A SOLVING METHOD FOR A SYSTEM OF PARTIAL DIFFERENTIAL EQUATIONS WITH AN APPLICATION TO THE BENDING PROBLEM OF A THICK PLATE

A theorem of solving a system of linear non-homogeneous differential equations through integrating and adding its basic solutions is put forward and proved, the mathematical role and physical nature of the theorem is interpreted briefly. As an example, the theorem is applied to solve the problem of thermo-force bending of a thick plate.

Anisotropy of localized corrosion in 7050-T7451 Al alloy thick plate

The corrosion anisotropy of 7050-T7451 A1 alloy thick plate in NaCI solution was investigated by immersion tests, slow strain rate testing （SSRT） technique, potentiodynamic and anode polarization measurements, optical microscropy （OM） and scanning electron microscopy （SEM） observations. The results show that the thick plate exhibits severe corrosion anisotropy due to the microstructure anisotropy. The observations of immersion surfaces together with the analysis of polarization curves reveal that the differences of the corrosion morphologies on various sections in this material are mainly related to the area fraction of the remnant second phase, and higher area fraction displays worst corrosion resistance. The stress corrosion cracking （SCC） susceptibility of different directions relative to the rolling direction is assessed by SSRT technique, ranked in the order： S direction 〉 L direction 〉 T direction. The result show that the smaller the grain aspect ratio, the better the corrosion resistance to SCC.

FINITE ELEMENT MODELING OF MODERATELY THICK COMPOSITE PLATE WITH PIEZOELECTRIC SENSORS AND ACTUATORS

A rectangular finite element for laminated plate with bonded and/or embedded piezoelectric sensors and actuators is developed based on the variational principle and the first order shear deformation theory. The element has four-node, 20-degrees-of-freedom with one potential degree of freedom for each piezoelectric layer to represent the piezoelectric behavior. The higher order derivation of deflection is obtained by using the normal rotation expressions to take the effects of transverse shear deformation into considerations. The finite element can accurately simulate the deformation of both thin and moderately thick plates. A Fortran program is written and a number of benchmark tests are exercised to verify its effectiveness. Results are compared well with the existing data. The unbalanced composite with piezoelectric layers is then analyzed by using the model. Results show that the changes of the ratio between the thickness of positive angle layers and the negative angle layers have an effect on the deformation of the structure under the same electric loading.

Construction and solution of strain model along thickness of aluminum alloy plate under plastic deformation

A thickness strain model of aluminium alloy plate under plastic deformation,based on thin plate assumption was proposed.It is found that when ratio of stress fractions is constant during in-plane loading,ratios of strain components under various loading conditions are linearly related and these points of ratios form a η-η line.Under these simple loadings,strains in thickness direction can be easily calculated by the η-η line equation without integral and differential work.When the plate is under more complicated loading conditions,the thickness can be computed by the proposed optimization and piecewise calculation model.Validation computations indicate that the relative error of the results of the presented model is less than 0.75% compared with the proven theories and FE simulation.Therefore,the developed model can be applied to engineering calculation,e.g.pre-stretching analysis of aerospace aluminium thick plate,with acceptable accuracy.

A new hole-flanging method for thick plate by upsetting process

Flange height and lip accuracy are generally restricted by the formability of sheet metals in the conventional hole-flanging operation. A new hole-flanging process, named upsetting-flanging process, was proposed to obtain a more substantial flange from thick plate. The finite element method （FEM） with DEFORM was utilized to simulate the novel upsetting-flanging process and the influence of geometric parameters on the flange height was studied in details. A series of flanging experiments with A1050P-O were carried out to validate the FEM results, and the variations of Vicker hardness in the plate section were discussed. The results showed that the newly upsetting-flanging process revealed higher flange height and better lip accuracy than the conventional hole-flanging process, and the results between FEM simulations and experiments showed good agreement. Besides, the hardness of the plate around the flange part increases due to the work hardening after the upsetting-flanging process, which reveals better superiority in strength for the subsequent machining or assembling processes.

SCATTERING OF FLEXURAL WAVES AND DYNAMIC STRESS CONCENTRATIONS IN MINDLIN'S THICK PLATES WITH A CUTOUT

Using the complex variable method and conformal mapping,scat- tering of flexural waves and dynamic stress concentrations in Mindlin's thick plates with a cutout have been studied.The general solution of the stress problem of the thick plate satisfying the boundary conditions on the contour of cutouts is obtained. Applying the orthogonal function expansion technique,the dynamic stress problem can be reduced into the solution of a set of infinite algebraic equations.As examples, numerical results for the dynamic stress concentration factor in Mindlin's plates with a circular,elliptic cutout are graphically presented in sequence.

Relationship between microstructure and mechanical properties of 5083 aluminum alloy thick plate

The microstructure and mechanical properties of 105 mm thick 5083 aluminum alloy hot rolled plate were investigated by metallurgical microscope, scanning electron microscope and tensile testing machine, and three major characteristic problems in mechanical properties inhomogeneity were explained. The results show that the mechanical properties of the rolled plate are inhomogeneous along the thickness direction. From the surface to the center, the strength shows an inverted "N" shape change and the elongation presents a semi "U" shape change. Several similar structural units composed of long fibrous grains(LFG) and short fibrous grains bands(SFGB) exist in a special layer(Layer 2) adjacent to the surface. This alternating layered distribution of LFG and SFGB is conducive to improving the plasticity by dispersing the plastic deformation concentrated on the boundary line(BL) between them. However, their different deformability will cause the alternation of additional stresses during the hot rolling, leading to the strength reduction. The closer the location to the center of the plate is, the more likely the recovery rather than the recrystallization occurs. This is the possible reason for the unnegligible difference in strength near the central region(Layer 4 and Layer 5).

Ageing behavior of an Al-Zn-Mg-Cu alloy pre-stretched thick plate

The ageing behavior of a pre-stretched thick plate of Al-Zn-Mg-Cu alloy was systemically studied including one-step ageing, two-step ageing, and retrogression and reageing treatment （RRA）. One-step ageing of the alloy resulted in peak ultimate tensile strengths of 595 and 575 MPa after 22 and 6 h at 120 and 135°C, respectively. The strengthening phase in peak aged （T6 temper） alloy contained GP zones and the η′ phase predominantly. After two-step ageing, the electrical conductivity was increased markedly, but the pre-stretched thick plate sacrificed a great loss of strength. RRA treatment provided a method for maintaining the strength close to that obtained by T6 temper and for obtaining the high electrical conductivity close to that obtained by T7 temper; the ultimate tensile strength and electrical conductivity were 583 MPa and 21.0 MS/m, respectively. TEM analysis of T7 and RRA specimens revealed two types of precipitates that contributed to age strengthening i.e. the η′ and η phases.

Calculation and experimental study on heating temperature field of super-high strength aluminum alloy thick plate

Stepped heating treatment has been applied to aluminum alloy thick plate to improve the mechanical performance and corrosion resistance.Accurate temperature control of the plate is the difficulty in engineering application.The heating process,the calculation of surface heat transfer coefficient and the accurate temperature control method were studied based on measured heating temperature for the large-size thick plate.The results show that,the temperature difference between the surface and center of the thick plate is small.Based on the temperature uniformity,the surface heat transfer coefficient was calculated,and it is constant below300°C,but grows greatly over300°C.Consequently,a lumped parameter method(LPM)was developed to predict the plate temperature.A stepped solution treatment was designed by using LPM,and verified by finite element method(FEM)and experiments.Temperature curves calculated by LPM and FEM agree well with the experimental data,and the LPM is more convenient in engineering application.

Effect of non-isothermal retrogression and re-ageing on through-thickness homogeneity of microstructure and properties of Al-8Zn-2Mg-2Cu alloy thick plate

In order to improve the through-thickness homogeneity and properties of aviation aluminum alloy thick plate.The effect of heating-cooling retrogression and re-ageing on the performance of Al-8Zn-2Mg-2Cu alloy thick plate was investigated by hardness tests, electrical conductivity tests and transmission electron microscopy(TEM) observation.Results revealed that, during retrogression heating, the fine pre-precipitates in surface layer dissolve more and the undissolved η′ or η phases are more coarsened than that of center layer. During slow cooling after retrogression,precipitates continue coarsening but with a lower rate and the secondary precipitation occurs in both layers. Finer precipitates resulting from the secondary precipitation are more in surface. However, the coarsening and secondary precipitation behaviors are restrained in both layers under quick cooling condition. The electrical conductivity and through-thickness homogeneity of precipitates increases while the hardness decreases with cooling rate decreasing. After the optimized non-isothermal retrogression and re-ageing(NRRA) including air-cooling retrogression, the throughthickness homogeneity which is evaluated by integrated retrogression effects has been improved to 94%. The tensile strength, fracture toughness and exfoliation corrosion grade of Al-8Zn-2Mg-2Cu alloy plate is 619 MPa, 24.7 MPa·m^(1/2)and EB, respectively, which indicates that the non-isothermal retrogression and re-aging(NRRA) could improve the mechanical properties and corrosion resistance with higher through-thickness homogeneity.

ELASTIC DYNAMIC ANALYSIS OF MODERATELY THICK PLATE USING MESHLESS LRPIM

A meshless local radial point interpolation method （LRPIM） for solving elastic dy-namic problems of moderately thick plates is presented in this paper. The discretized system equation of the plate is obtained using a locally weighted residual method. It uses a radial basis function （RBF） coupled with a polynomial basis function as a trial function,and uses the quartic spline function as a test function of the weighted residual method. The shape function has the properties of the Kronecker delta function,and no additional treatment is done to impose essen-tial boundary conditions. The Newmark method for solving the dynamic problem is adopted in computation. Effects of sizes of the quadrature sub-domain and influence domain on the dynamic properties are investigated. The numerical results show that the presented method can give quite accurate results for the elastic dynamic problem of the moderately thick plate.

Anisotropy of mechanical properties of 2297-T87 Al–Li alloy thick plates

The tensile properties of 2297-T87 Al–Li alloy thick plates at different thickness position and in different direction were analyzed via tensile testing,optical microscopy(OM),X-ray diffraction(XRD),scanning electron microscopy(SEM),energy dispersive spectrometry(EDS),and transmission electron microscopy(TEM).Results indicated that the ultimate tensile strength(UTS)and yield strength(YS)of the alloy decreased firstly and then increased from the 1/8T position to the 1/2T position,whereas elongation to failure(Ef)decreased gradually such that its value along the rolling direction(RD)was higher than those along the transverse direction(TD)at the same thickness position.From the 1/8T position to the 3/8T position of the alloy,the UTS and YS along the TD were higher than those along the RD.At the 1/2T position of the alloy,the UTS,YS,and Ef along the RD were the highest,whereas those along the normal direction(ND)were the lowest.Microstructural observations further revealed that the anisotropy of tensile properties was related to grain morphology,crystal texture,second-phase particles,and Li atom segregation.

An innovative joint interface design for reducing intermetallic compounds and improving joint strength of thick plate friction stir welded Al/Mg joints

Friction stir welding of dissimilar Al/Mg thick plates still faces severe challenges, such as poor formability, formation of thick intermetallic compounds, and low joint strength. In this work, two joint configurations, namely inclined butt(conventional butt) and serrated interlocking(innovative butt), are proposed for improving weld formation and joint quality. The results show that a continuous and straight intermetallic compound layer appears at the Mg side interface in conventional butt joint, and the maximum average thickness reaches about 60.1 μm.Additionally, the Mg side interface also partially melts, forming a eutectic structure composed of Mg solid solution and Al_(12)Mg_(17) phase.For the innovative butt joint, the Mg side interface presents the curved interlocking feature, and intermetallic compounds can be reduced to less than 10 μm. The joint strength of innovative butt joint is more than three times that of conventional butt joint. This is due to the interlocking effect and thin intermetallic compounds in the innovative joint.

GENERALIZED VARIATIONAL PRINCIPLES FOR VISCOELASTIC THIN AND THICK PLATES WITH DAMAGE

From the constitutive model with generalized force fields for a viscoelastic body with damage, the differential equations of motion for thin and thick plates with damage are derived under arbitrary boundary conditions. The convolution-type functionals for the bending of viscoelastic thin and thick plates with damage are presented, and the corresponding generalized variational principles are given. From these generalized principles, all the basic equations of the displacement and damage variables and initial and boundary conditions can be deduced. As an example, we compare the difference between the dynamical properties of plates with and without damage and consider the effect of damage on the dynamical properties of plates.

AN ANALYSIS FOR DIFFRACTION OF FLEXURAL WAVES AND DYNAMIC STRESS CONCENTRATIONS IN THICK PLATES WITH A CAVITY

By using the complex variable method and conformal mapping, the diffraction of flexural waves and dynamic stress concentrations in thick plates with a cavity have been studied. A general solution of the stress problem of the thick plate satisfying the boundary conditions on the contour of an arbitrary cavity is obtained. By employing the orthogonal function expansion technique, the dynamic stress problem can be reduced to the solution of an infinite algebraic equation series. As an example, the numerical results for the dynamic stress concentration factor in thick plates with a circular, elliptic cavity are graphically presented. The numerical results are discussed.

Dynamic stress concentrations in thick plates with two holes based on refined theory

Based on complex variables and conformal mapping, the elastic wave scat- tering and dynamic stress concentrations in the plates with two holes are studied by the refined dynamic equation of plate bending. The problem to be solved is changed to a set of infinite algebraic equations by an orthogonM function expansion method. As examples, under free boundary conditions, the numerical results of the dynamic moment concen- tration factors in the plates with two circular holes are computed. The results indicate that the parameters such as the incident wave number, the thickness of plates, and the spacing between holes have great effects on the dynamic stress distributions. The results are accurate because the refined equation is derived without any engineering hypothese.

Formation mechanism of transverse crack in repair welding for thick plate of Q345 high strength steel

Transverse cracks occur usually in repair welding for thick plate of high strength steel. It needs multiple times of repair welding. The quality of production and deliver deadline will be influenced. Therefore, it is very significant to investigate the cause and control of transverse crack in repair welding. In this paper, both ends restraint crack experiment is developed to produce delay transverse crack for high strength steel. Metallographic results show that four types of cracks are found in repair welding metal zone and heat affected zone. Large chevron transverse cracks are found in repair welding zone. Lots of micro transverse cracks are found in inter-layer repair welding metal zone, root HAZ and two ends of repair welding individually. The distribution character and formation mechanism of the transverse crack are further analyzed through hardness testing and residual stress measurement.

Dynamic Design of Thick Orthotropic Cantilever Plates with Consideration of Bimoments

The paper is devoted to dynamic design of thick orthotropic cantilever plates by applying the bimoment theory of plates, which takes into account the forces, moments and bimoments;and the theory takes into account nonlinear law of displacements distribution in cross section of the plate. The methods for constructing bimoment theory are based on Hooke’s Law, three-dimensional equations of the theory of dynamic elasticity and the method of displacements expansion into Maclaurin series. The article gives the expressions to determine the forces, moments and bimoments. Bimoment theory of plates is described by two unrelated two-dimensional systems with nine equations in each. On each edge of the plate, depending on the type of fastening, nine boundary conditions are given. As an example, the solution of the problem of dynamic bending of thick isotropic and orthotropic plate under the influence of transverse dynamic loads in the form of the Heaviside function is given. The equations of motion of the plate are solved by numerical method of finite differences. The numerical results are obtained for isotropic and orthotropic plate. The graphs of changes of displacements and stresses of faces surfaces of the plate are presented. Maximum values of these displacements are found and analyzed. It is shown that by Timoshenko theory numerical values of stresses are much smaller compared to the ones obtained by bimoment theory of plates. Maximum numerical values of generalized displacements, forces, moments, and bimoments are obtained and presented in tabular form. The analysis of numerical results is done and the conclusions are drawn.

CONSTRUCTION OF RECTANGULAR DISPLACEMENT-BASED ELEMENT FOR THICK/THIN PLATES

In this paper,a method of constructing displacement-based elment for thick/thin plates is devel- oped by using the technique of generalized compatibility,and a rectangular displacement-based element with 12 degrees of freedom for thick/thin plates is presented.This method enjoys a good accuracy with simple formulation and is free of shear- locking as the thickness of the plate approaches zero.

Micro porosity and its effect on fatigue performance of 7050 aluminum thick plates

Micro porosity in aluminum alloys may contribute to fatigue life degradation, which can largely limit the application of alloys. Therefore, the fatigue life of a commercial 7050-T7451 thick plate and an experimental plate with different porosities was compared in this study. The X-ray computed tomography(XCT) was utilized to characterize the size, number density and spatial distribution of porosity inside various samples, and the fracture surface of fatigued specimens was compared by using scanning electron microscope(SEM). The results showed that the fatigue cracks prefer to initiate from constituent particles in the commercial alloy. Whereas the micro porosity is the predominant site for crack nucleation and subsequent failure in the experimental one. The presence of micro porosity in experimental7050-T7451 thick plate may reduce the fatigue life by an order of magnitude or more compared with the defect-free alloy. The pores close to sample surface are the main fatigue crack initiation site, among which larger and deeper pore leads to a shorter fatigue life. The crack initiation is also affected by the pore geometry and direction. Besides, the overall porosity inside the bulk can affect the crack propagation during fatigue tests.