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# Is paper ductile?

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In my previous blog, I complained about colleagues developing constitutive models without having any notion about the specific nature of deformation and damage and their micromechanical mechanisms. Unfortunately, this happens more often than one might (or would like to) believe, as a recent example testifies.

P. Mäkelä and S. Östlund: Cohesive crack modelling of thin sheet material exhibiting anisotropy, plasticity and large-scale damage evolution. *Engineering Fracture Mechanics*, Vol. 79, 2012 pp. 50-60.

Looking at the title and the Abstract:

“In this work, a cohesive crack model suitable for static fracture mechanics analysis of thin sheet materials exhibiting anisotropy, plasticity, and large-scale damage evolution was developed”,

the reader might think of “metal sheets”. But far wrong: the authors talk about paper! And the unbelieving reader starts realising this at the end of the Introduction:“The elastic–plastic model was calibrated by tensile testing and the cohesive zone model was calibrated by stable tensile testing for one grade of paper material”,and in section 2.4, where the tests are described.

The theory of plasticity deals with the stress-strain and load deflection relationships for ductile materials. Is paper “ductile”, and what means “ductile”, actually? Talking about ductility, people commonly think of the behaviour of metals, and it were metals for which phenomenological plasticity has been developed. Metals have a crystalline structure, so the plastic flow occurs by sliding along crystallographic planes or by twinning. “A physical theory of plasticity starts with these microscopic details and attempts to explain why and how plastic flow occurs” (A.S. Khan & S. Huang: Continuum Theory of Plasticity, Wiley, 1995, p. 310). What are the mechanisms of deformation and damage in paper? The authors do not tell us!

They just present a pretty conventional phenomenological model of orthotropic plasticity based on a transformation of the stress tensor and the von Mises yield criterion with an associated flow rule, and apply this to paper specimens. The “excellent prediction” of the test results by the model, which the authors claim with respect to Fig. 6, is not at all impressive, as it just shows that the uniaxial stress-strain curves can be described by an exponential function as in Eq. (6). Finally, they combine this with some exponential cohesive softening law to describe the tearing of the paper and as the respective cohesive parameters were fitted to the test results, there is no reason why this should not “predict” failure of centre-cracked sheets for varying crack lengths, *a*/*W*, satisfactorily.

What a “large-scale damage evolution” announced in the title is supposed to be, remains obscure, as a cohesive zone describes localised and no “large scale” damage.

That the authors try to surprise us with repeated statements like

- “The accuracy of the cohesive crack model is largely dependent on accurate constitutive modelling.”
- “The performance of the cohesive crack model is generally most dependent on the accurate formulation and calibration of the cohesive zone model.”
- “The key to accurate cohesive crack modelling is constituted by the ability to accurately determine the cohesive material behaviour.”

underlines the lack of information about paper in the present manuscript.

Final remark: The present blogs intend to encourage discussion on fracture mechanics and related subjects. No reaction by the authors has yet come to any of them. Does this support the suspicion that publishing does not intend to contribute to science but just to increase the individual scoring of scientists?

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## Comments

## Plasticity modeling

I do agree with you on the fact that paper material is not proper to be modeled with the use of classical plasticity theory. In fact the elastic-plastic material model described through equation 6 (which is the same as eq. 7, the yield function) is a direct relation between stress and plastic strain, when one considers the simplest case of uniaxial monotonic loading. Thus, for the uniaxial case, the material exhibits zero elastic range, since the stress depends only on the plastic strain (so the model is not

ELASTO-plastic). I think that the model used is more of a curve-fitting formula and cannot be considered as a constitutive plasticity model. The model used in this publication (2012) is taken from a previous paper published in theInt. Journal of Solid and Structures (2003)(ref.16), therefore a closer look on this is deemed necessary for fully understanding the choice of such an approach.