#### Installation — business terrible - 1 part

September 8th, 2015

When a crack in a laboratory specimen may be guaranteed to behave in a linear elastic way, the experimental determination of Kjc or Gy is conceptually easy. The simplest way is to use a specimen in which the crack growth initiation coincides with the peak load (all the standard specimens belong to this category). In this case, one simply loads the specimen up to failure and records the peak loadPu. If LEFM conditions are fulfilled, the value of the stress intensity factor for this load coincides with Кjc:

Klc = KIu = ^fc(e) (3.4.1)

where a = a/D is the relative crack length at the beginning of the test.

The difficulties in this kind of testing arise at two different levels: (1) Specimen preparation (precracking), and (2) verification of LEFM conditions. These aspects are well defined for metals in most national standards, particularly in ASTM E 399. The crack is grown from a normalized starter notch by fatigue under controlled conditions. The LEFM conditions are verified in two ways. First, the nonlinearity of the load displacement curve before peak is limited (an ideally brittle material is completely linear up to failure). This is done as shown in Fig. 3.4. la by defining a kind of conventional (load) elastic limit P5 for which the secant stiffness is 95% of the initial tangent stiffness. Deviation from linearity is acceptable if either the peak load occurs before the elastic limit or the ratio Pu/P$ is less than 1.1 (see the standards for details).

Apart from this direct verification oflinearity, there is a further condition which verifies that the specimen thickness and size are large enough for the nonlinear zone at fracture to be negligible (for engineering purposes). Since the standard specimens are designed so that their thickness is one-half of their width or depth (6 = 0.5D) and the crack length is close to half the depth (a ~ 0.51)), the thickness and size conditions are expressed in a single condition:

b > 2.5 ("~)2 (3.4.2)

where <7C is a conventional flow stress (usually a value between the conventional 0.2% proof stress and the tensile strength). The origin of the foregoing equation is discussed in detail in the next chapter. Here, it is enough to say that the factor (/T/C/<7C)2 is proportionalto the size of the plastic zone, so the equation really places a limit on the extent of the plastic zone relative to the specimen size.

For materials other than metals, the situation is more complex. Cracks in polymers and structural ceramics cannot easily be grown using cyclic loading. For polymers, cracking by forcing a razor blade into the notch root has been chosen by ASTM standards (ASTM 1991). For fine ceramics, no standards are yet available, and round robins are being performed to compare toughness test results on specimens with different kinds of notches and cracks, as that promoted by ESIS TC 6 (Pastor 1993; Primas and Gstrein 1994). Specifications for the minimum size required for LEFM to apply have been set for polymers, and are similar to those previously stated for metals. No agreed limitations have been set yet for ceramics.

For concrete, it is generally accepted today that the sizes required for LEFM to apply are really huge (several meters or even tens of meter). Therefore, special purpose tests taking into account the nonlinear

Figure 3.4.1 Experimental determination of fracture properties: (a) Load-displacement curves and definition of the conventional limit P5 (After AS’I’M E 399, simplified); (b) determination of Gf from experiment.

fracture behavior of concrete have been set. They will be analyzed in the following chapters, where nonlinear models are introduced.

For any material, nonstandard tests may also be used to determine the fracture properties of the material, whenever the size of the specimen is large enough for LEFM to apply. One such method, based on an energetic analysis, consists of performing a stable test (controlling the displacement rather than the applied load) and simultaneously measuring the load, P, the load point displacement, и, and the crack length, a. Let the P — u curve be known between the points 1 and 2 at which the crack lengths were measured to be, respectively, ai and a2 (Fig. 3.4.1b). Then, according to Section 2.1.4, the energy consumed in fracture between points 1 and 2, is the area of the curvilinear triangle 012, while the area of the newly

formed crack is b(a2 — аі); hence,

The accuracy of the result depends on the accuracy of the individual measurements, which may be controlled to some extent by adequate experimental design, but it also depends on the degree of accuracy of the hypotheses underlying the equation above. The method becomes inaccurate, even invalid, if the inelastic zone ahead of the crack tip is so large that the hypothesis of negligible inelastic zone is no longer acceptable.

Determination of how large the inelastic zone is, relative to the specimen dimensions, and how large its size must be to stay reasonably close to LEFM is, to a great extent, one of the objectives of the various inelastic fracture mechanics approaches that will be analyzed in the remaining chapters. At this stage, we only list the most obvious conditions that the experimental outputs should fulfill:

*1. *Deviation from linearity prior to the peak load should be small. This applies to specimens where the і so-6 c’urves are monotonically decreasing as in Fig. 2.1.10a. The more rounded the peak, the farther the behavior is from LEFM. Quantitative criteria to ensure prepeak linearity can be formulated, similar to those previously given by ASTM E 399.

*2. *The P — u curve after the peak should be an iso-б curve. The most direct way to check this point is to take various arcs 1-2, 2-3, 3-4, and so on, and calculate a value of Gf for each of these arcs. They should be equal if LF. FM applies.

*3. *When unloading is performed, the unloading curve should be straight and unload to the origin. Deviation from this behavior indicates deviation from LEFM.

*3.8 *Find the expression for the energy release rate of the structure in (a) Fig. 3.2.1a, (b) Fig. 3.2.1c.

*3.9 *A brittle material may contain planar voids. If these voids are similar to penny-shaped cracks, determine the maximum diameter of the voids which allow the material to be used up to 90 percent of its elastic limit. Complementary tests delivered values of 55 MPa for the yield strength and 16 kJ/nr for the fracture energy.

Figure 3.4.2 Test output in an experimental determination ol" Gy. |

*3.10 *Use the triangular notch approximation of Planas and Elices to get the energy release rate of a center – cracked panel such as that depicted in Fig. 3.2.2. Show that for the limiting case a/D —> 0 the expression coincides with that obtained from the stress relief zone approximation in Section 3.2.2.

*3.1 *J The results of the finite clement calculation of Guinea (described in example 3.3.1) gave, for a/D — 0.5, the crack opening profile near the crack tip included in the table below, where r ami w ate, respectively, the distance to the crack tip and the crack opening (in appropriate length units; remember that in the computation b=D=,P — ,E — 1 and plane stress was used), (a) Show that!<i — lint,….о wE’ л/тт~/У2г. (b) Plot wEJrt/32r vs. r and get an estimate of К і for the computed case, (c) For the usual definition of on shown in Fig. 3.1.1, determine from the previous result an estimate of the shape factor lc(0.5). (d) Evaluate the error of the estimate in comparison with the more precise equations of example 3.1.1.

r |
0.01 |
0.02 |
0.03 |
0.04 |
0.05 |
0.06 |
0.07 |
0.08 |
0.09 |

w |
3.1669 |
4.5816 |
5.7928 |
6.8123 |
7.7424 |
8.6050 |
9.4201 |
10.198 |
10.947 |

*3.12 *Compliance tests have been performed on single-edge crack specimens of 25 mm thickness and 50 mm depth made of a material with an elastic modulus of E = 3 GPa. Nine specimens with crack lengths ranging between 23 and 27 mm were subjected to the same load, P = 500 N, and their displacement was measured. The results are shown in the following table. Give an estimate of the energy release rate, in J/m2, for a specimen of this particular shape, size, and material with a crack length of 25 mnt, for any load, P expressed in N.

a (mm) |
23.1 |
23.6 |
23.9 |
24.4 |
24.9 |
25.6 |
25.8 |
26.5 |
27.0 |

и (yim) |
270 |
291 |
289 |
310 |
317 |
347 |
348 |
369 |
397 |

*3.13 *A sped men geometrically similar to that in the preceding exercise has been loaded up to crack initiation. The specimen had a thickness of 10 mm and a depth of 100 mm, with an initial crack of 50 mm. The load at which the crack started to grow was 17.5 kN. Additional testing provided for the elastic modulus of that material, the value E = 100 GPa. Estimate the fracture energy under the assumption that LEFM applies.

*3.14 *A double-cantilever beam specimen subjected to opposite point loads at the ends of its arms has been tested under displacement control. The specimen thickness was 50 mm and the arm depth 30 mm. The resulting load-displacement curve is shown in Fig. 3.4.2b. If the initial crack length was 150 mm, (a) give an estimate of the elastic modulus of the material, (b) Give an estimate of Gy using the peak load value and the initial crack length, (c) Give estimates of the crack length at points А, В, C, and D. (d) Give average values of the crack growth resistance over arcs А В, BC, and DE. (e) Decide whether LEFM is a reasonably good approximation, and, if it is, give a final estimate for Gy.