Engineering Rock Mechanics

Soft, stiff and servo-controlled testing machines 91

of thinking of the machine stiffness is to consider its axial force+xtension
curve were we to replace the specimen with a hydraulic jack). As the speci-
men is loaded, the machine is also loaded, as indicated by the arrows in the
diagram. Also, as the specimen's load-bearing capability decreases in the post-
peak region, so the machine elastically unloads as the force is being reduced.
Hence, as indicated in Fig. 6.6, the machine can be soft or stiff, and in a
testing machine this stiffness will be a complex function of many of the
component parts of the machine: these include the loading platens, the
hydraulic system (the fluid, hoses and rams) and the frame. Were we to
consider all these as an equivalent cylinder of cross-sectional area A, depth
L and modulus E, the stiffness is given as AEIL. Thus, the machine stiffness
will increase with increasing area A, decreasing length L and increasing
modulus E. This means that the stiffness of the testing machine can be
altered via these values.
Figure 6.7 illustrates the same complete stress-strain curve for the rock.
Here we have superimposed the assumed linear behaviour of a soft testing
machine and a stiff testing machine at the point A, just beyond the peak
strength: this is to consider whether the machine can unload purely
elastically without any intervention from the operator. In the left-hand
diagram in Fig. 6.7, the unloading curve for the machine in the direction
AE is very similar to the deadweight mentioned earlier. The machine can
unload along this line because at all points the axial force associated with
the elastic unloading of the machine is greater than the specimen can
sustain, resulting in 'explosive' failure. The failure occurs because, in an
increment of axial displacement DC, the machine is capable of performing
the amount of work corresponding to the area DCEA, whereas the
maximum work the specimen can absorb is given by the area DCBA. This
work is utilized in the continuing microstructural disintegration that occurs
during the axial displacement increment DC. The work represented by the
area AEB is liberated as energy, manifested especially as kinetic energy:
particles of the specimen fly in all directions.
We can now compare this with the right-hand diagram in Fig. 6.7, where
the testing machine stiffness is represented by the steeper line AE. A similar
argument to the previous one can be used to predict the response of the
system. In this case, the machine cannot elastically unload of its own
volition along AE, because the specimen requires more work to be done
than is available. Consequently, the operator will have to increase the strain
in order to follow the post-peak portion of the curve.

Soft testing machine Stiff testing machine

Energy required = ABCD Energy required = ABCD

Axial Energy supplied = AECD Energy supplied = AECD

II

ID IC

Axial displacement Y--c Axial

ID 'C

c--c displacement

Figure 6.7 Machine stiffness and specimen stiffness in the post-peak region.