Traction-separation laws

Motivated by the Kramer and Berger [3] description of the crazing process, Tijssens et al. [9] proposed a viscoplastic crazing model within the fiamework of a cohesive zone methodology. The traction-separation law proposed in [9] comprises three parts corresponding to initiation, thickening and breakdown of the craze. 

The cohesive zone approach to fracture mechanics reduces the fracture resistance properties of a material to a traction-separation law. This Taw relates, o, the normal cohesive stress which resists the parallel separation of two internal planes which were initially very close together, to the current increase r] in their separation. The fracture resistance (and hence the fracture toughness K,.) is simply given by the area under the o,. ri) curve up to 6,.. The simplest form of traction separation law assumes 0 (77) to be constant up to a critical maximum separation 4. and this was developed analytically into a fracture model by Dugdale... 

Figure 8.4 A typical nonlinear interfacial traction-separation law. Source [66] Reproduced with permission from Elsevier...

Similar to the Mode I fracture test, the energy release rate /jj can be experimentally determined as a function of the crack tip slip Sq and the global shear force gj. Once the experimental /n o curves are obtained according to Equation (8.13), the Mode 11 interfacial traction-separation law t = t(Sq) can be experimentally determined as foUows ... 

Figure 8.14 Typical shapes of the interfacial traction-separation laws at different adhesive layer thicknesses ha- Source [63] Reproduced with permission from Springer...

Finally, Figure 8.18 shows the overall mixed mode traction-separation laws with various adhesive thicknesses. It is seen that the area under the T-cp curves increase as the thickness of the adhesive layer increases, which suggests that the energy release rate increases, as also does the fracture toughness. It is interesting to note that the peak traction is almost constant as the adhesive layer thickness increases. This is understandable because the peak traction represents the strength of the material under the complex stress condition. 

Zhu, Y., Liechti, K.M., and Ravi-Chandar, K. (2009) Direct extraction of rate-dependent traction-separation laws for polyurea/steel interfaces. International Journal of Solids and Structures, 46, 31-51. 

Figure 17.10 Implementation of the Cohesive Zone Model (CZM) (a) the Traction Separation Law (TSL), (b) Intrinsic TSL and (c) Extrinsic TSL. The cohesive laws are characterized by the strength, the critical opening and the critical energy release rate [145].

FEA has also been used to study interface adhesion between thin film and substrate under indentation. Liu et al. (2007) examined the interface delamination and buckling of thin film subjected to microwedge indentation. In their model, the interface adjoining the thin film and substrate is assumed to be the only site where cracking can occur. A traction—separation law with interface strength and interface energy as two major parameters was introduced to simulate the adhesive and failure behaviors of the interface between the film and the substrate. 

As discussed earlier, the area under the traction-separation curve Fi and Fiio) and the peak stress (a and f) are the important parameters that describe the cohesive tractions. The precise shape of the traction-separation law does not strongly influence the behavior of the system. For example, one generally useful form of a mode-I traction-separation law is shown schematically in Fig. 4. It should be appreciated that while the area and peak stress are the two important parameters from a mechanics point-of-view, they may not necessarily represent the fundamental parameters from a materials perspective. In some ways, the peak... 

Fig. 4. Schematic mode-1 traction-separation law that is generally useful in cohesive-zone models.

Elastic-foundation models have been used to analyze the effects of both the compliance [23,55] and the plastic deformation of an adhesive layer [55]. Cohesive-zone models in which the adhesive layer is replaced by cohesive-zone elements provide powerful techniques to analyze the phenomena and to couple them to the fracture process. Using the trapezoidal traction-separation law shown in Fig. 4, normalized load-displacement curves for a DCB specimen are shown in Fig. 12 [4]. The maximum loads supported by the DCB joints are of the form [29]... 

To model geometries that are not symmetrical, the traction-separation laws for modes I and II need to be determined (a third law would be required for mode-III problems). In particular, values for /I, flio. a and f need to be determined from mode-I and mode-II tests. These are then incorporated into the traction-separation... 

The corresponding traction-separation laws for the normal directions were assumed to occupy the same size of damage initiation displacement Di and failure displacement Df like in shear mode, but a doubled shear strength and energy release rate values. 

Tab e 1 Material paramel ters for traction-separation laws used in the model. ... 

The full stress-strain behaviour was plotted in figure 6. It shows the nonlinear behaviour caused by the cohesive zone elements and their traction-separation laws. Damage is clearly seen in the curve of interphase combination B, BA and BC. The stress-strain curve of B and BA are approximately coincident. 

A method of predicting failure based on the concepts of stress and fracture mechanics is the cohesive zone method. The cohesive zone model has been used increasingly in recent years to simulate crack initiation, propagation, and failure. The cohesive zone model allows multiple cracks to be modelled and the direction of crack propagation need not be known in advance however, cohesive zone elements need to be present at all possible crack paths. Cohesive zone models follow a traction-separation constitutive law to predict failure initiation, damage, and failure. Several shapes for the traction-separation law have been presented in the literature, with the bilinear, exponential, and trapezoidal shapes, as shown in Fig. 25.14, being the most commonly used for strength prediction. 

The shape of the traction-separation law is difficult to determine from experimental methods and is often assumed or simplified. The effect of the shape of traction-separation... 

Nossek and Marzi (2009) have developed a CZM base method to predict the impact strength of adhesive joints for car structures. A trapezoidal traction-separation law was adopted. In this case, they expanded the CZM model from mode I condition to mode II and combined both stress conditions. In this study the fracture toughness of adhesive joints was... 

Traction-separation law for mixed-mode criterion (Marzi et al. 2009c)... 

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