In this contribution,the microscopic fracture mechanism and extension criterion for mixed type crack in ductile material under plane mixed mode loading are investigated in details.A universal extension criterion for t...In this contribution,the microscopic fracture mechanism and extension criterion for mixed type crack in ductile material under plane mixed mode loading are investigated in details.A universal extension criterion for the mixed type crack,i.e.the crack propagates along the direction of the maximum gradient of equivalent stress,is suggested.This new criterion is used to predict the propagation direction of mixed type crack,showing a good agreement with other theories for different types of mode mixity.Moreover,the numerical verification is also carried out for the case of an edge crack with different mixed mode loadings.Finally,a potential application to three-dimensional fracture in the ductile material induced by holes is also discussed.展开更多
The aim of this work is to develop a three-dimensional model of deep groove ball bearing to investigate the loaded stresses and central displacements of bearing rings. The equivalent stresses and central displacements...The aim of this work is to develop a three-dimensional model of deep groove ball bearing to investigate the loaded stresses and central displacements of bearing rings. The equivalent stresses and central displacements of bearing rings are obtained based on the simulated analysis. Moreover, several parameters, such as load magnitude, raceway groove curvature radius(RGCR), thicknesses of outer and inner rings, are varied to investigate their effects on the equivalent stresses and central displacements of bearing rings. Research results provide useful guidelines for determining the design parameters.展开更多
Optimum utilization of the loading capability of engineering materials is an important and active contribution to protect nature's limited resources,and it is the key for economic design methods.In order to make u...Optimum utilization of the loading capability of engineering materials is an important and active contribution to protect nature's limited resources,and it is the key for economic design methods.In order to make use of the materials' resources,those must be known very well;but conventional test methods will offer only limited informational value.The range of questions raised is as wide as the application of engineering materials,and partially they are very specific.The development of huge computer powers enables numeric modelling to simulate structural behaviour in rather complex loading environments-so the real material behaviour is known under the given loading conditions.Here the art of material testing design starts.To study the material behaviour under very distinct and specific loading conditions makes it necessary to simulate different temperature ranges,loading speeds, environments etc.and mostly there doesn't exist any commonly agreed test standard.In this contribution two popular,non-standard test procedures and test systems will be discussed on the base of their application background,special design features as well as test results and typically gained information:The demand for highspeed tests up to 1000 s^(-1) of strain rate is very specific and originates primarily in the automotive industry and the answers enable CAE analysis of crashworthiness of vehicle structures under crash conditions.The information on the material behaviour under multiaxial loading conditions is a more general one.Multiaxial stress states can be reduced to an equivalent stress,which allows the evaluation of the material's constraint and criticality of stress state.Both discussed examples shall show that the open dialogue between the user and the producer of testing machines allows custom-tailored test solutions.展开更多
The finite element method(FEM) is used to analyze the effects of lead widths and pitches on reliability of soldered joints. The optimum simulation for QFP devices is also researched. The results indicate that when t...The finite element method(FEM) is used to analyze the effects of lead widths and pitches on reliability of soldered joints. The optimum simulation for QFP devices is also researched. The results indicate that when the lead pitches are the same, the maximum equivalent stress of the soldered joints increases with the increasing of lead widths, while the reliability of the soldered joints reduces. When the lead widths are the same, the maximum equivalent stress of the soldered joints doesn't decrease completely with the increasing of lead pitches, a minimum value of the maximum equivalent stress values exists in all the curves. Under this condition the maximum equivalent stress of the soldewed joints is relatively the least, the reliability of soldered joints is high and the assembly is excellent. The simulating results indicate the best parameter: The lead width is 0.2 mm and lead pitch is 0.3 mm (the distance between two leads is 0.1 mm), which are benefited for the micromation of QFP devices now. The minimum value of the maximum equivalent stress of soldered joints exists while lead width is 0.25 mm and lead pitch is 0.35 mm (the distance between two leads is 0.1 mm), the devices can serve for a long time and the reliability is the highest, the assembly is excellent. The simulating results also indicate the fact that the lead width is 0.15 mm and lead pitch is 0.2 mm maybe the limit of QFP, which is significant for the high lead count and micromation of assembly.展开更多
Three passes of plate rolling during vertical-horizontal rolling process are simulated with explicit dynamic finite element method and updating geometric method. The equivalent strain and stress fields, and shape chan...Three passes of plate rolling during vertical-horizontal rolling process are simulated with explicit dynamic finite element method and updating geometric method. The equivalent strain and stress fields, and shape change at the head and tail of slab during rolling are obtained. The calculated result of the shape at the head and tail of slab is in good agreement with the measured one. The explicit dynamic finite element method and updating geometric method can be used effectively to analyze the multipass vertical-horizontal (V-H) rolling process.展开更多
Under the surface peeling of Cu- Fe- P lead frame alloy larger Fe particles were observed by energy dispersive spectroscopy. By using the large strain two-dinension plane strain model and elastic plastic finite elemen...Under the surface peeling of Cu- Fe- P lead frame alloy larger Fe particles were observed by energy dispersive spectroscopy. By using the large strain two-dinension plane strain model and elastic plastic finite element method, the cause for peeling damage of Cu-Fe-P lead frame aUoy was investigated. The results show that when the content of Fe particles is more than 30% at local Fe-rich area the intense stress coacentration in the Fe particle would make the Fe particle broken up. The high equivalent stress mutation and the mismatch of equivalent strain 10% at the two sides of intefrace make it easy to develop the crack and peeling damage on finish rolling. The larger Fe particles in the Cu-Fe-P alloy should be avoided.展开更多
The maximum principal stress, von Mises equivalent stress, equivalent creep strain, stress triaxiality in dissimilar metal welded joints between austenitic(HR3C) and martensitic heat-resistant steel(T91) are simul...The maximum principal stress, von Mises equivalent stress, equivalent creep strain, stress triaxiality in dissimilar metal welded joints between austenitic(HR3C) and martensitic heat-resistant steel(T91) are simulated by FEM at 873 K and under inner pressure of 42.26 MPa. The results show that the maximum principal stress and von Mises equivalent stress are quite high in the vicinity of weld/T91 interface, creep cavities are easy to form and expand in the weld/T91 interface. There are two peaks of equivalent creep strains in welded joint, and the maximum equivalent creep strain is in the place 27-32 mm away from the weld/T91 interface, and there exists creep constrain region in the vicinity of weld/T91 interface. The high stress triaxiality peak is located exactly at the weld/T91 interface. Accordingly, the weld/T91 interface is the weakest site of welded joint. Therefore, using stress triaxiality to describe creep cavity nucleation and expansion and crack development is reasonable for the dissimilar metal welded joint between austenitic and martensitic steel.展开更多
The blade precision forging process is a forming process with high temperature and large plastic deformation. Interaction of deformation and heat conduction leads to large uneven distribution of temperature. The uneve...The blade precision forging process is a forming process with high temperature and large plastic deformation. Interaction of deformation and heat conduction leads to large uneven distribution of temperature. The unevenness of temperature distribution has a great effect on mechanical properties and the microstracture of materials. So it is necessary to consider the influence of temperature on the precision forging process of blades. Taking a blade with a tenon into consideration, a 3D mechanical model in precision forging is built up. The distribution laws of temperature field and the influence of the temperature on the equivalem stress in the process are obtained by using 3-D coupled thermo-mechanical FEM code developed by the authors Theresuits obtained illustrate that the influence of the temperature field on the blade forging process is considerable. The achievements of predicting microstructure and mechanical properties for forged blades is significant.展开更多
Grain growth behavior in a copper foil under nonproportional loading was investigated. The grain growth density decreased with increasing phase shift between normal stress and shear stress. Moreover, under this condit...Grain growth behavior in a copper foil under nonproportional loading was investigated. The grain growth density decreased with increasing phase shift between normal stress and shear stress. Moreover, under this condition, grains tended to grow in various directions. However, the relative frequency of the grain growth direction was slightly large in the maximum shear stress direction. Therefore, grain growth density is mainly dominated by the equivalent shear stress in Tresca theory in the case of nonproportional loading as well as proportional loading.展开更多
The axial compressive strength,axial tensile strength,elastic modulus,poisson ratio and stress-strain relationship of RPC are obtained by compressive tests and tensile tests of RPC prism.Mathematical models of axial c...The axial compressive strength,axial tensile strength,elastic modulus,poisson ratio and stress-strain relationship of RPC are obtained by compressive tests and tensile tests of RPC prism.Mathematical models of axial compressive and axial tensile stress-strain relationship are established,from which the equivalent coefficient of compressive and tensile stress diagram of cross-section in RPC beam are deduced.The results provide the theory basis for the design of RPC structure and the wider popularization of RPC.展开更多
The basic principle of corrode groove on outside of steel pipe during storage was analyzed in this paper, namely the water film on the contacted surface of steel pipe, which gathered from humidity in the air, rain or ...The basic principle of corrode groove on outside of steel pipe during storage was analyzed in this paper, namely the water film on the contacted surface of steel pipe, which gathered from humidity in the air, rain or gel, and the suspended particles in air, and the corrosive composition, such as SO2, CO2, O2 and NaCI, in addition to the inhomogeneity of the organization and composition, which lead to the corrosion cell reaction, so that cause the corrosion initial from the contact surface of the between steel pipes, so as to form the corrosion groove. At the same time, the corrosion groove with depth of 0.125t (t pipe wall thickness) on the pipe of φ 1016 mm×21 mm ×70 API SPEC 5L was simulated using the FEM (finite element method), and the stress and strain distribution of the defect area near corrosion groove were solved at the inner pressure of 12 MPa, 10 MPa, 8 MPa, 6 MPa, 4 MPa and 2 MPa, respectively, which showed that no matter the pressure values were, the maximum stress and strain were lied at the bottom of corrosion defects groove and were in good linear relationship with the internal pressure increasing from 2 MPa to 6 MPa. When the internal pres- sures were greater than 6 MPa, they felled into the nonlinear model and to be yielded or even to be destroyed. In addition, the residual strength and the limit operation pressure of the corrode pipe with the defects groove of 0.125t were calculated or simulated according to the theoretical calculation, the finite element method based on the stress, the finite element method based on strain, DNV-RP-F101, ASME B31G and experimental methods respectively. The results showed that the residual strength and the limit operation pressure of the defective parts solved by the finite element method based on stress were 424 MPa, and 15.34 MPa, respectively, which was very close to that of experimental method, the residual strength was 410 MPa and the limit operation pressure 14.78 MPa. Besides, the results also showed that it was feasible and effective to simulate the residual strength of the structure with corrosion defects using the finite element method.展开更多
In the centrifugal separations of drilling fluid,screw conveyor is a critical component to push and separate the sediment.The work performance and structural parameters of conveyor are immediately related to the produ...In the centrifugal separations of drilling fluid,screw conveyor is a critical component to push and separate the sediment.The work performance and structural parameters of conveyor are immediately related to the production capability,the working life and the separating effect of the centrifuge.The existing researches always use the theoretical calculation of the approximate loads to analyze the strength of conveyor,and it cannot reflect the stress situations accurately.In order to ensure the precise mastery of the working performance,this article obtained pressure distribution under working conditions from CFX evaluation and gained equivalent stress and deformation under several load conditions by using the ANSYS Workbench platform to check the strength of conveyor.The results showed that the influence of centrifugal hydraulic pressure was less than that of centrifugal force on the strength and deformation of conveyor.Besides,the maximum equivalent stress occurred at the inside of the feed opening,while the maximum deformation occurred at the conveyor blade edge of taper extremity.Furthermore,whether considered the feed opening or not,the computing model had a great influence on the analysis results,and the simplified loads had a great influence on the deformation analysis results.The methods and results from this article can provide reference for the design and the improvement of screw conveyor.展开更多
The equivalent stress fundamental solution for the elastoplastic dynamic plane strain problem is proposed to transform the virtual work in the third direction to the plane.Subsequently,based on Betti reciprocal theore...The equivalent stress fundamental solution for the elastoplastic dynamic plane strain problem is proposed to transform the virtual work in the third direction to the plane.Subsequently,based on Betti reciprocal theorem,by adopting the time dependent fundamental solutions in terms of displacement,traction and equivalent stress,the boundary integral equations for dynamic elastoplastic analysis for the plane strain problem are established.The establishment procedures for the displacement and the stress boundary integral equations,together with the stress equation at boundary points,are presented in details,while the standard discretization both in time and space under the frame of time domain boundary element method(TD-BEM)and the solution of the algebraic equations are also briefly stated.Two verification examples are presented from different viewpoints,for elastic and elastoplastic analysis,for 1-D and 2-D geometries,and for finite and infinite domains.The TD-BEM formulation for dynamic elastoplastic analysis is presented for the plane strain problem as an example,where the formulation is also applicable for the plane stress problem by properly transforming the elastic constants and adopting the corresponding fundamental solutions.展开更多
基金supported by the Natural Science Foundation of Shannxi (No. 2005A19)the Open Foundation of Engineering Key Laboratory of Disaster Prevention and Structural Safety,Guangxi Univ.(No. 2008TMKF004)the National Natural Science Foundation (Nos. 10932007 and 11172228)
文摘In this contribution,the microscopic fracture mechanism and extension criterion for mixed type crack in ductile material under plane mixed mode loading are investigated in details.A universal extension criterion for the mixed type crack,i.e.the crack propagates along the direction of the maximum gradient of equivalent stress,is suggested.This new criterion is used to predict the propagation direction of mixed type crack,showing a good agreement with other theories for different types of mode mixity.Moreover,the numerical verification is also carried out for the case of an edge crack with different mixed mode loadings.Finally,a potential application to three-dimensional fracture in the ductile material induced by holes is also discussed.
基金the National Natural Science Foundation of China (No. 51605354)the Fundamental Research Funds for the Central Universities (WUT: 2015IVA021)+1 种基金the Innovative Research Team Development Program of Ministry of Education of China (No. IRT13087)the High-end Talent Leading Program of Hubei Province (No. 201286)
文摘The aim of this work is to develop a three-dimensional model of deep groove ball bearing to investigate the loaded stresses and central displacements of bearing rings. The equivalent stresses and central displacements of bearing rings are obtained based on the simulated analysis. Moreover, several parameters, such as load magnitude, raceway groove curvature radius(RGCR), thicknesses of outer and inner rings, are varied to investigate their effects on the equivalent stresses and central displacements of bearing rings. Research results provide useful guidelines for determining the design parameters.
文摘Optimum utilization of the loading capability of engineering materials is an important and active contribution to protect nature's limited resources,and it is the key for economic design methods.In order to make use of the materials' resources,those must be known very well;but conventional test methods will offer only limited informational value.The range of questions raised is as wide as the application of engineering materials,and partially they are very specific.The development of huge computer powers enables numeric modelling to simulate structural behaviour in rather complex loading environments-so the real material behaviour is known under the given loading conditions.Here the art of material testing design starts.To study the material behaviour under very distinct and specific loading conditions makes it necessary to simulate different temperature ranges,loading speeds, environments etc.and mostly there doesn't exist any commonly agreed test standard.In this contribution two popular,non-standard test procedures and test systems will be discussed on the base of their application background,special design features as well as test results and typically gained information:The demand for highspeed tests up to 1000 s^(-1) of strain rate is very specific and originates primarily in the automotive industry and the answers enable CAE analysis of crashworthiness of vehicle structures under crash conditions.The information on the material behaviour under multiaxial loading conditions is a more general one.Multiaxial stress states can be reduced to an equivalent stress,which allows the evaluation of the material's constraint and criticality of stress state.Both discussed examples shall show that the open dialogue between the user and the producer of testing machines allows custom-tailored test solutions.
基金This project is supported by Provincial Six Kind Skilled Personnel Project of Jiangsu,China(No.06-E-020).
文摘The finite element method(FEM) is used to analyze the effects of lead widths and pitches on reliability of soldered joints. The optimum simulation for QFP devices is also researched. The results indicate that when the lead pitches are the same, the maximum equivalent stress of the soldered joints increases with the increasing of lead widths, while the reliability of the soldered joints reduces. When the lead widths are the same, the maximum equivalent stress of the soldered joints doesn't decrease completely with the increasing of lead pitches, a minimum value of the maximum equivalent stress values exists in all the curves. Under this condition the maximum equivalent stress of the soldewed joints is relatively the least, the reliability of soldered joints is high and the assembly is excellent. The simulating results indicate the best parameter: The lead width is 0.2 mm and lead pitch is 0.3 mm (the distance between two leads is 0.1 mm), which are benefited for the micromation of QFP devices now. The minimum value of the maximum equivalent stress of soldered joints exists while lead width is 0.25 mm and lead pitch is 0.35 mm (the distance between two leads is 0.1 mm), the devices can serve for a long time and the reliability is the highest, the assembly is excellent. The simulating results also indicate the fact that the lead width is 0.15 mm and lead pitch is 0.2 mm maybe the limit of QFP, which is significant for the high lead count and micromation of assembly.
基金Item Sponsored by State Ministry of Science and Technology of China (G2000067208-4)
文摘Three passes of plate rolling during vertical-horizontal rolling process are simulated with explicit dynamic finite element method and updating geometric method. The equivalent strain and stress fields, and shape change at the head and tail of slab during rolling are obtained. The calculated result of the shape at the head and tail of slab is in good agreement with the measured one. The explicit dynamic finite element method and updating geometric method can be used effectively to analyze the multipass vertical-horizontal (V-H) rolling process.
基金Funded by the National "863" Plan of China ( No.2002AA331112) ,the Doctorate Foundation of Northwestern Poly-technical University,andthe Science Research Foundation of HenanUniversity of Science and Technology(No.2006ZY041)
文摘Under the surface peeling of Cu- Fe- P lead frame alloy larger Fe particles were observed by energy dispersive spectroscopy. By using the large strain two-dinension plane strain model and elastic plastic finite element method, the cause for peeling damage of Cu-Fe-P lead frame aUoy was investigated. The results show that when the content of Fe particles is more than 30% at local Fe-rich area the intense stress coacentration in the Fe particle would make the Fe particle broken up. The high equivalent stress mutation and the mismatch of equivalent strain 10% at the two sides of intefrace make it easy to develop the crack and peeling damage on finish rolling. The larger Fe particles in the Cu-Fe-P alloy should be avoided.
基金Funded by the National Natural Science Foundation of China(No.51374154)
文摘The maximum principal stress, von Mises equivalent stress, equivalent creep strain, stress triaxiality in dissimilar metal welded joints between austenitic(HR3C) and martensitic heat-resistant steel(T91) are simulated by FEM at 873 K and under inner pressure of 42.26 MPa. The results show that the maximum principal stress and von Mises equivalent stress are quite high in the vicinity of weld/T91 interface, creep cavities are easy to form and expand in the weld/T91 interface. There are two peaks of equivalent creep strains in welded joint, and the maximum equivalent creep strain is in the place 27-32 mm away from the weld/T91 interface, and there exists creep constrain region in the vicinity of weld/T91 interface. The high stress triaxiality peak is located exactly at the weld/T91 interface. Accordingly, the weld/T91 interface is the weakest site of welded joint. Therefore, using stress triaxiality to describe creep cavity nucleation and expansion and crack development is reasonable for the dissimilar metal welded joint between austenitic and martensitic steel.
基金supported by the Aeronautical Science Foundation of China(No.02H53061)the National Science Found of China for Distinguished Young Scholar(No.50225518)the Shaan'xi Provincial Natural Science Foundation of China(No.2001CS0401)
文摘The blade precision forging process is a forming process with high temperature and large plastic deformation. Interaction of deformation and heat conduction leads to large uneven distribution of temperature. The unevenness of temperature distribution has a great effect on mechanical properties and the microstracture of materials. So it is necessary to consider the influence of temperature on the precision forging process of blades. Taking a blade with a tenon into consideration, a 3D mechanical model in precision forging is built up. The distribution laws of temperature field and the influence of the temperature on the equivalem stress in the process are obtained by using 3-D coupled thermo-mechanical FEM code developed by the authors Theresuits obtained illustrate that the influence of the temperature field on the blade forging process is considerable. The achievements of predicting microstructure and mechanical properties for forged blades is significant.
文摘Grain growth behavior in a copper foil under nonproportional loading was investigated. The grain growth density decreased with increasing phase shift between normal stress and shear stress. Moreover, under this condition, grains tended to grow in various directions. However, the relative frequency of the grain growth direction was slightly large in the maximum shear stress direction. Therefore, grain growth density is mainly dominated by the equivalent shear stress in Tresca theory in the case of nonproportional loading as well as proportional loading.
文摘The axial compressive strength,axial tensile strength,elastic modulus,poisson ratio and stress-strain relationship of RPC are obtained by compressive tests and tensile tests of RPC prism.Mathematical models of axial compressive and axial tensile stress-strain relationship are established,from which the equivalent coefficient of compressive and tensile stress diagram of cross-section in RPC beam are deduced.The results provide the theory basis for the design of RPC structure and the wider popularization of RPC.
基金supported by the National Natural Science Foundation of China(Nos.51101127 and 51171154)Soar Star of Northwestern Polytechnical University(2011)Fundamental Research Foundation of Northwestern Polytechnical University(No.JC201213)
文摘The basic principle of corrode groove on outside of steel pipe during storage was analyzed in this paper, namely the water film on the contacted surface of steel pipe, which gathered from humidity in the air, rain or gel, and the suspended particles in air, and the corrosive composition, such as SO2, CO2, O2 and NaCI, in addition to the inhomogeneity of the organization and composition, which lead to the corrosion cell reaction, so that cause the corrosion initial from the contact surface of the between steel pipes, so as to form the corrosion groove. At the same time, the corrosion groove with depth of 0.125t (t pipe wall thickness) on the pipe of φ 1016 mm×21 mm ×70 API SPEC 5L was simulated using the FEM (finite element method), and the stress and strain distribution of the defect area near corrosion groove were solved at the inner pressure of 12 MPa, 10 MPa, 8 MPa, 6 MPa, 4 MPa and 2 MPa, respectively, which showed that no matter the pressure values were, the maximum stress and strain were lied at the bottom of corrosion defects groove and were in good linear relationship with the internal pressure increasing from 2 MPa to 6 MPa. When the internal pres- sures were greater than 6 MPa, they felled into the nonlinear model and to be yielded or even to be destroyed. In addition, the residual strength and the limit operation pressure of the corrode pipe with the defects groove of 0.125t were calculated or simulated according to the theoretical calculation, the finite element method based on the stress, the finite element method based on strain, DNV-RP-F101, ASME B31G and experimental methods respectively. The results showed that the residual strength and the limit operation pressure of the defective parts solved by the finite element method based on stress were 424 MPa, and 15.34 MPa, respectively, which was very close to that of experimental method, the residual strength was 410 MPa and the limit operation pressure 14.78 MPa. Besides, the results also showed that it was feasible and effective to simulate the residual strength of the structure with corrosion defects using the finite element method.
基金This research is supported by the Major Cultivation Foundation of Sichuan Education Department(13ZA0179).
文摘In the centrifugal separations of drilling fluid,screw conveyor is a critical component to push and separate the sediment.The work performance and structural parameters of conveyor are immediately related to the production capability,the working life and the separating effect of the centrifuge.The existing researches always use the theoretical calculation of the approximate loads to analyze the strength of conveyor,and it cannot reflect the stress situations accurately.In order to ensure the precise mastery of the working performance,this article obtained pressure distribution under working conditions from CFX evaluation and gained equivalent stress and deformation under several load conditions by using the ANSYS Workbench platform to check the strength of conveyor.The results showed that the influence of centrifugal hydraulic pressure was less than that of centrifugal force on the strength and deformation of conveyor.Besides,the maximum equivalent stress occurred at the inside of the feed opening,while the maximum deformation occurred at the conveyor blade edge of taper extremity.Furthermore,whether considered the feed opening or not,the computing model had a great influence on the analysis results,and the simplified loads had a great influence on the deformation analysis results.The methods and results from this article can provide reference for the design and the improvement of screw conveyor.
基金The authors would like to acknowledge the financial support provided by Hebei Education Department(Grant QN2020135)the National Key R&D Program of China(Grants 2019YFC1511105 and 2019YFC1511104)the National Natural Science Foundation of China(Grant 51778193).
文摘The equivalent stress fundamental solution for the elastoplastic dynamic plane strain problem is proposed to transform the virtual work in the third direction to the plane.Subsequently,based on Betti reciprocal theorem,by adopting the time dependent fundamental solutions in terms of displacement,traction and equivalent stress,the boundary integral equations for dynamic elastoplastic analysis for the plane strain problem are established.The establishment procedures for the displacement and the stress boundary integral equations,together with the stress equation at boundary points,are presented in details,while the standard discretization both in time and space under the frame of time domain boundary element method(TD-BEM)and the solution of the algebraic equations are also briefly stated.Two verification examples are presented from different viewpoints,for elastic and elastoplastic analysis,for 1-D and 2-D geometries,and for finite and infinite domains.The TD-BEM formulation for dynamic elastoplastic analysis is presented for the plane strain problem as an example,where the formulation is also applicable for the plane stress problem by properly transforming the elastic constants and adopting the corresponding fundamental solutions.