Further studies on dynamic crack branching

The newly derived dynamic-crack-branching criterion with its modifications is verified by the dynamicphotoelastic results of dynamic crack branchings in thinpolycarbonate, single-edged crack-tension specimens. Successful crack branching was observed in four specimens and unsuccessful branching in another. Crack branching consistently occurred when the necessary conditions ofK _I =K _{I b} =3.3 MPa \(\sqrt m\) and the sufficiency condition ofr _o =r _c =0.75 mm were satisfied simultaneously. In the unsuccessful branching test, the necessary condition was not satisfied sinceK _I was always less thanK _{I b} .     

An experimental investigation of crack-path directional stability

Using the criterion that a crack will extend perpendicular to the maximum circumferential stress,σ _θ, we show that the directional stability of crack growth is governed by the location of microcrack initiation ahead of the crack tip. At distances greater than a geometrical radiusr _o, the maximum value ofσ _θ deviates from the position of symmetry. Thus, if we assume that the physical processes involved in fracture lead to crack initiation at a distancer _c ahead of the crack tip, the criterion for directional stability isr _o>r _c. Experimental and theoretical values ofr _o verify that, asr _o becomes small, the crack's directional stability deteriorates. Observing that a lengthwise compressive stress increasesr _o, a center-cracked specimen was developed which allows the application of controlled lengthwise compression independently of the opening-mode load. A detailed photoelastic analysis of the specimen has provided the value ofr _o as a function of the crack length. The value ofr _o is then compared with the expected microcrack initiation distances in ductile fracture. By applying sufficient lengthwise compression, we are able to make the crack grow straight and obtain numerous data points from this specimen which would otherwise be directionally unstable. The results indicate that, as the total lengthwise tensile stress at the crack tip increases, the fracture toughness also increases. Using this information we can then adjustK _Ic for zero lengthwise loading and obtain a geometry independent fracture toughness.     

Asymptotic analysis of linearly elastic shells. III. Justification of Koiter's shell equations

We consider as in Parts I and II a family of linearly elastic shells of thickness 2?, all having the same middle surfaceS=?(?)?R ³, whereω?R ² is a bounded and connected open set with a Lipschitz-continuous boundary, and? ∈ ?³ (?;R ³). The shells are clamped on a portion of their lateral face, whose middle line is?(γ ₀), whereγ ₀ is a portion of?ω withlength γ ₀>0. For all?>0, let $\zeta _i^\varepsilon$ denote the covariant components of the displacement $u_i^\varepsilon g^{i,\varepsilon }$ of the points of the shell, obtained by solving the three-dimensional problem; let $\zeta _i^\varepsilon$ denote the covariant components of the displacement $\zeta _i^\varepsilon$ a ⁱ of the points of the middle surfaceS, obtained by solving the two-dimensional model ofW.T. Koiter, which consists in finding $$\zeta ^\varepsilon = \left( {\zeta _i^\varepsilon } \right) \in V_K (\omega ) = \left\{ {\eta = (\eta _\iota ) \in {\rm H}^1 (\omega ) \times H^1 (\omega ) \times H^2 (\omega ); \eta _i = \partial _v \eta _3 = 0 on \gamma _0 } \right\}$$ such that $$\begin{gathered} \varepsilon \mathop \smallint \limits_\omega a^{\alpha \beta \sigma \tau } \gamma _{\sigma \tau } (\zeta ^\varepsilon )\gamma _{\alpha \beta } (\eta )\sqrt a dy + \frac{{\varepsilon ^3 }}{3} \mathop \smallint \limits_\omega a^{\alpha \beta \sigma \tau } \rho _{\sigma \tau } (\zeta ^\varepsilon )\rho _{\alpha \beta } (\eta )\sqrt a dy \hfill \\ = \mathop \smallint \limits_\omega p^{i,\varepsilon } \eta _i \sqrt a dy for all \eta = (\eta _i ) \in V_K (\omega ), \hfill \\ \end{gathered}$$ where $a^{\alpha \beta \sigma \tau }$ are the components of the two-dimensional elasticity tensor ofS, $\gamma _{\alpha \beta }$ (η) and $\rho _{\alpha \beta }$ (η) are the components of the linearized change of metric and change of curvature tensors ofS, and $p^{i,\varepsilon }$ are the components of the resultant of the applied forces. Under the same assumptions as in Part I, we show that the fields $\frac{1}{{2_\varepsilon }}\smallint _{ - \varepsilon }^\varepsilon u_i^\varepsilon g^{i,\varepsilon } dx_3^\varepsilon$ and $\zeta _i^\varepsilon$ a ⁱ , both defined on the surfaceS, have the same principal part as? → 0, inH ¹ (ω) for the tangential components, and inL ²(ω) for the normal component; under the same assumptions as in Part II, we show that the same fields again have the same principal part as? → 0, inH ¹ (ω) for all their components. For “membrane” and “flexural” shells, the two-dimensional model ofW.T. Koiter is therefore justified.     

Eulerian–Lagrangian bridge for the energy and dissipation spectra in isotropic turbulence

We study, numerically and analytically, the relationship between the Eulerian spectrum of kinetic energy, E _E(k, t), in isotropic turbulence and the corresponding Lagrangian frequency energy spectrum, E _L(ω, t), for which we derive an evolution equation. Our DNS results show that not only E _L(ω, t) but also the Lagrangian frequency spectrum of the dissipation rate ${\varepsilon_{\rm L} (\omega, t)}$ has its maximum at low frequencies (about the turnover frequency of energy-containing eddies) and decays exponentially at large frequencies ω (about a half of the Kolmogorov microscale frequency) for both stationary and decaying isotropic turbulence. Our main analytical result is the derivation of equations that bridge the Eulerian and Lagrangian spectra and allow the determination of the Lagrangian spectrum, E _L (ω) for a given Eulerian spectrum, E _E (k), as well as the Lagrangian dissipation, ${\varepsilon_{\rm L}(\omega)}$ , for a given Eulerian counterpart, ${\varepsilon_{\rm E} (k)=2\nu k^2 E_{\rm E}(k)}$ . These equations were derived from the Navier–Stokes equations in the sweeping-free coordinate system (intermediate between the Eulerian and Lagrangian frameworks) which eliminates the effect of the kinematic sweeping of the small eddies by the larger eddies. We show that both analytical relationships between E _L (ω) and E _E (k) and between ${\varepsilon_{\rm L} (\omega)}$ and ${\varepsilon_{\rm E} (k)}$ are in very good quantitative agreement with our DNS results and explain how ${\varepsilon_{\rm L} (\omega, t)}$ has its maximum at low frequencies and decays exponentially at large frequencies.     

A moving-load controlled-displacement fracture-toughness testing machine

In conventional fracture-toughness testing, the line of application of the loads remains fixed with respect to specimen geometry. In this testing machine, the load moves with the advancing crack front, and displacement is used as the controlled variable to propagate and arrest a crack. The energy-release rate at the onset of crack propagation and, hence, the plane-strain fracture toughnessK _Ic can be measured directly without compliance calibration or stress-intensity evaluation. The specimen is in the form of a flat plat 25 by 50 cm which is simple to machine and provides about 30 values ofK _Ic. The versatility of the machine is demonstrated by making a statistical analysis ofK _Ic for 7075-T6 Al by showing the effect of plate thickness on the fracture toughnessK _c using a tapered specimen, and by evaluatingK _c in 7075 Al as a function of aging temperature in a thermal-gradient-treated specimen.     

Anomalies associated with energy release parameters for cracks in piezoelectric materials

For a central crack in a piezoelectric plate, the mode-I stress intensity factor (K_I), electric displacement intensity factor (K_D), energy release rates (G, G_M) and energy density factor (S) are obtained from the finite element results. For the impermeable crack, the numerical results of K_I and K_D are coupled; this error is contrary to the uncoupled analytical solutions. The error has little effect on the total energy release rate G and energy density factor S, but in some cases, large errors in the mechanical energy release rate G_M are observed. G is global while SED is local. Also G is negative which defies physics where energy cannot be created while crack attempts to extend as implied by G. Computations should be made for the J-integral and also show that J becomes negative. What this shows is that the global fracture energy criterion is not suitable to address the local release of energy because it includes the overall energy which are irrelevant to fracture initiation being a local behavior. In addition, the case study shows that the energy density theory is the better fracture criterion for the piezoelectric material. According to the results of S, it retards the crack growth when the external electric field and piezoelectric poling are on opposite directions. This conclusion agrees with analytical and experimental evidence in the past references.     

The Problems of the Turbulent Burning Velocity

The paper reviews the practical problems in measuring a turbulent burning velocity that gives the mass rate of burning. These largely centre on identifying an appropriate flame surface to associate with the turbulent burning velocity, u _t , and the density of the unburned mixture. Such a flame surface has been identified, in terms of the mean reaction progress variable, $\bar {c}$ , for explosive flame propagation in a fan-stirred bomb. Measurement of $\bar {c}$ makes possible an estimation of the flame surface density, ??, from the relationship ${\it \Sigma} =k\bar {c}\left( {1-\bar {c}} \right)$ . It is shown that in such explosions, mass rates of burning derived from the measured total flame surface area agreed well with those found from the measured turbulent burning velocity. Flamelet considerations identify appropriate dimensionless correlating parameters for u _t . As a result, correlations of turbulent burning velocity divided by the effective rms turbulent velocity, are plotted against the turbulent Karlovitz stretch factor, K, for different values of the Markstein number for flame strain rate, Ma_sr. These plots cover a wide range of variables, including pressure and fuels, and are indicative of different regimes of turbulent combustion. At the lower values of K, there is some evidence of increases in u _t and k due to high-frequency flame surface wrinkling arising from flame instabilities. These increase as Ma_sr becomes more negative. It is found from the developed value of the mean flame surface density throughout the flame brush that, to a first approximation, an increase in u _t for a given mixture is accompanied by a proportional increase in the volume of the brush. The analysis shows that the volume fraction of the turbulent flame brush that is reacting is quite small.     

Microcracking and Fracture Process in Cement Mortar and Concrete: A Comparative Study Using Acoustic Emission Technique

This article reports the acoustic emission (AE) study of precursory micro-cracking activity and fracture behaviour of quasi-brittle materials such as concrete and cement mortar. In the present study, notched three-point bend specimens (TPB) were tested under crack mouth opening displacement (CMOD) control at a rate of 0.0004 mm/sec and the accompanying AE were recorded using a 8 channel AE monitoring system. The various AE statistical parameters including AE event rate $ \left( {\frac{dn }{dt }} \right) $ , AE energy release rate $ \left( {\frac{dE }{dt }} \right) $ , amplitude distribution for computing the AE based b-value, cumulative energy (ΣE) and ring down count (RDC) were used for the analysis. The results show that the micro-cracks initiated and grew at an early stage in mortar in the pre peak regime. While in the case of concrete, the micro-crack growth occurred during the peak load regime. However, both concrete and mortar showed three distinct stages of micro-cracking activity, namely initiation, stable growth and nucleation prior to the final failure. The AE statistical behavior of each individual stage is dependent on the number and size distribution of micro-cracks. The results obtained in the laboratory are useful to understand the various stages of micro-cracking activity during the fracture process in quasi-brittle materials such as concrete & mortar and extend them for field applications.     

Finite-dimensional asymptotic behavior of some semilinear damped hyperbolic problems

We prove that the solution semigroup $$S_t \left[ {u_0 ,v_0 } \right] = \left[ {u(t),u_t (t)} \right]$$ generated by the evolutionary problem $$\left\{ P \right\}\left\{ \begin{gathered} u_{tt} + g(u_t ) + Lu + f(u) = 0, t \geqslant 0 \hfill \\ u(0) = u_0 , u_t (0) = \upsilon _0 \hfill \\ \end{gathered} \right.$$ possesses a global attractorA in the energy spaceE _o=V×L ²(Ω). Moreover,A is contained in a finite-dimensional inertial setA attracting bounded subsets ofE ₁=D(L)×V exponentially with growing time.     

SIF-based fracture criterion for interface cracks

The complex stress intensity factor K governing the stress field of an interface crack tip may be split into two parts, i.e.,■ and s~(-iε), so that K = ■ s~(-iε), s is a characteristic length and ε is the oscillatory index. ■ has the same dimension as the classical stress intensity factor and characterizes the interface crack tip field. That means a criterion for interface cracks may be formulated directly with■, as Irwin(ASME J. Appl. Mech. 24:361–364, 1957) did in 1957 for the classical fracture mechanics. Then, for an interface crack,it is demonstrated that the quasi Mode I and Mode II tip fields can be defined and distinguished from the coupled mode tip fields. Built upon SIF-based fracture criteria for quasi Mode I and Mode II, the stress intensity factor(SIF)-based fracture criterion for mixed mode interface cracks is proposed and validated against existing experimental results.     

Integrability of the constrained rigid body

The integrability theory for the differential equations, which describe the motion of an unconstrained rigid body around a fixed point is well known. When there are constraints the theory of integrability is incomplete. The main objective of this paper is to analyze the integrability of the equations of motion of a constrained rigid body around a fixed point in a force field with potential U(γ)=U(γ ₁,γ ₂,γ ₃). This motion subject to the constraint 〈ν,ω〉=0 with ν is a constant vector is known as the Suslov problem, and when ν=γ is the known Veselova problem, here ω=(ω ₁,ω ₂,ω ₃) is the angular velocity and 〈?,?〉 is the inner product of $\mathbb{R}^{3}$ . We provide the following new integrable cases. (i) The Suslov’s problem is integrable under the assumption that ν is an eigenvector of the inertial tensor I and the potential is such that $$U=-\frac{1}{2I_1I_2}\bigl(I_1\mu^2_1+I_2 \mu^2_2\bigr), $$ where I ₁,I ₂, and I ₃ are the principal moments of inertia of the body, μ ₁ and μ ₂ are solutions of the first-order partial differential equation $$\gamma_3 \biggl(\frac{\partial\mu_1}{\partial\gamma_2}- \frac{\partial\mu_2}{\partial \gamma_1} \biggr)- \gamma_2\frac{\partial \mu_1}{\partial\gamma_3}+\gamma_1\frac{\partial\mu_2}{\partial \gamma_3}=0. $$ (ii) The Veselova problem is integrable for the potential $$U=-\frac{\varPsi^2_1+\varPsi^2_2}{2(I_1\gamma^2_2+I_2\gamma^2_1)}, $$ where Ψ ₁ and Ψ ₂ are the solutions of the first-order partial differential equation where $p=\sqrt{I_{1}I_{2}I_{3} (\frac{\gamma^{2}_{1}}{I_{1}}+\frac{\gamma^{2}_{2}}{I_{2}}+ \frac{\gamma^{2}_{3}}{I_{3}} )}$ . Also it is integrable when the potential U is a solution of the second-order partial differential equation where $\tau_{2}=I_{1}\gamma^{2}_{1}+I_{2}\gamma^{2}_{2}+I_{3}\gamma^{2}_{3}$ and $\tau_{3}=\frac{\gamma^{2}_{1}}{I_{1}}+\frac{\gamma^{2}_{2}}{I_{2}}+ \frac{\gamma^{2}_{3}}{I_{3}}$ . Moreover, we show that these integrable cases contain as a particular case the previous known results.     

Simultaneous measurements of stress intensity and toughness for fast-running cracks in steel

Simultaneous measurements of the dynamicstress-intensity factorK _I ^dyn and the dynamic-fracture toughnessK _ID were made in a high-strength steel to investigate the relation between energy delivered to and energy absorbed by rapidly propagating cracks. Values ofK _I ^dyn were obtained intermittently during the propagation history by the shadow optical method of caustics from high-speed photographs of the moving crack tips. Values ofK _ID were calculated from temperature maxima recorded by thermocouples near the crack path. The results indicate that for fast-running cracks, the change in energy available at the crack tip can be significantly less than the energy absorbed in crack extension, suggesting that currently used dynamic-energy-balance methods for determining dynamic-fracture toughnesses may provide erroneous values.     

Nondispersive solutions to the L 2-critical Half-Wave Equation

We consider the focusing L ²-critical half-wave equation in one space dimension, $$i \partial_t u = D u - |u|^2 u$$ , where D denotes the first-order fractional derivative. Standard arguments show that there is a critical threshold ${M_{*} > 0}$ such that all H ^1/2 solutions with ${\|u\|_{L^2} < M_*}$ extend globally in time, while solutions with ${\|u\|_{L^2} \geq M_*}$ may develop singularities in finite time. In this paper, we first prove the existence of a family of traveling waves with subcritical arbitrarily small mass. We then give a second example of nondispersive dynamics and show the existence of finite-time blowup solutions with minimal mass ${\|u_0\|_{L^2} = M_*}$ . More precisely, we construct a family of minimal mass blowup solutions that are parametrized by the energy E ₀ > 0 and the linear momentum ${P_0 \in \mathbb{R}}$ . In particular, our main result (and its proof) can be seen as a model scenario of minimal mass blowup for L ²-critical nonlinear PDEs with nonlocal dispersion.     

On Saint-Venant’s Problem with Concentrated Loads

The classical Saint-Venant problem is to find a solution of the traction problem of elastostatics in a finite cylinder ?? loaded over its bases. We prove that the problem has a unique solution for equilibrated surface forces $\hat{ \boldsymbol { s}}\in W^{-1,q}(\partial\Omega)$ , with q??(2?? ₀,+??) for some positive ? ₀ depending on ??. Hence $\hat{ \boldsymbol { s}}$ can model force acting on ???, concentrated on sets of zero Lebesgue surface measure of ???. Moreover, if $\hat{ \boldsymbol { s}}$ is equilibrated on each basis, we give a simple proof of the Toupin estimate expressing Saint-Venant??s principle.     

Advective Transport in Heterogeneous Formations: The Impact of Spatial Anisotropy on the Breakthrough Curve

Water flow and solute transport take place in formations of spatially variable conductivity K. The logconductivity Y?= ln K is modeled as a random stationary space function, of normal univariate pdf (of mean In K _G and variance ${\sigma_{Y}^{2}}$ ) and of axisymmetric autocorrelation of integral scales I _h,I _v (anisotropy ratio f?=?I _v/I _h?<?1). The head gradient and the velocity are uniform in the mean, parallel to bedding, and of constant and given as J and U, respectively. Transport is ruled by advection, which typically overwhelms pore scale dispersion in the breakthrough curve (BTC) determination. In the present study we analyze the impact of anisotropy f on the BTC of a passive solute, which is related to the mass flux??? (t, x) at a control plane at x. While a considerable body of literature dealt with weakly heterogeneous formations ( ${\sigma _{Y}^{2} <1 }$ ), the present study addresses the case of ${\sigma _{Y}^{2} >1 }$ , which is of interest for many aquifers and is more difficult to solve either numerically or by approximations. We approach the three dimensional problem by modeling the structure as an ensemble of densely packed oblate spheroids of semi-major and semi-minor axis R and f R, respectively, and independent lognormal K, submerged in a matrix of uniform conductivity K _ef, the effective conductivity of the ensemble. The detailed numerical simulations of transport show that the BTC is insensitive to the value of the anisotropy ratio f, i.e.,??? (t, x) I _h/U depends only on ${\sigma _{Y}^{2}}$ (except for small differences in the tail). This important result implies that transport, as quantified by BTCs or spatial longitudinal mass distributions, can be modeled accurately by the much simpler solutions developed in the past for isotropic media, like e.g., the semi-analytical self-consistent approximation.     

A bar impact tester for dynamic fracture testing of ceramics and ceramic composites

A bar impact test was developed to study the dynamic fracture responses of precracked ceramic bars, Al₂O₃ and 15/29-percent volume SiC_w/Al₂O₃. Crack-opening displacement was measured with a laser-interferometric displacement gage and was used to determine the crack velocity and the dynamic stress-intensity factorK _I ^dyn . The crack velocity andK _I ^dyn increased with increasing impact velocity while the dynamic-initiation fracture toughness,K _Id, did not vary consistently with increasing impact velocities.Paper was presented at the 1992 SEM Spring Conference on Experimental Mechanics held in Las Vegas on June 8–11.     

An experimental analysis of displacement field of mixed mode crack

In this paper, the characteristic properties ofv (y-direction displacement) field surrounding the tip of a mixed mode crack are studied. These properties can be used to evaluate the rigid body rotation of the crack tip, theK _I SIF and the ratio ofK _II SIF toK _I.The authors employ a film to record the displacement information based on the technique of moire interferometry with sticking films. By using the data taken from the moire pattern and treating them with the damping least square method, all of the parameters of the crack can be obtained accurately.     

Dynamic crack curving—A photoelastic evaluation

A dynamic-crack-curving criterion, which is valid under pure Mode I or combined Modes I and II loadings and which is based on either the maximum circumferential stress or minimum strain-energy-density factor at a reference distance ofr ₀ from the crack tip, is verified with dynamic-photoelastic experiments. Directional stability of a Mode I crack propagation is attained when \(r_o= \frac{1}{{128\pi }}(\frac{{K_r }}{{\sigma _{ox} }})^2 V_o^2 (C,C_1 ,C_2 ) > r_c \) wherer _c =1.3 mm for Homalite-100 used in the dynamic-photoelastic experiments.     

Asymptotic analysis of linearly elastic shells. I. Justification of membrane shell equations

We consider a family of linearly elastic shells with thickness 2?, clamped along their entire lateral face, all having the same middle surfaceS=φ(?ω) ?R ³, whereω ?R ² is a bounded and connected open set with a Lipschitz-continuous boundaryγ, andφ ∈l ³ ( $\overline \omega$ ;R ³). We make an essential geometrical assumption on the middle surfaceS, which is satisfied ifγ andφ are smooth enough andS is “uniformly elliptic”, in the sense that the two principal radii of curvature are either both>0 at all points ofS, or both<0 at all points ofS. We show that, if the applied body force density isO(1) with respect to?, the fieldtu(?)=(u _i(?)), whereu _i (?) denote the three covariant components of the displacement of the points of the shell given by the equations of three-dimensional elasticity, one “scaled” so as to be defined over the fixed domain Ω=ω×]?1, 1[, converges inH ¹(Ω)×H ¹(Ω)×L ²(Ω) as?→0 to a limitu, which is independent of the transverse variable. Furthermore, the averageξ=1/2ε _?1 ¹ u dx ₃, which belongs to the space $$V_M (\omega ) = H_0^1 (\omega ) \times H_0^1 (\omega ) \times L^2 (\omega ),$$ satisfies the (scaled) two-dimensional equations of a “membrane shell” viz., $$\mathop \smallint \limits_\omega a^{\alpha \beta \sigma \tau } \gamma _{\sigma \tau } (\zeta )\gamma _{\alpha \beta } (\eta ) \sqrt \alpha dy = \mathop \smallint \limits_\omega \left\{ {\mathop \smallint \limits_{ - 1}^1 f^i dx_3 } \right\}\eta _i \sqrt a dy$$ for allη=(η _i) εV _M(ω), where $a^{\alpha \beta \sigma \tau }$ are the components of the two-dimensional elasticity tensor of the surfaceS, $$\gamma _{\alpha \beta } (\eta ) = \frac{1}{2}\left( {\partial _{\alpha \eta \beta } + \partial _{\beta \eta \alpha } } \right) - \Gamma _{\alpha \beta }^\sigma \eta _\sigma - b_{\alpha \beta \eta 3} $$ are the components of the linearized change of metric tensor ofS, $\Gamma _{\alpha \beta }^\sigma$ are the Christoffel symbols ofS, $b_{\alpha \beta }$ are the components of the curvature tensor ofS, andf ⁱ are the scaled components of the applied body force. Under the above assumptions, the two-dimensional equations of a “membrane shell” are therefore justified.     

Characterization of crack-tip strain singularity in brittle-microcracking materials by means of the photoelastic-coating method

The strain fields ahead of crack tips in rock, mortar, and graphite were measured using a photoelastic coating method. A transparent ferroelectric ceramic was used as a coating material to observe the photoelastic fringe pattern. A coating plate of 110–150 μm thick was placed on single-edge-notch specimens, and photoelasticity experiments were conducted in three-point bending under a polarizing microscope. The results show that the principal-strain difference ahead of the crack tip is given by $$\Delta \in = \Delta \in _o [(J/\sigma _{ult} )/r]^m $$ whereσ _ult is the ultimate tensile strength,r is the distance from the crack tip, and9? _o andm are material constants. Based on this observation, the use of theJ _Ic concept in determining the fracture toughness of brittle-microcracking materials is discussed.