The terms “second order” and “third order” corresponds to presence in (5.35) the Lame elastic constants in the second order summands and the Murnaghan

5 SHORT DESCRIPTION OF NONLINEAR THEORY OF

Note 5.11. The terms “second order” and “third order” corresponds to presence in (5.35) the Lame elastic constants in the second order summands and the Murnaghan

elastic constants in the third order summands.

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which are distinguishing from (4.36) and using instead

, ,

A B C different notations

² ³

1 2 3 1 1 2 3 1 2 2 1

( , , ) 2 2 2 ,

2 3

l m W I I I O P I PI nI mI I I

2 3

1 2 3 1 1 2 4 3 3 2 1 2 1 1 1

( , , ) ,

2 3 6

W I I I OI PI Q I Q I I Q I

2 3

1 2 3 1 1 2 3 1 2 1

( , , ) .

2 3 3

c a

W I I I OI PI I bI I I

The main invariants

Ik are linked with the algebraic ones

Ik by the formulas

1 1, 2 1 2 1 2 , 3 1 6 1 31 2 2 .3

I I I I I I I I I I

(5.37) The Murnaghan potential can be meant as the classical one in the nonlinear theory of elasticity, it describes a large class of industrial materials, is widely used and is thoroughly commented in the fundamental books on nonlinear solid mechanics.

It seems to be the place to show a few different modifications of Murnaghan potential from different theories of elastic deformations. Six modifications can be listed here.

M1. Guz’s modification of Murnaghan potential

This model of deformation process can be referred to the classical theory of elasticity. The new potential has the form

1 2 3 3 1 2 13

( , , ) 1 2 _{iklm ik lm} 3 3 .

W I I I K H H c I bI I a I (5.38)

Here,

H_ik is the Green strain tensor,

K_iklm is the fourth rank tensor of the second order elastic constants.

This potential consists of two different nonlinear parts - quadratic and cubic. According to the author, the quadratic part characterizes anisotropy of material properties in the unloaded state of a body and corresponds to the potential of a linearly elastic anisotropic body. The cubic part corresponds to isotropic body.

The Guz’s potential was used for studying the regularities of wave propagation in poly- crystalline bodies having weak anisotropy of properties in a natural state (so-called quasi- isotropic bodies).

M2. Mindlin-Eringen’s modification of Murnaghan potential

This modification forms the basis for nonlinear micromorphic theory proposed by Mindlin and Eringen and goes out the framework of classical elasticity.

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The micromorphic medium is described kinematically by the macrodisplacements vector ^u^G and the 2^nd rank asymmetric tensor of microdisplacements

<. A process of deformation is defined by three independent tensors –

tensor of macrostrains

, , , ,

ik 1 2 ui k uk i u um i m k

H , (5.39)

tensor of relative distortions

, ,

ik ui k ik um k mk

J < < , (5.40)

tensor of macrodistortions

, , ,

ikl ik l um k mk l

N < < . (5.41)

This theory is built similarly to the classical theory of hyperelasticity. Therefore, the internal energy is assumed to be the analytical function of tensors (5.39)-(5.41) and so on.

The most important for the next consideration is that the micromorphic continuum turned out the direct generalization of three well-known microstructural continua:

Cosserat continuum, Cosserat pseudo-continuum, and Le Roux gradient continuum.

M3. Cosserat’s modification of Murnaghan potential

In the Cosserat theory of asymmetric elasticity, some simplifications in the kinematic picture (4.39)-(4.41) should be done. Only six independent parameters are used here: the vector of macrodisplacements

uG and the vector of macrorotations

<G. Then two tensors only define a process of deformation

, , 1 2 ,

ik ui k ikm Rm m Rm ikm i ku

J < , (5.42)

ikm ikm k m,

N < , (5.43)

where

mik is the conventional Levi-Civita tensor.

The internal energy is assumed as the function of two tensors:

the asymmetric tensor of relative distortion

J_ik

and the tensor of bending-torsion

*_km <_k,m.

The next form of the internal energy is used in the Cosserat theory and can be meant as some modification of the classical Murnaghan potential

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, 1 2 _{ik ik} 1 2 _{ik ki} 2 _kk ²

W J * P D J J P D J J O J

1 2 O H _{ik ik} 1 2 O H _{ik ki} E 2 _kk

* * * * *

Q1 6 J_kk ³ 1 2 Q2 G J J J1 _{ik ik kk} 1 2 Q2 G J J J1 _{ik ki kk} 1 3 2 Q3 G J J J2 _{ik km mi}

3 2 1 2 3

1 3 2Q G J J J_{ik km im} V _kk J_mm V _{ik ki mm}J V _{ik ik mm}J .

* * * * * (5.44)

Here

, ,c

O P are the Lame elastic constants,

1, ,2 3

Q Q Q are the Murnaghan elastic constants,

, , .

D E J H are the linear constants of a micropolar medium,

1, , , ,2 1 2 3

G G V V V are the nonlinear constants of a micropolar medium.

Historical sketch. The first important step from the classical theory of elasticity to the theory of asymmetric elasticity has been taken by Voigt, which proposed to use the tensor of couple stresses in addition to the conventional tensor of force stresses.

The general theory of asymmetric elasticity has been developed about more than one hundred years ago by brothers Cosserat. It is wide adopted that this theory was anew discovered in 1960 by Truesdell and Toupin. A lot of high quality and well- known mechanicians have been involved in studies of asymmetric elasticity that and the next decades. Most of the problems were solved and well commented. For exa mple, in Ukraine the well-known publications of Savin and his colleagues can be noted. Analogously, in Italy the high level publications of Grioli, Ferrarese and others should be mentioned. It can be noted also many efforts to the development of asymmetric elasticity have been made by Polish scientists (Nowacki, Wozniak).

M.4. Modification of Murnaghan potential in the Cosserat pseudo-continuum

The transition to Cosserat pseudo-continuum assumes a reduction of some independent kinematic quantities - the classical vector of macrodisplacements is only saved. The rotation in this continuum is restri- cted. Therefore, the vectors of macro-rotations and micro-rotations are identical

1 2 ,

m m mik i k

R < u . But the pseudo-continuum safes the couple stresses and asymmetry of force stresses. All physical constants of the Cosserat continuum are saved also.

The internal energy is written using the vector

uG only

, 1 2 1 2 2

1 2 1 2 2

ik ik ik ik ik ki kk

ik ik ik ki kk

W H P D H H P D H H O H

O H O H E

* * * * *

Q₁ 6 H_kk ³ 1 2 Q₂ G H H H₁ _{ik ik kk} 1 2 Q₂ G H H H₁ _{ik ki kk} 1 3 2Q₃ G H H H₂ _{ik km mi}

3 2 1 ² 2 3

1 3 2Q G H H H_{ik km im} V _kk H_mm V _{ik ki mm}H V _{ik ik mm}H .

* * * * * (5.45)

The macro-strain tensor

H_ik is meant as nonlinear one, the macro-rotation tensor should be assumed nonlinear, too. But traditionally the macro-rotation tensor is used in the linear form. Therefore, the bending-torsion tensor is written as

1 2 ,

ik kmnum ni

* .

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M5. Modification of Murnaghan potential in the Le Roux gradient theory

The next modification is associated with Le Roux variant of the moment theory of elasticity.

Only two tensors define the deformations picture:

the macrostrains tensor H_ik 1 2

u_{i k}, u_{k i}, u u_{m i m k}, ,

, the microdistortion tensor Nikm ui km, .

The proposed new potential modification in the Le Roux’s approach is slightly similar to the Guz’s modification because it consists in separating the linear part and nonlinear part of the deformation process.

Coupling the process is reflected only in the linear part, which can be explained as the influence of coupling is very weak. So, the Le Roux’s potential can be written in the form

_ik, _ikm _{ik ik} 1 2 _ii ² 2 ² _{ikm ikm} _{ikm kim}

W H N PH H O H PM N N QN N

4 2 2

1 3 1 3

ik mk im ik ki mm mm

mm ik mk mi nn ik ki nn ik ki

A B C

D G H J

H H H H H H H H H H H H H H H H H

(5.46)

Here

, ,c

O P are Lame elastic constants,

M,Q are new microstructural constants, are elastic microstructural constants of higher orders.

M6. Modification of Murnaghan potential in the elastic mixture theory

Usually, two modifications of Murnaghan potential are used in the microstructural theory of mixtures. This theory can be treated as the direct generalization of the classical theory of elasticity in the case of multi-continuum media. The main hypothesis is that a material is modeled by the set of interacting and interpenetrating continua.

Consider further the two-phase (two-component) elastic mixtures. Then the kinematic picture of a deformation process is described using two partial displacements vectors

( ) 1,2

uG^D D . The mixture in whole and its written for the mixture as whole internal energy (potential) are described using two different kind kinematic parameters - partial strain tensors

H_ik⁽^D⁾ and relative displacements vector

(1) (2)

v uG G uG (but other variants can be used; as an example can be pointed the Tiersten variant)

ik⁽¹⁾^, ik⁽²⁾^,

W W H H v^G (5.47)

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(as a rule, the Latin indexes take values 1;2;3 and Greek indexes – 1;2).

The first modification of Murnaghan potential follows from (5.47), when a cross- influence is taken into account both in quadratic nonlinear and in cubic nonlinear parts of the potential (that is, in the linear and quadratic nonlinear parts of constitutive equations)

2 2

(1) (2) ( ) ( ) ( ) ( ) ( ) ( )

3 3

( _ik , _ik , )_k _ik 2 _ik _ik 1 2 _mm _{mm mm}

W H H v P H_D ^D P H H^D ^G O H_D ^D O H H^D ^G

2 3

( ) ( ) ( ) ( ) ( ) ( )

1 3 A_DH H H_ik^D _im^D _km^D B_DH_mm^D H_ik^D 1 3 C_D H_mm^D

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 3

3 3 3

1 3 AH H H_ik^D _im^G _km^G 2BH H H_mm^G _ik^G _ik^D CH_mm^D H_mm^G E( )v_k 1 3 Ec( ) .v_k

(5.48)

This potential includes seven elastic constants

k, ,k

O P E of the second order and ten elastic constants

, , ,

k k k

A B C Ec of the third order.

The second mixture modification of Murnaghan potential is simpler than the modification (5.48):

the cross-influence is taken into account in the quadratic nonlinear parts of the potential and is neglected in the cubic nonlinear ones (that is, it is taken into account in the linear part of constitutive equations and is neglected in the quadratic nonlinear one)

2 2

(1) (2) ( ) ( ) ( ) ( ) ( ) ( )

3 3

2 3

( ) ( ) ( ) ( ) ( ) ( ) 2 3

( , , ) 2 1 2

1 3 1 3 ( ) ( )

k mm mm mm

ik ik ik ik ik

im mm mm k k

ik km ik

W v

A B C v v

D D G D D G

D D

D D D D D D

D D D

H H P H P H H O H O H H

H H H H H H E E

c (5.49)

This potential includes seven elastic constants

k, ,k

O P E of the second order and seven elastic constants

, , ,

A B C_D _D _D Ec of the third order.

A shown above plenty of potentials is characteristic for the nonlinear theory of elasticity

and differs this theory from the linear one where one potential is only used.

At the end, let us repeat the Note that seems important.

Note 5.12. The constitutive equations are evaluated by formulas (3.21),(3.22) for all kinds of hyperelastic potentials. Therefore, knowledge of the potential means a knowledge of constitutive equations. The equations of motion (equilibrium), equations (3.19),(3.20) for the strain tensors, and the depending on the form of elastic potential constitutive equations form the close system of the equations of the nonlinear theory of hyperelastic materials.

Comments

Comment 5.1. An effect of nonlinear dependence of decreasing the shear stresses when the torsion angle (deformation) to the level of non-small values is called “the Poynting effect”

owing to Poynting pub-li-cation of 1909, where this effect was described. At that, Poynting does not mentioned the results of Coloumb (1784), Wertheim (1857), Kelvin (1865), Bauschinger (1881), Tomlinson (1883), where this effect was also described in one way or another. But only within the framework of the nonlinear theory of elasticity that admits the finite elastic deformations, that was developed in 20 century, this effect was satisfactorily explained by Rivlin in 1951. He used the model of nonlinear elastic deformation which now is termed “the Mooney-Rivlin model”.

Comment 5.2. The most part of publications on hyperelastic materials realize an approach based on postulation of the elastic potential (explicit nonlinear dependence of internal energy) on the nonlinear strain tensors (their invariants or the corresponding principal stretches) that admits the large strains. Practically all the variants of potentials are tailored with use of the speculative considerations, can be associated with theoretical modeling of phenomenological character and assume ultima analysi the next determination of the presenting in the potentials physical constants by the ways, which are not linked with phenomenological considerations (mainly, by the experimental ways).

In studies of hyperelastic materials, the Murnaghan potential is used most often. This can be explained partially by that presence in this potential of the third algebraic invariant permits to take into account a row of significant nonlinear effects. Also, it became clear with time that both Murnaghan and his successors thought (being the mechanicians) very important the experiments on determination of the presented in the potential elastic constants (two Lame constants and three Murnaghan constants). Therefo- re. to date these constants are known for many tens of engineering materials.

But not all authors of potentials had so luck as Murnaghan. This can be associated with the adopted in the theoretical studies practice of speculative analysis and access to experiment at the final stage of studying the potential. Seeming- ly, not all the initiators of potential were realized an importance of determination of physical constants. It is known the sentence of Signorini about the time of starting with experiments – “the later, the better”.

May be, therefore Signorini has been not left an apparent trace of analysis of new constant from the potential named afterwards the Signorini potential. Despite of some advantage of its own potential (the potential includes one only constant of the third order, whereas, for example, the Murnaghan potential includes three constants of the third order), Signorini lost the struggle against Murnaghan for sympathy of next investigators. The well-known fact of

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advantage has the approach, which is using the less number of physical constants, Signorini has been not used. Later after many tens years of oblivion, the Signorini potential is used to describe the nonlinear waves in elastic materials.

Further reading

5.1. Arruda, EM & Boyce, MC 2000, Constitutive models of rubber elasticity: a review.

Rubber Chemistry and Technology 72, 504–523

5.2. Arruda, EM & Boyce, MC 1993, A three-dimensional constitutive model for the large stretch behavior of rub ber elastic materials. J. Mech. Phys. Solids 41, 389–412.

5.3. Bartenev, GM & Khazanovich, TN 1960, On the law of high-elastic deformations of the net polymers. High-molecular Compaunds, 2, N1, 20-28.

5.4. Blatz, PJ & Ko, WL 1962, Application of finite elastic theory to the deformation of rubber materials. Trans. Soc. Rheology, 7, N6, 223-251.

5.5. Chernykh, KF & Litvinenkova, ZN 1988, Theory of large elastic deformations. Leningrad University Press, Leningrad. (In Russian)

5.6. Dynayev, IM 1975, Generalized elastic potential for analysis of structures from resilient polymers. Izvestiya vuzov. Stroitelstvo i arkhitektura, N2, 52-59. (In Russian)

5.7. Fu, YB 2001, Nonlinear Elasticity: Theory and Applications. London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge.

5.8. Fung, YC 1993, Biomechanics. Mechanical Properties of Living Tissues. Springer, New York.

5.9. Gent, AN 1996, A new constitutive relation for rubber, Rubber Chemistry and Technology, 69, N1, 59-61.

5.10. Green, AE & Adkins, JE 1960, Large Elastic Deformations and Nonlinear Continuum Mechanics. Oxford University Press, Clarendon Press, London.

5.11. Guz, AN 1986, Elastic Waves in Bodies with Initial (Residual) Stresses. In 2 vols. V.1.

General Problems. V.2. Regularities of Propagation. – Kyiv: Naukova Dumka. (In Russian).

5.12. Hanyga, A 1983, Mathematical Theory of Nonlinear Elasticity. Elis Horwood, California.

5.13. Holzapfel, GA 2000, Nonlinear Solid Mechanics: A Continuum Approach for Engineering.

Birkhauser, Zurich.

5.14. John, F. 1960, Plane strain problems for a perfectly elastic material of harmonic type.

Comms. Pure Appl. Math. 13, N2, 239-296.

5.15. Kappus, R 1939, Zur Elastizitattheorie endlicher Verschiebungen. ZAMM,19,271- 285, 344-361.

5.16. Lur’e, AI 1990, Nonlinear Theory of Elasticity. North-Holland Series in Applied Mathematics and Mechanics, North-Holland, Amsterdam.

5.17. Mooney, M 1940, A theory of large elastic deformation. J. Appl. Phys. 11, 582–596.

5.18. Murnaghan, FD 1951,1967, Finite Deformation in an Elastic Solid. John Wiley, New York.

5.19. Novozhilov, VV 1948, Osnovy nielinieinoi uprugosti (Foundations of nonlinear elasticity).

Gostekhizdat, Moscow. (In Russian)

5.20. Novozhilov, VV 1953, Foundations of the nonlinear theory of elasticity. Graylock Press, Rochester, New York.

5.21. Ogden, RW 1972, Large deformation isotropic elasticity—on the correlation of theory and experiment for incompressible rubberlike solids. Proc. R. Soc. Lond. A Math. Phys.

Sci. A326, N1567, 565–584.

5.22. Ogden, RW 1997, Non-Linear Elastic Deformations. Dover Civil and Mechanical Engineering, Mineola.

5.23. Rivlin, RS 1948, Some applications of the elasticity theory to rubber engineering.

Collected papers of R.S. Rivlin. in 2 vols. 1997, Springer, Berlin.

5.24. Rivlin, RS 1948, Large elastic deformations of isotropic materials. IV. Further developments of the general theory. Philos. Trans. Roy. Soc. A241, 379-397.

5.25. Rivlin, RS 1949, Large elastic deformations of isotropic materials. V. The problem of flexure. Proc. Roy. Soc. Lond. A Math. Phys. R. Soc. A195, 463-473.

5.26. Rivlin, RS 1949, Large elastic deformations of isotropic materials. VI. Further results in the theory of torsion, shear and flexure. Philos. Trans. R. Soc. A242, 173–195.

5.27. Rivlin, RS & Saunders, DW 1951, Large elastic deformations of isotropic materials.

VII. Experiments on the deformation of rubber. Philos. Trans. Roy. Soc. A 243, 865, 251–288.

5.28. Rivlin, RS 1953, The solution of problems in second order elasticity theory. Arch.

Rat. Mech. Anal. 2, 53-81.

5.29. Rushchitsky, JJ 2014, Nonlinear Elastic Waves in Materials. Series “Foundations of Engineering Mechanics”. Springer, Heidelberg.

5.30. Seth, BR 1935, Finite strain in elastic problems. Phil. Trans. Roy. Soc. London, A234, 231-264.

5.31. Signorini, A 1943, Transformazioni termoelastiche finite. Annali di Matematica Pura ed Applicata, Serie IV, 22, 33-143.

5.32. Signorini, A 1949, Transformazioni termoelastiche finite. Annali di Matematica Pura ed Applicata, Serie IV. 30,1-72.

5.33. Signorini, A 1955, Transformazioni termoelastiche finite. Solidi Incomprimibili. A Mauro Picone nel suo 70 ane compleano. Annali di Matematica Pura ed Applicata, Serie IV, 39, 147-201.

5.34. Signorini, A 1959, Questioni di elasticite non linearizzata. Edizioni Cremonese, Roma.

5.35. Signorini, A 1959, Questioni di elasticite non linearizzata e semilinearizzata. Rendiconti di Matematica 18 (1-2), 95-139.

5.36. Signorini, A 1960, Transformazioni termoelastiche finite. Solidi Vincolati. A Giovanni Sansone nel suo 70 ane compleano. Annali di Matematica Pura ed Applicata, Serie IV, 51, 320-372.

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SHORT DESCRIPTION OF NONLINEAR THEORY OF ELASTICITY. PART 2

5.38. Treloar, LRG 1975, The Physics of Rubber Elasticity. Clarendon Press, Oxford.

5.39. Valanis, KS & Landel, RF 1967, The strain-energy function of a hyperelastic materials in terms of the exten sion ratios. J. Appl. Phys, 38, 2997-3002.

5.40. Yeoh, OH 1993, Some forms of the strain energy function for rubber. Rubber Chemistry and Technology, 66, N5,754-771.

Questions

5.1. Consider the Seth model and formulate why the existence of the internal energy elastic potential is important for the models of nonlinear elastic deformation.

5.2. Find the experimental facts that show the applicability of the Neo-Hookean model for the small strains. Note that, as a rule, the nonlinear models describe well the deformation process for the large strain and not so well for the small strain.

5.3. Write the constitutive equations that correspond to the John potential. Separate the nonlinear summands and comment on this nonlinearity.

5.4. Find the experiments that link the new Signorini constant c with the real material.

May be, Signorini wrote some comments in his publications?

5.5. Explain the popularity of the Treloar potential despite its simplicity. Which level of strains can be described by this potential?

5.6. Describe suitability of statistical approaches as compared with the phenomenological app roaches that dominate in the nonlinear mechanics of materials. Estimate the work of statistically substantiated models in the case of small strains.

5.7. Write the list of modern materials which are deformed nonlinearly and admit the large strains.

5.8. Find the experimental data that show the value of the new constant from the Blatz-Ko potential. Estimate also the effect of the ratio

D on the process of deformation.

5.9. Comment absence of the first invariant in the Knowles-Sternberg potential.

5.10. Comment absence of the second invariant in the Lurie potential.

5.11. Write the constitutive equations that correspond to the Gent and Fung models and draw the picture stress-strain for the unidirectional tension. Comment a possibility to model the soft tissue by the elastic material.

5.12. Rivlin proposed a few nonlinear models for elastic materials. Make a list and compare these models from point of view of the number of constants.

5.13. Estimate the rationality of introduction into the nonlinear models the new functions (usually the model includes the new constants). Use for comments the shown above models that include the new function.

5.14. Consider the multi-constant models and comment on the possibility to determine from experiment five and more elastic constants.

5.15. Try to add to the six variants of modification of the Murnaghan potential some new modifications that are reported in the new publications and not shown here.

Dalam dokumen Foundations of mechanics of materials: Part 1 (Halaman 93-103)