the half cone angleyand of the cone massmaffect the value ofLpfor a givensy
value? What are the advantages of the test?
Answer
As the cone penetrates the material, it will at first exert a stress of considerable
magnitude; but this stress becomes ever smaller as the cone goes deeper, and when
the stress is decreased to the value ofsythe cone will stop. The force exerted by the
cone would approximately equalm?gand it acts, in first approximation, over an
areapR^2 p, whereRpis the radius of the cone at a distanceLpfrom its tip. Hence
sy¼mg=pR^2 p. SinceRp¼Lptany, we would obtainsy!L^2 p. Moreover, for a
given value ofsythenLp!m0.5andLp!coty.
These quantitative relations are very approximate. The force exerted by the
cone involves momentum as well as weight: the cone falls into the sample. The
reaction force of the material involves friction exerted by the material and depends
on the volume of material that does yield, which may not depend in a simple manner
onLp; also the buoyancy force of the material on the ‘‘submersed’’ part of the cone
may not be negligible; finally, the phenomenon of stress overshoot (see Figure 17.6c)
may upset the relations. Careful comparison for a number of margarine samples with
a well-defined test for the yield stress led to the relationsy!Lp1.6.
The advantages of cone penetration include the simplicity of the test; that the
test piece can be used without any previous deformation (which would affect the
value ofsy); and that the deformation time (of the order of a second) is about the
same as during the spreading of margarine. Altogether, the test results correlate well
with the subjectively evaluated spreadability.
17.1.2 Fracture Mechanics
Failure. When a stress, however small, is applied to a Newtonian
liquid, it flows, implying that all bonds between the constituting molecules
frequently break, while new ones are formed. When an increasing stress is
applied to a piece of an elastic solid, it becomes deformed, and when a
certain stress is reached, the test piece starts to fracture. Macroscopic
fractureis characterized by
simultaneous breaking in one or more macroscopic planes throughout
the specimen of all bonds between structural elements of the solidwhich results in the specimen falling into pieces. This implies that (most of)
the broken bonds do not reform. The structural elements can be atoms,
molecules, or particles, and the bonds that break are generally those
between the largest elements. The word macroscopic means that the size of
the fracture plane is much larger than that of these structural elements: