Could someone acquaint me with experimental works providing the applied stress at failure of rectangular specimens, horizontally in a wall on one end and vertically loaded at the other end, as a function of the specimen dimensions? We are studying theoretically this problem.
Our recent study (to be published) gives this very simple result: SigmaF/SigmaT = e / 6L, SigmaF is the applied stress (applied force divided by specimen section) at failure in bending; SigmaT is the applied stress at failure in tension; e and L are the thickness and length of the specimen in the bending test (the thickness is the dimension of the specimen measured in the direction of the applied force). Experimental results are wanted and may be included in a common report.
Hello Patrick, would you please mention what exactly you used for SigmaF and SigmaT ? What Kind of load and beam cross-section do you assume?
Hello Patrick, I think you mixed up a bending Problem description and a normaly loaded tensile description. This doesn't make sense to me. what is the ultimate Goal of your Analysis?
Thanks for your interest. Please take perfect elastic homogeneous isotropic materials without defects. In tension tests, measure the stress at failure: this is SigmaT. In bending tests with same specimens with rectangular cross sections, horizontally in a wall on one end and vertically loaded at the other end, measure the stress at failure: this is SigmaF. Our modelling suggests that SigmaF/SigmaT = e / 6L. This relation comes naturally in the crack analysis in bending through the relation G = 2 Gamma, where G is the crack extension force and Gamma, the surface energy. We consider large specimens and complete failure (i.e. macroscopic crack propagation distances), not initiation of crack motion.
G is calculated at the tip of the crack that moves through the compression region of the bending specimen. Hence, in fracture tests, specimens may contain initial crack in the compression region, close to the wall.
My goal is to have an understanding of crack propagation under general bending.
Hello Patrick, your analytical expression is true and run out from well known relation for inertia moment for rectangular bar: J=be**3/12 . It may be useful for correlation of bending and shear (because SigmaT = F/cross section of the bar) strength of material. But three-point bending test is seemed more convenient than console bar test. Best regards.
Hello Serge, shall I understand that this relation has been obtained from other considerations than G = 2 Gamma?
Yes, Patrick, this relation has been obtained from primary expression for maximal bending stress Sigma = M/W, where M is bending moment and W=J/(0,5e).
If the Load conditions are Tension and Bending, then the tensile stress is = F/A where F is the applied load in tension and A is the original cross-section of rectangular (b*h). The bending stress = My/I where M is max. applied bending moment and in this case is equal PL where L is the length of the rectangular beam and P is the applied load for bending, y is equal h/2 where h is high of the rectangular cross-section and I is the moment of inertia about the neutral axis and for this case I= bh**3/12 where b is the width of the rectangular. Then, the maximum applied stress due to this loading conditions (Tension + Bending) is = F/A + My/I. = F/bh + PL*0.5h/bh**3/12 = F/bh+ 6PL/bh**2 . This formula is a function of specimen dimensions (h,b and L)
If the Load conditions are Tension and Bending, then the answer of Hassan is a good answer.
Hi Patrick, your modelling to the inertia effect of cross section is Ok, provided that you work on bending effect of the applied load. I=(b*h**3)/12 for bending about X-axis.
Serge Shil'ko
Oh, I see, I know this formula.
In our view, there are two different expressions of the crack extension force: the crack extension force G1 at the tip of the crack that moves in the tensile region of the bending specimen and the crack extension force at the tip of the crack G2 that moves in the compression region (the complementary part of the tensile region in the specimen) of the bending specimen under load.
Our modelling assumes that G2 = 2 Gamma is the controlling condition for complete failure of the bending specimen. Work is in progress.
We have a work that provides relations between SigmaF, SigmaT and fractured specimen dimensions and welcome any comment and information regarding experimental findings. Please see our contribution " A theory of the fracture of rectangular bars bent by terminal transverse load and couple" .
- Badges
- Science topic
- Similar topics
- Physical Sciences
- Materials Science
- Materials
- Inorganic Materials
- Metals
- Alloys
- Steel