to simulate the mechanical behavior of carbon-black filled rubber vulcanizates with the typical stiffening effect in the large strain domain. Published data for filled rubbers (see KAWABATA and KAWAl [1977] and SEKI et al. :(.1987]) sug·gest .that 0'11/8!2 .is numerically c.Iose to zero. Yeah .made a simplifying assumption th.at 8\J! /812 .is equa1 to zero and proposed a three-term strain-energy function where the second strain invariant does not appear. ]t has the specific form

(6.134)

where <~.1., c2 , c:1 are material constants which ·mus.t....~atisfy certain restrictions.

Since by (6..5) the strain-energy function \JI is either zero (in which case whas

configuration). Hence, the (convex) strain-energy function increases monotonically with / 1 and 8\J! / 811 = 0 has no real roots. From the discriminants of the respective cubic and quadrntic equations in (11 - .3) the appropriate restrict.ions on the values for c., c2, c:1 may be determined.

Recall. the simple shear deformation example of a filled rubber from above once more. With the strain energy (6.134) we now conclude from eq. (6.133)1 that

(6.135)

The shear modulus p. involves first-order and second-order terms in (JL -3) and approximates the observed nonlinear physical behavior with satisfying accuracy (provided c2 < 0 and c1 > 0, ca > 0).

and K and (3 denote the constant bulk modulus in the reference configuration and an

(empirical) coefficient, respectively. The strain energy (6.137) satisfies the normal-

ization condition, '1'v(1) = 0. Note that this empirical function ·meets experimental

01

results with excellent accuracy (see OGDEN -[197.2b, Figure l]), indicating that rubberlike materials are (slightly) compressible. In particular, for /3 = 9, the distribution of

the hydrostatic pressure is in good agreement with experimental data of ADAMS and

GIBSON [1930] and BRIDGMAN [1945].

An alternative version of (6.137h, due to SIMO and MI.EH.E :.[.1992], is obtained by

The second part of th.e decoupled strain energy, i.e. Wjsq(~1, .,\2 , .,\3 ), describes the purely isochoric elastic response in terms of modified princip~I. stretches Aa

1-1/ 3 Aa, a= 1, 2, 3. We have

···--·---...---~.':""·-·.. ---..,.,~-.~··-·-~.___,__~ ..~-...·-~.'"":'----..~.·""~~.-.....-.....~.~

"EXAMPLE 6.7 Consider the decomposed structure (6.137) .and (6.139) of the strain energy and the additive split of the second Pio.la-Kirchhoff stress tensor (6.88)2. With

SpecificatiOOS (6.} ~7h and (6.13.8) find the purely VO/UmetriC COllfribttfiOIJ s\.'O) Of the

stress response in the explicit form. In addition, with (6.139h, find the spectral decomposition of Siso' i.e. the purely isochoric contribution.

Solutjon. In order to particularize the volumetric stress Svot it is only necessary to derive the term d'11vor(J)/dJ (see -eq. (6.9l)i). From eq. (6.89) we .find, using (6.91)1 and the relation for the purely volumetric elastic response in the :form of (6.13 7)1, that

in which, with the strain-energy functions (6.137h and (6..138), we obtain the specifi-

·Cation

As a second step we particularize the isochoric stress Siso in respect of the strain energy (6.139). Before proceeding it is first necessary to provide the relation

246 6 Hyperelastic IVlaterials ·

Recall (6.81), i.e. Aa = J-I/a An, a = 1, 2, 3, and relation (6.82h which, when formulated in principal stretches, reads DJ/fJ.,\1 = J ;\;1 . Thus, we have

which is relation (6.83) expressed through the modified principal stretches /\1 •

Hence, by analogy with (6.52), we obtain the isochoric stress response in terms of principal values for the general case /\ 1 =f 1\2 # /\1 -I A1, namely

(6."143)

where siHCHt'll = 1, 2, 3~ are the principal values -of the second Piola-Kirchhoff stress tensor Siso and Na, a = 1, 2_, 3, denote the principal referential directions.

By use of the chain rule and relation (6..J42h, a straightforward computation fron1

(6.143}i gives the explic.it expressions

for the priiici.pal'iSochork Stress-·va"lues,..(eompare_w,l.t.h...Q.aDEN [ 1997, Section 7.2.31)~

The summation symbol (which could be omitted) emphasizes that the index b is re~ peated, meaning summation ove.r 1, 2, 3. Howeve~, there is no summation over the index a. Using· the relation for the purely isocho.ric elastic response in the form of {6..139), we achieve finally the term D\J!1so/[),,\a, a. = 1, 2, 3, in the specific form

a= 1, 2, 3 (6..145)

(s.ee also the derivation by SIMO and TAYLOR [199laJ).

The complete stress response, as given through (6..140), (6."141) and the spectral decomposition (6.143)-(6.145), serves as a meaningfu] basis for finite element ana1yses

of constitutive models for isotropic hyperelastic materials at finite strains. •

Similarly to the compressible version of Ogden's .model we can reformulate the

Mooney-Rivlin, neo-Hookean, Vm:I:a, ·Yeoh, Arruda and Boyce .models, .i.e. (6.127_)-

suitable voiumetric response function '11v01 , for example, (6.. l.37h or (6.13'8).

For example, the decoupled strain-energy function for the Mooney-Rivlin model

has the form

{6.146)

However, material mode]s are often presented in a .coupled form. The compressible ·Mooney-Rivlin model, for example, may be given as

(6.147)

where c is a material constant and d de.fines a (dependent) parameter with certain re- strictions. By recalling the assumption that the reference configuration is stress-free we may deduce from (6.1.47) that d = 2(c1 + 2c2 ). The first two tenns in (6..147) were proposed by C.IARLET and GEYMONAT U98~] .in a slightly different :form (see also

ClARLET [19.88, Section 4.1.0]). ·

Another example is the coupled form of the compressible neo-Hookean model

given by the strain-energy function

compressible BLATZ and Ko .[.1962] and OGDEN [1972b] proposed a strain-e.nergy function which combines theoretical arguments and .experimental data (performed on certain solid polyurethane rubbers and foamed polyurethane e'lastomers). It is based on a coupled function of volumetric and isochoric parts according to

eq. (6.149) reduces to the Mooney-Rivlin form :introduced in eq. (6.127) (with the constants c1 = f Jt/2 .and c2 = (1 - f)J.i/2).

Another specfri'I case of the strain energy (6.149) may be found by taking f = 1, leading to the compressible neo-Hookean model ·introduced in eq. (6.148) (with la = J 2 and the constant J.L/2 = An interesting description ·Of the Blatz and Ko mode]

was presented by BEATTY and STALNAKER [1986] and BEATTY [1987].

EXERCISES

1.For the description of isotropic hyperelastic materials at finite strains we recall the important class of strain-energy functions \JI in terms of principal invariants.. Study some models suitable to describe compressible materials and particularize the .associated stress relations.

(a) Firstly, we consider the coupled form of the compressible Mooney-Rivli11, neo-Hookean, Blatz and Ko models according to the _given strain energies

(6.147)-(6.149), respectively. By means of (6.32) deduce the stress relation

-28w /812 , /'J = 2138\P /fJI..1 for the second Piola-Kirchhoff stress tensor

S as specified in Table 6.1.

Table 6.1 Specified coefficients for the constitutive equations of some ma- terials "in the coupled .form..

For notational simplicity we ·have introduced the .non-negative parameter

~ = 2µ (1 - f) / ! 3 • Note .that the response coefficients ')'1, / 2 , ")' for the neo-

3

Hooke.an model may be found as a special case of the Blatz and Ko model by taking~= 0 (with the constant µf = 2c1).

(b)Secondly, we consider the .decoupled fa.rm of the compressible Mooney-

Rivlin model (6.146) and the compressible neo-Hookean model (obtained by setting c2 = 0 in the Mooney-Rivlin model). In .addition, we consider

the decoupled versions of the Yeoh model and the Arruda and Boyce model,

.

i.e.