Propagation of horizontally polarized transverse waves in a solid with a periodic distribution of cracks

The propagation of time-harmonic waves in a solid containing a periodic distribution of cracks is investigated in a two-dimensional configuration. The cracks are parallel to the x-axis, and their centers are located at positions x = md, y = lh(m, l = 0, ±1, ±2,…). The wave motion is polarized in the z-direction and propagates in the y-direction (normal to the cracks). The theory of Floquet or Bloch waves, together with an appropriate Green's function and the condition of vanishing traction on the crack faces leads to a system of singular integral equations, which provides the basis for the derivation of an exact dispersion equation. Numerical results are presented for the wave number as a function of the frequency. The frequency spectrum shows a pattern of passing and stopping bands. The exact results are compared with the frequency spectrum according to a simplified theory which considers the arrays of collinear cracks in the planes y = lh (l= 0 ±1, ±2,…) as planes of homogeneous transmission and reflection. Good agreement is observed between exact and approximate results.     

A Partially Built-in Plate under Uniform Load

A thin plate has the form of the infinite strip ?∞<x<∞, 0≤y≤aand has the edge y=abuilt-in. The edge y=0 has its right half 0<x<∞ built-in while the left half ?∞<x<0 is free. The whole plate is now subjected to a uniform load p ₀applied to its upper surface. What is the resulting deflection of the plate and what are the induced moment and shear resultants? We present a solution to this classical problem based on eigenfunction expansions. In the right and left halves of the strip, the deflection can be expanded as separate eigenfunction expansion series, but these are difficult to match across the line x=0 because of the singularity at (0,0) induced by the boundary conditions. We adopt the novel technique of expanding the field near the centre of the strip in its correct form as a series of Williams polar eigenfunctions, and then linking this expansion to the right and left eigenfunction expansions by using a special form of elastic reciprocity. These right and left reciprocity conditions give two infinite systems of linear equations satisfied by the polar expansion coefficients, and we prove that these equations are sufficient to determine these coefficients. Further applications of reciprocity give closed form expressions for the right and left eigenfunction expansion coefficients so that the whole solution is then determined. The method yields accurate results using small systems of linear equations. We present numerical results for the deflection of the plate and the induced moment and shear resultants.     

Mixed-mode interfacial adhesive strength of a thin film on an anisotropic substrate

The mixed-mode interfacial adhesion strength between a gold (Au) thin film and an anisotropic passivated silicon (Si) substrate is measured using laser-induced stress wave loading. Test specimens are prepared by bonding a fused silica (FS) prism to the back side of a 〈1 0 0〉 Si substrate with a thin silicon nitride (Si_xN_y) passivation layer deposited on the top surface. A high-amplitude stress wave is developed by pulsed laser ablation of a sacrificial absorbing layer on one of the lateral surfaces of the FS prism. Due to the negative non-linear elastic properties of the FS, the compressive stress wave evolves into a decompression shock with fast fall time. Careful selection of the incident angle between the pulse and the FS/Si interface generates a mode-converted shear wave in refraction, subjecting the Si_xN_y/Au thin film interface to dynamic mixed-mode loading, sufficient to cause interfacial fracture. A detailed analysis of the anisotropic wave propagation combined with interferometric measurements of surface displacements enables calculation of the interfacial stresses developed under mixed-mode loading. The mixed-mode interfacial strength is compared to the interfacial strength measured under purely tensile loading.     

Time-domain finite difference calculations for interaction of an ultrasonic wave with a surface-breaking crack

Four two-dimensional configurations are considered in this paper. The first two concern a homogeneous slab (0⩽y⩽H, −∞<x<∞), with a surface-breaking crack (x=0, 0⩽y⩽a), and without such a crack. The other two configurations concern semi-infinite slabs of different mechanical properties which are in welded contact over x=0, 0⩽y⩽H. One of these has a surface-breaking crack in the interface (x=0, 0⩽y⩽a), and the other has perfect contact over the whole interface. Results are presented for diffraction and corner reflection of an ultrasonic displacement pulse. Time-domain calculations have been carried out bu the use of the finite difference method. The results are presented as full-field snapshots of the displacement fields at specified times, and as time histories of the particle velocity at the midpoint of the transducer-specimen interface at x=−H, y=H.     

Scattering of sound by two parallel semi-infinite screens

A plane sound wave is incident upon two semi-infinite rigid plates, lying along y = 0, x > 0 and y = -h, x < 0, respectively, where (x, y) are two-dimensional Cartesian coordinates. The problem is formulated into a matrix Wiener-Hopf equation which is uncoupled by the introduction of an infinite sum of poles. The exact solution is then easily obtained in terms of the coefficients of the poles, where these coefficients are shown to satisfy a linear system of algebraic equations. The far-field solution is obtained and an asymptotic approximation to the total potential is determined in the limit as h, the plate spacing, becomes small compared to the wavelength of the incident wave. The algebraic system is solved numerically in this limit and the results are shown to agree with those obtained by the method of matched asymptotic expansions.     

Piston Stokes flow in a semi-infinite channelL'écoulement de Stokes autour d'un piston dans un conduit semi-infini

The piston flow is bounded by rigid walls at y=±1, x>0 and generated by the uniform translation of the end wall x=0. After Katopodes, Davis and Stone [3] constructed a solution in terms of biorthogonal eigenfunctions, Meleshko and Krasnopolskaya [1] used a variation of an asymptotic technique developed by Meleshko and Gomilko [2] to examine the pointwise convergence of the non-orthogonal series. However, they overlooked the nonuniqueness of their solution and the consequent solvability condition which is shown here to necessitate a minor modification without significant harm to their contribution. To cite this article: A.M.J. Davis, C. R. Mecanique 330 (2002) 457–459.     

Stress and finite-strain analysis of a circular ring under diametral compression

Using a moiré, large-strain analysis method, a complete solution is shown in this paper of the fields of strain and stress for a circular ring subjected to diametral compression between two flat platens. The isotheticsu andv, obtained using 1000-lines-per-inch gratings, were differentiated photographically by the shifting technique (moiré-of-moiré) to determine ?u/?x, ?v/?y, ?u/?y and ?v/?x. Using the exact finite strain-displacement relationship, the Eulerian strains ? _x ^E , ? _y ^E and γ _xy ^E were computed. From these, the principal Eulerian strains were obtained. These results were verified with the isochromatics obtained from a large-deformation photoelasticity analysis. The ring was made of a polyurethane rubber which exhibits a linear relationship between natural strain and a newly introduced concept of “natural stress”. The Eulerian strains were converted to natural strains, and from these natural stresses were computed using the newly developed concept. Results are presented graphically for the whole field of the ring.     

Wave propagation in elastic lattices subjected to a local harmonic loading. I. A quasi-one-dimensional problem

The anti-plane dynamics of infinite (?∞ < x < ∞, ?∞ < y < ∞) material-bond rectangular lattices subjected to a uniform monochromatic excitation of the x = 0 line nodes is studied. A quasi-one-dimensional model is formulated: the original lattice is considered as an infinite waveguide in the x-direction with periodically joined bonds bounded in the y-direction. In such a structure, the wave pattern consists of waves propagated along x-axis and standing waves along y-axis. Steady and unsteady processes are investigated. Dispersion relations are analyzed and resonance points are detected. A combined analytical–numerical approach is used to describe (i) the quasi-steady propagation of waves when the source frequency is within the pass-band, (ii) development of resonance waves, and (iii) percolation of perturbations to the periphery when the excitation frequency is within the stop-band. Long-wave and short-wave components of solutions are compared with those for a simplified 1D mass-spring lattice (MSL) model.     

On the energy conservation and critical velocities for the propagation of a ＂steady-shock＂ wave in a bar made of cellular material

The propagation of shock waves in a cellular bar is systematically studied in the framework of continuum solids by adopting two idealized material models, viz. the dynamic rigid, perfectly plastic, locking （D-R-PP-L） model and the dynamic rigid, linear hardening plastic, locking （D-R-LHP-L） model, both considering the effects of strain-rate on the material properties. The shock wave speed relevant to these two models is derived. Consider the case of a bar made of one of such material with initial length L 0 and initial velocity v i impinging onto a rigid target. The variations of the stress, strain, particle velocity, specific internal energy across the shock wave and the cease distance of shock wave are all determined analytically. In particular the ＂energy conservation condition＂ and the ＂kinematic existence condition＂ as proposed by Tan et al. （2005） is re-examined, showing that the ＂energy conservation condition＂ and the consequent ＂critical velocity＂, i.e. the shock can only be generated and sustained in R-PP-L bars when the impact velocity is above this critical velocity, is incorrect. Instead, with elastic deformation, strain-hardening and strain-rate sensitivity of the cellular materials being considered, it is appropriate to redefine a first and a second critical impact velocity for the existence and propagation of shock waves in cellular solids. Starting from the basic relations for shock wave propagating in D-R-LHP-L cellular materials, a new method for inversely determining the dynamic stress-strain curve for cellular materials is proposed. By using e.g. a combination of Taylor bar and Hopkinson pressure bar impact experimental technique, the dynamic stress-strain curve of aluminum foam could bedetermined. Finally, it is demonstrated that this new formulation of shock theory in this one-dimensional stress state can be generalized to shocks in a one-dimensional strain state, i.e. for the case of plate impact on cellular materials, by simply making proper replacements of the elastic and plastic constants.     

Multiple scattering of flexural waves in a semi-infinite thin plate with a cutout

The multiple scattering of flexural waves and dynamic stress concentration in a semi-infinite thin plate with a cutout are investigated, and the expressions of this problem are obtained. The analytical solutions of wave fields are expressed by employing the wave function expansion method and the expanded mode coefficients are solved by satisfying the boundary condition of the cutout. The image method is used to satisfy the traction free boundary condition of the plate. As an example, the numerical results of dynamic stress concentration factors are graphically presented and discussed. Numerical results show that the analytical results of the scattered waves and dynamic stress in semi-infinite plates are significantly different from those in infinite plates when the ratio of distance b/a is relatively little. In the region of low frequency and long wavelength, the maximum dynamic stress concentration factors occur on the illuminated side of the scattering body with θ = π, but not at the edge of the cutout with θ = π/2. As the incidence frequency increases (the wavelength becomes short), the dynamic stress on the illuminated side of the cutout decreases, however, the dynamic stress on the shadow side increases.     

Magneto elastic stress subjected to uniform magnetic field over a thin infinite plate containing an elliptical hole with an edge crack

Two dimensional solutions of the magnetic field and magneto elastic stress are presented for a magnetic material of a thin infinite plate containing an elliptical hole with an edge crack subjected to uniform magnetic field. Using a rational mapping function, each solution is obtained as a closed form. The linear constitutive equation is used for these analyses. According to the electro-magneto theory, only Maxwell stress is caused as a body force in a plate. In the present paper, it raises a plane stress state for a thin plate, the deformation of the plate thickness and the shear deflection. Therefore the magneto elastic stress is analyzed using Maxwell stress. No further assumption of the plane stress state that the plate is thin is made for the stress analysis, though Maxwell stress components are expressed by nonlinear terms. The rigorous boundary condition expressed by Maxwell stress components is completely satisfied without any linear assumptions on the boundary. First, magnetic field and stress analyses for soft ferromagnetic material are carried out and then those analyses for paramagnetic and diamagnetic materials are carried out. It is stated that those plane stress components are expressed by the same expressions for those materials and the difference is only the magnitude of the permeability, though the magnetic fields H_x, H_y are different each other in the plates. If the analysis of magnetic field of paramagnetic material is easier than that of soft ferromagnetic material, the stress analysis may be carried out using the magnetic field for paramagnetic material to analyze the stress field, and the results may be applied for a soft ferromagnetic material. It is stated that the stress state for the magnetic field H_x, H_y is the same as the pure shear stress state. Solutions of the magneto elastic stress are nonlinear for the direction of uniform magnetic field. Stresses in the direction of the plate thickness and shear deflection are caused and the solutions are also obtained. Figures of the magnetic field and stress distribution are shown. Stress intensity factors are also derived and investigated for the crack length.     

End reflections in a layered piezoelectric circular cylinder

End reflection phenomenon in a semi-infinitely long layered piezoelectric circular cylinder is constructed with modal data from a spectral decomposition of the differential operator governing its natural vibrations. These modal data consist of all propagating modes and edge vibrations and they constitute the basis for a wave function expansion of the reflection of waves arriving at the traction-free end of the cylinder. Without any other external stimulus, a passive reflection event occurs. This traction-free end condition is enforced at the Gaussian integration points over the end cross-section on the combination of incoming and reflected wave fields. Reflections due to monochromatic incoming axisymmetric (m = 0) and flexural (m = 1) waves are studied and two numerical examples are presented.For an incoming axisymmetric wave, there is a particular frequency that induces an end resonance, which is characterized by high (but finite) amplitudes of end displacements vis-a-vis those of neighboring (i.e., slightly different) frequencies. This phenomenon is illustrated in the two cylinder examples.It is possible to modify the passive reflection event by imposing some voltage distribution over the free end. For an oscillating end voltage that is out-of-phase with the incoming wave, it is possible to extract electrical energy from it, i.e., energy harvesting. Examples of such an oscillating voltage with a particular radial distribution are given, that illustrate the amount of extracted energy as a function of the frequency of the incident monochromatic wave.     

Phenomenological study of parabolic and spherical indentation of elastic-ideally plastic material

A phenomenological study of parabolic and spherical indentation of elastic ideally plastic materials was carried out by using precise results of finite elements calculations. The study shows that no “pseudo-Hertzian” regime occurs during spherical indentation. As soon as the yield stress of the indented material is exceeded, a deviation from the, purely elastic Hertzian contact behaviour is found. Two elastic–plastic regimes and two plastic regimes are observed for materials of very large Young modulus to Yield stress ratio, E/σ_y. The first elastic–plastic regime corresponds to a strong evolution of the indented plastic zone. The first plastic regime corresponds to the commonly called “fully plastic regime”, in which the average indentation pressure is constant and equal to about three times the yield stress of the indented material. In this regime, the contact depth to penetration depth ratio tends toward a constant value, i.e. h_c/h = 1.47. h_c/h is only constant for very low values of yield strain (σ_y/E lower than 5 × 10^?6) when aE¹/Rσ_y is higher than 10,000. The second plastic regime corresponds to a decrease in the average indentation pressure and to a steeper increase in the pile-up. For materials with very large E/σ_y ratio, the second plastic regime appears when the value of the non-dimensional contact radius a/R is lower than 0.01. In the case of spherical and parabolic indentation, results show that the first plastic regime exists only for elastic-ideally plastic materials having an E/σ_y ratio higher than approximately 2.000.     

Consequences of the Translation Invariance on the Darcy Free Convection Flow Past a Vertical Surface

It is shown that the governing equation for the stream function of the Darcy free convection boundary layer flows past a vertical surface is invariant under arbitrary translations of the transverse coordinate y. The consequences of this basic symmetry property on the solutions corresponding to a prescribed surface temperature distribution T _w (x) are investigated. It is found that starting with a “primary solution” which describes the temperature boundary layer on an impermeable surface, infinitely many “translated solutions” can be generated which form a continuous group, the “translation group” of the given primary solution. The elements of this group describe free convection boundary layer flows from permeable counterparts of the original surface with a transformed temperature distribution \({\tilde {T}_w \left( x \right)}\), when simultaneously a suitable lateral suction/injection of the fluid is applied. It turns out in this way that several exact solutions discovered during the latter few decades are in fact not basically new solutions, but translated counterparts of some formerly reported primary solutions. A few specific examples are discussed in detail.     

Use of equivalent body forces for acoustic emission from a crack in a plate

A method to determine acoustic emission of surface waves from a crack near the free edge of a plate, is presented, in terms of the function f(t), which defines the time dependence of the crack opening process, the crack opening volume per unit thickness of the plate, and the elastic constants of the plate. The determination of the time-varying displacement is based on the use of equivalent body forces, which are shown to be two double forces. The acoustic emission of the crack, or the equivalent radiation from the double forces, has been obtained by a novel use of the elastodynamic reciprocity theorem. It is of interest that the normal surface-wave displacement at a position x₀ of the free edge comes out as depending on df/dt evaluated at x₀ for t > x₀/c_R, where c_R is the velocity of surface waves on the free edge.     

Theoretical and experimental studies in fracture mechanics of crack-like defects under biaxial loading

It is a common point of view in fracture mechanics that, for any geometry of the body with a crack and any boundary conditions for the loading acting in the body plane, the stress and displacement components near the crack tip can be approximated in the framework of the theory of elasticity by a one-parameter or one-term representation, i.e., strictly in terms of the stress intensity coefficients K _I and K _II for an arbitrary failure crack [1, 2]. The authors of [2] specified the Westergaard function of the singular solution for a central crack under the biaxial loading of a plate. This approximate two-component solution has satisfactory accuracy. It is clear from [2] that this method cannot be admitted as a general statement [1], although it has long been assumed to be correct. The cause is that one cannot reasonably justify neglecting the second term in the Williams representation of the stress components in the plane case in the form of eigenfunction series; the contribution of this term in the rectangular coordinate system x, y is independent of the distance from the crack tip. This method may result in a serious mistake, from both the qualitative and quantitative viewpoints, in the prediction of local stresses, displacements, and related variables that are of interest. Apparently, this can best be demonstrated by an example of biaxial loading of a plate with a crack [1]. The unfounded neglect of the second term (whose contribution is independent of the distance from the crack tip) in the series representing the stress components is the source of the above-mentioned difficulties. In this problem, the influence of the load applied in the direction parallel to the crack plane manifests itself only in the second term of the series [3]. Therefore, this term should be clearly determined and studied in detail in the case of technological welding defects (faulty fusions, incomplete fusions, undercuts, and slag inclusions) and crack-like defects (scratches and cuts) in the base metal. The influence of the stress σ _OX along the crack axis on the stress tensor σ _x , σ _y , τ _xy and on the displacements u _x and u _y is confirmed by experimental studies of cracks by the photoelasticity method [4].     

A new analytical theory for earing generated from anisotropic plasticity

Commercial canmaking processes include drawing, redrawing and several ironing operations. It is experimentally observed that during the drawing and redrawing processes earing develops, but during the ironing processes earing is reduced. It is essential to understand the earing mechanism during drawing and ironing for an advanced material modeling. A new analytical approach that relates the earing profile to r-value and yield stress directionalities is presented in this work. The analytical formula is based on the exact integration of the logarithmic strain. The derivation is for a cylindrical cup under the plane stress condition based on rigid perfect plasticity while force equilibrium is not considered. The earing profile is obtained solely from anisotropic plastic properties in simple tension. The earing mechanism is explained from the present theory with explicit formulae. It has been proved that earing is the combination of the contributions from r-value and yield stress directionalities. From a directionality (y-axis) vs. angle from the rolling (x-axis) plot, the earing profile is generated to be a scaled mirror image of the r-value directionality with respect to 90° (x = 90) and also a scaled mirror image of the yield stress directionality with respect to the reference yield stress (y = 1). Three different materials (Al-5% Mg alloy, AA 2090-T3 and AA 3104 RPDT control coil) are considered for verification purposes. This approach provides a fundamental basis for understanding the earing mechanism. In practice, the present theory is also very useful for the prediction of the earing profile of a drawn and iron cup and its related convolute cut-edge design for an earless cup.     

The effect of multi-plate array spacing on grid-generated turbulence

This study compares measurements of the streamwise integral length scale, the root mean square (r.m.s.) of the streamwise component of velocity, and the r.m.s. of the normal component of velocity obtained at the exit of a plate array with measurements obtained at the same position for the “open pipe case”. The “open pipe case” is defined as the empty tube, without the plates in place, i.e., the apparatus becomes grid flow entering an unobstructed pipe. In general, this study finds that the length scale in the streamwise direction decreases with increasing plate spacing while the r.m.s. velocity in the streamwise direction increases as the plate spacing increases for fixed values of x/M (i.e., the streamwise direction to mesh-spacing ratio). These measured trends are consistent with a simple model based on vortex elements and conservation of angular momentum.     

Long Wave Run-up on Plane and “Non-Reflecting” Slopes

The long wave run-up on two types of slopes is investigated numerically within the framework of nonlinear shallow water theory using the CLAWPACK software. One of the slopes represents a plane slope widely used in the laboratory and numerical experiments; the second is the so-called “non-reflecting” slope (h ~ x^4/3, where h is the basin depth and x is the distance from the shoreline). In the case of very low wave amplitudes when there is no wave breaking, the run-up height is greater on the non-reflecting beach than that on the plane slope. As the wave amplitude increases, the breaking effects have the stronger impact in the case of non-reflecting beach and the run-up height becomes smaller.     

Planar Vector Fields with a Given Set of Orbits

We determine all the \({\mathcal{C}^1}\) planar vector fields with a given set of orbits of the form y ? y(x) = 0 satisfying convenient assumptions. The case when these orbits are branches of an algebraic curve is also study. We show that if a quadratic vector field admits a unique irreducible invariant algebraic curve \({g(x, y) = \sum_{j=0}^S a_j(x) y^{S-j}= 0}\) with S branches with respect to the variable y, then the degree of the polynomial g is at most 4S.