Adhesives Design Toolkit

The crack growth in adhesive bond specimens can proceed in two ways:

Slow-stable extension where the crack is dictated by the displacement (i.e. cross-head) rate (e.g. rubber modified film adhesives tested at sub-zero temperatures).
Run-arrest extension where a stationary crack abruptly grows at a rate that exceeds the cross-head speed (e.g. modified epoxies at 730C).

Terms associated with run-arrest crack growth are given below:

Opening mode fracture toughness, G_Ic: The value of G_I just prior to the onset of rapid fracture is determined from the load required to initiate crack growth.

Opening mode crack arrest toughness, G_Ia: The value of G_I just after arrest of a run-arrest segment of crack extension is determined from the load required for crack arrest.

Double cantilever beam

Double Cantilever Beam (DCB): Although, this test has attracted considerable academic interest, it is not widely used in industry. The method is covered in both ASTM D 3433 [58] and BS 7991 [59]. The test is used to measure the initiation and propagation values of G_I under static and cyclic loading conditions. A tensile load is applied to a specimen with an embedded through-width insert (i.e. debond) at the specimen mid-plane. The tensile force acts in a direction normal to the crack surface. Specimens are typically 25 mm wide and 356 mm long. The adherend thickness is typically 6.35 mm (0.25 inches). Crack length is measured using either a travelling microscope, a crack gauge or video camera. The use of a crack gauge enables crack measurement to be automated. For static tests, the coefficient of variation in G_Ic is typically 20% or higher.

The critical strain-energy release rate or fracture toughness G_I is calculated as follows [60]:

where P is applied load, E is the flexural modulus of adherend in the longitudinal direction, b is the specimen width, a is the crack length and h is the adherend thickness.

The analysis assumes linear elastic behaviour and no large displacement effects. To determine G_Ic and G_Ia values, the corresponding applied load values P_MAX and P_MIN are substituted into the above equation. Mode I beam theory equation above does not take into account the following corrections [60]:

χ—Shear deformation and deflection at the crack tip
F—Large displacement of DCB
N—End blocks stiffening effect (shortening of moment arms and the tilting of the loading blocks)

The corrected beam theory equation taking all these effects into account is given by [60]:

Shear deformation and deflection at the crack tip can be determined experimentally by plotting compliance C as a function of crack length a.

Compliance versus crack length

Experimental compliance method equation for G_Ic is:

Non-linear load-displacement response may result from inelastic material behaviour and/or sub-critical damage formation in front of the planar crack (i.e. micro-cracking and extensive deformation) or by large beam deflection. The latter can be associated with materials with low flexural stiffness or loss in flexural stiffness resulting from substantial crack growth.

Fabrication and testing of DCB specimens is straightforward and relatively inexpensive. Testing can be conducted using standard mechanical test frames. Specimen fabrication is identical to that employed for wedge cleavage specimens. Reusable aluminium loading blocks are recommended. Both static and fatigue loading can be used with these specimens. Tests may also be conducted under simulated service environments such as hot humid environments.

Cyclic Fatigue: A form of the Paris Law can be used to relate crack growth rate da/dN per cycle to the maximum value of the applied strain-energy release-rate G_MAX [61]:

where C and n are material constants.

Alternatively, the crack growth rate can be expressed as a function of the range of strain-energy release-rate DG by:

where A and q are constants, and DG is the difference between maximum and minimum strain-energy release rate per cycle (G_MAX—G_MIN).

The above relationships apply only to the linear portion of the logarithmic-logarithmic plots of G_MAX or DG versus da/dN. Values for both sets of constants (i.e. C and n, and A and q) can be determined using linear regression fit to the linear region of the logarithmic-logarithmic plots.

Regions

Region I—Threshold region (G_TH) associated with low crack growth rate da/dN and G_MAX values (G_TH » 0.1G_C).

Region II—Linear region defined by the Paris Law given by Equation (4).

Region III—Value of G_MAX approaches the adhesive fracture toughness G_c measured under monotonic loading conditions.

Typical log-log crack growth rate versus G_MAX plot

Generally, the relationship between log G_MAX and log DG and log da/dN is S-shaped (i.e. sigmoidal). This relationship can be described as follows [62, 63]:

G_TH is the minimum (or threshold) value of the adhesive fracture energy, G_c, and A, n, n₁ and n₂ are material constants. G_c is determined from constant rate of displacement tests (i.e. monotonic fracture energy).

As with static tests, the crack extension is measured using either a travelling microscope, a crack gauge or video camera. The uncertainty in G_Ic is far greater for fatigue tests where the power-law relationship is sensitive to small errors in applied load and crack length.

Advantages	Disadvantages
Yields Mode I fracture toughness Compatible with metals and PMCs Straightforward/economic Specimen fabrication Testing Data reduction BS 7991/ASTM D 3433 Suitable for cyclic/environmental testing	Limited to rigid adherends Loading tabs and test fixture required Limited ability for generating design data Analysis required to account for End (loading) block tilting Large beam deflection Non-linear load-displacement Crack extension measurements difficult Moderate to large uncertainties in measurements

Tapered Double Cantilever Beam

Tapered Double-Cantilever Beam (TDCB): The purpose of this test geometry is to make the measurement of fracture toughness G_I independent of the crack length (i.e. constant compliance). The specimen design is well suited to tests where the crack length a is difficult to measure, especially environmental testing. To develop a linear compliance specimen, the height is varied so that the quantity 3a²/h³ + 1/h is constant. The method is covered in both ASTM D 3433 [58] and BS 7991 [59].

The critical strain-energy release rate or fracture toughness G_Ic is calculated as follows [64]:

where P is applied load, E is the flexural modulus of adherend in the longitudinal direction, b is the specimen width, a is the crack length and h is the adherend thickness.

The above relationship has been corrected for root-rotation and for the initial beam profile:

The TDCB test specimen has been used to determine the rate of crack growth under various cyclic loading and environmental conditions [6]. The main disadvantage is the relatively high cost associated with specimen fabrication. Specimens are typically 241 mm long and 25 mm wide. Adherend thickness has a nominal maximum thickness of 32 mm. Variations of TCDB test are being developed to test thin metal sheet or composite laminates.

Advantages	Disadvantages
Yields Mode I fracture toughness Compatible with metals Adaptable to thin adherends Constant compliance Straightforward/economic Specimen fabrication Testing Data reduction BS 7991/ASTM D 3433 Suitable for cyclic/environmental testing	Limited to rigid adherends Large specimens required Special test fixture required Limited ability for generating design data Special bonding fixture required Analysis required to account for Root-rotation Initial beam profile Moderate to large uncertainties in measurements

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