When a material is tested in flexure, i.e. under bending; tensile stresses are produced on the top layers; zero stress at neutral axis and compressive stresses in layers below the neutral axis. In isotropic materials, when the material fails, the corresponding load is taken for calculation of flexural strength. Since the isotropic material fails in tensile portion; the strength is nothing but its tensile strength. Hence when tested using tensile mode on a UTM;; we designate it as tensile strength and when tested on flexural loading set-up, we call it as flexural strength. Ideally both should be same. In fact for ceramic materials; tensile strength is obtained using 3 point bending set-up because tensile specimens for ceramic materials cannot be prepared. Only difference is if any defect is present in the specimen; irrespective of its location; it affects tensile strength in the same manner because uifom tensile stresses are produced across the whole cross section. In bending situation however; defects will affect the strength differently depending upon their location. A defect nearer top or bottom surface will have significant effect as compared to the same type of defect located nearer or on the neutral axis. Hence the strength in two modes of testing may differ. .   

In my opinion you can not give a universal formula for the relationship between tensile stength / modulus and flexural strength / modulus. This relationship depends on the type of the material and geometry of the sample. Theoretically it should be the same.

The difference between the modules I would explain by influence of complex stress state of bent sample. If the material has different properties in tension / compression and to do so is not perfectly linear, the module computed using the linear theory is a certain "averaging".

In case of differences in flexural strength and tensile strength, I think that can be affected by scale effects (which can also be explained by statistical effects). The maximum stresses occur on a small area of the bending section (theoretically only on the edge). While the entire cross section of the sample is under the maximum stresses during the tension test. Therefore, it is more likely to find a "weak point" from which the destruction starts. Similarly, smaller samples show greater strength than bigger ones.

I join opinion of the previous commentator about durability. A ratio between  strength at a bend and strength  at extention it is difficult to receive the general. Theoretically the problem can be solved having a big data file on strength  at extention. then it is possible to use the WeybulI-Gnedenko approach in case of the linear chart of extention and to consider large-scale effect, that at extention more defects get to a zone of the maximum deformations.

As for the elasticity module, my opinion that it is identical at a bend and at extention and at compression. The methodical difference arises first because of deformations of shift at a bend, secondly because of a sample deforamtion in points of application of loading. For isotropic bodies influence of shift can be considered on the basis of Tymoshenko's textbook (it has to be in English)

For homogenous material or where modulus in tension & compression is same, theoretically both moduli should match.

But when tensile & compressive moduli are different then ratio of flexural modulus by tensile modulus can be derived as 4/(1+sqrt(a))2 where a is ratio of modulus in tension by modulus in compression.

Flexural modulus should be used for comparison of stiffness of materials as it may be average as stated by Karol & also that it has dependency upon factors such as test sample thickness. Nevertheless, It may be used for FEA of parts which have flexural mode of deformation to estimate closer values of deformation.

 Many years ago I had many data on stretching and on a bend of samples carbon - a carbon composite and I tried to carry out such accounting of large-scale effect, but the computer then very weak and could be programmed only on BASIK languge and work remained not complete as there was no main.

You didn't define clearly the isotropic material, for example a thermoplastic material or the thermoplastic materials filled by particles. For a thermoplastic material, you may find out the relationship. But for the thermoplastic materials filled by particles, you wouldn't find the relationship. If the particles have a high Young's modulus, tensile modulus of the filled thermoplastics is higher than flexural modulus.

An isotropic material (similar to ideal gas!) does not exist in the nature. It is only an abstract model. All the real materials are anisotropic (!), if only you are doing your measurements properly and thoroughly. I am researching graphite/graphene containing materials for more than 40 years; so, I can predict with the high degree of probability that your so-called "epoxy resin which is reinforced with graphene nano platelets" is a transversely isotropic material. Concerning the similar materials I can refer you to Chapter 2 of the 1st Volume of my book on the UHTM (recently published). You can find there the ratios between tensile, flexural and compressive strength characteristics as well, but also many other useful, in my opinion, things for your present and future studies. The full-text of the chapter is available on my account in the ResearchGate. However, be sure that the characteristics are different for two main directions. This is the specifity of all the transversely isotropic materials. I don't know what you have termed as "flexural modulus"... Maybe, Coulomb's (shear) modulus...??? And it is too long story about the elastic characteristics of graphene containing materials, but the main idea you need to understand clearly in this case that these characteristics depend on the volume fraction of graphite/graphene inclusions extremely sharply. Good luck to you on your way from unknowledge to the knowledge...!

What is the relation between tensile strength and flexural strength in isotropic material?

I think that this question can be solved by clearly defining a model: failure by crack propagation under these two different types of applied loading; brittle crack, crack with localized plastic zone ahead of the crack-tip or crack with plastic yielding. But I don't know if this has been done previously.

Hi, there

The relationship is really exist for an isotropic and homogeneous material, but both the plastic and elastic property might be absolutely same. The flexural loading distributes in the sample at the summit section in a particular mode, say, below the central plane (line) it is a linear tension stress from 0 to Xmax (as you calculated as the fletural strength), and in the upper side of the section, it is a compression stress field: from -Xmax to 0. stress 0 is at just the central line (plane), -Xmax at the upper surface of the sample. Moreover, because of the stress gap between yielding (upsetting) stress and the fracture stress (tensile or compressive strength), at the the frature time point there is 2 certain layers (of course thinner than the half-height of the sample, located at the upper side and lower side the the section of the sample), and upper side layer has been yielded by compression, and lower side layer by tension, the left layer (sentral layer besides the central line) is still in elastic mode. So you see, the supporting force from the full section of sample is calculated by you into a surface stress called as "flexural strength", and this calculation is defined with elastic method, thus you get a "fletural strength", it is really a nominated stress, not real strength, and it must be much higher than the tension or compression strength.  At a ideal situation, you can make a formula to correlate these two parameters. For polymer, because the the tension behavior and the compression are quite different, so the formula should by very complicated.

Dear All,

Sorry..., my advice to you to be very careful to the text you are writiing to answer the question. I have just found very silly things here... So, I think, for some participants of the discussion it is necessary to go to a library to read textbooks on the fundamentals of mechanics, but also on materials engineering...

First of all, I would like to remind you that about 50-60 years ago (or maybe even earlier!) in the PRACTICE of materials engineering it was found that for BRITTLE materials the approximate ratios between tensile , flexural (bending) and compressive ultimate strengths are following:

: : = 1 : 2 : 6.667.

For some materials with probable high anisotropy the approximate ratios are bit different, for example for commercial graphites (see my book on UHTM)

: : = 1 : 2 (or from 1.5 to 2.1) : 4 (or from 3.2 to 5.8).

These ratios are of great importance, but only for the practical mechnical testing methodology and PRACTICE of materials engineering in general.

Secondly, for the fundamental mechanics, however, all these things are not interesting at all, as there are only "plus" or "minus" in mathematics, and so only "tension" and "compression" in mechanics (NO THIRD sign or operator!). Flexure (bending) is only one from many-...-many types of so called complex stress states, where the combinations of "tension" and "compression" are realised in various forms. Everything connected with flexural (bending) mechanical testing is of great interest only for materials science and engineering, and only for BRITTLE materials (absolutely NOT for metals, alloys or non-glassy polymers!). 

So, dear colleagues, please do not mix the fundamental mechanics and materials engineering, as these areas are rather different scientific disciplines with different terminology as well...

Thank you all for your nice comments and i appreciate professor Igor L Shabalin for his perfect answers

it is my new questions i will be happy to see your comments on my new question:

https://www.researchgate.net/post/which_one_is_affected_by_interface_in_nanocomposites_flexural_or_tensile_properties?_tpcectx=qa_overview_asked&_trid=5Eo5I1WHp46velmYNgzBd9WE_

Refer this link:http://rsta.royalsocietypublishing.org/content/310/1511/139.short

Refer this link:http://ipn2.epfl.ch/CHBU/papers/ourpapers/Salvetat_APA99.pdf

Refer this link:http://onlinelibrary.wiley.com/doi/10.1111/j.1151-2916.1997.tb03074.x/abstract

When a material is tested in flexure, i.e. under bending; tensile stresses are produced on the top layers; zero stress at neutral axis and compressive stresses in layers below the neutral axis. In isotropic materials, when the material fails, the corresponding load is taken for calculation of flexural strength. Since the isotropic material fails in tensile portion; the strength is nothing but its tensile strength. Hence when tested using tensile mode on a UTM;; we designate it as tensile strength and when tested on flexural loading set-up, we call it as flexural strength. Ideally both should be same. In fact for ceramic materials; tensile strength is obtained using 3 point bending set-up because tensile specimens for ceramic materials cannot be prepared. Only difference is if any defect is present in the specimen; irrespective of its location; it affects tensile strength in the same manner because uifom tensile stresses are produced across the whole cross section. In bending situation however; defects will affect the strength differently depending upon their location. A defect nearer top or bottom surface will have significant effect as compared to the same type of defect located nearer or on the neutral axis. Hence the strength in two modes of testing may differ. .   

I understand that we are dealing with large specimens so that initiation of crack motion does not imply failure. Ravi Prakash says" Since the isotropic material fails in tensile portion...". To my opinion, he means that the initiation of crack motion is in the tensile portion of the specimen. But, failure of the material will require necessarily the propagation of the crack in the complementary (to the tensile portion) region of the specimen. The ultimate strength in flexure is connected to the conditions for crack propagation in the non-tensile region (at the begining of crack propagation)  of the specimen. I am discovering ( if this has not been already solved) an interesting fracture problem.

in the flexural strength there a portion of shear strength. particularly if the span long is small

It appears to me that there are two different expressions of the crack extension force: the crack extension force of the tip of the crack moving in the direction of the tensile region of the specimen and the crack extension force of the tip of the crack that moves in the opposite direction towards  the complementary ( to the tensile region) portion of the specimen. The latter expression is the controlling physical quantity of the ultimate flexural strength. Now, don't speak to much, just do the job and give the result. See you latter

Sir,

    refer this link

http://web.mit.edu/course/3/3.225/book.pdf

We have opened the Q&A " What are the relations between failure stresses in brittle isotropic specimens tested in tension, compression and bending?". In this forum, it is indicated that this is the condition G2 = 2 ϒ that controls the complete failure of the fractured specimen in bending tests, where G2 is the crack extension force (per unit length of the crack front) pertaining to the tip of the crack that moves through the compression part (2) of the specimen and ϒ is the surface energy; the other part (1) of the specimen is in tension.

This G2 = 2 ϒ condition (uncommonly reported), and associated treatments, provides a straightforward description of the relations between failure stresses in tension, compression and bending of brittle isotropic broken specimens; please refer to this "Q&A" to avoid excessive repetition.

to Dr. P. N. B. Anongba: Could you give us any experimental/practical examples of isotropic materials? My practical experience show that so-called "isotropic materials" are mainly connected with poor measuring technique. Any thorough and accurate measurements, especially for polycrystalline bodies, lead us to the conclusion that all the materials are anisotropic, even the single crystals of cubic symmetry. So, please, explain the reason of the condition mentioned by yourself for materials science and engineering practice.

I am an experimentalist (as testified by my initial publications). I have worked personally with aluminium, the intermetallic compound CoSi2 of the fluorite structure, for instance; both are quite isotropic.

Do you think that it is correct to propose a failure stress for a material under bending with no reference to the compressive part (2) of that specimen? No matter if the material is isotropic or not! You have to justify why G2 = 2 ϒ is not taken into account.

It is quite evident (see our Q&A quoted above) that this condition is the controlling physical quantity of the failure stress in large bent specimens. This condition is general for homogeneous materials with large dimensions in practice.