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Updated: 22 hours 10 min ago

yes, there is when sigma = ft

Thu, 2021-04-01 10:49

In reply to Then, there is no instability

yes, there is when sigma = ft for AT1/PF-CZM.

BFGS, quasi-Newton

Thu, 2021-04-01 10:42

In reply to Journal Club For April 2021: Variational phase-field modeling of brittle and cohesive fracture

Thank you Phu for this nice overview. It is probably worth emphasising that quasi-Newton methods such as the BFGS algorithm enable robust and efficient implementations of phase field fracture also for AT2 and AT1 models:

Then, there is no instability

Thu, 2021-04-01 10:37

In reply to Note that d only needs to

Then, there is no instability and strength of material.

Note that d only needs to

Thu, 2021-04-01 10:31

In reply to Sorry again. d=0 cannot "hold

Note that d only needs to satisfy Q <=0, not necessarily Q = 0. So d = 0 is the solution. No trick at all.

Sorry again. d=0 cannot "hold

Thu, 2021-04-01 10:14

In reply to yes, d= 0 holds for AT1 and

Sorry again. d=0 cannot "hold". It must be solution of Q(d)=0 in any theory. If that is not the case then you do not have theory - you have a tricky algorithm instead.

yes, d= 0 holds for AT1 and

Thu, 2021-04-01 10:09

In reply to Sorry, I do not understand

yes, d= 0 holds for AT1 and PF-CZM, but not for AT2. For the AT2 the damage is actiavated immediately once the load is applied no matter how small it is.

The damage is not activated until sigma = ft for AT1 and PF-CZM, but in the AT1 sigma is dependent on the length scale while the PF-CZM. I suggest you to read our AAM paper for the differences between AT1/AT2/PF-CZM.

Yes, grad(d)=0 before the

Thu, 2021-04-01 09:42

In reply to crack nucleation

Yes, grad(d)=0 before the crack nucleation but d is not zero! It deviates from zero and this deviation leads to material instability (with the subsequent localization of damage into crack).

Sorry, I do not understand

Thu, 2021-04-01 09:34

In reply to Hi Kosta,

Sorry, I do not understand transition from (9.2) to (9.3). You should solve (9.2)_2 for d: Q(d)=0. You cannot assume d=0 in advance!

Hi Kosta,

Thu, 2021-04-01 09:28

In reply to Dear Phu

Hi Kosta,

See this for details.

In short, PF-CZM is similar to CZM: initially the bar is linear elastic, the stress is homogeneous and its magnitude is smaller than the material strength f_t. This is the case until the stress equals f_t, a cohesive crack is initiated, and the stress is decreasing. 

PF-CZM is just a geometrically regularised CZM, in which the phase-field is used to represent the sharp cohesive cracks. 

Hope it helped.



crack nucleation

Thu, 2021-04-01 09:12

In reply to strength

Dear Kosta,

I asked Phu to upload a picture (I don't know how to do it) and you will understand why the PF-CZM is able to model crack nucleation.



Dear Phu

Thu, 2021-04-01 08:10

In reply to Hi Kosta,

Dear Phu,

My analysis is correct for any model :))

The phase field must be a solution of the phase field equation. If the solution is identically zero, then you cannot initiate the localization at all.


Thank you

Wed, 2021-03-31 06:47

In reply to Nice work!

Hi Roy,

Many thanks for your interest!



Elliptic integral

Wed, 2021-03-31 06:45

In reply to VdW foundation

Hi Zhaohe,

Good point! We tried to calculate the integration of atom-atom interaction between a cylinderical monolayer and a cylindrical core, but it is in fact the so-called elliptic integral. This kind of integral cannot be written in an explicit expression in terms of elementary functions (Conrad, Impossibility theorems for elementary integration, Academy Colloquium Series, 2005; Rosenlicht, Integration in Finite Terms, Am. Math. Mon. 79, 963-972, 1972). Hence, we consider the interaction between an atomic layer and a semi-infinite substrate instead. Similar strategy works well in macroscopic models of both cylindrical and spherical film/substrate systems. For modest curvature, such Winkler-type foundation (only in function of w) describes the interaction between the surface layer and an infinite thick substrate, while Zhao et al., J. Mech. Phys. Solids 73, 212, 2014; Xu and Potier-Ferry, J. Mech. Phys. Solids 94, 68, 2016 theoretically and numerically showed that this assumption works well for shallow curved substrates, as well as doubly curved spherical substrate (Xu et al., J. Mech. Phys. Solids 137, 103892, 2020; Zhao et al., J. Mech. Phys. Solids 135, 103798, 2020). In the present work, results obtained from our continuum model agree with MD simulations, and the radius effect of the substrate considered in our cases appears insignificant.



Dear Ahmad

Tue, 2021-03-30 14:11

In reply to Dear Jie,

Dear Ahmad,

Thanks for your inspiring thoughts. 

For 2), I think it is possible if one can carefully design the inhomoenenous cut patterns without cuting holes. The in-plane mismatch deformation could lead out of plane buckling to form 3D shapes. 

For 3), it is a great idea to have krigami patterns embedded in soft layers. In that case, it could be doable by acutating the boundary of kirigami patterns to form target shapes.    

position filled

Mon, 2021-03-29 10:44

In reply to Opening for a postdoc position at UT Austin

This position has been filled.

You consider uniform tension.

Mon, 2021-03-29 06:14

In reply to Journal Club For April 2021: Variational phase-field modeling of brittle and cohesive fracture

Dear Phu,

You consider uniform tension. The deformation should be uniform until the limit point on the stress-strain curve. The limit point is called strength. Strength in your theory (or any phase field formulation) will depend on the characteristic length (b).


Dear Dr Volokh,

Mon, 2021-03-29 05:54

In reply to  Dear Dr. Nguyen,

Dear Dr Volokh,

Thanks for your comment. According to Ref. [26], K. Pham, H. Amor, J.-J. Marigo, and C. Maurini. Gradient damage models and their use to approximate brittle fracture. International Journal of Damage Mechanics, 20:618–652, 2011, who used a second-order stability condition to show that the softening stage of the homogeneous solution is stable only if the bar is short enough. 

So it is possible to have localized damage even for a homogenous state.

Best regards,


 Dear Dr. Nguyen,

Mon, 2021-03-29 04:01

In reply to Journal Club For April 2021: Variational phase-field modeling of brittle and cohesive fracture

 Dear Dr. Nguyen,

This topic is interesting. Fig. 2 is not clear. You say you consider homogeneous tension. Then, how localization of damage can occur? You should consider truly homogeneous case in which failure is also homogeneous. This will give you strength.

Best regards,


Nice work!

Sat, 2021-03-27 13:23

In reply to Nanosleeves: Morphology transitions of infilled carbon nanotubes


This is an important topic.


Very sad to hear this loss.

Sat, 2021-03-27 04:14

In reply to A tribute to Arthur Peter “Art” Boresi

Very sad to hear this loss. May his soul be at peace with the Heavenly Father.

I use his book "Adavanced Mechanics of Materials" for teaching my PG Students.


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