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# Multi-phase hyperelasticity with interface energy effect

Recently, J. Wang, L. Sun and I have formulated some ideas about the effective properties of heterogeneous materials with surface/interface energy effect, which are shown in the attached file.

Papers in the attached file can be viewed as a two-part paper, called “Multi-phase hyperelasticity with interface energy effect” if it is standalone. Part one of this topic is covered in “A theory of hyperelasticity of multi-phase media with surface/interface energy effect”, which provides theoretical background. Part two is covered in “Size-dependent effective properties of a heterogeneous material with interface energy effect: from finite deformation theory to infinitesimal strain analysis”, with more emphasis on application.

There are several new results in these papers and they are different from those in the existing literature, e.g.,(1) the constitutive equations of the surface/interface at finite deformation are proposed in part I, and the Shuttleworth equation and the Gurtin & Murdoch equation are only valid for small deformation; (2) in the framework of finite deformation, both Lagrangian and Eulerian descriptions of the generalized Young-Laplace equations are derived in part I;(3)the symmetric interface stress was used in the existing literature, however, in order to predict the overall properties of a structure or a composite material, asymmetric surface/interface stress has to be used even in the case of linear elasticity. (4) in the existing literature, it is believed that the residual surface/interface tension does not influence the effective moduli of the composite material, however, it is shown that residual surface/interface tension does affect the effective moduli of the composite material.

The preprints are attached here ( Part I and Part II ).

And DOI links are:

Attachment | Size |
---|---|

Part I by Huang and Wang.pdf | 140.22 KB |

Part II by Huang and Sun.pdf | 227.29 KB |

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## Comments

## Prof. Huang, your group's work is the very essential part.

Dear Prof. Huang.

Your group's work is the very essential part of the theory about the effect of surface energy on the mechanical properity of solids.

It is helpful to clarify the confusions, for example, if the interfacial residual stress has influence on the effective moduli.

Recently Prof. T.J. Wang's group has investiged the effect of surface energy on the deformation of elliptical hole(Appl. Phys. Lett. 89, 231923),the diffraction of plane compressional elastic waves by a nanosized circular hole. (Appl. Phys. Lett. 2006 in press) and the plastic deformation of solids with nano-inhomogeneities (Doctal Dissertation of W.X. Zhang)

With regards

## Dear Weixu, Can I have a

Dear Weixu,

Can I have a copy of the apper you mentioned, which is to appear in Appl. Phys. Lett.

I am very interested in wave diffraction in particulte composite materials.

Thanks.

## Dear Henry

Now all our papers can be downloaded online now.

W.X. Zhang and T.J. Wang, 2007,Effect of surface energy on the yield strength of nanoporous materials, Applied Physics Letters. 90, (063104)

G.F. Wang, T.J. Wang and X.Q. Feng, 2006, Surface effects on the diffraction of plane compressional waves by a

nanosized circular hole. 89,(23193)

G.F. Wang and T.J. Wang,2006, Deformation around a nanosized elliptical hole with surface effect. 89, (161901)

Thank you

## Further comments on the effect of surface/interface energy

Dear Weixu,

Thank you for your comments. There are several posts and a lot of comments about the effect of surface/interface energy (and surface/interface stress) recently, such as

http://www.imechanica.org/node/555

http://www.imechanica.org/node/219

http://www.imechanica.org/node/591 .

Here I want to make further comments on this topic.

There are two kinds of fundamental equations in the solution of boundary-value problems for stress field with surface/interface energy effect. The first is the surface/interface constitutive relations, and the second is the discontinuity conditions of the stress across the surface/interface, namely, the Young-Laplace equations.

It should be noted that, even for linear elasticity (where an infinitesimal analysis is employed), the above mentioned equations should be established within the framework of finite deformation in the first place. The reasons are:

(1) In the study of the mechanical behavior of composites, what we are concerned with is the mechanical response from the reference configuration to the current configuration. During the deformation process, the size and the shape of the surface/interface change, and thus change the curvature tensor in the governing equations. This means that the deformation will change the residual elastic field induced by the surface/interface energy. The effect of surface/interface energy manifests itself precisely through the change of the residual elastic field due to the change of configuration. Therefore, this is essentially a finite deformation problem. (Please note that in linear elasticity, only small quantities with the same order as displacement gradient are retained. We do not consider the change of configuration due to the fact that there is no residual stress in traditional linear elasticity, or the residual stress does not change when the configuration changes. However, the change of the curvature tensor in the Young-Laplace equation has the

same orderof the displacement gradient. Therefore, the residual elastic field induced by the surface/interface energy is “configuration dependent”).(2) It can be seen that even if an infinitesimal deformation approximation is used, the first kind and second kind Piola-Kirchhoff stresses as well as the Cauchy stress of the surface/interface are not the same, since there exists residual surface/interface stress in the reference configuration. Expressions of these surface/interface stresses can be found in the second part of the “two-part paper” in the attached file. This situation is completely different from that in the three-dimensional analysis in linear elasticity, in which there is no residual stress in the reference configuration. This indicates that in the study of effect of surface/interface energy on the composite properties, only starting from a finite-deformation theory can we correctly choose an appropriate infinitesimal surface/interface stress to be used in the governing equations in linear elasticity.

(3) There should be a residual elastic field due to the presence of the surface/interface energy (and surface/interface stresses) in materials, even though there is no external loading. In order to take into account the change of the residual elastic field due to the change of configuration, the Lagrangian description of the Young-Laplace equation (in the framework of finite deformation) has to be used, thus the influence of the liquid-like surface tension on the effective properties of composite materials can also be included.

Now the problem becomes how to consider the effect of surface/interface energy in linear elasticity. The answer is to

the two kinds of fundamental equations in the case of finite-deformation framework. The procedures are:linearize(1) Use the Lagrangian description of the generalized Young-Laplace equation; i.e., the discontinuity conditions of the stress across the surface/interface are expressed in terms of the first kind Piola-Kirchhoff stress of the surface/interface in the first place, and then linearize it. (This is because in Lagrangian description, all quantities are based on one configuration, i.e., the reference configuration, so these quantities are comparable).

(2) linearize the constitutive relation of the surface/interface. This relation should be expressed in terms of the asymmetric surface/interface stress.

## Thank you, Zhuping, for tips on superficial elasticity

I've just read your comments with great interest. Your comments on using Lagrangian description is right on. A student of mine has just begun to work on a related problem. We'll benefit from your teaching.

Incidentally, in teaching a graduate course on solid mechanics, I began to read your 2003 book, Fundamentals of Continuum Mechanics (in Chinese). It is a great book! Scholarly written, and with insight that can only come from many years of experience. I didn't have enough time this year to incorporate materials into my course, but I'll try next time.

Perhaps you can write a post to let other people know how they can order the book. If it is possible, I'll have our library order a copy. As you probably know, many of our students and iMechanicians read Chinese.

Best wishes for the new year, and happy every day!

## Many thanks for your comment on my book

Hi, Zhigang,

Many thanks for your comment on my book. I will contact the publisher to figure out how to buy the book.The problem may be the mail expense. I will let you know as soon as I get the news from the publisher.

Zhuping

## dimensionless curvature tensor

I missed one word in my comments in point (1) of the reasons. It should be "The change of the dimensionless curvature tensor in the Young-Laplace equation has the same order of the displacement gradient."

## Size effect in nanocomposites

Dear Zhuping, nice papers. In the concluding and discussion section of the Part II article, you mentioned that the results in this paper can find application in the studies of nanocomposites. Could you let me know your plan for this further direction? Are you considering the effect of different radius sizes for nanotube composite materials?

For example, the radius of the armchair (18,18), (12,12) and (6,6) carbon nanotubes are different, about 1.25, 0.83 nanometer, respectively. The macroscopic stress-strain relations are completely different for a composite material consists of those different nanotubes.

## Dear Henry, Thank you for

Dear Henry,

Thank you for your comment. At present, I am not working on the carbon nanotubes, and nanotube composite materials. But I think, part II article can be directly applied to the study of effective properties of nanocomposite materials filled with ellipsoidal (or spheroidal) particles.

## Zhuping Huang: Fundamentals of Continuum Mechanics

Dear Zhuping: Your excellent textbook, Fundamentals of Continuum Mechanics, has now been cataloged in Harvard Library. As you know, many of our students can read Chinese, and will benefit from your work. The book is so well written and so finely produced. It brings pleasure every time I read it. Thank you and your publisher.