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### another quick elementary reading is this:

*In reply to Fracture and singularities of the mass-density gradient field*

The Theory of Critical Distances (TCD) is a bi-parametrical approach suitable for predicting, under both static and high-cycle fatigue loading, the non-propagation of cracks by directly post-processing the linear-elastic stress fields, calculated according to continuum mechanics, acting on the material in the vicinity of the geometrical features being assessed. In other words, the TCD estimates static and high-cycle fatigue strength of cracked bodies by making use of a critical distance and a reference strength which are assumed to be material constants whose values change as the material microstructural features vary. Similarly, Gradient Mechanics postulates that the relevant stress fields in the vicinity of crack tips have to be determined by directly incorporating into the material constitutive law an intrinsic scale length. The main advantage of such a method is that stress fields become non-singular also in the presence of cracks and sharp notches. The above idea can be formalized in different ways allowing, under both static and high-cycle fatigue loading, the static and high-cycle fatigue assessment of cracked/notched components to be performed without the need for defining the position of the failure locations *a priori*.

The present paper investigates the existing analogies and differences between the TCD and Gradient Mechanics, the latter formalized according to the so-called Implicit Gradient Method, when such theories are used to process linear-elastic crack tip stress fields.

Askes, H., Livieri, P., Susmel, L., Taylor, D., & Tovo, R. (2013). Intrinsic material length, Theory of Critical Distances and Gradient Mechanics: analogies and differences in processing linear‐elastic crack tip stress fields. *Fatigue & Fracture of Engineering Materials & Structures*, *36*(1), 39-55.

### "critical distance" in few words

*In reply to Fracture and singularities of the mass-density gradient field*

The critical distance in few words (so that you can avoid reading the entire Taylor book!)

1) point method:- this has the standard singularity of the classical solution, but neglects it! So it simply considers a stress value at some point ahead of the crack --- of a given material constant distance

2) line method: there is some averaging of stress here: results are very similar to PM above, and perhaps this becomes "non-local".

3) area method and volume method --- the average is taken over area and volume, but again no big deal

4) quantized fracture mechanics: the average is taken of the stress intensity factor

What do you think of these?

### Re: no offence in "much simpler"

*In reply to Fracture and singularities of the mass-density gradient field*

Dear Michele,

I did not take offence at all (it takes a lot to get me offended in a technical discussion; hopefully never).

While I have not yet looked at the papers you mentioned (I definitely want to have a look - I like to take a look at everything I can!), I have a reasonable sense of the basic idea of what goes on in peridynamics. My model is a pde model and not nonlocal (ok this depends on what variables one is working with) and so there are some big differences in that respect..... an interesting question that comes to mind is what is the limiting form of the perdiynamic model in the limit of zero-horizon. That might be a pde model and looking at the connection of that to my model may make sense. However, in the limit model I am not sure if there can be a length scale, and my pde model certainly does have at least one intrinsic length scale which will be operative even in equilibrium solutions, so....

### no offence in "much simpler"!

*In reply to Fracture and singularities of the mass-density gradient field*

dear Amit

when I said that the "critical distance" approaches are perhaps "much simpler", I wasn't meaning to say that your theory is too complicated! The critical distance approaches are very empirical, not evolved in general, and I do not think they have a solid background. Yet, they work quite well, so I am happy you say you want to have a look.

Perhaps as you mention the "peridynamics approach of Silling", you could consider that "critical distance" methods are similar: they both define what Silling calls an "horizon".

Have you made any further comparison with peridynamics? If so, you are quite close to "critical distances".

### Re: Interesting theory...

*In reply to interesting theory Amit*

Dear Michele,

Thank you for your comment and the references. I was not aware of the works by Taylor & Susmel and Pugno and Ruoff you mention, but now will be, thanks. I need a little time to look through these and try to form an opinion, at which time I will get back to you.

For me avoiding the singularity is important in so far as models for evolution of defects inevitably require objects like stress and energy density at the defect for driving force and in nonlinear theories it is very difficult if not impossible to separate out `my stress' from the defect itself from 'your stress' of other defects and boundary conditions.....if I was only interested in energies and minimizing it for results, this would not bother me so much except if global energies of finite bodies started becoming infinite, which happens for dislocations, but not for cracks or disclinations.

On the question of simplicity - I guess it is in the eye of the beholder. I have spelled out my goals in the paper, and am interested in a conceptual framework for dealing with evolving singularities in continuum mechanics that can lead to practical advances (and for me this means through robust computation). I also want to be faithful to the different kinematics related to different types of singularities and in understanding how to accommodate such in the energy. Curiously enough, this directly connects to the `technology' of gauge theories in physics, where they pull out these kinematic fields from the gauge principle. I find it much more satisfying to find such things emerge simply from trying to understand singularities, which is what the deeper message is, in my opinion, in the works of Weingarten, Volterra and Mura and DeWit. And if the principle of gauge invariance is not satisfied, so be it.....

Anyway, I am probably rambling here, so I will stop. But simplicity for me is defined as a model that

1) has generality, so it does something for you beyond an immediate problem you want to solve, preferably shows some new way of thinking (since one has to work hard at the problem anyway), i.e. does not make one miss the forest for the trees

2) you can command an electronic device with not much intelligence to execute, once given precise instructions. The precise instructions may require very detailed algorithms related to subjects like nonlinear wave prorpagation etc. which in itself takes a lot of learning.

I don't think the current state of affairs related to dealing with evolving singularities in continuum mechanics is at a stage of theoretical and algorithmic development where this claim can be made, but progress is happening.

### nonlocal intensity factors from Claudio Montebello

*In reply to moving singularities (crack analogues) in fretting fatigue*

I received feedback from collegues in France

A very interesting thesis on related problems

https://tel.archives-ouvertes.fr/tel-01238905/document

Analysis of the stress gradient effect in Fretting-Fatigue through a description based on nonlocal intensity factors Claudio Montebello

*In reply to Fracture and singularities of the mass-density gradient field*

Taylor, D. (2010). The theory of critical distances: a new perspective in fracture mechanics. Elsevier.

Taylor, D. (2008). The theory of critical distances. Engineering Fracture Mechanics, 75(7), 1696-1705.

Susmel, L., & Taylor, D. (2008). The theory of critical distances to predict static strength of notched brittle components subjected to mixed-mode loading. Engineering Fracture Mechanics, 75(3), 534-550.

Susmel, L. (2008). The theory of critical distances: a review of its applications in fatigue. Engineering Fracture Mechanics, 75(7), 1706-1724.

Cornetti, P., Pugno, N., Carpinteri, A., & Taylor, D. (2006). Finite fracture mechanics: a coupled stress and energy failure criterion. Engineering Fracture Mechanics, 73(14), 2021-2033.

Pugno, N. M., & Ruoff, R. S. (2004). Quantized fracture mechanics. Philosophical Magazine, 84(27), 2829-2845.

### interesting theory Amit

*In reply to Fracture and singularities of the mass-density gradient field*

dear Amit, interesting theory, although I am not an expert in the field to judge completely.

I have always been reluctant to accept singularities, and in the end, many theories take singularities, but then return to non-singular solutions, for example in fatigue do you know of the so-called "Theory of critical distances" which has been popularized by David Taylor but is in fact existent much before him? There is also a version of "Quantized Fracture Mechanics" by Nicola Pugno which is similar.

Perhaps your theory can be connected with this (much simpler) ideas?

### and they added Davide Bigoni also!

*In reply to top italians scientists*

Somebody told me that also Davide Bigoni, who was Katia supervisor in Italy, was forgotten from the list. He is now there.

### I see they added Katia now!

*In reply to top italians scientists*

She is towards the end of the list.

So there are 15 in the USA, 4 in switzerland, one in Canada, one in Norway, one in France, but all the rest is in home country!

I wonder if there is any such list for chinese origin, or indians, or such? Does anybody know of similar lists?

### Katia Bertoldi from Harvard should be there for example

*In reply to top italians scientists*

https://scholar.google.com/citations?user=Jt1sCk8AAAAJ&hl=en

I will contact the list managers.

### Big congratulations for the

*In reply to Rational design of reconfigurable prismatic architected materials*

Big congratulations for the very interesting work!

### Material Jacobian Matrix for hyperelastic materials

*In reply to Calculating material jacobian matrix for ABAQUS UMAT subroutine*

Although the methods for calculating jacobian matrix for well-known hyperelastic materials such as Neo-Hookean model is available online, there is no appropriate and comprehensive source for the more complicated formulations.

One of the most famous formulations recently is following:

Strain Stored energy (W)=W(λi)

while λ is the eigenvalue of deformation gradient or principle stretch and **n **is the eigenvector of deformation gradient or principle direction

then one can find the stress using following equation:

** σ**=(λi ∂W/∂λi **n**i ×**n**i) /**J**

while × is dyadic product and **J **is λ1 *λ2 *λ3

Now I have the following questions:

1- what is the explicit equation in terms λ to find jacobian matrix?

2- do we need to subtract the hydrostatic pressure from the **σ **and then use new stress tensor and calculate material jacobian regarding new stress tensor?

### Research

*In reply to Molecular dynamics simulations of mechanical behavior in nanoscale ceramic –metallic multilayer composites*

This work is the starting point for developing continuum viscoplastic models for potential use in macro-scale anaylsis.

### Post-doc in Experimental Non-linear Dynamics

*In reply to Post-doc in Non-linear Dynamics*

Employment as an assistant researcher (Post-doc) at the project OPUS from National Centre of Science.

Details: http://www.perlikowski.kdm.p.lodz.pl/project.html

Description of project tasks:

• Analytical and numerical investigations of existing implementations of TMDs

• Bifurcation analysis of the influence of TMD with non-linear spring, non-linear damper and inerter on mitigation of vibrations.

• Investigation of coexisting solutions and existence of rare attractors in system with proposed TMDs.

• Design of the experimental rig based on existing device.

Requirements

• Education: PhD in mechanical/civil engineering, physics or related disciplines,

• Laboratory experience: dynamical measurements,

• Programming experience: C++, Matlab, Mathematica,

• Knowledge of path-following toolbox: Auto/MatCont/Coco

• Publications: at least one publication,

• Languages: good command of English.

Procedures for the recruitment

The recruitment procedure has a two-stage formula and it will consist of the following stages:

1st Stage – an assessment of written applications. Applicants should submit:

• A motivating cover letter stating the current/future interests,

• Full CV,

• A list of publications,

• Letter of recommendation.

• Confirmation of PhD degree

2nd Stage – a qualifying interview with selected candidates

Terms and Conditions

The contract of employment (full-time) for a period of 1 year (with a possible extension for 2 years.) Remuneration 4622 zł per month (gross).

Applications will be accepted until 14.11.2016 at the Secretariat of the Department of Dynamics at the Faculty of Mechanical Engineering Technical University of Lodz, address: Stefanowskiego 1/15, 90-924 Lodz, (A22 building, room 122) and the e-mail address: przemyslaw.perlikowski@p.lodz.pl

### Post-doc in Experimental Non-linear Dynamics

*In reply to Post-doc in Non-linear Dynamics*

Employment as an assistant researcher (Post-doc) at the project OPUS from National Centre of Science.

Details: http://www.perlikowski.kdm.p.lodz.pl/project.html

Description of project tasks:

• Analytical and numerical investigations of existing implementations of TMDs

• Bifurcation analysis of the influence of TMD with non-linear spring, non-linear damper and inerter on mitigation of vibrations.

• Investigation of coexisting solutions and existence of rare attractors in system with proposed TMDs.

• Design of the experimental rig based on existing device.

Requirements

• Education: PhD in mechanical/civil engineering, physics or related disciplines,

• Laboratory experience: dynamical measurements,

• Programming experience: C++, Matlab, Mathematica,

• Knowledge of path-following toolbox: Auto/MatCont/Coco

• Publications: at least one publication,

• Languages: good command of English.

Procedures for the recruitment

The recruitment procedure has a two-stage formula and it will consist of the following stages:

1st Stage – an assessment of written applications. Applicants should submit:

• A motivating cover letter stating the current/future interests,

• Full CV,

• A list of publications,

• Letter of recommendation.

• Confirmation of PhD degree

2nd Stage – a qualifying interview with selected candidates

Terms and Conditions

The contract of employment (full-time) for a period of 1 year (with a possible extension for 2 years.) Remuneration 4622 zł per month (gross).

Applications will be accepted until 14.11.2016 at the Secretariat of the Department of Dynamics at the Faculty of Mechanical Engineering Technical University of Lodz, address: Stefanowskiego 1/15, 90-924 Lodz, (A22 building, room 122) and the e-mail address: przemyslaw.perlikowski@p.lodz.pl

### Pure coincidence DMT approx work in Pastewka-Robbins model?

*In reply to On Pastewka and Robbins' Criterion for Macroscopic Adhesion of Rough Surfaces*

This paper may help shed some ligth why the very numerous crude approximations in PR model may have coincidentally led to reasonable agreement in their set of parameters. But not for any extrapolation fundamental purpose! The paper is submitted but probably siib accepted, as reviewers were positive.

1. arXiv:1701.04300 [pdf, ps, other]

On the use of DMT approximations in adhesive contacts, with remarks on random rough contactsMichele CiavarellaComments: 11 pages, 5 figuresSubjects: Materials Science (cond-mat.mtrl-sci)

### Is there a video?

*In reply to Speech of Acceptance of the 2016 Timoshenko Medal by Ray Ogden*

Thanks for sharing the speech. Can we also watch the video somewhere? Perhaps it can be uploaded by the committee on websites like YouTube!

### Homogenization in ABAQUS

*In reply to Homogenization in ABAQUS*

The homogenization technique in ABAQUS for a RVE is presented in detail in the following paper:

Shahzamanian, M., et al., *Representative Volume Element Based Modeling of Cementitious Materials.* Journal of Engineering Materials and Technology, 2014. **136**(1): p. 011007.

### Candidates will be shortlisted soon

*In reply to Positions in computational solid mechanics at Institute of High Performance Computing, A*STAR, Singapore*

Thank you all for submitting the applications, we will be shortlisting the candidates and come back to you soon.

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