A model explaining the Rule for Calculating the Break-up Time of homogeneous ductile metals

It was shown in a previous paper⁽¹⁾ that the break-up time of homogenious ductile metals which elongate due to explosions, is equal to the sample's smallest initial dimension (thickness) divided by the velocity V_pl which characterizes the metal. It is attempted to explain this rule by suggesting a break-up mechanism where holes caused by a pile-up of vacancies are formed at the metal surface and gradually increase until breaking it. This model also predicts the existence of a strain rate threshold below which other mechanisms dominate the break-up process. Its validity is supported by experimental evidence which is also presented.     

On the Break-Up of Shaped Charge Jets

It is shown from dimensional arguments that the semi-empirical jet break-up formula of Hirsch⁽¹⁾ is consistent with physical models characterized by ideal plasticity (yield stress σ₀), the material initial density (ϱ₀), strain rate (η₀), and jet radius (r₀). Similar arguments applied to a one-dimensional jet stretching model yields the existence of a maximum unstable perturbation wavelength, and fixes its value to within a numerical constant. Within the class of physical models considered the plastic velocity appearing in the Hirsch formula, the incremental velocity between successive fragments of a particulated jet, and the velocity (σ₀/ϱ₀)^1/2 are all related, as originally conjectured by Hirsch.     

Scaling of the Shaped Charge Jet Break‐Up Time

Since the formula for the shaped charge jet break‐up time was published on 1979 many attempts were made to interpret it and to make its use more efficient. It is shown herein that the V_pl parameter depends on the ratio of the liner thickness to the charge explosive diameter by the formula: 1/V_pl=13.9−101⋅(T_L/CD), where T_L is the liner thickness, CD is the explosive charge diameter and the numbers are for a published set of measurements with an OFE copper liner driven by COMP‐B explosive. To find how the numbers used in this formula change with the liner material and its metallurgical state and with the type of explosive, measurements should be made as prescribed herein. An attempt to begin explaining this formula is made in the discussion.     

Damage mechanisms of 2.5D SiO2f/SiO2 woven ceramic matrix composites under compressive impact

Fiber-reinforced SiO_2f/SiO₂ woven ceramic matrix composites (CMCs) are widely applied in the aeronautics and astronautics field due to their many physical and chemical advantages. However, compressive impact loads affect many application scenarios; thus, the damage morphologies and failure modes of these composites under compressive impact should be studied, particularly in the through-thickness direction. In this study, a comparative analysis using the split Hopkinson pressure bar (SHPB) experiment and finite element analysis (FEA) model revealed the damage mechanisms. The compressive impact was simulated in Abaqus/Explicit, and the cohesive element was selected to simulate cohesion of the interface according to the Hashin criterion for multi-mode failure. The results show that 2.5-dimensional SiO_2f/SiO₂ woven CMCs do not have sufficient plasticity to restrain the propagation of micro-cracks under high strain rates after the elastic stage under compressive impact. Many micro-cracks propagated and formed large cracks in the adiabatic shear zone. The adiabatic shear zones were generated when a compressive impact load was applied to the SiO_2f/SiO₂ CMCs at a high strain rate, and these impacts deformed the SiO_2f/SiO₂ CMCs. The adiabatic shear zone occurred at a high strain rate at the weakest point inside the fields of the SiO_2f/SiO₂ CMCs. The micro-cracks propagated and accumulated in this zone. Plastic fracture is the major failure mode for the SiO_2f/SiO₂ CMC specimens; this failure was characterized by fiber yarn fracture and the formation of micro-voids and micro-cracks due to matrix fracture. The large cracks, new voids, and interfacial debonding lead to the failure of the SiO_2f/SiO₂ CMCs. Combined action due to micro-crack formation and its propagation in the adiabatic shear band leads to a softening mechanism of the strain rate.     

Transversal thermal patterns in packed‐bed reactors with simple kinetics: Bifurcation criterion and simulations

We derive a new criterion for transversal instability of planar fronts based on the bifurcation condition dV_f/dK|_K=0 = 0, where V_f and K are the front velocity and its curvature, respectively. This refines our previously obtained condition, which was formulated as α = (ΔT_adPe_T)/(ΔT_mPe_C) > 1 to α > 1 + |δ|, where ΔT_ad and ΔT_m are the adiabatic and maximal temperature rise, respectively, Pe_C and Pe_T are the axial mass and the heat Pe numbers, respectively, and δ is a small parameter. The criterion is based on approximate relations for ΔT_m and V_f, which account for the local curvature of a propagating front in a packed bed reactor with a first‐order activated kinetics. The obtained relations are verified by linear stability analysis of planar fronts. Simulations of a simplified 2D model in the form of a thin cylindrical shell are in good agreement with the critical parameters predicted by dispersion relations. Three types of patterns were detected in simulations: “frozen” multiwave patterns, spinning waves, and complex rotating–oscillating patterns. We map bifurcation diagrams showing domains of different modes using the shell radius as the bifurcation parameter. The possible translation of the 2D cylindrical shell model results to the 3D case is discussed. © 2010 American Institute of Chemical Engineers AIChE J, 2011     

Effects of interlaminar stresses on damping of 0-degree unidirectional laminated composites

The principal aim of this paper is to formulate a general model for predicting damping in composites on the basis of the concept of strain energy-weighted dissipation. In this model, the effects of interlaminar stresses on damping have been included in addition to the effects of in-plane extension/compression and in-plane shear. Validation of the model was confirmed by performing damping measurements on 0° unidirectional composite beams with varying length and thickness. The results of theoretical predictions of damping in laminated composites were found to compare favorably with experimental data. The transverse shear (σ_xz) reveals to have a considerable effect on the damping mechanisms in 0° unidirectional polymer composites. However, the other interlaminar stresses (σ_yz, σ_z) were shown to have little influence on damping in composite beam.     

On critical criteria for shifting towards plastic strain localization during explosive cladding of Inconel 625 on low-carbon steel

The present study aims at detecting the critical criteria and corresponding critical impact energy for initiation of strain localization during explosive cladding of the Inconel 625 superalloy as a cladding material and low-carbon steel as a substrate. The results do not reveal adiabatic shear bands, which are the main signs of strain localization, within the superalloy in all studied impact energies up to 205 kJ. At impact energies greater than 78–114 kJ, strain localization is observed in low-carbon steel, and microcracks develop within the adiabatic shear bands. The Johnson-Cook model is used to explains the results obtained and to study the thermomechanical behavior of materials.     

Kinetics of bio-oxidation of metal sulphides

The microbiological oxidation of cadmium, cobalt, nickel and zinc sulphides has been investigated using a pure strain of Thiobacillus ferrooxidans. Kinetic parameters (V_m and K) have been derived regarding the effect of initial total surface area of these substrates. A relationship is suggested to exist between the rate of sulphide oxidation and the solubility product of the respective metal sulphide.     

A physical model of the pressure dependence and biaxial mechanical properties of solid polymers

We give here a model for the pressure dependent, biaxial mechanical behavior of glassy polymers based on the thermally activated growth of deformation zones (Somigliana dislocation loops). The Coulomb criterion of plasticity, σ_c = S ? mσ_n, is found as the critical threshold needed to propagate Somigliana loops, in the same way as yield in crystals is found as the stress to move Volterra dislocation loops. While S is the shear strength, it is proposed that m follows basically from chain spacing fluctuations in the polymer glass; the temperature dependences of both parameters are derived. Application to tensile and compressive tests under a confinement pressure P is developed, with the aim to derive the pressure dependent (biaxial) strain-rate law. In particular, the pressure effect on dislocation density, that is, on plasticity defect nucleation, is shown to have a definite role in the plasticity of these solids. It introduces in the strain-rate law a normal stress dependent term (exp Dσ_n), which may have a decisive importance in a number of situations like multiaxial solicitations, solid state polymer shaping, second phase effects in polymer blends, and so on. Finally, a set of constant strain rate experiments is presented on an unsaturated polyester resin crosslinked with styrene. Measurements fit reasonably well with the predictions of the above model up to ～50 K below the glass transition, at which collective molecular motions invalidate its basic assumptions. The fit includes: (i) the Coulomb Criterion and its temperature dependence; and (ii) the dilative and shear apparent activation volumes at yield at all pressures.     

Effect of the Secondary Transition on the Behaviour of Epoxy Adhesive Joints at High Rates of Loading

The shear yield behaviour of a modified epoxy joint has been investigated over a wide range of strain rates (γ˙ = 10^–2 s^–1 – 10⁴s^–1) and at different temperatures (–30°C, 24°C, 60°C, 80°C).

Assuming that high polymers exhibit pure viscous yield, the sharp increase of the yield stress in the strain rate sensitivity at high strain rates is explained in terms of a difference in relaxation times at low strain rates and high strain rates (α and β). The Bauwens's approach, which is a modification of the Ree-Eyring theory, gives an acceptable fit to the data. The yield behaviour of the modified epoxy joint, above a critical strain rate γ˙_β(T), may be described by the sum of the partial stresses τ_α and τ_β required to free the different kinds of molecular motions implied in the deformation process.

A good correlation between high impact resistance and the presence of the β mechanical loss peak in the range of the explored strain rates is established.

At very low temperature (-30°C), the data do not accurately fit the Ree-Eyring equation, meaning a heterogeneous deformation process characterized by the formation of local adiabatic shear bands and a permanent evolution of the molecular structure.     

The elastic stresses generated during fiber formation by wet-spinning

The pressure required to cause a spinning solution to flow through the spinnerette hole in a wet-spinning process has been found to decrease as the take-up velocity V₁ is increased. The amount of pressure reduction from free extrusion, where V₁ = V_f, to V₁ denoted by p(V_f)–p(V₁) was found to correlate well with the term (V₁–V_f)/V_f. The pressure reduction is interpreted as the change in axial tensile stress of the solution within the capillary brought about by take-up of the filament, and the latter term is interpreted as a meaure of the strain state of the filament. For large hole diameters, plots of these measures of stress and strain are linear and independent of hole diameter, capillary length and configuration, and shear rate but depend on factors that affect basic rheological and coagulation characteristics of the system. At smaller hole sizes, the situation is somewhat more complex. Aspects of the fiber formation process are discussed in light of these observations.     

Specific features of the transformation of spall cracks to localized shear bands

Investigations of specific features of the microstructure of the region where a spall crack transforms to an adiabatic shear band are based on a spall model of strain localization, which implies that adiabatic shear bands are induced by interference of unloading waves, and the value of the negative stress in the expansion region of these waves does not exceed the dynamic strength of the material. It is shown that the transformation region contains a tremendous number of dislocation ensembles, which is much greater than the number of dislocation ensembles generated by a shock wave. Detection of micrometer-sized fracture sites in the region of interference of unloading waves implies that small fracture sites are formed in a polycrystalline material on dislocations arising in the course of dynamic tension.     

A Formula for the Shaped Charge Jet Breakup-Time

It was found that the breakup-time of jets formed by shaped charges of cylindrical symmetry is given very accurately by the formula: This formula is obtained from a general principal applied here for the first time which says that the breakup-time of homogeneous ductile metals under very high strain rates is equal to the smallest initial dimension of the elongating metal divided by V_PL . When applied to simple configurations such as a pipe which expands due to internal explosion this principal leads to a correct prediction of the average formed fragment dimensions. This principal provides for the first time an explanation to the experimental fact that metals' ductility can increase by an order of magnitude when the strain rate increases from 10⁻² to 10⁵ pro second.     

Atomistic insight into ideal strength,fracture toughness,deformation, and failure mechanisms of magnetocaloric Fe2AlB2

The thermal and magnetic cycling of a magnetocaloric material degrades its mechanical properties and device performance. We used ab initio tensile and shear simulations to investigate the mechanical properties such as ideal strength, fracture toughness and deformation and failure mechanisms of Fe₂AlB₂ at finite strain. The weakest direction of Fe₂AlB₂ is [010], and the weakest slip system is (010)[100]. The ideal tensile strength (σ_m = 12.51 GPa) of Fe₂AlB₂ is less than its ideal shear strength (τ_m = 13.32 GPa). The strain energy difference (ΔE = −13 eV/f.u.) of Fe₂AlB₂ confirms cleavage fracture as its most plausible failure mode. The concomitant changes in the c-lattice parameter and Al–Al bond along the c-axis determine the ideal tensile strength of Fe₂AlB₂. Likewise, the subtle changes in the a-lattice parameter and Al–Al bond along the a-axis specify its ideal shear strength. The tensile strain induces a magnetic to nonmagnetic transition in Fe₂AlB₂ at the critical tensile strain (ε_c = 0.08). A similar transition occurs at the critical fracture strain (ε_cf = 0.48) due to shear deformation. The brittle nature of Fe₂AlB₂ is predicted by its anisotropic Poisson's ratios, strength ratio, and failure mode. The fracture toughness of Fe₂AlB₂ for mode I fracture is (K_Ic = 2.17 MPa m^1/2), mode II fracture is (K_IIc = 1.33 MPa m^1/2), and mode III fracture is (K_IIIc = 1.16 MPa m^1/2). The failure mechanism of Fe₂AlB₂ due to the tensile deformation is marked by the sharp and appreciable changes in the lattice parameters, bonding characteristics, and magnetic moment of Fe at the critical fracture strain (ε_cf = 0.44). This study provides a fundamental understanding of the mechanical behavior of Fe₂AlB₂ at the finite strain relevant to the cycling stability of the magnetocaloric Fe₂AlB₂.     

Strong spin–orbit splitting in graphene with adsorbed Au atoms

Spin–orbit (SO) splitting in graphene with adsorbed Au atoms is investigated by using a first-principles method. Considerable (～200 meV) Rashba-type SO splitting can be achieved in the graphene π bands. When a Au atom is adsorbed above a C–C bond, Dresselhaus-type SO splitting is found to be present due to the absence of inversion symmetry and the substantial contribution of Au 5d_xz components. The influence of strains in graphene on SO splitting is also explored. A slight strain with the strength of ?5% to 5% usually does not change much the SO splitting. The variation of SO splitting versus strain strength is rationalized through structural relaxation and effective hybridization between C 2p_z and certain Au 5d states. Our study predicts a new way to increase the SO splitting in graphene and provides useful understanding of the mechanism.     

Comprehensive understanding of horizontal and vertical crack effects on failure mechanism of lamellar structured thermal barrier coatings

The local spalling induced by the propagation and coalescence of cracks in the ceramic layer is the fundamental reason for the thermal barrier coatings (TBCs) failure. To clarify the effects of horizontal and vertical cracks on the coating failure, an integrated model combining dynamic TGO growth and ceramic sintering is developed. The effects of cracks on the normal and shear stress characteristics are analyzed. The driving force and propagation ability of cracks under different configurations are evaluated. The interaction between horizontal and vertical cracks is explored by analyzing the variation of the crack driving force. The results show that TGO growth causes the ratcheting increase of σ₂₂ tensile stress above the valley, and the σ₁₂ shear stress is on both sides of the peak. Ceramic sintering mainly contributes to the ratcheting increase of σ₁₁ tensile stress. There is minimum strain energy when the horizontal crack extends to the peak. The vertical cracks on the surface of the ceramic layer are easier to propagate through the coating than that of other locations. When the horizontal and vertical cracks simultaneously appear near the valley, they can promote the propagation of each other. The present results can offer theoretical support for the design of an advanced TBC system in the future.     

Application of critical state to uniaxial compaction

From the radial stress σ_R and the normal stress σ_A, measured continuously during uniaxial loading and unloading on three compactable (sodium chloride, polyethylene and tartaric acid) and two non-compactable (polypropylene and polystyrene plastics) materials, characteristic compaction profiles of (σ_A ? σ_R) versus (σ_A + σ_R) can be observed. The uniaxial loading stress pathways for both compactable and non-compactable materials validated the assumption that the Coulomb yield criterion, which is usually applicable for the shear testing of soils, can be applied to the uniaxial compression of particulate materials. In addition, the unloading stress profiles for the compactable materials produced two characteristic parameters: a normal stress value at zero shear (σ_A + σ_R)_o and a minimum shear stress value (σ_A ? σ_R)_min. Correlation of (σ_A + σ_R)_o and (σ_A ? σ_R)_min values with either the tensile strength f_c or the Vickers hardness number H_V from the resultant compacts showed a linear logarithmic relationship. No such relationship was found, however, with non-compactable materials.     

New criterion for craze initiation

Existing criteria for craze initiation are reviewed, and their limitations are discussed. The most obvious problem is that they are formulated simply in terms of principal stresses, making no provision for the known effects of small inclusions and surface imperfections. To solve this problem, a new criterion is proposed, which is based on linear elastic fracture mechanics. Craze initiation is treated as a frustrated fracture process rather than a yield mechanism. Calculations show that the strain energy release rate, G_I(nasc), required to generate a typical 20 nm thick nascent craze, is less than 1 J m⁻². This explains why flaws less than 1 μm in length are capable of nucleating crazes at stresses of 20-30 MPa. Subsequent craze propagation is dependent upon two flow rates, one relating to fibril drawing at the craze wall and the other to shear yielding at the craze tip. Under biaxial stress, the second principal stress σ₂ affects craze tip shear yielding but not fibril drawing. This model is used in conjunction with the von Mises yield criterion to derive a new expression for the crazing stress σ₁(craze), which provides a good fit to data on visible crazes obtained by Sternstein, Ongchin and Myers in biaxial tests on cast PMMA [Sternstein SS, Ongchin L, Silverman A. Appl Polym Symp 1968;7:175; Sternstein SS, Ongchin L. Polym Prepr Am Chem Soc Div Polym Chem 1969;10:1117; Sternstein SS, Myers FA. J Macromol Sci Phys 1973;B8:539].     

Thermal residual stresses in an adhesively-bonded functionally graded single-lap joint

This study investigates three-dimensional thermal residual stresses occurring in an adhesively-bonded functionally graded single-lap joint subjected to a uniform cooling. The adherends are composed of a through-the-thickness functionally graded region between Al₂O₃ ceramic and Ni metal layers. Their mechanical properties were calculated using a power law for the volume fraction of the metal phase and a 3D layered finite element was implemented. In a free single-lap joint the normal stress σ_xx was dominant through the overlap region of the upper and lower adherends and along the adhesive free edges, whereas the transverse shear stress σ_xy concentrations appeared only along the free edges. The peel stress σ_yy and the transverse shear stress σ_xy became dominant along the free edges of the adhesive layer. In addition, the von Mises stress decreased uniformly through the adherend thickness from compressive in the top ceramic-rich layer to tensile in the bottom metal-rich layer. In addition, the layer number had only a minor effect on the through-the-thickness stress profiles after a layer number of 50, except for the peak stress values in the ceramic layer. In a single-lap joint fixed at two edges both adherends underwent considerable normal stress σ_xx concentrations varying from compressive in the top ceramic-rich layer to tensile in the bottom metal-rich layer along the free edges of both adherend–adhesive interfaces, whereas the peel stress σ_yy and transverse shear stress σ_xy reached peak levels along the left and right free edges of the adhesive layer. The layer number and the compositional gradient exponent had only minor effects on the through-the-thickness von Mises stress profiles but considerably affected the peak stress levels. The free edges of adhesive–adherend interfaces and the corresponding adherend regions are the most critical regions, and the adherend edge conditions play more important role in the critical adherend and adhesive stresses. Therefore, the first initiation of the joint failure can be expected along the left and right free edges of the upper and lower adherend–adhesive interfaces.