Stress wave fields in plates weakened by curvilinear holes with edge cracks

Malezhik, M. P.; Malezhik, O. P.; Zirka, A. I.; Chernyshenko, I. S.

doi:10.1007/s10778-006-0075-2

Cited by 6 publications

(9 citation statements)

References 7 publications

(9 reference statements)

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“…It should be noted that if problems are solved in a physically nonlinear formulation for not only plates, but also shells with a hole, then, as shown in [10][11][12][13], successive approximations (solutions) decay with distance from the hole.…”

Section: Introductionmentioning

confidence: 99%

Stress state of flexible plates with a hole that are subject to bending

Kayuk

2007

Int Appl Mech

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539.3This paper proposes and mathematically proves a method to regularize singular iterations used to solve geometrically nonlinear problems of bending of thin plates with a hole. It is established that the new approximations for deflections and stress functions satisfy the boundary conditions exactly and Karman's equations asymptotically exactly. The regularization method and the approximations for solutions are based on new methods of summing functional series and sequences Introduction. A linear solution to stress problems for thin plates with holes bent by moments at infinity was obtained by Savin in the monograph [7]. An attempt to solve these problems in a geometrically nonlinear formulation was made in [2][3][4]. It was proposed to expand the external moment load in powers of a dimensionless parameter.It appeared that the values of iterations at infinite points of the plate increase, especially with increasing number of approximation, etc. Therefore, the need arose to develop special methods for the regularization (summation) of series in a parameter.It should be noted that if problems are solved in a physically nonlinear formulation for not only plates, but also shells with a hole, then, as shown in [10-13], successive approximations (solutions) decay with distance from the hole.Issues of theoretical importance to be resolved are associated with more complete proof of the new regularization (summation) method. These are the permanence (continuity) of the summation methods and the satisfaction of boundary conditions and equilibrium equations, etc. by special approximations for desired solutions. We will consider these issues for the case where the problem posed is solved by the simple iteration method rather than the parameter expansion method.1. Problem Formulation. Mechanical Subject of Study. Let us consider an undeformed (configuration C 0 ) homogeneous isotropic rectangular plate of thickness h with an arbitrary curvilinear hole with boundary G 1 and no corner points. Assume that the external boundary G 2 is rather far from G 1 .Let distributed bending moments of constant intensity M be specified on a section of the boundary G 2 . Figure 1 shows a rectangular plate with a hole whose parallel faces experience the action of bending moments.In C 0 , the median surface p is described by curvilinear orthogonal coordinates a 1 and a 2 . We also assume that the boundary of the hole coincides with one of the coordinate lines, say a 2 (Fig. 1).The bending moments acting on G 2 cause the plate to deform, i.e., to go over to a configuration C t . If the deflections are small, i.e. w max / h < 1, then the stress state of the bent plate is described as a linear boundary-value problem for the deflection function. The perturbation of the state caused by the hole is described by a homogeneous boundary-value problem for the biharmonic equation. The characteristics of this state are then used to determine the concentration of moments near the boundary G 1 .

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Section: Introductionmentioning

confidence: 99%

Stress state of flexible plates with a hole that are subject to bending

Kayuk

2007

Int Appl Mech

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“…1); p and q are the principal axes of orthotropy. Considering the available developments and results [5,[16][17][18][19][20], we will study the diffraction of these stress waves by the hole.We used the dynamic photoelastic method [8] to test model plates ( Fig. 1) with the following dimensions (in mm): à = 300, b = 120, h = 3, c 1 90 = , c 2 110 = , where a, b, and h are the length, width, and thickness of the plates; ñ i (³ = 1, 2) is the distance from the center of the hole to the upper and left edges of the plate; r = r 0 is the radius of the hole (Fig.…”

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Diffraction of stress waves by a free or reinforced hole in an orthotropic plate

2007

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Photoelastic plates made of an orthotropic material are used to model the dynamic stress state near free and reinforced circular holes under blast loading. The diffraction of stress waves by holes in a thin-walled plate is studied. Experimental data are used to analyze the dynamic stress concentration in a plate with a hole in which quasilongitudinal and quasitransverse waves propagate Keywords: dynamic photoelasticity, orthotropic plate, blast loading, free and reinforced circular holes, dynamic stress concentration Introduction. The stress concentration near holes of different shapes induced by longitudinal waves of different durations was studied in [12,14,15] using the photoelastic method. In some experiments, the effect of a transverse wave on the hole was also taken into account, but the experiment parameters were such that the stress state around it was determined by both transverse and longitudinal waves. The diffraction of transverse waves by a circular hole was experimentally studied only in [5], which is due to the difficulty of obtaining a pure S-wave. The diffraction of elastic waves in multiply connected bodies is addressed in [2]; and some dynamic problems for structural members are solved in [3,10].Data on the distribution of dynamic stresses around a mine working in a rock mass are of importance for the strength analysis of underground structures subject to blast loads [7,11,13]. Different waves superimposed at the obstacle produce a complicated stress pattern varying in space and time, which gives rise to certain difficulties in solving linear stationary diffraction problems for waves in isotropic and reinforced multiply connected bodies [7,11,13]. To study wave stress fields in bodies of complex geometry, both a general approach and an efficient practical procedure are needed. Note that in modeling a rock mass with a mine working, natural anisotropy should be taken into account [1].The present paper analyzes the dynamic stress concentration around free and reinforced holes in thin anisotropic plates in which plane quasilongitudinal and quasitransverse waves propagate. The wave stress field near the boundary of an anisotropic plate subject to an impulsive load is studied in [6].1. Problem Formulation and Experiment Description. Consider a thin rectangular plate with a circular hole free or reinforced with a ring of certain shape. The plate is made of an orthotropic material with given mechanical characteristics and is subjected to a blast load that generates plane quasilongitudinal P-wave and plane quasitransverse S-wave (Fig. 1); p and q are the principal axes of orthotropy. Considering the available developments and results [5,[16][17][18][19][20], we will study the diffraction of these stress waves by the hole.We used the dynamic photoelastic method [8] to test model plates ( Fig. 1) with the following dimensions (in mm): à = 300, b = 120, h = 3, c 1 90 = , c 2 110 = , where a, b, and h are the length, width, and thickness of the plates; ñ i (³ = 1, 2) is the distance from the center of the h...

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“…Results on the nonstationary behavior of punched thin plates are given in [4,9]. The photoelastic method was used in [11,12] to study the wave stress field in a plate with a circular hole and in a plate with a curvilinear hole with an edge crack under an impulsive load.Here we will study the influence of a notch on the deformation of thin glass-fiber-reinforced plastic plates under the action of a normally incident shock wave described by a step function. A special technique will be developed.…”

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Nonstationary deformation of an elastic plate with a notch under the action of a shock wave

2007

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The behavior of a thin elastic plate with a rectilinear notch under the action a weak shock wave in air is studied experimentally. A technique is developed for this purpose. The effect of the notch on the strain state of the plate is analyzed Keywords: elastic plate, rectilinear notch, shock wave, experiment technique Introduction. Structural members in the form of plates and shells with various special-purpose or process-induced inhomogeneities (holes, cutouts, notches, cracks etc.) are widely used in many areas of technology. This necessitates studying the influence of such inhomogeneities on the strain state, stability, and fracture of structural members. These issues are well covered in the scientific literature. For example, the influence of an edge crack and a hole on the fracture of plates under tension was examined in [8,10]. In [7], applied theories of shells are used to study the nonstationary dynamics of structurally inhomogeneous shell elements, and experimental results on the effect of shock waves on them are presented. Results on the nonstationary behavior of punched thin plates are given in [4,9]. The photoelastic method was used in [11,12] to study the wave stress field in a plate with a circular hole and in a plate with a curvilinear hole with an edge crack under an impulsive load.Here we will study the influence of a notch on the deformation of thin glass-fiber-reinforced plastic plates under the action of a normally incident shock wave described by a step function. A special technique will be developed. We will experimentally study the nonstationary deformation and analyze the qualitative effect of the notch on the strain state of the plate at the time of maximum deflection caused by the shock wave.1. Technique of Study. The dynamic tests are schematized in Fig. 1, which shows test plate 1with notch 2; shock wave front 3 propagating along shock tube 4 in the direction indicated by arrows; piezoelectric transducers 5, which are elements of a wavefront velocity sensor; oscilloscope 6 recoding the moments the wavefront passes the piezoelectric transducers 5; resistance strain gages 7; computer 8.The test plates are made of SF-1-150 glass-cloth-base laminate (GOST 10316-78) with elastic modulus Å = 2.6×10 10 Pa and density r =1.7 g/cm 3 . The experiments [13] showed that rod and flat structural members of this material do not change their elastic properties under repeated (up to several hundred times) action of weak shock waves causing strains of no larger than 3×10 -3 . Rectangular plates with thickness h = 2.5 mm and sides a = 140 mm and b = 210 mm were rigidly fixed on the short sides between two steel frames 10 mm thick each. The inside dimensions of the frames are 210´140 mm, which coincide with the effective cross-section of the diaphragm-type shock tube; the outside dimensions are 300´230 mm. The plates were fixed with screws and the restrained surfaces were glued with cold-setting epoxy adhesive. The gaps between the long sides of the test plate and the inside of the frame were 0.5 mm appro...

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Stress wave fields in plates weakened by curvilinear holes with edge cracks

Cited by 6 publications

References 7 publications

Stress state of flexible plates with a hole that are subject to bending

Stress state of flexible plates with a hole that are subject to bending

Diffraction of stress waves by a free or reinforced hole in an orthotropic plate

Nonstationary deformation of an elastic plate with a notch under the action of a shock wave

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