Science topic

# Fracture Mechanics - Science topic

Fracture mechanics is the study of the influence of loading, crack size, and structural geometry on the fracture resistance of materials containing natural flaws and cracks. When applied to design, the objective of the fracture mechanics analysis is to limit the operating stress level so that a preexisting crack would not grow to a critical size during the service life of the structure.

Questions related to Fracture Mechanics

Hello All,

I'm looking into measuring crack growth rate using the DCPD method and was wondering if it is possible to use a "bench top power supply" and a multimeter with higher accuracy? Or do I need to have specialized equipment for DCPD measurement?

The materials I'm interested are iron alloys and aluminum alloys.

Compact tensions specimen thickness =< 13mm

Cheers,

Rashiga

I'd like some opinions. What are the major challenges in fatigue and fracture mechanics?

I need elaboration in easy language to grasp the concept.

I discovered two distinct phenomena when cracks begin to form at α and β phases in titanium alloys. How does this difference mechanism come about?

It is difficult to capture the descending branch in the

**stress- strain diagram**of the material in an experimental direct tension protocol (see**Figure (a)**: tension behavior). The flexibility of the loading system causes premature specimen collapse. This is due to the**release of stored strain energy**in the loading equipement. It has been argued that it is feasible to measure this**softening behavior**when a sufficiently controlled**strain rate**is used. Given these facts, the following questions arise:**Is it possible to measure the softening behavior of plain concrete in an experiment?****Is there any experimental evidence of this measurement being possible when using a sufficiently controlled strain rate?**

**What are the good research Areas/topics in Structural Engineering for Ph.D.?**- Can anyone suggest to me research topics related to
**Fatigue and fracture mechanics of Concrete**? - Kindly suggest me a good
**review article regarding the studies performed in Structural engineering related to Fatigue life analysis of fiber reinforced concrete/ Flexural fatigue performance of concrete made with recycled materials**? - Kindly suggest me research topics related
**Flexural-Fatigue Properties of Sustainable Concrete Pavement Material ?**

Dear experts in fracture mechanics, I need help in choosing a model to track crack propagation using numerical simulations. If anyone could provide me with any suggestions, articles, tutorials, or reference books where I can study (to understand the basic concept of J-integral and SIF) and which model to choose and how to implement it numerically, it will be highly appreciated.

Looking forward to your precious responses.

Thank you

Izaz Ali

Can I be assisted with guidelines to linear elastic fracture mechanics in timbers?

Can an elliptic crack (small enough to remain a single entity, with

**no internal pressure or shear force**) inside an isotropic material (no boundary effect) be expanded in its own plane under**externally applied shearing stresses**only?If yes, how did you show that? Do we have experimental evidence for the process?

In fracture mechanics, we accept that quasi-brittle materials such as concrete have a strain-softening branch under tensile stresses. The branch that can be obtained from a Direct Tensile Test. Accordingly to CEB-FIP, it is possible to characterize this branch by means of the fracture energy. Recently, our research team has been trying to find experimentally this branch in hollow concrete blocks (HCB) employing displacement-controlled tests, at very slow velocities (as low as 0.0005 mm/s), in a servo-hydraulic testing machine. However, it has not been possible to capture this softening branch.

The experimental results that we have had the opportunity to review, which are reported in the specialized literature, make us think that there is a problem with the testing equipment (inertia and stiffness of the machine) and that it could not be an intrinsic characteristic of the material.

After these arguments, we would appreciate your help to get some insight on the following:

1) Is it possible to obtain experimentally a strain-softening branch for quasi-brittle materials, using the Direct Tensile Test, particularly for plain concrete or HCBs?

2) Is this softening branch really a property of the material? Or is it just an apparent behavior generated by the testing machine and the measurement devices used, I mean, due to the way these devices work?

If it exists, does the speed at which it occurs need to be of an order of magnitude lower than 0.0005 mm/s? And finally, if that’s the case, does it make sense to use it in the interesting research problems?

In linear elastic fracture mechanics (LEFM), only the stress intensity factor seems to be used. LEFM implies the same r

^{−1/2}singularities of stress and strain at the crack tip. Very few articles on fatigue use the strain intensity factor. Formally, the two factors could be related by Hooke's law. What is the reason why the strain intensity factor is not used in the same way as the stress intensity factor? A search for strain intensity factor returns only stress intensity factor. Why is the strain intensity factor more or less ignored?R.M. Christensen states that "fracture mechanics in the brittle range ... require(s) formulations in terms of stress." https://www.failurecriteria.com/isitstressorstra.html

Is anything wrong with the strain intensity factor?

I am trying to find out how to identify specified parameters of ductile damage material in its stress-strain curve in Abaqus. I could identify the Young's modulus and the yield stress, but not the fracture energy. How do I identify this in my results?

For now I am simulating a simple tensile test in y-direction on a one-element model (1x0.2x1).

Cast iron is metal, not ceramic. But it is brittle, and quite strong. What can be additives that can be added to increase its strength, corrosion resistance and high temperature stability so that it can be used in place of hard ceramics in structural application? on can it not be qualified as a ceramic even if its microstructure is predominantly intermetallic?

I sam earching a preporcessor program for fracture mechanics analysis using abaqus, I am doubting between zencrack and feacrack, I wonder if someone has experience with these programs and can help me to select which one (or another program) is better.

I want to use it for 3D studies

Looking for a motivated Ph.D. candidate to work in the field of additive manufacturing with the background of mechanical engineering and material science.

Deadline for application: February 14, 2020

How to model delamination results from matrix cracking with XFEM?

Dear all,

I need to do 2D multicrack propagation in a composite laminate (GFRP) in tensile load. I have 3 parts namely (0-90-0 degrees plies). And I already have existing cracks which you can see below in attached pictures. Cracks propagate well, but when it comes to the boundary I couldn’t see correct delamination behavior. Is it possible to simulate delamination because of matrix cracking in Abaqus? I both tried cohesive surface and elements. With the elements it says I cant define XFEM crack zone for the cohesive element. With the surface, during loading I see some overclosures and clearances.

Here are my material properties.

Cohesive Surface:

Knn=Kss=Ktt= 1e6 N/mm3 (which I am not so sure)

Damage Initiation: Maxs damage initiation criteria Normal=Shear1=Shear2=3.4MPa.

Damage evolution criteria: Mixed mode behavior=BK(Power=2.2)

Fracture energy: Normal=Shear1=Shear 2= 0.27 N/mm2.

Also, frictionless tangential behavior, and hard contact.

For 0 degrees ply

Damage Initiation: Maxps=807MPa

Damage evolution criteria: Mixed mode behavior=BK(Power=2.2)

Fracture energy: 0.5, 0.5, 0.01 N/mm

Elastic, type=ENGINEERING CONSTANTS 35100.,9600.,9600., 0.3, 0.3, 0.3, 4000.,4000.4000.,

For 90 degrees ply, I just swap E11 and E33 values, and maxps=15MPa

Can someone suggest me how I can model it and if my material properties are reasonable?

Regards

Berkay

if plastic zone (rp) is obtained from maximum stress intensity (=del. K/(1-R)). How to determine plastic zone (rp) from log-log da/dN

I know this question has been asked in this forum but want to rephrase it as there have been some features that came for Ansys. Can anyone please tell from their experience which one is better? For example, I have used Ansys and felt it has some limitations e.g. in XFEM. As it can not be directly used using UI in the benchmark. Also in 3D, it is unstable. Also, all of them in Ansys fail in Anisotropic material. But I don't have experience in ABAQUS.

Kindly let us know what are your insights.

With regards

Biswabhanu Puhan

Peel stress singularities occur at the bi-material interface of the free edges of, for example, a steel-adhesive-steel sandwich specimen of say 500 mm in length, 100 mm in width, and 10 mm thickness (5 mm adhesive thickness) under longitudinal cyclic tension due to the difference in Young's modulus and Poisson's ratio of the two materials.

The singularity can be relatively economically investigated in a very fine cross-sectional finite element (FE) model representing the as manufactured geometry using the generalized plain strain approach (ANSYS), for example, see attached picture. The finer the mesh gets the larger gets the local peel stress peak at the bi-material free edge interface. The graph shows the equivalent stress according to Beltrami (similar to von Mises).

The load amplitude is chosen from a fatigue load spectrum of a cyclic test and is thus that small that linear elasticity holds for the average stress according to classical laminated plate theory (CLPT) in the adhesive. Now, this local stress peak, however, gets already as large (factor of 10 of the average equivalent stress) that it could be dedicated to a plastic deformation and failure. In reality, i.e., in the experiment, we would not see such a failure after say the first 100 load cycles. Thus, there is a discrepancy between linear elasticity theory and reality. The material seems to withstand these peaks.

For the static strength analysis one could use a non-linear stress-strain curve to circumvent the "problem" and lower the peaks and use then fracture mechanics to quantitatively evaluate such situations, but this requires the modeling and relatively expensive analysis of possible crack orientations in all possible dimensions due to the multi-axial stress state. Therefore, fracture mechanics does not seem to be an applicable method to evaluate the fatigue damage for a load spectrum whose stress vector components vary over time and thus with them the critical crack planes .

Another technique is to extract the stress value at a critical distance from the edge and then extrapolate a "more realistic" stress peak. But how to define such a critical distance on a physically sound basis and how does the extrapolation function look?

Do you have any other ideas how to extract a realistic stress value from the finite element model for a fatigue analysis?

Hi all,

I'm interested in testing the Hydrogen Embrittlement of several metals. I

Interms of fracture testing and fatigue crack growth rate testing I want to know the difference between the following.

1. Long time H2 exposure in an environmental chamber -> conducting tests in ambient enviroment.

2. Long time H2 exposure in an environmental chamber -> conducting tests in a H2 exposed environment.

Are there design equations for concrete sectors based on fracture mechanics parameters?

using the fracture parameters KIc or Gf to design concrete section as using compressive strength in design equation

Does a single-valued description exist for isotropic materials?

I'm trying to study cracked structures under a compressive load taking into account contact between the crack lips.

I would like to know if the method of extrapolation by displacement remains valid to calculate the stress intensity in this case or other valid methods.

And if you can send me some references containing the same study.

What is the best finite element modeling software packages for modeling non-linear and fracture analysis of CONCRETE? I've had some with ABAQUS standart/explicit, but I would like to learn if there is a better software for non-linear behaviour and fracture mechanics of concrete according to your experiences.

Best regards.

Hello everyone,

I am trying to develop an

**analytical model**to study the propagation of fracture in**reinforced concrete structures**. I would really appreciate it if anyone could suggest to me the optimal analytical model to study the crack propagation.Thanks in advance

Regards,

Pawan

How to define a crack at symmetry plane? Is it right idea? Crack seam we cant specify at symmetry edge (As crack seam can be specified at interior face of 2D). How to handle this type of situation? I need to find KI for different crack lengths. Its possible to find without define seam. But I m not confident about the results. If same crac length defined inside the face (other than symmetry edge) with partition, I can get crack tip stress field. But at symmetry without seam Im not getting. (Of course we cant get, That I can understand) What may be alternate for this? Anyone have Idea, pl share.

Recently, I came into contact with a new discipline named chemo-mechanical coupling. I found that chemo-mechanical coupling phenomena exist in many research areas, ranging from development of advanced batteries, biomechanical engineering, hydrogen embrittlement, and high temperature oxidation, etc. Although it is very important in engineering field, I can't know the main mechanisms of coupled chemical and mechanical interactions. Can you give me some suggestions? Such as, some related publications or research project. Thanks very much.

I am planning to use the

**Discrete Element Method**for**propagation of Fracture**in**Reinforced concrete Structures**. Which book would be the best start to study**Discrete Element Method**from the beginning?Dear All,

I want to model crack growth in a plate using ABAQUS XFEM method. Here is
my question about it: How to calculate Stress intensity factor in XFEM
when crack growth in ON?

At this time, when I using ABAQUS XFEM with allowing for crack growth and request for SIF through History output request,

It says that SIF could not be produced when crack growth in XFEM is ON.

Best,

adel,

I want to simulate the growth of a crack due to fatigue. My model consists of shell elements that are non-planar. I wanted to use the XFEM available in Abaqus, but in its documentation was written that:

" XFEM is available only for three-dimensional solid and two-dimensional planar models; three-dimensional shell models are not supported. "

So the question is that are there any software to simulate crack growth in non-planar parts modeled with shell elements?

I've heard of some software like FRANC3D or ZenCrack, but I'm not sure if they're capable of doing this.

Regards

Good afternoon!

FE software ANSYS provides SIFs calculation in fracture mechanics for 3 modes by CINT command for isotropic elastic material for example. Does command CINT calculate SIFs for orthotropic material or anisotropic material correctly or not? Does calculation method with CINT command in ANSYS take into account anisotropy or orthotropy of material?

In engineering, the damage that can be detected is usually called crack. But how to describe the small cracks which cannot be measured in the early stage of damage? Can we give a universal definition? Greatly appreciate your review.

I have reviewed some books about fracture mechanics, but I cannot explain microstructure damage with engineering fracture mechanics theory, such as fatigue damage of metals with micro-defects. Is there any outstanding work in this field? If so, can you share it with me? Greatly appreciate your answer and providing good practices in engineering.

Many thanks!

It has been established that the tetragonal to monoclinic phase transformation of zirconia can be used to improve the toughness. The driving force for this transformation is the temperature gradient, which results in a change in the crystal structure of zirconia from tetragonal to monoclinic.

But, at room temperature Metastable inclusions of tetragonal Zirconia dispersed in a ceramic
matrix will transform to the thermodynamically stable monoclinic form on the application of an external tensile stress, what is the driving force for such transformation to occur?

FE software ANSYS provides SIFs calculation in fracture mechanics for 3 modes by CINT command for isotropic elastic material for example. Does command CINT calculate SIFs for orthotropic material or anisotropic material correctly or not? Does calculation method with CINT command in ANSYS take into account anisotropy or orthotropy of material?

Hello everyone,

I'm working on crack propagation using a meshless method in linear elastic fracture mechanics and I want to use the Maximum Tensile Stress Criterion. I need some references or help in order to implement numerically this criterion ? And is it possible to use it without evaluating the stress intensity factors?

I am working on impact of proppant shape on fracture conductivity. I have found some articles and I know that some shapes as spherical will improve fracture conductivity comparing to other shapes. But now I want to simulate or calculate or model this effect when all other conditions of fracturing remains unchanged. I want to know is it possible? and if yes, what is this software?

I am performing probabilistic fatigue analysis and need to find the distributions which could be used for material constants C and m used in Paris law.

I have a scenario in which I need to do crack initiation & propagation study . Based on the load which is applied on the component, crack needs to be initiated . Once it is initiated, the same crack needs to be propagate if the external loads applied continuously . How to simulate this scenario using ABAQUS?

Dear connections ,

What is the

**scientific relationship**between stress intensity factor**(SIF)**and crack tip opening displacement**(CTOD)**with specimen double edge cracked.So really I need your help if it is possible.

Regards,

When a planar crack is loaded in tension (mode I) in brittle materials, the velocity V of the crack is observed to increase from zero to a terminal velocity VT, then follows crack branching on further increasing V. VT falls well below the Rayleigh wave velocity VR.

To our knowledge, one attempt has been made to estimate theoretically VT. This attempt makes use of Mott (1948) extension of the Griffith concept to dynamic fracture. This is expounded by Lawn (1993). This estimate gives VT ≈ 0.38cl where cl is the velocity of longitudinal sound wave.

Does any other estimate exist?

I have read in some articles about unstable crack growth which is sometimes referred to as RCP, however it seems that there is a clear difference between unstable crack growth and RCP. According to a paper from Leevers, P. (2001; ISBN: 0-08-0431526 📷 pp. 3322±3329 ;see attached image), he differentiated between an unstable crack growth which follows a slow crack Growth (SCG) when a critical value K1c is reached and an RCP which also follows SCG, however in a different way. I am grateful for any kind of explanation.

For a perfect column, the critical load is equal to that found analytically, but when I introduce a crack, the critical load increases, but in reality it should decrease.

Fracture mechanics

Structural dynamics

Earthquake Engineering

Structural Health monitoring

Dear connections,

I encountered a problem in that the results of the

**stress intensity factor SIF**in the two methods did not match. And I think the fault lies in**abaqus (EFM intégral J)**.But I do not know where exactly!So really I need your help if it is possible.

You find in attachment file which contains more information.

In order to check the accuracy of trapezoidal form in tabular form in Abaqus, I did a single 3D element tensile example which named 3D element in attached inp file. The result seems good. Please refer to the attached plot of the Damage (SDEG) Vs Separation and Traction Vs Separation.

However, when I define same trapezoidal CZM for ductile carbon fiber reinforced thermoplastic material to simulate Mode I delamination fracture process in Abaqus with surface-based cohesive contact. The displacement-force curve are totally different with experiment results. The FEM and experimental are shown in attached file. I also attached the DCB-trapezoidal. inp file, please check it.

So anyone can help me check the reason? How to get the saw-tooth displacement-force curve like expereimental result using CZM?

Thank you!

Good mourning, could any one give me some references to standard methode used to determin fracture parameters from SENT specimens, some articles do not give any referance to a standard method that give impression that these specimens are not machined according to any standards. thanks

I am a new user of Abaqus. I want to ask whether it's possible or not to perform fatigue life analysis using standard abaqus (without sub-routine). If possible, then how to define their loadings and where can I get their results. I hope that somebody can share their experiences here. Many thanks.

I am looking about any useful review paper about fracture mechanics (brittle and ductile)....Have one you any suggestions?

I try to model fatigue crack growth in aluminium 2024-T3 alloy.

I have plate specimens with finite width 'W' and a circular cut-out with an additional notch, as shown in Figure.

1. The number of fatigue cycles for crack initiation could be predicted by Stress Concentration Factor, but how to find a suitable equation for that complicated notch?

2. When calculating fatigue crack growth rate by Paris-Erdogan equation:

da/dN = C * (dK) ^m

the difference in Stress Intensity Factor (dK) in a specified fatigue cycle should be calculated for crack length caused by fatigue crack growth 'a_CG'? Or as a total crack length, taking into account also initial crack length ('a_CG' + 'a_ini') ?

Due to its fined microstructure, HSC possesses different mechanical properties compared to normal strength concrete. One common characteristic is its increased brittleness. Many approaches have been used to judge the material brittleness. One method is using the ratio between the tension strength to compressive strength: the lower the ratio, the more brittle. Another is the linear portion in the stress-strain curve observed in uniaxial compressive concrete. The larger the linear portion, the more brittle. In fracture mechanics, some brittleness quantifiers have been defined according to different models In fictitious crack model (FCM), a characteristic length (lch) that is defined by combining the fracture energy (Gf) with elastic modulus E and the tensile strength of material (ft). The smaller the value lch, the more brittle the material. Another common used brittleness indicator is the critical crack extension length (ac). the larger the value ac, the less brittle. When it decreases with an increase in compressive strength, we can say the brittleness increases with strength. Apart from these, what are the other approaches that can be used to define brittleness of HSC?

Based on the principles of fracture mechanics, a crack propagates when K

_{I }>= K_{IC}, i.e. stress intensity factor is larger than fracture toughness, where K_{I}=σ√(πa) for a central crack. Theoretically, as crack propagates and crack length (a) becomes larger, K_{I}increases too. Therefore, when a crack starts to grow, it never stops! However, it is not the case in many geomechanics applications in underground structures. For example, a propagating crack in a tunnel surface will stop at a distance from the tunnel wall eventually.In numerical modeling, a stress relaxation method may be used to lower the acting stress on the crack to model crack propagation more realistically. what are other efficient ways to model this phenomenon realistically? Good references are appreciated.

Thanks.

Any papers, books,codes, or sites as a start? Thank you.

ASTM E-1820 recommend side grooving after fatigue pre-cracking. But I think side grooving before fatigue pre-cracking is better because the fatigue pre-crack front will be more uniform. Also, crack growth during fatigue pre-crack, will follow side grooved region.

Please give your comment and and share your experiences.

I'm looking for JIC ralated to J-integral method in fracture mechanic for Al 2024-T3 at different thickness. Who knows the reference which this parameter has been mentioned for this material at variable thickness?

Fracture surface after Uni-axial tensile testing of pure aluminium sub-size tensile specimen shows the presence of elongated dimples instead of equi-axed dimples. What could be the reason behind this? Does it imply that the material failed by shear?

Suppose avg. grain size, crystal structure, young's Modulus, fracture strength, velocity of sound and surface roughness of both the crushed crystals (before and after crushing ) and the crushing surfaces, as well as load on grinding surface, static/dynamic friction coefficients are known. Then, is it possible to estimate the crushing sound of the crystals?

Conversely, if crushing sound of the crystals are analyzed, is it possible to find any mathematical relation between the variables outlined?

I am asking the questions since crushing minerals and recording-analyzing the sound require no sophisticated instrument at all.

Please provide relevant research links.

Hi. I want to plot R-curve diagram analytically for Al 2024 and evaluate nonlinear fracture mechanics of it.

I know there is ASTM standard E561 for obtaining this graph experimentally.

Is there any document that I can find this graph for this material?

I'll be grateful if you can give me any hint.

I am modelling an interfacial zone by cohesive elements in ABAQUS. In order to conduct a parametric study, I need to change thickness of the cohesive elements in a range (assume in the range of 0 to 0.8 mm). In a manual, it has been written that when a unit thickness is specified during defining section in the property module , penalty stiffness is the same as the elastic modulus and thickness of the layer is kind of ignored.

Herein, there are two questions;

1- Is it rational to use non-zero thickness cohesive elements at all (non-zero thickness in the model geometry and unit thickness in the cohesive section)?

2- How the penalty stiffness should be defined for the model with zero thickness up to that with 0.8 mm?

- With a good fault and fracture dataset (displacement and length), how can I extract a mathematical data, strain rate, so to speak? Maybe programs or even formula? Do you have any indications of papers/authors or program?

(initial, minimum, maxium arc length increment and estimated total arc length) to get more accurate results?

A simple crack system (

*Figure***1**) can be readily studied to estimate the Hertzian conoidal crack angle and length, and also the stress intensity factor.This is a 3

*-D*brittle elastic half-space on the flat boundary*Ox*1*x*3 of which a rectilinear contact pressure along*Ox*3 is exerted by a cylinder whose axis is parallel to*x*3; the cylinder lies along Ox3 on the flat boundary. A planar straight-front crack inclined by an angle*θ*with respect to*x*1*x*3 is present under the action of the load along*x*2 due to the cylinder. The relevance of this modelling may be understood as follows. A slab of cylinder with thickness*dx*3 at spatial position*O’*(0*,*0*, x*3) exerts elastic fields (displacement and stress) proportional to those of a point load at*O’*(proportionality coefficient*dx*3). Physically, this corresponds to the action of a spherical indenter to which is associated a conoidal fracture surface for sufficiently large load (Roesler (1956) as quoted by Frank and Lawn (1967)). The coalescence of conoidal cracks from different slabs of cylinder along*Ox*3 would produce planar fracture surface envelops parallel to*x*3 at large crack lengths. Therefore, we expect the modelling in**Figure****1**to provide the experimentally observed fracture surface inclination angle*θ*and crack length*l*as a function of critical load*P*by both a spherical indenter and cylinder. This is the essence of the modelling depicted schematically in**Figure****1**.I am planning to teach a new class and wanted to use this specimen. Note that the black-ish line on the specimen in the LHS picture was originally along the axis of the cylinder, indicating a bulk shear strain on the order of 1. The smooth failure surface at around 50 deg from the axis in the RHS picture is thought to occur first. Note the small similar failure surfaces initiating under the blue arrow in the LHS picture. Also note the this smooth surface is predominantly under compression with the torque direction indicated. I believe there is something about molecular alignment in the tensile direction creating a weaker material in the plane perpendicular...approx. corresponding to the smooth fracture surface.

I am using ductile damage in FE model for steel material. the damage evolution law should be specified in terms of equivalent plastic displacement (U

_{pl}) or in terms of fracture energy dissipation (G_{f})Is the correct to use the following relation for calculating equivalent plastic displacement (U

_{pl})?Where L is the characteristic length of element, "epsilon.f.pl" is the equivalent plastic strain at failure and "epsilon.0.pl" is the equivalent plastic strain at the onset of damage.

or it should be calculated using this equation?

U

_{pl}=L(epsilon.f.pl)based on abaqus documentation the fracture energy can be calculated using the following relation

G

_{f}=(U_{pl}*sigma_{yo})/2where sigma

_{yo}is the value of the yield stress at the time when the failure criterion is reachedHello,

I am trying to simulate crack propagation between two composite plies in an in-house finite element code. I use high-order elements for better convergence and to reduce computational cost, but i have never seen in literature to release nodes instead of the whole elements in any fracture mechanics theory. Is there any specific reason based on theory that forbids this fact?

Thanks

- Generally, fracture mechanics parameters and terms (CTOD, J, C*, stress, load line displacement, plastic zone size) have relevant & respective dimensional units. The physical significance of their units is easily understandable like, energy per unit area or energy rate per unit area of crack growth. In case of K, why its (meter)
^{1/2 }i.e. square root of crack size, a and not simple a or 1/a or a^{-2 }? What is the physical significance of including square root of crack size, a in K formula?

In modeling concrete fracture, I could use fixed or rotating smeared crack model. But when could I use each of them.

Also how could I identify the pre-transition and post-transition parts of strain increment?