Energy release or absorption due to simultaneous expansion of many interacting nanoholes in elastic materials

The energy release or absorption due to simultaneous expansion of many interacting nanoholes in elastic materials under plane strain deformation is studied as influenced by the surface effect along rims of nanoholes. The M-integral classically used in macro mechanics with defects is extended to treat the problem with many interacting nanoholes. After some manipulations, the energy change due to the simultaneous expansion of many nanoholes represented by the M-integral is evaluated. Four different arrays of many nanoholes under a monotonically increasing tensile loading are considered. Attention is focused on the influence induced from the surface tension, the surface Lamé constants, and the interaction among many nanoholes on the M-integral. It is concluded that the surface tension yields significant influence on the M-integral, whereas the surface Lamé constants offer much smaller influence, which could be neglected with some relative errors less than 2%. It is found that, unlike those in macro mechanics with defects, the simultaneous expansion could either release energy (the positive value of the M-integral) or absorb energy (the negative value of the M-integral), depending on the loading levels. There is a neutral loading point, at which the M-integral transforms from a negative value to a positive value in all arrays of nanovoids under consideration. It is also found that the interaction among multiple nanoholes influences the value of the neutral loading point significantly because the mutual influence induced from both the interacting effect and the surface effect yields a quite different feature from those induced from the interacting effect only. That is, the surface effect always inhibits the influence of the interacting effect on the M-integral.     

Energy release or absorption due to simultaneous expansion of many interacting nano-holes in elastic materials

The energy release or absorption due to simultaneous expansion of many interacting nano-holes in elastic materials under plane strain deformation is studied as influenced by the surface effect along rims of nano-holes. The M-integral classically used in macro mechanics with defects is extended to treat the problem with many interacting nano-holes. After some manipulations, the energy change due to the simultaneous expansion of many nano-holes represented by the M-integral is evaluated. Four different arrays of many nano-holes under a monotonically increasing tensile loading are considered. Attention is focused on the influence induced from the surface tension, the surface Lamé constants, and the interaction among many nano-holes on the M-integral. It is concluded that the surface tension yields significant influence on the M-integral, whereas the surface Lamé constants offer much smaller influence, which could be neglected with some relative errors less than 2%. It is found that, unlike those in macro mechanics with defects, the simultaneous expansion could either release energy (the positive value of the M-integral) or absorb energy (the negative value of the M-integral), depending on the loading levels. There is a neutral loading point, at which the M-integral transforms from a negative value to a positive value in all arrays of nano-voids under consideration. It is also found that the interaction among multiple nano-holes influences the value of the neutral loading point significantly because the mutual influence induced from both the interacting effect and the surface effect yields a quite different feature from those induced from the interacting effect only. That is, the surface effect always inhibits the influence of the interacting effect on the M-integral.     

The area contraction and expansion for a nano-void under four different kinds of loading

This paper provides an explanation for the authors’ recent conclusions as why the M-integral could be negative for a nano-void (hole) in elastic materials under tensile, bi-axes tensile, and bi-axes tensile-compress loadings (Hui T, Chen YH, in ASME J Appl Mech 77:021019-1-9, 2009; 024505-1-5, 2009). Attention is focused on whether the area of the nano-void under each of four different kinds of loading is contractive when compared to the original geometrical area without loading. The four kinds of loading are as follows: (1) simple tensile loading; (2) bi-axes tensile loading; (3) bi-axes tensile-compression loading; and (4) purely shear loading. It is concluded that, unlike those for a macro/micro hole, the area of the nano-void under relatively lower amplitude of the loadings (1) and (2) is always smaller than the original area of the same void before loading. This reduction in the nano-void area is induced from the surface effect including the surface tension and the surface Lamé constants along the nano-void rim. Thus, these two kinds of loading can either decrease or increase the nano-void area depending on the amplitude of the loading. Under a relatively lower tensile loading, the area contraction occurs but not always corresponding to the negative value of the M-integral, whereas under a relatively higher tensile loading, the area expansion occurs but not always corresponding to the positive value of the M-integral. This again verifies that the positive value of the M-integral does not always correspond to the energy release due to the nano-hole expansion, rather, the area contraction might yield the energy release either. Under bi-axes tensile compression or purely shear loading, this feature disappears.     

M-integral analysis for a two-dimensional metal/ceramic bimaterial solid with extending subinterface microcracks

Summary  The problem of the extension of subinterface microcracks in an infinite metal/ceramic bimaterial solid is studied. For the microcrack growth, the values of the M-integral are calculated under the assumption of a self-similar growth. First, the role that the M-integral plays in a metal/ceramic bimaterial solid with growing subinterface cracks is analyzed. It is concluded that an inherent relation exists between the value of the M-integral and the decrease of the effective elastic moduli for a bimaterial solid with growing subinterface microcracks. Second, it is concluded that mutual amplification and shielding effects exist during the microcrack extension, while they are substantially dependent on the increment of the microcrack length as well as the geometry of the microcrack arrangement under given loads. This strong mutual shielding effect of interacting microcracks makes the microcrack extension become increasingly difficult, and may stop the growth of the microcracks even under constant loads. Also, it is concluded that for a certain microcrack growth, the value of the M-integral in metal/ceramic bimaterial solid is always larger than that in homogeneous brittle solid for the same crack configuration. This means that the same microcrack growth in the former case shows lower stability than that in the latter one, due to the existence of a ductile phase. Received 3 May 2001; accepted for publication 27 June 2002 This work was supported by the Chinese National Nature Science Foundation (Grant 19472053) and supported by the Doctorate Foundation of Xi'an Jiaotong University (Grant DFXJU2000-15).     

Analysis of L-integral and theory of the derivative stress field in plane elasticity

In this paper, analysis of the L-integral in plane elasticity is present. An infinite plate with any number of inclusions and cracks and with any remote tractions is assumed in analysis. Arbitrary forces are applied on the cracks, inclusions or at a point of the infinite medium. To study the problem, the concept of the derivative stress field is introduced, which is derived from a physical stress field. The mutual work difference integral (MWDI) is also introduced, which is defined as a difference of mutual works done by each other from the physical stress field and the derivative field. It is proved that the L(CR) (L-integral on a large circle) is equal to a particular MWDI. General expression for the L(CR) is obtained. For a given stress field, the variation of the L(CR) is studied when the coordinates have a translation or rotation. It is found that the L(CR) is an invariant with respect to the rotation of coordinates, and it has a variation when the coordinates have a translation.     

Energy dissipation and contour integral characterizing fracture behavior of incremental plasticity

Jep -integral is derived for characterizing the frac- ture behavior of elastic-plastic materials. The J ep -integral differs from Rice’s J-integral in that the free energy density rather than the stress working density is employed to define energy-momentum tensor. The J ep -integral is proved to be path-dependent regardless of incremental plasticity and deformation plasticity. The J epintegral possesses clearly clear physical meaning: (1) the value J ep tip evaluated on the infinitely small contour surrounding the crack tip represents the crack tip energy dissipation; (2) when the global steadystate crack growth condition is approached, the value of J ep farss calculated along the boundary contour equals to the sum of crack tip dissipation and bulk dissipation of plastic zone. The theoretical results are verified by simulating mode I crack problems.     

Computational and experimental study of supersonic flow over axisymmetric cavities

Laminar and turbulent computations are presented for annular rectangular-section cavities, on a body of revolution, in a Mach 2.2 flow. Unsteady ‘open cavity flows’ result for all laminar computations for all cavity length-to-depth ratios, L/D (1.33, 10.33, 11.33 and 12.33). The turbulent computations produce ‘closed cavity flows’ for L/D of 11.33 and 12.33. Surface pressure fluctuations at the front corner of the L/D = 1.33 cavity are periodic in some cases depending on the cavity length and depth, the boundary layer at the cavity front lip and the cavity scale. The turbulent computations are supported by experimental schlieren images, obtained using a spark light source, and time-averaged surface pressure data.     

Stress Magnification due to Stretching and Bending of Thin Ligaments between Voids

Stress magnification in thin ligaments between small and large cylindrical voids is obtained by matching the inner field approximation by beam theory to the outer rigid-body field in the bulk of the material. A void between two larger voids is modeled as a large hole within a strip of straight edges (boundaries of the holes with infinite radii of curvature). Both stretching and bending types of loading are applied to the strip. Comparison of different orders of stress magnification for different geometries and loading conditions is made. It is shown that the order of stress magnification in thin ligaments is (R/δ) ⁿ , where n=1/2 in the ligament between one small and one large void, n=1 in the ligament between one small void and two large voids, or between two small and two large voids, and n=2 in the ligament between a large void and a small void coalescing with another large void. The relevance of these results for the study of material failure by void growth and coalescence is discussed.     

Influence of flow path modification on oscillation of cavity shear layer

This study investigates the influence on the oscillating characteristics of a cavity shear layer by introducing either a sloped bottom or a flow path modifier at the bottom of the cavity. All the experiments are performed in a recirculating water channel. The laser Doppler velocimetry system and the laser sheet technique are employed to perform the quantitative velocity measurements and the qualitative flow visualization, respectively. The Reynolds number, based on the momentum thickness at the upstream edge of the cavity, is kept at about Re _{θ 0}=194 ± 3.4. It is found that, in addition to the feedback effect, the upstream moving part of the recirculating flow inside the cavity also plays an important role in changing the oscillating characteristics of the unstable shear layer. As the bottom of the cavity is either negatively or positively sloped, the oscillating characteristics of the cavity shear layer are modified to different extents. Significant reduction of the oscillating amplitude within the cavity is found while the bottom slope increases up to d/L=± 2/5. As the bottom slope further increases up to d/L=± 1/2, the self-excited oscillation is completely suppressed. In addition, the ability to suppress the self-excited oscillation by the negative bottom slopes is superior to that in the case of a positive bottom slope. Depending upon the fence locations, the upstream moving part of the recirculating flow will perturb the unstable shear layer at different x/L locations, leading to different oscillating amplitudes. The ability to promote the enlarged oscillating amplitude of the unstable shear layer is better for a fence inclined at a positive angle than for one at a negative angle. Received: 31 May 2000/Accepted: 11 January 2001     

The explicit formulations for theL-integral

In this paper, a simple but inberent relation between theL-integral and the Bueckner work conjugate integral is proved for crack problems in isotropic, anisotropic, and dissimilar materials, respectively. It is found that, in the above-mentioned three cases, theL-integral, from the mathematical point of view as well as in principle, arises from Betti's reciprocal theorem. This means that the Bueckner work conjugate integral is a more general path-independent integral than the others since any other path-independent integrals could be derived by using the Bueckner integral while choosing a different subsidiary stress-displacement field.     

Inequalities relating to Lp-version of Petty＇s conjectured projection inequality

Petty＇s conjectured projection inequality is a famous open problem in the theory of convex bodies. In this paper, it is shown that an inequality relating to Lp-version of the Petty＇s conjectured projection inequality is developed by using the notions of the Lp-mixed volume and the Lp-dual mixed volume, the relation of the Lp-projection body and the geometric body Г-pK, the Bourgain-Milman inequality and the Lp-Bnsemann-Petty inequality. In addition, for each origin-symmetric convex body, by applying the Jensen inequality and the monotonicity of the geometric body Г-pK, the reverses of Lp-version of the Petty＇s conjectured projection inequality and the Lp-Petty projection inequality are given, respectively.     

Edge fracture in cone-plate and parallel plate flows

Edge fracture is an instability of cone-plate and parallel plate flows of viscoelastic liquids and suspensions, characterised by the formation of a `crack' or indentation at a critical shear rate on the free surface of the liquid. A study is undertaken of the theoretical, experimental and computational aspects of edge fracture. The Tanner-Keentok theory of edge fracture in second-order liquids is re-examined and is approximately extended to cover the Criminale-Ericksen-Filbey (CEF) model. The second-order theory shows that the stress distribution on the semi-circular crack is not constant, requiring an average to be taken of the stress; this affects the proportionality constant, K in the edge fracture equation −N _2c = KΓ/a, where N _2c is the critical second normal stress difference, Γ is the surface tension coefficient and a is the fracture diameter. When the minimum stress is used, K = 2/3 as found by Tanner and Keentok (1983). Consideration is given to the sources of experimental error, including secondary flow and slip (wall effect). The effect of inertia on edge fracture is derived. A video camera was used to record the inception and development of edge fracture in four viscoelastic liquids and two suspensions. The recorded image was then measured to obtain the fracture diameter. The edge fracture phenomenon was examined to find its dependence on the physical dimensions of the flow (i.e. parallel plate gap or cone angle), on the surface tension coefficient, on the critical shear rate and on the critical second normal stress difference. The critical second normal stress difference was found to depend on the surface tension coefficient and the fracture diameter, as shown by the theory of Tanner and Keentok (1983); however, the experimental data were best fitted by the equation −N _2c = 1.095Γ/a. It was found that edge fracture in viscoelastic liquids depends on the Reynolds number, which is in good agreement with the inertial theory of edge fracture. Edge fracture in lubricating grease and toothpaste is broadly consistent with the CEF model of edge fracture. A finite volume method program was used to simulate the flow of a viscoelastic liquid, obeying the modified Phan-Thien-Tanner model, to obtain the velocity and stress distribution in parallel plate flow in three dimensions. Stress concentrations of the second normal stress difference (N ₂) were found in the plane of the crack; the velocity distribution shows a secondary flow tending to aid crack formation if N ₂ is negative, and a secondary flow tending to suppress crack formation if N ₂ is positive. Received: 4 January 1999 Accepted: 19 May 1999     

Numerical and experimental investigation of variable phase transformation number effect in porous media during freezing process

A model is proposed to investigate heat and moisture transfer in porous media during freezing process based on Luikov’s model by considering the effect of variation of phase transformation number, ε. This parameter has been mostly used as a constant by researchers. Three-dimensional Luikov’s equations are considered and solved numerically. The model is compared with obtained experimental data. It is shown that the effect of variable phase transformation number is noticeable in heat and moisture transfer process.     

Positivity and Almost Positivity of Biharmonic Green’s Functions under Dirichlet Boundary Conditions

In general, for higher order elliptic equations and boundary value problems like the biharmonic equation and the linear clamped plate boundary value problem, neither a maximum principle nor a comparison principle or—equivalently—a positivity preserving property is available. The problem is rather involved since the clamped boundary conditions prevent the boundary value problem from being reasonably written as a system of second order boundary value problems. It is shown that, on the other hand, for bounded smooth domains W ì \mathbbRⁿ{\Omega \subset\mathbb{R}^n} , the negative part of the corresponding Green’s function is “small” when compared with its singular positive part, provided n\geqq 3{n\geqq 3} . Moreover, the biharmonic Green’s function in balls B ì \mathbbRⁿ{B\subset\mathbb{R}^n} under Dirichlet (that is, clamped) boundary conditions is known explicitly and is positive. It has been known for some time that positivity is preserved under small regular perturbations of the domain, if n = 2. In the present paper, such a stability result is proved for n\geqq 3{n\geqq 3} .     

Vortex ring generation due to the coalescence of a water drop at a free surface

 Results are presented of an experimental investigation of vortex ring formation by a fluid drop contacting a free surface with negligible velocity. The pool fluid is mixed with fluorescein dye, and a laser sheet is used to illuminate a plane of the flow. A series of representative images is recorded by a CCD camera and speculation is made regarding specific sources of vorticity flux through the free surface. Two scaling analyses previously presented by other investigators are demonstrated to be equivalent under the assumptions of this experiment, and they provide the motivation for a series of test runs in which the duration of the coalescence process, τ_*, is related to variations in drop diameter L and fluid surface tension σ. Experimental results are in agreement with the analyses, showing τ_*∼σ^-1/2 and τ_*∼L ^3/2. Received: 22 December 1995 / Accepted: 15 October 1996     

Non-linear rheology and flow-induced transition of a lamellar-to-vesicle phase in ternary systems of alkyldimethyl oxide/alcohol/water

 The time-dependent transformation of an ionically charged lamellar phase (L _α-phase) into a vesicle phase under the influence of shear is investigated using rheological and conductivity measurements. The L _α-phase consists of the zwitterionic surfactant tetradecyldimethylaminoxide (C₁₄DMAO), hexanol, oxalic acid and water. The experiments were carried out on the L _α-phase in a well defined state. It was prepared by a special route from the neighbouring L ₃-phase that consists of 100 mM C₁₄DMAO, 250 mM hexanol and 5 mM oxalicdiethylester (OEE). The OEE hydrolyses in the L ₃ -phase to oxalic acid and ethanol. The result is a virgin L _α-phase which consists of stacked bilayers and which has not been exposed to shear. When this low-viscous phase is subjected to shear it is transformed into a highly viscous vesicle phase. The transformation of the L _α-phase into vesicles under constant shear was monitored by recording the viscosity and conductivity with time. It is observed that at least three different time constants can be distinguished in the transformation process. The conductivity passes through a minimum (τ₁) in the direction of shear. The viscosity first passes through a minimum (τ₂) and then over a maximum (τ₃). It is concluded that τ₁ belongs to the complete alignment of the bilayer parallel to the wall, τ₂ to the beginning of the break-up of the bilayers to the vesicles and τ₃ to the complete transformation of the L _α- to the vesicle phase. When the shear rate was varied, it was noted that the product of the time constants and shear is constant. Received: 30 June 1999/Accepted: 30 August 1999     

Well-posedness and Stability of a Multi-dimensional Tumor Growth Model

We study a moving boundary problem modeling the growth of in vitro tumors. This problem consists of two elliptic equations describing the distribution of the nutrient and the internal pressure, respectively, and a first-order partial differential equation describing the evolution of the moving boundary. An important feature is that the effect of surface tension on the moving boundary is taken into account. We show that this problem is locally well-posed for a large class of initial data by using analytic semi-group theory. We also prove that if the surface tension coefficient γ is larger than a threshold value γ _* then the unique flat equilibrium is asymptotically stable, whereas in the case γ  < γ _* this flat equilibrium is unstable.     

A physico-mechanical approach to modeling of metal forming processes—Part II: damage analysis in processes with plastic flow of metals

A damage analysis is presented for the extrusion of a case-shaped cylindrical part by using a physico-mechanical approach for modeling metal forming processes. Two integral measures related to the hydrostatic and deviatoric parts of the damage tensor are used for the calculation of strain damage. The combined use of two damage measures in contrast to only one allows us to assess not only a risk of macro-fracture of the deformed material but also the stage of formation of large cavernous defects due to coalescence of ellipsoidal voids. Such a refined prediction of the actual quality of the material’s micro-structure is important when producing metalware that is supposed to operate under intense loading and thermal conditions. In case study of this paper the kinetic equations of damage are solved by using mutually consistent fields of stresses, flow velocities, and strains. It is shown that the predicted damage is less than its permissible value since a high hydrostatic pressure in the plastic zone heals the micro-defects, prevents their growth, and, thereby, increases the processing ductility of deformed metals during extrusion.      

High-order asymptotic analysis for the crack in nonlinear material

Accurate high-order asymptotic analyses were carried out for Mode II plane strain crack in power hardening materials. The second-order crack tip fields have been obtained. It is found that the amplitude coefficientk ₂ of the second term of the asymptotic field is correlated to the first order field as the hardening exponentn<n ^* (n ^*≈5), but asn≥n ^*,k ₂ turns to become an independent parameter. Our results also indicated that, the second term of the asymptotic field has little influence on the near-crack-tip field and can be neglected whenn<n ^*. In fact,k ₂ directly reflects the effects of triaxiality near the crack tip, the crack geometry and the loading mode, so that besidesJ-integral it can be used as another characteristic parameter in the two-parameter criterion. The project supported by National Natural Science Foundation of China     

Soret effect and slow mass diffusion as a catalyst for overstability in Marangoni-Bénard flows

We study the onset of time dependent Marangoni-Bénard convection in binary mixtures subject to Soret effect by numerical computation of linear instability thresholds in infinite fluid layers and two-dimensional boxes. The calculations are done for positive Marangoni numbers (Ma > 0) and negative Marangoni Soret parameters S _M = –(D _S γ _c )/(Dγ _T ) where D _S and D are the Soret and mass diffusion coefficients, respectively, and γ _T , γ _c are the first derivatives of the surface tension with respect to temperature and concentration. Our purpose is to understand why for particular choices of Prandtl and Schmidt numbers, the increase of the stabilizing solutal contribution leads to a decrease of the critical temperature difference, a phenomenon already reported by Chen & Chen [5] and Skarda et al. [12] For various choices of Prandtl and Schmidt numbers we analyze the evolution of the critical Marangoni number Ma _c , critical wavenumber k _c and angular frequency ω _c with S _M and compute the corresponding eigenvectors. We next propose a physical mechanism which explains how the stabilizing solutal contribution acts as a catalyst for overstability. Finally, we extend our results to two dimensional boxes of small aspect ratio.