Energy Density Failure Criterion Research Articles

Study on Rock Failure Criterion Based on Elastic Strain Energy Density

Uniaxial and five conventional triaxial compression tests were conducted on sandstone to obtain the evolution laws of the input energy density, elastic strain energy density, and dissipative energy density. The input and dissipative energy densities increased with increasing axial strain; the elastic strain energy density increased with increasing axial strain at the pre-peak stage and decreased after the peak. According to the linear change rule between the peak elastic strain energy density and confining pressure, the energy density failure criterion of sandstone was established, and the criterion has high precision and few parameters, and the parameters have clear physical meaning. Moreover, the expression of the energy density failure criterion was similar to the classical Hoek-Brown criterion, but its adaptability was more extensive. The strength calculation results for seven different rocks under different confining pressures calculated using the energy density failure criterion were consistent with the experimental values, and the calculation error was smaller than that of the Mohr–Coulomb criterion and Drucker–Prager criterion, verifying the accuracy and applicability of the criterion.

A manifold Lagrangian integration point method for large deformation failure modelling of rock slopes

A manifold Lagrangian integration point method is proposed by combining the manifold coverage algorithm and a Lagrangian integral point finite element method, aiming to simulate the large deformation failure in slope engineering. The model is established based on the dual-discrete method with a fixed Eulerian background mesh and moving material particles. The method adopts the Eulerian mesh as the fixed background mesh and describes the deformation behaviour of macroscopic objects via particle motion between meshes, thus avoiding the problem of grid distortion in the calculation process and eliminating particle motion constraints in the set region. A strain softening strain energy density failure criterion is introduced for the first time to determine the failure of the continuum, such that the process of discontinuous failure can be addressed by a continuous method. Finally, the method is verified by simulating the post-excavation failure evolution of a slope engineering case for highway reconstruction and expansion. The results found that the obtained slope slip surface is essentially consistent with the theoretical solution.

On large deformation, damage and failure of ductile plates to blast loading

This paper concerns the irreversible deformation and failure of metal plates to brief, but intense, dynamic (impulsive) load encountered in an air-blast. Hitherto, nearly all of the dynamic structural plasticity models of ‘fully-clamped’ plate only calculates its inelastic (mode I) deformation but have largely ignored damage and failure – exemplified by the partial and/or complete detachment (modes II*, II and III) of a plate from its support through material rupture – that occurs when the blast load is sufficiently intense. An analytical model that captures all three modes of the plate deformation is developed in this paper where emphasis shall be on providing the simplest formulation that contains all the essence of a material and deformation effect needed to quantify the problem. The proposed model adopts an energy density failure criterion of Shen and Jones [19] to delineate the various inelastic failure modes; employs the constitutive framework of limit analysis for simplicity; accounts for the simultaneous influence of bending, membrane stretch and transverse shear through an interaction yield criterion; and, include the effects of strain rate on the material flow stress through the Cowper–Symonds relation. Accuracy of model predictions for square mild-steel plates are assessed through comparison with results from finite element analysis, and experimental data compiled from the literature for a range of impulsive load intensities – they will be shown to be in good general agreement. Design maps delineating the different deformation régimes for different combinations of blast impulse and length versus thickness ratio are constructed for plates of equal mass.

Ductile failure prediction of thin notched aluminum plates subjected to combined tension-shear loading

The main goal of the present research is to check the suitability of the combined Equivalent Material Concept-Averaged Strain Energy Density (ASED) failure criterion, called EMC-ASED, in predicting the load-carrying capacity (LCC) of notched aluminum plates subjected to mixed mode I/II loading. For this purpose, first, a set of experimental results on the LCC of thin V-notched rectangular Al 7075-T6 plates, that fail by large-scale yielding (LSY) regime, are taken from the open literature. Then, Al 7075-T6, which is a ductile material, is equated with an equivalent linear-elastic isotropic material by means of the EMC. Finally, the EMC is linked to the well-known ASED criterion to predict the LCC of the notched Al 7075-T6 plates. It is revealed that the experimental results could successfully be predicted by means of the combined EMC-ASED criterion without requiring complex and time-consuming elastic-plastic failure analyses.

Comment on “An extended strain energy density failure criterion by differentiating volumetric and distortional deformation” by Yujie Wei (Int. J. Solids Struct. 49 (9), 1117–1126)

Comment on “An extended strain energy density failure criterion by differentiating volumetric and distortional deformation” by Yujie Wei (Int. J. Solids Struct. 49 (9), 1117–1126)

Response to discussion on “An extended strain energy density failure criterion by differentiating volumetric and distortional deformation”

Response to discussion on “An extended strain energy density failure criterion by differentiating volumetric and distortional deformation”

An extended strain energy density failure criterion by differentiating volumetric and distortional deformation

We extend Sih’s strain energy density criterion (Sih, 1974) for crack kinks and material failure by weighting differently the volumetric and distortional parts in the extended strain energy density factor. The work is inspired by the factor that failure by microscopic shearing governed by distortion and microscopic separation controlled by hydrostatic tension represent distinct deformation processes, and should be treated differently as we count their influences to material failure. With the weight parameter introduced to the extended strain energy density factor criterion, we explain satisfactorily several critical experiments which reported crack kink in samples subjected to mixed-mode loading. The extended strain energy density idea is also used to derive a generalized pressure-dependent yielding criterion, which supplies a theoretical basis for those novel strength criteria for materials like bulk metallic glasses. Corresponding methods to determine the two material parameters, the critical strain energy density factor and the weight parameter quantifying the relative contribution by distortion over volumetric deformation, are discussed.

Mixed mode axisymmetric annular cracks in a finite layer: Off angle crack initiation

The solutions of axisymmetric Volterra type climb and glide edge dislocations are obtained in a layer by means of the Hankel transforms. Utilizing the same procedure, Green’s function solution is obtained for a layer under self-equilibration normal ring traction. The distributed dislocation technique is used to construct integral equations for a system of co-axial annular cracks where the layer is under axisymmetric normal loads. These equations are solved numerically to obtain dislocation density on the cracks surfaces. The results are employed to determine stress intensity factors for annular and penny-shaped cracks and the interaction between two co-axial penny-shaped cracks is studied. Moreover, the stress intensity factors of the interacting cracks are determined such that they can be further used in conjunction with strain energy density (SED) failure criterion to obtain the possible direction of crack initiation that may not be apparent under mixed mode conditions.

Thermo-Mechanical Failure Criterion at the Micron Scale in Electronic Devices

Thermo-mechanical failures may occur in the passivation layer of micro-electronic devices during the fabrication process. These are in form of cracks which initiate at keyhole corners. In order to predict and eventually prevent these cracks a failure criterion is presented, based on an average value of the elastic strain energy in the vicinity of a reentrant corner of any angle. The proposed strain energy density (SED) failure criterion is validated by a test including 24 full size wafers which have been fabricated with different parameters: the interconnects (metal lines) height, the passivation thickness, and the passivating plasma power which was shown to correlate with the mechanical properties of the passivation layer. For each wafer, a FE model has been constructed, and the SED computed. It has been clearly shown, that above the critical value of SEDcr[R=0.15μm]≈1000 [J/m3], all wafers manufactured were cracked. The SED criterion seems to correlate well with the empirical observations, and may be used as a standard tool for the mechanical design of failure free micro-electronic devices.

Strain energy density failure criterion

The initiation of macroscopic material failure is associated with the collective disruption of atomic bonds, which is driven by the potential energy stored in the atomic bonds. This potential energy, which is represented by specified elastic strain energy density in a mechanical system, is releasable. Thus, a universal macroscopic material failure criterion in a mechanical system can be defined by a specified elastic strain energy density together with its critical value that is determined by preceding irreversible deformation process and current environmental state. A dissipative function based on continuum mechanics and irreversible thermodynamics is proposed to represent the irreversible deformation process. The increase of this dissipative function due to material inelastic deformation, damage and other possible intrinsic dissipative mechanisms in a mechanical system leads to the reduction of material strength. When the material failure is dominated by the dissipation, a dissipative energy density failure criterion can be defined by using the dissipative function. On the other hand when the intrinsic dissipation is negligible during the deformation process before failure, the specified elastic strain energy density and its critical value, which is determined by the initial material bond strength, can be used to define material brittle failure. It also shows the possibility to set up a relationship between fracture mechanics and failure criterion. The proposed method to represent failure criteria is based on continuum mechanics and irreversible thermodynamics and retrieves previously successful failure criteria.

Numerical and failure analysis on the drawing of a circular cylindrical cup

The incremental theory of plasticity is applied in conjunction with finite elements for modeling of the circular cylindrical cup. A punch/die system is prescribed such that a circular blank is displaced at constant increments of 0.005 mm until a cup 16 mm deep is formed. Based on the strain energy density failure criterion, failure initiation accompanied by large dilatation is located by the maximum of the minimum local strain energy density function, ( dW dV ) max min, while yielding initiation accompanied by large distortion is located by the maximum of the maximum strain energy density function, ( dW dV ) max min. For a punch displacement of 16 mm, ( dW dV ) max min is near the critical value ( dW dV ) c of the metal sheet which suggests the possibility of fracture initiation. The influence of material drawability and punch movement on the rate of cup formation could be evaluated from analytical simulation of the deep drawing process. Computer software developed in the present study can also be applied to evaluate the drawing characteristics of non-axisymmetric cups.

Dynamic plastic response and failure of a clamped beam struck transversely by a mass

A theoretical analysis is presented which examines the dynamic response of a rigid-perfectly plastic clamped beam struck transversely by a mass at any point on the span. The analysis employs an interaction yield surface which combines the bending moment, membrane force and transverse shear force required for plastic flow. Material strain rate sensitive effects are examined with the aid of the Cowper-Symonds equation. Good agreement is obtained between the theoretical predictions for the maximum permanent transverse displacements and the corresponding experimental results. An energy density failure criterion is used to predict the dynamic inelastic failure of this impact problem.

Repeated impact response of XT-75 20 mm cannon support

Unlike vibration and fatigue, repeated impact has not received the attention it deserves. Response of many of the military equipment falls into this category. It involves not only high intensity time-dependent disturbances but also damage accumulation that can lead to premature failure if these effects are not properly accounted for in the design. To this end, the Ordnance Production Service (OPS) of the Republic of China (ROC) has made an effort to develop the methodology for analyzing the dynamic wave propagation history and life expectancy of structural components subjected to repeated impact. This capability will be demonstrated in this work for the XT-75 20 mm cannon support. The design of a cannon support involves knowing the transient load transmission characteristics. In particular, both the location and time corresponding to the maximum dynamic stress and/or energy must be determined. A dynamic finite element computer program using mode superposition is applied in conjunction with the strain energy density failure criterion for determining the structural integrity of the cannon support. More specifically, dynamic stresses, displacements and shape modes are obtained for nine (9) time steps starting from t = 3 × 10 −4 s to 2.7 × 10 −3 s inclusfive and firing angles of −15° to 85° with reference of the 0° horizontal line. Contours of constant volume energy density dW/ dV are plotted to illustrate how the locations of maximum and minimum dW/ dV shift with time. The most detrimental position corresponds to a firing angle of −15° and occurs at approximately seven o'clock on the outer wall of the pivot hole where the recoil force gives rise to high energy intensification. Based on accumulative energy density calculation for an aluminum support, an impact load can be repeated about 90 × 10 3 time for a cannon firing at 1500 rounds/min. A more conservative estimate would be to assume the existence of a small defect of 10 −2 cm in size. This reduces the number of impact to approximately 9 × 10 3 time.

Crack growth initiation in elastic-plastic materials

The problem of initiation of crack growth in an elastic-plastic material under idealized conditions of plane stress or plane strain was studied. The stress analysis of the cracked plate was performed by using an elastic-plastic finite element computer program based on the J2 flow theory of plasticity and a singular crack tip element. A method for determining the critical quantities at the moment of crack growth was presented. The strain energy density failure criterion was used according to which the crack starts to grow when material elements ahead of the crack absorb a critical amount of stored strain energy density which remains after subtracting the amount of irreversible energy dissipated by permanent deformation. The critical stress and the plastic zone at crack initiation were determined. No restrictions regarding the amount of plastic deformation at the crack tip and the form of the stress-strain diagram of the material in tension were imposed in the analysis. Results were given for a center cracked plate specimen for various crack lengths and ultimate stresses of the material in tension. The dependence of the critical stress on these factors and on the idealized conditions of plane stress or plane strain was established.

Path dependent nature of fatigue crack growth

Fatigue crack propagation is an inherently nonlinear phenomena governed by the material response to the load history. The accurate prediction of fatigue crack growth necessitates the modelling of the irreversible material damage in the neighborhood of the crack tip region. In this investigation, a two-dimensional center-cracked panel undergoing cyclic applied loading is analyzed to address the load history or path-dependency of fatigue crack growth. The crack growth process is simulated by predicting a series of crack growth steps which are assumed to occur when the material elements ahead of the crack tip accumulate a critical amount of stored strain energy density, ( dW/ dV) c . It is shown that the fatigue process is very sensitive to the choice of growth step size made in the analysis. The existing crack propagation prediction schemes based on linear elasticity are applied to the results and their inadequacies discussed. Crack growth predictions based on the nonlinear stress analysis and the Strain Energy Density Failure Criterion produce physically realistic results and also provide a method for determining the range of applicability of the linear growth laws (which are more practical for design usage). Final failure of the specimen is predicted when the strain energy density factor, S, reaches a critical value, S c ( dW/ dV) c and S c are material constants, independent of the loading and geometry. Thus, the fatigue process is characterized by these two material parameters. The crack tip yield zones (or plastic enclaves) are also shown to demonstrate the extent and influence of plasticity.

Brittle fracture considerations of two external notches under combined loading

The interaction behavior of two external notches, modeled by plane hyperbolas, is analyzed using the strain energy density failure criterion[1,2]. It is shown that the pure shear and normal loadings may be combined to the case of inclined or off-axis loading of this double-notch geometry. The crack trajectory emanating from the notch surface is assumed for most cases to follow a path along which the material elements experience more volume change than shape change. This is determined by taking the minimum values of the strain energy density function. The failure loads are obtained by holding the critical strain energy density function constant and are shown to vary with the angle between the inclined load and the major axis of the notch. The variations will depend on the aspect ratios of the notches. Graphical plots of the crack trajectories for different loading angles are displayed. The double-notch geometry is such that only local trajectory information can be gleaned from the initial state. Numerical results of failure load, fracture angle, etc. are given for notches with different degrees of bluntness.

The strain energy density failure criterion applied to notched elastic solids

Application of the strain energy density failure criterion is made to plane notch problems, where the crack now becomes a special case of a more generalized approach to failure. The specific case considered is that of the plane elliptical cavity under remote tension and compression. Both failure loads and fracture trajectories are discussed. It is shown that an additional characteristic dimension provides satisfactory agreement of the theory with available data. Finally, known characteristics of fracture trajectories from a notch tip are shown to be predicted for unstable fracture conditions.