This paper presents a mathematical model consisting of conservation and balance laws (CBL) of classical continuum mechanics (CCM) and ordered rate constitutive theories in Lagrangian description derived using entropy ...This paper presents a mathematical model consisting of conservation and balance laws (CBL) of classical continuum mechanics (CCM) and ordered rate constitutive theories in Lagrangian description derived using entropy inequality and the representation theorem for thermoviscoelastic solids (TVES) with rheology. The CBL and the constitutive theories take into account finite deformation and finite strain deformation physics and are based on contravariant deviatoric second Piola-Kirchhoff stress tensor and its work conjugate covariant Green’s strain tensor and their material derivatives of up to order m and n respectively. All published works on nonlinear dynamics of TVES with rheology are mostly based on phenomenological mathematical models. In rare instances, some aspects of CBL are used but are incorrectly altered to obtain mass, stiffness and damping matrices using space-time decoupled approaches. In the work presented in this paper, we show that this is not possible using CBL of CCM for TVES with rheology. Thus, the mathematical models used currently in the published works are not the correct description of the physics of nonlinear dynamics of TVES with rheology. The mathematical model used in the present work is strictly based on the CBL of CCM and is thermodynamically and mathematically consistent and the space-time coupled finite element methodology used in this work is unconditionally stable and provides solutions with desired accuracy and is ideally suited for nonlinear dynamics of TVES with memory. The work in this paper is the first presentation of a mathematical model strictly based on CBL of CCM and the solution of the mathematical model is obtained using unconditionally stable space-time coupled computational methodology that provides control over the errors in the evolution. Both space-time coupled and space-time decoupled finite element formulations are considered for obtaining solutions of the IVPs described by the mathematical model and are presented in the paper. Factors or the physics influencing dynamic response and dynamic bifurcation for TVES with rheology are identified and are also demonstrated through model problem studies. A simple model problem consisting of a rod (1D) of TVES material with memory fixed at one end and subjected to harmonic excitation at the other end is considered to study nonlinear dynamics of TVES with rheology, frequency response as well as dynamic bifurcation phenomenon.展开更多
This paper presents the mathematical model consisting of conservation and balance laws (CBL) of classical continuum mechanics (CCM) and the constitutive theories derived using entropy inequality and representation the...This paper presents the mathematical model consisting of conservation and balance laws (CBL) of classical continuum mechanics (CCM) and the constitutive theories derived using entropy inequality and representation theorem for thermoviscoelastic solids (TVES) matter without memory. The CBL and the constitutive theories take into account finite deformation and finite strain deformation physics. This mathematical model is thermodynamically and mathematically consistent and is ideally suited to study nonlinear dynamics of TVES and dynamic bifurcation and is used in the work presented in this paper. The finite element formulations are constructed for obtaining the solution of the initial value problems (IVPs) described by the mathematical models. Both space-time coupled as well as space-time decoupled finite element methods are considered for obtaining solutions of the IVPs. Space-time coupled finite element formulations based on space-time residual functional (STRF) that yield space-time variationally consistent space-time integral forms are considered. This approach ensures unconditional stability of the computations during the entire evolution. In the space-time decoupled finite element method based on Galerkin method with weak form for spatial discretization, the solutions of nonlinear ODEs in time resulting from the decoupling of space and time are obtained using Newmark linear acceleration method. Newton’s linear method is used to obtain converged solution for the nonlinear system of algebraic equations at each time step in the Newmark method. The different aspects of the deformation physics leading to the factors that influence nonlinear dynamic response and dynamic bifurcation are established using the proposed mathematical model, the solution method and their validity is demonstrated through model problem studies presented in this paper. Energy methods and superposition techniques in any form including those used in obtaining solutions are neither advocated nor used in the present work as these are not supported by calculus of variations and mathematical classification of differential operators appearing in nonlinear dynamics. The primary focus of the paper is to address various aspects of the deformation physics in nonlinear dynamics and their influence on dynamic bifurcation phenomenon using mathematical models strictly based on CBL of CCM using reliable unconditionally stable space-time coupled solution methods, which ensure solution accuracy or errors in the calculated solution are always identified. Many model problem studies are presented to further substantiate the concepts presented and discussed in the paper. Investigations presented in this paper are also compared with published works when appropriate.展开更多
This paper proposes a geometrically nonlinear total Lagrangian Galerkin meshfree formulation based on the stabilized conforming nodal integration for efficient analysis of shear deformable beam.The present nonlinear a...This paper proposes a geometrically nonlinear total Lagrangian Galerkin meshfree formulation based on the stabilized conforming nodal integration for efficient analysis of shear deformable beam.The present nonlinear analysis encompasses the fully geometric nonlinearities due to large deflection,large deformation as well as finite rotation.The incremental equilibrium equation is obtained by the consistent linearization of the nonlinear variational equation.The Lagrangian meshfree shape function is utilized to discretize the variational equation.Subsequently to resolve the shear and membrane locking issues and accelerate the computation,the method of stabilized conforming nodal integration is systematically implemented through the Lagrangian gradient smoothing operation.Numerical results reveal that the present formulation is very effective.展开更多
It is noted that the behavior of most piezoelectric materials is temperature dependent and suchpiezo-thermo-elastic coupling phenomenon has become even more pronounced in the case of finite deforma-tion.On the other h...It is noted that the behavior of most piezoelectric materials is temperature dependent and suchpiezo-thermo-elastic coupling phenomenon has become even more pronounced in the case of finite deforma-tion.On the other hand,for the purpose of precise shape and vibration control of piezoelectric smart struc-tures,their deformation under external excitation must be ideally modeled.This demands a thorough study ofthe coupled piezo-thermo-elastic response under finite deformation.In this study,the governing equations ofpiezoelectric stractures are formulated through the theory of virtual displacement principle and a finite elementmethod is developed,it should be emphasized that in the finite element method the fully coupled piezo-ther-mo-elastic behavior and the geometric non-linearity ate considered.The method developed is then applied tosimulate the dynamic and steady response of a clamped plate to heat flux acting on one side of the plate tomimic the behavior of a battery plate of satellite irradiated under the sun.The results obtained are comparedagainst classical solutions,whereby the thermal conductivity is assumed to be independent of deformation.Itis found that the full-cnupled theory predicts less transient response of the temperature compared to the clas-sic analysis.In the steady state limit,the predicted temperature distribution within the plate for small heatflux is almost the same for both analyses.However,it is noted that increasing the heat flux will increase thedeviation between the predictions of the temperature distributinn by the full coupled theory and by the classicanalysis.It is concluded from the present study that,in order to precisely predict the deformation of smartstructures,the piezo-thermo-elastic coupling,geometric nou-linearity and the deformation dependent thermalconductivity should be taken into account.展开更多
In this paper, based on the finite deformation S_R decomposition theorem, the definition of the body moment is renewed as the sum of its internal and external. The expression of the increment rate of the deformation e...In this paper, based on the finite deformation S_R decomposition theorem, the definition of the body moment is renewed as the sum of its internal and external. The expression of the increment rate of the deformation energy is derived and the physical meaning is clarified. The power variational principle and the complementary power variational principle for finite deformation mechanics are supplemented and perfected.展开更多
According to the basic idea of classical Yin-Yang complementarity and modern dual-complementarity,in a simple and unified new way proposed by Luo,the unconventional Gurtin-type variational prinicples for finite deform...According to the basic idea of classical Yin-Yang complementarity and modern dual-complementarity,in a simple and unified new way proposed by Luo,the unconventional Gurtin-type variational prinicples for finite deformation elastodynamics can be established systemati-cally.In this paper,an important integral relation in terms of convolution is given,which canbe considered as the expression of the generalized principle of virtual work for finite deformationdynamics.Based on this relation,it is possible not only to obtain the principle of virtual work forfinite deformation dynamics,but also to derive systematically the complementary functionals forfive-field,three-field,two-field and one-field unconventional Gurtin-type variational principles bythe generalized Legendre transformations given in this paper.Furthermore,with this approach,the intrinsic relationship among various principles can be clearly explained.展开更多
Localized deformation and instability is the focal point of research in mechanics. The most typical problem is the plastic analysis of cylindrical bar neckingand shear band under uniaxial tension. Traditional elasto-p...Localized deformation and instability is the focal point of research in mechanics. The most typical problem is the plastic analysis of cylindrical bar neckingand shear band under uniaxial tension. Traditional elasto-plastic mechanics of infinitesimal deformation can not solve this problem successfully. In this paper, on the basis of S(strain) -R(rotation) decomposition theorem, the authors obtain the localstrain distribution and progressive state of axial symmetric finite deformation of cylindrical bar under uniaxial tension adopting nonlinear gauge approximate method and computer modelling technique.展开更多
The overbroken rock mass of gob areas is made up of broken and accumulated rock blocks compressed to some extent by the overlying strata. The bearing pressure of the gob can directly affect the safety of mining fields...The overbroken rock mass of gob areas is made up of broken and accumulated rock blocks compressed to some extent by the overlying strata. The bearing pressure of the gob can directly affect the safety of mining fields, formation of road retained along the next goaf and seepage of water and methane through the gob. In this paper, the software RFPA’2000 is used to construct numerical models. Especially the Euler method of control volume is proposed to solve the simulation difficulty arising from plastically finite deformations. The results show that three characteristic regions occurred in the gob area: (1) a naturally accumulated region, 0–10 m away from unbroken surrounding rock walls, where the bearing pressure is nearly zero; (2) an overcompacted region, 10–20 m away from unbroken walls, where the bearing pressure results in the maximum value of the gob area; (3) a stable compaction region, more than 20 m away from unbroken walls and occupying absolutely most of the gob area, where the bearing pressures show basically no differences. Such a characteristic can explain the easy-seepaged “O”-ring phenomena around mining fields very well.展开更多
On the basis of classical linear theory on longitudinal,torsional and fiexural waves in thin elastic rods,and taking finite deformation and dispersive effects into con- sideration,three kinds of nonlinear evolution eq...On the basis of classical linear theory on longitudinal,torsional and fiexural waves in thin elastic rods,and taking finite deformation and dispersive effects into con- sideration,three kinds of nonlinear evolution equations are derived.Qualitative analysis of three kinds of nonlinear equations are presented.It is shown that these equations have homoclinic or heteroclinic orbits on the phase plane,corresponding to solitary wave or shock wave solutions,respectively.Based on the principle of homogeneous balance,these equations are solved with the Jacobi elliptic function expansion method.Results show that existence of solitary wave solution and shock wave solution is possible under certain conditions.These conclusions are consistent with qualitative analysis.展开更多
Based on the general solution given to a kind of linear tensor equations,the spin of asymmetric tensor is derived in an invariant form.The result is applied to find the spins of the left and thetight stretch tensors a...Based on the general solution given to a kind of linear tensor equations,the spin of asymmetric tensor is derived in an invariant form.The result is applied to find the spins of the left and thetight stretch tensors and the relation among different rotation rate tensors has been discussed.According towork conjugacy,the relations between Cauchy stress and the stresses conjugate to Hill’s generalized strains areobtained.Particularly,the logarithmic strain,its time rate and the conjugate stress have been discussed in de-tail.These results are important in modeling the constitutive relations for finite deformations in continuum me-chanics.展开更多
By using the logarithmic strain, the finite deformation plastic theory, corresponding to the infinitesimal plastic theory, is established successively. The plastic consistent algorithm with first order accuracy for th...By using the logarithmic strain, the finite deformation plastic theory, corresponding to the infinitesimal plastic theory, is established successively. The plastic consistent algorithm with first order accuracy for the finite element method (FEM) is developed. Numerical examples are presented to illustrate the validity of the theory and effectiveness of the algorithm.展开更多
This paper deals with finite deformation problems of cantilever beam with variable sec-tion under the action of arbitrary transverse loads.By the use of a method of variable replacement,the nonlinear differential equa...This paper deals with finite deformation problems of cantilever beam with variable sec-tion under the action of arbitrary transverse loads.By the use of a method of variable replacement,the nonlinear differential equation with varied coefficient for the problem can be transformed into anequation with variable separable.The exact solution can be obtained by the integration method.Some examples are given in the paper,and the results of these examples show that this exact solutionincludes the existing solutions in references as special cases.展开更多
By combining grain boundary(GB)and its influence zone,a micromechanic model forpolycrystal is established for considering the influence of GB.By using the crystal plasticity theory andthe finite element method for fin...By combining grain boundary(GB)and its influence zone,a micromechanic model forpolycrystal is established for considering the influence of GB.By using the crystal plasticity theory andthe finite element method for finite deformation,numerical simulation is carried out by the model.Calculated results display the microscopic characteristic of deformation fields of grains and are in quali-tative agreement with experimental results.展开更多
This paper concerns the dynamic plastic response of a circular plate resting on fluid subjectedto a uniformly distributed rectangular load pulse with finite deformation.It is assumed that the fluid isincompressible an...This paper concerns the dynamic plastic response of a circular plate resting on fluid subjectedto a uniformly distributed rectangular load pulse with finite deformation.It is assumed that the fluid isincompressible and inviscous,and the plate is made of rigid-plastic material and simply supported along itsedge.By using the method of the Hankel integral transformation,the nonuniform fluid resistance is de-rived as the plate and the fluid is coupled.Finally,an analytic solution for a circular plate under a mediumload is obtained according to the equations of motion of the plate with finite deformation.展开更多
Based on the Reddy’s theory of plates with the effect of higher-order shear deformations, the governing equations for bending of orthotropic plates with finite deformations were established. The differential quadratu...Based on the Reddy’s theory of plates with the effect of higher-order shear deformations, the governing equations for bending of orthotropic plates with finite deformations were established. The differential quadrature (DQ) method of nonlinear analysis to the problem was presented. New DQ approach, presented by Wang and Bert (DQWB), is extended to handle the multiple boundary conditions of plates. The techniques were also further extended to simplify nonlinear computations. The numerical convergence and comparison of solutions were studied. The results show that the DQ method presented is very reliable and valid. Moreover, the influences of geometric and material parameters as well as the transverse shear deformations on nonlinear bending were investigated. Numerical results show the influence of the shear deformation on the static bending of orthotropic moderately thick plate is significant.展开更多
With symmetries measured by the Lie group and curvatures revealed by differential geometry, the continuum stored energy function possesses a translational deformation component, a rotational deformation component, and...With symmetries measured by the Lie group and curvatures revealed by differential geometry, the continuum stored energy function possesses a translational deformation component, a rotational deformation component, and an ellipsoidal volumetric deformation component. The function, originally developed for elastomeric polymers, has been extended to model brittle and ductile polymers. The function fits uniaxial tension testing data for brittle, ductile, and elastomeric polymers, and elucidates deformation mechanisms. A clear distinction in damage modes between brittle and ductile deformations has been captured. The von Mises equivalent stress has been evaluated by the function and the newly discovered break-even stretch. Common practices of constitutive modeling, relevant features of existing models and testing methods, and a new perspective on the finite elasticity-plasticity theory have also been offered.展开更多
A new numerical approach is presented to compute the large deformations of shell-type structures made of the Saint Venant-Kirchhoff and Neo-Hookean materials based on the seven-parameter shell theory.A work conjugate ...A new numerical approach is presented to compute the large deformations of shell-type structures made of the Saint Venant-Kirchhoff and Neo-Hookean materials based on the seven-parameter shell theory.A work conjugate pair of the first Piola Kirchhoff stress tensor and deformation gradient tensor is considered for the stress and strain measures in the paper.Through introducing the displacement vector,the deformation gradient,and the stress tensor in the Cartesian coordinate system and by means of the chain rule for taking derivative of tensors,the difficulties in using the curvilinear coordinate system are bypassed.The variational differential quadrature(VDQ)method as a pointwise numerical method is also used to discretize the weak form of the governing equations.Being locking-free,the simple implementation,computational efficiency,and fast convergence rate are the main features of the proposed numerical approach.Some well-known benchmark problems are solved to assess the approach.The results indicate that it is capable of addressing the large deformation problems of elastic and hyperelastic shell-type structures efficiently.展开更多
文摘This paper presents a mathematical model consisting of conservation and balance laws (CBL) of classical continuum mechanics (CCM) and ordered rate constitutive theories in Lagrangian description derived using entropy inequality and the representation theorem for thermoviscoelastic solids (TVES) with rheology. The CBL and the constitutive theories take into account finite deformation and finite strain deformation physics and are based on contravariant deviatoric second Piola-Kirchhoff stress tensor and its work conjugate covariant Green’s strain tensor and their material derivatives of up to order m and n respectively. All published works on nonlinear dynamics of TVES with rheology are mostly based on phenomenological mathematical models. In rare instances, some aspects of CBL are used but are incorrectly altered to obtain mass, stiffness and damping matrices using space-time decoupled approaches. In the work presented in this paper, we show that this is not possible using CBL of CCM for TVES with rheology. Thus, the mathematical models used currently in the published works are not the correct description of the physics of nonlinear dynamics of TVES with rheology. The mathematical model used in the present work is strictly based on the CBL of CCM and is thermodynamically and mathematically consistent and the space-time coupled finite element methodology used in this work is unconditionally stable and provides solutions with desired accuracy and is ideally suited for nonlinear dynamics of TVES with memory. The work in this paper is the first presentation of a mathematical model strictly based on CBL of CCM and the solution of the mathematical model is obtained using unconditionally stable space-time coupled computational methodology that provides control over the errors in the evolution. Both space-time coupled and space-time decoupled finite element formulations are considered for obtaining solutions of the IVPs described by the mathematical model and are presented in the paper. Factors or the physics influencing dynamic response and dynamic bifurcation for TVES with rheology are identified and are also demonstrated through model problem studies. A simple model problem consisting of a rod (1D) of TVES material with memory fixed at one end and subjected to harmonic excitation at the other end is considered to study nonlinear dynamics of TVES with rheology, frequency response as well as dynamic bifurcation phenomenon.
文摘This paper presents the mathematical model consisting of conservation and balance laws (CBL) of classical continuum mechanics (CCM) and the constitutive theories derived using entropy inequality and representation theorem for thermoviscoelastic solids (TVES) matter without memory. The CBL and the constitutive theories take into account finite deformation and finite strain deformation physics. This mathematical model is thermodynamically and mathematically consistent and is ideally suited to study nonlinear dynamics of TVES and dynamic bifurcation and is used in the work presented in this paper. The finite element formulations are constructed for obtaining the solution of the initial value problems (IVPs) described by the mathematical models. Both space-time coupled as well as space-time decoupled finite element methods are considered for obtaining solutions of the IVPs. Space-time coupled finite element formulations based on space-time residual functional (STRF) that yield space-time variationally consistent space-time integral forms are considered. This approach ensures unconditional stability of the computations during the entire evolution. In the space-time decoupled finite element method based on Galerkin method with weak form for spatial discretization, the solutions of nonlinear ODEs in time resulting from the decoupling of space and time are obtained using Newmark linear acceleration method. Newton’s linear method is used to obtain converged solution for the nonlinear system of algebraic equations at each time step in the Newmark method. The different aspects of the deformation physics leading to the factors that influence nonlinear dynamic response and dynamic bifurcation are established using the proposed mathematical model, the solution method and their validity is demonstrated through model problem studies presented in this paper. Energy methods and superposition techniques in any form including those used in obtaining solutions are neither advocated nor used in the present work as these are not supported by calculus of variations and mathematical classification of differential operators appearing in nonlinear dynamics. The primary focus of the paper is to address various aspects of the deformation physics in nonlinear dynamics and their influence on dynamic bifurcation phenomenon using mathematical models strictly based on CBL of CCM using reliable unconditionally stable space-time coupled solution methods, which ensure solution accuracy or errors in the calculated solution are always identified. Many model problem studies are presented to further substantiate the concepts presented and discussed in the paper. Investigations presented in this paper are also compared with published works when appropriate.
基金supported by the National Natural Science Foundation of China (10972188)the Program for New Century Excellent Talents in University from China Education Ministry (NCET-09-0678)
文摘This paper proposes a geometrically nonlinear total Lagrangian Galerkin meshfree formulation based on the stabilized conforming nodal integration for efficient analysis of shear deformable beam.The present nonlinear analysis encompasses the fully geometric nonlinearities due to large deflection,large deformation as well as finite rotation.The incremental equilibrium equation is obtained by the consistent linearization of the nonlinear variational equation.The Lagrangian meshfree shape function is utilized to discretize the variational equation.Subsequently to resolve the shear and membrane locking issues and accelerate the computation,the method of stabilized conforming nodal integration is systematically implemented through the Lagrangian gradient smoothing operation.Numerical results reveal that the present formulation is very effective.
基金the National Natural Science Foundation of China (Nos.10132010 and 50135030)the Foundation of In-service Doctors of Xi'an Jiaotong University
文摘It is noted that the behavior of most piezoelectric materials is temperature dependent and suchpiezo-thermo-elastic coupling phenomenon has become even more pronounced in the case of finite deforma-tion.On the other hand,for the purpose of precise shape and vibration control of piezoelectric smart struc-tures,their deformation under external excitation must be ideally modeled.This demands a thorough study ofthe coupled piezo-thermo-elastic response under finite deformation.In this study,the governing equations ofpiezoelectric stractures are formulated through the theory of virtual displacement principle and a finite elementmethod is developed,it should be emphasized that in the finite element method the fully coupled piezo-ther-mo-elastic behavior and the geometric non-linearity ate considered.The method developed is then applied tosimulate the dynamic and steady response of a clamped plate to heat flux acting on one side of the plate tomimic the behavior of a battery plate of satellite irradiated under the sun.The results obtained are comparedagainst classical solutions,whereby the thermal conductivity is assumed to be independent of deformation.Itis found that the full-cnupled theory predicts less transient response of the temperature compared to the clas-sic analysis.In the steady state limit,the predicted temperature distribution within the plate for small heatflux is almost the same for both analyses.However,it is noted that increasing the heat flux will increase thedeviation between the predictions of the temperature distributinn by the full coupled theory and by the classicanalysis.It is concluded from the present study that,in order to precisely predict the deformation of smartstructures,the piezo-thermo-elastic coupling,geometric nou-linearity and the deformation dependent thermalconductivity should be taken into account.
文摘In this paper, based on the finite deformation S_R decomposition theorem, the definition of the body moment is renewed as the sum of its internal and external. The expression of the increment rate of the deformation energy is derived and the physical meaning is clarified. The power variational principle and the complementary power variational principle for finite deformation mechanics are supplemented and perfected.
基金Project supported by the National Natural Science Foundation of China(Nos.10172097,19902022 and 19672074)
文摘According to the basic idea of classical Yin-Yang complementarity and modern dual-complementarity,in a simple and unified new way proposed by Luo,the unconventional Gurtin-type variational prinicples for finite deformation elastodynamics can be established systemati-cally.In this paper,an important integral relation in terms of convolution is given,which canbe considered as the expression of the generalized principle of virtual work for finite deformationdynamics.Based on this relation,it is possible not only to obtain the principle of virtual work forfinite deformation dynamics,but also to derive systematically the complementary functionals forfive-field,three-field,two-field and one-field unconventional Gurtin-type variational principles bythe generalized Legendre transformations given in this paper.Furthermore,with this approach,the intrinsic relationship among various principles can be clearly explained.
文摘Localized deformation and instability is the focal point of research in mechanics. The most typical problem is the plastic analysis of cylindrical bar neckingand shear band under uniaxial tension. Traditional elasto-plastic mechanics of infinitesimal deformation can not solve this problem successfully. In this paper, on the basis of S(strain) -R(rotation) decomposition theorem, the authors obtain the localstrain distribution and progressive state of axial symmetric finite deformation of cylindrical bar under uniaxial tension adopting nonlinear gauge approximate method and computer modelling technique.
基金Projects 2005CB221502 supported by the Vital Foundational 973 Program of China, 50225414 by the National Outstanding Youth Foundation,20040350222 by China Postdoctoral Science FoundationBK 2004033 by Jiangsu Natural Science Foundation
文摘The overbroken rock mass of gob areas is made up of broken and accumulated rock blocks compressed to some extent by the overlying strata. The bearing pressure of the gob can directly affect the safety of mining fields, formation of road retained along the next goaf and seepage of water and methane through the gob. In this paper, the software RFPA’2000 is used to construct numerical models. Especially the Euler method of control volume is proposed to solve the simulation difficulty arising from plastically finite deformations. The results show that three characteristic regions occurred in the gob area: (1) a naturally accumulated region, 0–10 m away from unbroken surrounding rock walls, where the bearing pressure is nearly zero; (2) an overcompacted region, 10–20 m away from unbroken walls, where the bearing pressure results in the maximum value of the gob area; (3) a stable compaction region, more than 20 m away from unbroken walls and occupying absolutely most of the gob area, where the bearing pressures show basically no differences. Such a characteristic can explain the easy-seepaged “O”-ring phenomena around mining fields very well.
基金Project supported by the National Natural Science Foundation of China (No.10772129)the Youth Science Foundation of Shanxi Province of China (No.2006021005)
文摘On the basis of classical linear theory on longitudinal,torsional and fiexural waves in thin elastic rods,and taking finite deformation and dispersive effects into con- sideration,three kinds of nonlinear evolution equations are derived.Qualitative analysis of three kinds of nonlinear equations are presented.It is shown that these equations have homoclinic or heteroclinic orbits on the phase plane,corresponding to solitary wave or shock wave solutions,respectively.Based on the principle of homogeneous balance,these equations are solved with the Jacobi elliptic function expansion method.Results show that existence of solitary wave solution and shock wave solution is possible under certain conditions.These conclusions are consistent with qualitative analysis.
基金The project is supported by the National Natural Science Foundation of Chinathe Chinese Academy of Sciences(No.87-52)
文摘Based on the general solution given to a kind of linear tensor equations,the spin of asymmetric tensor is derived in an invariant form.The result is applied to find the spins of the left and thetight stretch tensors and the relation among different rotation rate tensors has been discussed.According towork conjugacy,the relations between Cauchy stress and the stresses conjugate to Hill’s generalized strains areobtained.Particularly,the logarithmic strain,its time rate and the conjugate stress have been discussed in de-tail.These results are important in modeling the constitutive relations for finite deformations in continuum me-chanics.
文摘By using the logarithmic strain, the finite deformation plastic theory, corresponding to the infinitesimal plastic theory, is established successively. The plastic consistent algorithm with first order accuracy for the finite element method (FEM) is developed. Numerical examples are presented to illustrate the validity of the theory and effectiveness of the algorithm.
基金Projects Supported by the Science Foundation of the Chinese Academy of Sciences.
文摘This paper deals with finite deformation problems of cantilever beam with variable sec-tion under the action of arbitrary transverse loads.By the use of a method of variable replacement,the nonlinear differential equation with varied coefficient for the problem can be transformed into anequation with variable separable.The exact solution can be obtained by the integration method.Some examples are given in the paper,and the results of these examples show that this exact solutionincludes the existing solutions in references as special cases.
文摘By combining grain boundary(GB)and its influence zone,a micromechanic model forpolycrystal is established for considering the influence of GB.By using the crystal plasticity theory andthe finite element method for finite deformation,numerical simulation is carried out by the model.Calculated results display the microscopic characteristic of deformation fields of grains and are in quali-tative agreement with experimental results.
文摘This paper concerns the dynamic plastic response of a circular plate resting on fluid subjectedto a uniformly distributed rectangular load pulse with finite deformation.It is assumed that the fluid isincompressible and inviscous,and the plate is made of rigid-plastic material and simply supported along itsedge.By using the method of the Hankel integral transformation,the nonuniform fluid resistance is de-rived as the plate and the fluid is coupled.Finally,an analytic solution for a circular plate under a mediumload is obtained according to the equations of motion of the plate with finite deformation.
文摘Based on the Reddy’s theory of plates with the effect of higher-order shear deformations, the governing equations for bending of orthotropic plates with finite deformations were established. The differential quadrature (DQ) method of nonlinear analysis to the problem was presented. New DQ approach, presented by Wang and Bert (DQWB), is extended to handle the multiple boundary conditions of plates. The techniques were also further extended to simplify nonlinear computations. The numerical convergence and comparison of solutions were studied. The results show that the DQ method presented is very reliable and valid. Moreover, the influences of geometric and material parameters as well as the transverse shear deformations on nonlinear bending were investigated. Numerical results show the influence of the shear deformation on the static bending of orthotropic moderately thick plate is significant.
文摘With symmetries measured by the Lie group and curvatures revealed by differential geometry, the continuum stored energy function possesses a translational deformation component, a rotational deformation component, and an ellipsoidal volumetric deformation component. The function, originally developed for elastomeric polymers, has been extended to model brittle and ductile polymers. The function fits uniaxial tension testing data for brittle, ductile, and elastomeric polymers, and elucidates deformation mechanisms. A clear distinction in damage modes between brittle and ductile deformations has been captured. The von Mises equivalent stress has been evaluated by the function and the newly discovered break-even stretch. Common practices of constitutive modeling, relevant features of existing models and testing methods, and a new perspective on the finite elasticity-plasticity theory have also been offered.
文摘A new numerical approach is presented to compute the large deformations of shell-type structures made of the Saint Venant-Kirchhoff and Neo-Hookean materials based on the seven-parameter shell theory.A work conjugate pair of the first Piola Kirchhoff stress tensor and deformation gradient tensor is considered for the stress and strain measures in the paper.Through introducing the displacement vector,the deformation gradient,and the stress tensor in the Cartesian coordinate system and by means of the chain rule for taking derivative of tensors,the difficulties in using the curvilinear coordinate system are bypassed.The variational differential quadrature(VDQ)method as a pointwise numerical method is also used to discretize the weak form of the governing equations.Being locking-free,the simple implementation,computational efficiency,and fast convergence rate are the main features of the proposed numerical approach.Some well-known benchmark problems are solved to assess the approach.The results indicate that it is capable of addressing the large deformation problems of elastic and hyperelastic shell-type structures efficiently.
