Energy Approach to Fracture

A more general formulation of the energy approach to fracture of graphite has been reviewed by Sakai et al. [48]. Assume a graphite body containing a single crack of area A. Under conditions of stable crack growth, the global energy balance may be written as  [c.499]

Energy Approach to Fracture  [c.121]

The second approach to fracture is different in that it treats the material as a continuum rather than as an assembly of molecules. In this case it is recognised that failure initiates at microscopic defects and the strength predictions are then made on the basis of the stress system and the energy release processes around developing cracks. From the measured strength values it is possible to estimate the size of the inherent flaws which would have caused failure at this stress level. In some cases the flaw size prediction is unrealistically large but in many cases the predicted value agrees well with the size of the defects observed, or suspected to exist in the material.  [c.120]

An alternative energy approach to the fracture of polymers has also been developed on the basis of non-linear elasticity. This assumes that a material without any cracks will have a uniform strain energy density (strain energy per unit volume). Let this be IIq. When there is a crack in the material this strain energy density will reduce to zero over an area as shown shaded in Fig. 2.65. This area will be given by ka where )k is a proportionality constant. Thus the loss of elastic energy due to the presence of the crack is given by  [c.125]

The usual energy balance approach to fracture (see Section 8.9)-by equating the strain energy released to the energy consumed in creating new surface and in achieving plastic deformation-needs modification where corrosion processes are involved to take account of the chemical energy released, and it is the latter that distinguishes stress corrosion from other modes of fracture not involving environmental interaction  [c.1147]

On the other hand, it has also been recognized that the energy of newly created fracture surface must come from the work done on the body by the loading forces, and theories of fragment size have been developed based on energy balance ideas, quite independent of inherent flaw concepts. This connection was suggested early on by Rittinger (1867) and can be inferred from later studies (Poncelet, 1946 Ivanov and Mineev, 1980). An explicit energy balance approach to the prediction of fragment size has been proposed by Grady (1982).  [c.278]

Two theoretical approaches relating to the nominal fragment size achieved in dynamic fracture events have been discussed. The first is based on the inherently reasonable concept of existing fracture-producing flaws within the fragmenting body, where fragment size (or number) is necessarily correlated with the description of the flaw distribution. A second approach is based only on energy balance concepts, where no recourse to a pre-existing flaw distribution is made. Considering the diverse applications for which this latter approach provides reasonable fragment size predictions, energy balance ideas clearly play an important role in the fragmentation process. In outward appearance, these two theoretical descriptions do not seem compatible.  [c.293]

The continuum models based on the statistical nucleation and growth of brittle and ductile fracture appear to be an attractive approach, especially within a framework which provides for some form of a continuum cumulative-damage description of the evolving fracture state. Despite the importance of inherent flaws in the dynamic fracture of solids, some of the concepts introduced in this section clearly indicate the effectiveness of energy principles in determining dynamic fracture strengths and fragment sizes. Qualitative arguments were advanced which suggested that both inherent flaw and energy concepts play important roles in the dynamic fracture and fragmentation process, the relative importances depending on material conditions and loading intensities. These ideas need to be pursued further, and incorporated into computational modeling concepts.  [c.318]

JKR theory consists of a combination of the Hertz and flat punch problems. The flat punch part must be addressed in the context of fracture mechanics describing the viscoelastic stress and displacement near the crack tip. Again, with stationary cracks, the elastic solutions can be related to the viscoelastic problems via a classical correspondence principle. However, for the moving cracks only under certain restrictions, similar to those mentioned above for the counterpart of the Hertz problem, an extended correspondence principle proposed by Graham [105] can be used. In the case of a Griffith-type of crack (the JKR theory), Greenwood and Johnson [109] conclude that the apparent adhesion energy for opening cracks is overestimated by a rate-independent factor k = E(0)/E(oo) where (0) and E(oo) are the instantaneous (glassy) and relaxed shear moduli, respectively. This approach is not able to predict the rate dependency of the adhesion energies observed by experimentalists.  [c.124]

Peel energies of viscoelastic polymers are typically an order of magnitude greater than the intrinsic fracture energy of the interface, as discussed above. The influxes provide sufficient connectivity at the A/B interface to generate a visco-plastic deformation zone at the crack tip. Solutions to this problem are exceedingly complex and usually done on supercomputers when sufficient information is known about the constitutive properties of both materials and the nature of the deformation zone and interface. The J-integral theory is used to determine the fracture energy and this approach accounts for all the energy dissipated in the deformation zone. Detailed studies of bi-material interfaces normally consider differences in moduli, sample thickness, Poisson ratios, yield points, deformation zone evolution, tri-axial stresses, and fracture mode mixing (tensile, shear and twist). Let us assume that the yield strength of the A material is less than B, and that most of the visco-plastic processes occur in the A layer. The traction stresses cr, in the deformation zone will cause either fracture of the influxes at the interface, or cohesive fracture in the zone itself by the vector percolation process (Section 5). To simplify matters further, we can safely assume that the influxes control the evolution of the traction stresses.  [c.374]

Sometimes the failure occurs by propagation of a crack that starts at the top and travels downward until the interface is completely debonded. In this case, the fracture mechanics analysis using the energy balance approach has been applied [92] in which P, relates to specimen dimensions, elastic constants of fiber and matrix, initial crack length, and interfacial work of fracture (W,).  [c.831]

Another approach to modeling the dynamic fragmentation process focuses on energy principles to determine the nominal fragment size (Grady, 1982 Glenn and Chudnovsky, 1986 Grady, 1988). In this approach, energy introduced into the body by the dynamic load is considered to be the important energy fueling the fracture process and is balanced against new surface energy resisting crack growth during failure. The model is thought to be most reliable in extremely catastrophic fragmentation events however, in application it has been found quite useful over a fairly broad range of loading rates.  [c.282]

It is clear, however, that a strictly statistical approach ignores the dynamics of the fragmentation event. In the actual situation the partitioning boundaries represent growing, propagating, bifurcating, stress-relieving, energyconsuming, interacting cracks and fractures. These physical processes will most probably influence the fragment size statistics. A restricted set of those physical fracture features were considered in the studies of Mott (1947), and have been pursued more recently (Grady, 1981a, b Kipp and Grady, 1986 Grady, 1990). In this approach more significance is attributed to the dynamics of fracture activation and growth, including the nucleation process and the influence of material deformation properties. Mott (1947), in considering a restricted geometry, combined the spatial randomness of the fracture process with the growth of plastic tensile release waves and predicted fragment distributions dependent on both dynamic and material properties. More recent studies (Shockey et al., 1974 Dienes, 1978 Margolin, 1983) have focused on developing physically founded laws governing the nucleation, growth, and coalescence of fracture during one- and two-dimensional stress-wave propagation. A different approach to the statistics of fragmentation has been proposed by Griffith (1943), where particle fracture surface energy is related to the distribution through a unique application of classical Boltzmann statistics concepts. Application of percolation theory to characterize the statistics of fragmentation has also been explored, providing a method to account for residual internal cracks in fragments (e.g., Englman et al., 1984 Yatom and Ruppin, 1989).  [c.296]

A somewhat different approach to stress-wave induced fracture is represented in the work of Butkovich (1976), developed to calculate underground explosive fracture and induced permeability in coal. The method is more akin to conventional elastic-plastic calculations in that stress-space surfaces of yield or failure are established to determine onset of fracture. Fracture due to shearing is explicitly treated and two parameters are associated with fracture damage a shearing related to distortional strain and a tensile-induced cracking or porosity which is related to the permeability. A similar plasticity model of dynamic fracture has been described by Johnson (1978) and applied to explosive fracture in oil shale. A scalar fracture damage parameter is related to the damage-induced reduction in the unconfmed yield stress of the material, although the parameter is a mathematical concept rather than a measured property. Damage growth is related to the stress in excess of a pressure-dependent yield surface, with no damage growth above a brittle-ductile transition point on the yield surface. Computer simulations of explosives placed in boreholes provided successful descriptions of extent and regions of fracture damage and dependence on explosive energy and geometrical features.  [c.314]

Recognizing the need for an improved fracture model, Tucker, Rose and Burchell [15] investigated the fi acture of polygranular graphites and assessed the performance of several failure theories when applied to graphite. These theories included the Weibull theory, the Rose-Tucker model, Fracture Mechanics, Critical Strain Energy, Critical Stress and Critical Strain Theories. While no single criterion could satisfactorily account for all the situations they examined, their review showed that a combination of the fracture mechanics and a microstmcturally based fracture criterion might offer the most versatile approach to modelling fracture in graphite. Evidently, a necessary precursor to a successful fracture model is a clear understanding of the graphite fracture phenomena. Several approaches have been applied to examine the mechanism of fracture in graphite, including both direct microstmctural observations and acoustic emission monitoring as previously discussed and reported elsewhere [7,16-18].  [c.489]

A more general approach to quantitying the fracture criterion for a cracked body was first developed by Griffith [50] more than thirty-five years before the introduction of linear-elastic fracture mechanics. The Griffith criterion is based on conservation of energy and is illustrated in Fig. 4 for an infinite plate of unit thickness with a central transverse crack of length 2a. The plate is subjected to a remotely applied stress and is fixed at its ends. The elastic energy in this stressed body is represented by the area under the load versus displacement diagram. As the crack extends over an increment, da, some elastic energy is released and is  [c.497]

Adhesion generally requires the polymer(s) involved to be above their glass transition temperature, so that polymer diffusion (reptation) can proceed. Polymers can diffuse not only into other polymers but also, for instance, into slightly porous metal surfaces. The details have been effectively studied by Brown (1991, 1995) one approach is to use a diblock copolymer and deuteraie one of the blocks, so that after interdiffusion the location of residual deuterium (heavy hydrogen) can be assessed. It turns out that according to the length of the chains, the adhesive layer fractures either by pullout or by scission at the join between the blocks. Another aspect of the behaviour of adhesive layers depends on the energy required to develop and propagate crazes at the interface, which has been intensively studied by E.J. Kramer and others. When an adhesive has the right elastomeric character, it may be possible to generate very weak bonds by simple finger pressure, readily reversible without damage to the surface this is the basis of the well-known Post-it notes.  [c.332]

In Situ Retorting. True in situ retorting has been considered as a means of avoiding the costs of mining, crushing, and surface disposal of spent shale, and the associated environmental impacts of AGR. However, the impervious nature of the oil shale formation and the overburden pressures have prevented tme in situ operations. Shale oil yields, the amount of oil produced divided by the theoretical amount estimated to be in the oil shale rock, for in situ retorting are usually half that experienced with AGR retorting. A tme in situ experiment, using drilling and resource fracturing procedures typical of conventional petroleum development, was tried by the Energy Research Development Administration (a foreninner of the U.S. Department of Energy) in 1975 in Rock Springs, Wyoming. No significant yields of shale oil were produced (22). Other tme in situ tests were conducted using the Equity BX superheated steam process in Colorado, and Dow hot air process in Michigan neither produced significant yields of oil shale. It appears that tme in situ retorting is not a practical approach for the thick strata of oil shale normally situated deep below the surface.  [c.351]

Cr ck- Bridging. Crack-wake bridging occurs when matrix cracks, upon encountering a bridging entity, deflect around the entity, leaving it iatact ia the wake of the crack. Continued crack propagation requires further iacreases ia the appHed load to overcome the bridge closure forces. When the crack opening displacement at the bridging site is large enough to pull the bridge out of the matrix or to break the bridge, a steady state bridging 2one develops, which then moves with the crack. New bridges are created at the crack front, and the bridges farthest away from the crack front become iaactive. Such processes are energy dissipative and impart some degree of nonlinear stress—strain behavior to the composite fracture becomes tougher with the possibihty of developiag large strains before final failure. The phenomenon of crack-wake bridgiag leads to T-curve behavior. In terms of a toughness approach, ia which stress iatensity factors can be linearly added, the far-field stress iatensity factor is equal to the stress iatensity factor due to the bridging terms K added to the stress iatensity factor associated with the crack tip iCy  [c.53]

In the latter energy-balance approach, some tacit assumption of an adequate flaw distribution must be implied. Otherwise, tensile loading to the theoretical strength of the body would be achieved before fragmentation could proceed. Consequently, the new fragment surface area density. A, which determines the energy in Fig. 8.12 should be viewed as a coordinate which will seek an equilibrium value through a minimum energy principle, only if a sufficient fracture-producing flaw distribution exists. A less than sulficient distribution will result in nonequilibrium values of A.  [c.293]

Elastic-plastic fracture mechanics encompasses a number of quantitative approaches developed for characterizing fracture problems which lie beyond the applicability of LEFM. This class of problems generally involves nonlinear material behavior, large scale crack tip inelasticity, small crack sizes, and net-section stresses approaching that required for irreversible deformation. We will not attempt here to re-derive the various approaches but will define the important parameters and provide a description of the physical basis for each approach. There are two basic approaches for quantifying elastic-plastic fracture of graphite. The first is an energy balance approach similar to Griffith s theory of brittle fracture [50] but extended to account for irreversible processes associated with deformation and fracture. The second general approach seeks to develop parameters, similar to K, which describe the crack driving force in a manner consistent with elastic-plastic material response.  [c.497]

Using the Hutchinson [5] theory of fracture for EPZ, such as crazes at crack tips, the fracture energy is approximated using the J-integral approach as shown in Fig. 13. The EPZ model addresses small scale yielding at the crack tip and describes the work of separation as a function of rate, yield stress and the stress as a function of displacement 8 in the zone. The vector percolation model describes the maximum stresses attainable in the deformation zone. However, the crack opening displacements 8 are determined by the drawability or ductility of the matrix. This process is exceedingly complex and for polymers involves the non-Newtonian flow and plastic deformation of the strain hardened craze-like material in the zone. The craze fibrillar structure evolves from a combination of Saffmann-Taylor meniscus instability combined with cavitation processes, depending on rate and molecular weight [1]. Thus, the deformation zone initiates when the stress field at the crack tip exceeds the yield stress <7y and propagates with increasing traction stresses a up to the maximum stress a. As the zone breaks down, the stresses decrease but the displacements continue to increase and finally the crack advances at c- The traction stresses have a displacement function cr(3) as shown in Fig. 13 such that the fracture energy G(c is the integral of a(8) over the displacement range 3 = 0 to 3 = 3c- Solutions to the EPZ integrals are complex  [c.384]

There is archaeological evidence that the earliest stone tools were used in Africa more than two million years ago. These pebble tools were often made from flat, ovoid river stones that fit in the palm of one s hand. Another stone was used to chip off a few adjacent flakes to form a crude edge. However crude the edge, it could cut through the thick hide of a hunted or scavenged mammal when fingernails and teeth could not, and thus provide the group with several dozen to several hundred pounds of meat—a caloric and protein windfall. It is a common misconception that stone tools arc crude and dull. Human ancestors learned to chip tools from silicon-based stones such as flint and obsidian that have a glass-like fracture. Tools made from such stones arc harder and sharper than finely-honed knives of tool steel. As early as 500,000 years ago, well-made hand axes were being used in Africa, Europe and Asia. There is evidence that people were using fire as early as one million years ago. With the advent of the use of fire, it is believed that the amount of energy used by early man doubled from 2,000 kilocalories per person per day (energy from food) to about 4,000 kilocalories per person per day (Figure 1). Thus, fire and eventually the apparatus employed to make it were important energy liberating tools. Cooking food allowed humans to expand their diet. Controlled fire could also be used to scare game into the open to be hunted. Present-day hunters and gatherers burn beri-y patches in a controlled way, to encourage new plants and more berries. Fire also subsidized body heat, allowing people to colonize colder regions.  [c.71]

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