Marine thin plates are susceptible to welding deformation owing to their low structural stiffness.Therefore,the efficient and accurate prediction of welding deformation is essential for improving welding quality.The t...Marine thin plates are susceptible to welding deformation owing to their low structural stiffness.Therefore,the efficient and accurate prediction of welding deformation is essential for improving welding quality.The traditional thermal elastic-plastic finite element method(TEP-FEM)can accurately predict welding deformation.However,its efficiency is low because of the complex nonlinear transient computation,making it difficult to meet the needs of rapid engineering evaluation.To address this challenge,this study proposes an efficient prediction method for welding deformation in marine thin plate butt welds.This method is based on the coupled temperature gradient-thermal strain method(TG-TSM)that integrates inherent strain theory with a shell element finite element model.The proposed method first extracts the distribution pattern and characteristic value of welding-induced inherent strain through TEP-FEM analysis.This strain is then converted into the equivalent thermal load applied to the shell element model for rapid computation.The proposed method-particularly,the gradual temperature gradient-thermal strain method(GTG-TSM)-achieved improved computational efficiency and consistent precision.Furthermore,the proposed method required much less computation time than the traditional TEP-FEM.Thus,this study lays the foundation for future prediction of welding deformation in more complex marine thin plates.展开更多
In this paper,we propose a three-term conjugate gradient method for solving unconstrained optimization problems based on the Hestenes-Stiefel(HS)conjugate gradient method and Polak-Ribiere-Polyak(PRP)conjugate gradien...In this paper,we propose a three-term conjugate gradient method for solving unconstrained optimization problems based on the Hestenes-Stiefel(HS)conjugate gradient method and Polak-Ribiere-Polyak(PRP)conjugate gradient method.Under the condition of standard Wolfe line search,the proposed search direction is the descent direction.For general nonlinear functions,the method is globally convergent.Finally,numerical results show that the proposed method is efficient.展开更多
The acuurate prediction of the time-dependent mechanical behavior and deformation mechanisms of second-phase reinforced alloys under size effects is critical for the development of high-strength ductile metals and all...The acuurate prediction of the time-dependent mechanical behavior and deformation mechanisms of second-phase reinforced alloys under size effects is critical for the development of high-strength ductile metals and alloys for dynamic applications.However,solving their responses using high-fidelity numerical methods is computationally expensive and,in many cases,impractical.To address this issue,a dual-scale incremental variational formulation is proposed that incorporates the influence of plastic gradients on plastic evolution characteristics,integrating a strain-rate-dependent strain gradient plasticity model and including plastic gradients in the inelastic dissipation potential.Subsequently,two minimization problems based on the energy dissipation mechanisms of strain gradient plasticity,corresponding to the macroscopic and microscopic structures,are solved,leading to the development of a homogenization-based dual-scale solution algorithm.Finally,the effectiveness of the variational model and tangent algorithm is validated through a series of numerical simulations.The contributions of this work are as follows:first,it advances the theory of self-consistent computational homogenization modeling based on the energy dissipation mechanisms of plastic strain rates and their gradients,along with the development of a rigorous multi-level finite element method(FE2)solution procedure;second,the proposed algorithm provides an efficient and accurate method for evaluating the time-dependent mechanical behavior of second-phase reinforced alloys under strain gradient effects,exploring how these effects vary with the strain rate,and investigating their potential interactions.展开更多
The increasing integration of small-scale structures in engineering,particularly in Micro-Electro-Mechanical Systems(MEMS),necessitates advanced modeling approaches to accurately capture their complex mechanical behav...The increasing integration of small-scale structures in engineering,particularly in Micro-Electro-Mechanical Systems(MEMS),necessitates advanced modeling approaches to accurately capture their complex mechanical behavior.Classical continuum theories are inadequate at micro-and nanoscales,particularly concerning size effects,singularities,and phenomena like strain softening or phase transitions.This limitation follows from their lack of intrinsic length scale parameters,crucial for representingmicrostructural features.Theoretical and experimental findings emphasize the critical role of these parameters on small scales.This review thoroughly examines various strain gradient elasticity(SGE)theories commonly employed in literature to capture these size-dependent effects on the elastic response.Given the complexity arising from numerous SGE frameworks available in the literature,including first-and second-order gradient theories,we conduct a comprehensive and comparative analysis of common SGE models.This analysis highlights their unique physical interpretations and compares their effectiveness in modeling the size-dependent behavior of low-dimensional structures.A brief discussion on estimating additional material constants,such as intrinsic length scales,is also included to improve the practical relevance of SGE.Following this theoretical treatment,the review covers analytical and numerical methods for solving the associated higher-order governing differential equations.Finally,we present a detailed overview of strain gradient applications in multiscale andmultiphysics response of solids.Interesting research on exploring the relevance of SGE for reduced-order modeling of complex macrostructures,a universal multiphysics coupling in low-dimensional structures without being restricted to limited material symmetries(as in the case of microstructures),is also presented here for interested readers.Finally,we briefly discuss alternative nonlocal elasticity approaches(integral and integro-differential)for incorporating size effects,and conclude with some potential areas for future research on strain gradients.This review aims to provide a clear understanding of strain gradient theories and their broad applicability beyond classical elasticity.展开更多
The changes in strain gradient induced by grain boundaries are crucial for enhancing the plasticity of gradient magnesium(Mg)alloys.The change of strain distribution influence by grain boundaries during plastic deform...The changes in strain gradient induced by grain boundaries are crucial for enhancing the plasticity of gradient magnesium(Mg)alloys.The change of strain distribution influence by grain boundaries during plastic deformation of the gradient structure was examined.In this paper,the gradient structure AZ31 Mg-alloy plate with the surface fine grain(FG)to the center coarse grain(CG)was fabricated using hard plate rolling(HPR).The microstructure and strain distribution of Mg-alloy with a gradient structure were analyzed by electron backscatter diffraction(EBSD)and Digital image correlation(DIC)during uniaxial tensile.The findings indicate that the gradient structure sample(GS sample)displays a uniform strain distribution during the tensile process.Coarse-grain sample(CG sample)have obvious strain concentration,which leads to premature fracture.Based on EBSD characterization,low-angle grain boundaries(LAGBs)accumulates in the CG during plastic deformation.Orientation of CG tends to the(0001)basal.At the same time,the density of geometrically necessary dislocations(GNDs)inside CG has changed,which improves the Heterogeneous deformation induced(HDI)stress of gradient structure.During the uniaxial tensile,LAGBs accumulates in CG and changes the strain distribution of the gradient structure,which induces the accumulation of GNDs,and hence improving the properties of the GS Mg-alloy.These findings unveil the mechanism of strength-plasticity synergism of GS alloys from a new perspective and offer insights into the application of GS in Mg-alloys.展开更多
This paper presents a new criterion for determining the unloading points quantitatively and consistently in a multi-stage triaxial test.The radial strain gradient(RSG)is first introduced as an arc tangent function of ...This paper presents a new criterion for determining the unloading points quantitatively and consistently in a multi-stage triaxial test.The radial strain gradient(RSG)is first introduced as an arc tangent function of the rate of change of radial strain to time.RSG is observed to correlate closely with the stress state of a compressed sample,and reaches a horizontal asymptote as approaching failure.For a given rock type,RSG value at peak stress is almost the same,irrespective of the porosity and permeability.These findings lead to the development of RSG criterion:Unloading points can be precisely determined at the time when RSG reaches a pre-determined value that is a little smaller than or equal to the RSG at peak stress.The RSG criterion is validated against other criteria and the single-stage triaxial test on various types of rocks.Failure envelopes from the RSG criterion match well with those from single-stage tests.A practical procedure is recommended to use the RSG criterion:an unconfined compression or single-stage test is first conducted to determine the RSG at peak stress for one sample,the unloading point is then selected to be a value close to the RSG at peak stress,and the multi-stage test is finally performed on another sample using the pre-selected RSG unloading criterion.Generally,the RSG criterion is applicable for any type of rocks,especially brittle rocks,where other criteria are not suitable.Further,it can be practically implemented on the most available rock mechanical testing instruments.展开更多
This paper extends the one-dimensional(1D)nonlocal strain gradient integral model(NStraGIM)to the two-dimensional(2D)Kirchhoff axisymmetric nanoplates,based on nonlocal strain gradient integral relations formulated al...This paper extends the one-dimensional(1D)nonlocal strain gradient integral model(NStraGIM)to the two-dimensional(2D)Kirchhoff axisymmetric nanoplates,based on nonlocal strain gradient integral relations formulated along both the radial and circumferential directions.By transforming the proposed integral constitutive equations into the equivalent differential forms,complemented by the corresponding constitutive boundary conditions(CBCs),a well-posed mathematical formulation is established for analyzing the axisymmetric bending and buckling of annular/circular functionally graded(FG)sandwich nanoplates.The boundary conditions at the inner edge of a solid nanoplate are derived by L'H?spital's rule.The numerical solution is obtained by the generalized differential quadrature method(GDQM).The accuracy of the proposed model is validated through comparison with the data from the existing literature.A parameter study is conducted to demonstrate the effects of FG sandwich parameters,size parameters,and nonlocal gradient parameters.展开更多
By means of a comprehensive theory of elasticity, namely, a nonlocal strain gradient continuum theory, size-dependent nonlinear axial instability characteristics of cylindrical nanoshells made of functionally graded m...By means of a comprehensive theory of elasticity, namely, a nonlocal strain gradient continuum theory, size-dependent nonlinear axial instability characteristics of cylindrical nanoshells made of functionally graded material(FGM) are examined. To take small scale effects into consideration in a more accurate way, a nonlocal stress field parameter and an internal length scale parameter are incorporated simultaneously into an exponential shear deformation shell theory. The variation of material properties associated with FGM nanoshells is supposed along the shell thickness, and it is modeled based on the Mori-Tanaka homogenization scheme. With a boundary layer theory of shell buckling and a perturbation-based solving process, the nonlocal strain gradient load-deflection and load-shortening stability paths are derived explicitly. It is observed that the strain gradient size effect causes to the increases of both the critical axial buckling load and the width of snap-through phenomenon related to the postbuckling regime, while the nonlocal size dependency leads to the decreases of them. Moreover, the influence of the nonlocal type of small scale effect on the axial instability characteristics of FGM nanoshells is more than that of the strain gradient one.展开更多
In this paper,we analytically study vibration of functionally graded piezoelectric(FGP)nanoplates based on the nonlocal strain gradient theory.The top and bottom surfaces of the nanoplate are made of PZT-5H and PZT-4,...In this paper,we analytically study vibration of functionally graded piezoelectric(FGP)nanoplates based on the nonlocal strain gradient theory.The top and bottom surfaces of the nanoplate are made of PZT-5H and PZT-4,respectively.We employ Hamilton’s principle and derive the governing differential equations.Then,we use Navier’s solution to obtain the natural frequencies of the FGP nanoplate.In the first step,we compare our results with the obtained results for the piezoelectric nanoplates in the previous studies.In the second step,we neglect the piezoelectric effect and compare our results with those obtained for the functionally graded(FG)nanoplates.Finally,the effects of the FG power index,the nonlocal parameter,the aspect ratio,and the lengthto-thickness ratio,and the nanoplate shape on natural frequencies are investigated.展开更多
In this pap er, a novel size-dep endent functionally graded (FG) cylindrical shell model is develop ed based on the nonlocal strain gradient theory in conjunction with the Gurtin-Murdoch surface elasticity theory . Th...In this pap er, a novel size-dep endent functionally graded (FG) cylindrical shell model is develop ed based on the nonlocal strain gradient theory in conjunction with the Gurtin-Murdoch surface elasticity theory . The new model containing a nonlocal parameter, a material length scale parameter, and several surface elastic constants can capture three typical typ es of size e ects simultaneously , which are the nonlocal stress ef- fect, the strain gradient e ect, and the surface energy e ects. With the help of Hamilton’s principle and rst-order shear deformation theory , the non-classical governing equations and related b oundary conditions are derived. By using the prop osed model, the free vibra- tion problem of FG cylindrical nanoshells with material prop erties varying continuously through the thickness according to a p ower-law distribution is analytically solved, and the closed-form solutions for natural frequencies under various b oundary conditions are obtained. After verifying the reliability of the prop osed model and analytical method by comparing the degenerated results with those available in the literature, the in uences of nonlocal parameter, material length scale parameter, p ower-law index, radius-to-thickness ratio, length-to-radius ratio, and surface e ects on the vibration characteristic of func- tionally graded cylindrical nanoshells are examined in detail.展开更多
The mode I plane strain crack tip field with strain gradient effects is presented in this paper based on a simplified strain gradient theory within the framework proposed by Acharya and Bassani.The theory retains the ...The mode I plane strain crack tip field with strain gradient effects is presented in this paper based on a simplified strain gradient theory within the framework proposed by Acharya and Bassani.The theory retains the essential structure of the incremental version of the conventional J_2 deformation theory.No higher-order stress is introduced and no extra boundary value conditions beyond the conventional ones are required.The strain gradient effects are considered in the constitutive relation only through the instantaneous tangent modulus.The strain gradient measures are included into the tangent modulus as internal parameters.Therefore the boundary value problem is the same as that in the conventional theory.Two typical crack problems are studied:(a)the crack tip field under the small scale yielding condition induced by a linear elastic mode-I K-field and(b)the complete field for a compact tension specimen.The calculated results clearly show that the stress level near the crack tip with strain gradient effects is considerable higher than that in the classical theory.The singularity of the strain field near the crack tip is nearly equal to the square-root singularity and the singularity of the stress field is slightly greater than it.Consequently,the J-integral is no longer path independent and increases monotonically as the radius of the calculated circular contour decreases.展开更多
It has not been a simple matter to obtain a sound extension of the classical J2 flow theory of plasticity that incorporates a dependence on plastic strain gradients and that is capable of capturing size-dependent beha...It has not been a simple matter to obtain a sound extension of the classical J2 flow theory of plasticity that incorporates a dependence on plastic strain gradients and that is capable of capturing size-dependent behaviour of metals at the micron scale. Two classes of basic extensions of classical J2 theory have been proposed： one with increments in higher order stresses related to increments of strain gradients and the other characterized by the higher order stresses themselves expressed in terms of increments of strain gradients. The theories proposed by Muhlhans and Aifantis in 1991 and Fleck and Hutchinson in 2001 are in the first class, and, as formulated, these do not always satisfy thermodynamic requirements on plastic dissipation. On the other hand, theories of the second class proposed by Gudmundson in 2004 and Gurtin and Anand in 2009 have the physical deficiency that the higher order stress quantities can change discontinuously for bodies subject to arbitrarily small load changes. The present paper lays out this background to the quest for a sound phenomenological extension of the rateindependent J2 flow theory of plasticity to include a de- pendence on gradients of plastic strain. A modification of the Fleck-Hutchinson formulation that ensures its thermo- dynamic integrity is presented and contrasted with a comparable formulation of the second class where in the higher or- der stresses are expressed in terms of the plastic strain rate. Both versions are constructed to reduce to the classical J2 flow theory of plasticity when the gradients can be neglected and to coincide with the simpler and more readily formulated J2 deformation theory of gradient plasticity for deformation histories characterized by proportional straining.展开更多
Abstract For an infinite slab of strain gradient sensitive material subjected to plane-strain tensile loading, compu- tation established and analysis confirmed that passivation of the lateral boundaries at some stage ...Abstract For an infinite slab of strain gradient sensitive material subjected to plane-strain tensile loading, compu- tation established and analysis confirmed that passivation of the lateral boundaries at some stage of loading inhibits plastic deformation upon further loading. This result is not surprising in itself except that, remarkably, if the gradient terms contribute to the dissipation, the plastic deformation is switched off completely and only resumes at a clearly defined higher load, corresponding to a total strain ec, say. The analysis presented in this paper confirms the delay of plastic deformation following passivation and determines the exact manner in which the plastic flow resumes. The plastic strain rate is continuous at the exact point ec of resumption of plastic flow and, for the first small increment Ae = e - ec in the imposed total strain, the corresponding increment in plastic strain, AeP, is proportional to （Ae）2. The constant A in the relation AeP（0） = A（Ae）2, where AeP（0） denotes the plastic strain increment at the centre of the slab, has been determined explicitly; it depends on the hardening modulus of the material. The presence of energetic gradient terms has no effect on the value of ec unless the dissipative terms are absent, in which case passivation reduces the rate of plastic deformation but introduces no delay. This qualitative effect of dissipative gradient terms opens the possibility of experimen- tal discrimination of their presence or absence. The analysisemploys an incremental variational formulation that is likely to find use in other problems.展开更多
In this paper, multi-scale modeling for nanobeams with large deflection is conducted in the framework of the nonlocal strain gradient theory and the Euler-Bernoulli beam theory with exact bending curvature. The propos...In this paper, multi-scale modeling for nanobeams with large deflection is conducted in the framework of the nonlocal strain gradient theory and the Euler-Bernoulli beam theory with exact bending curvature. The proposed size-dependent nonlinear beam model incorporates structure-foundation interaction along with two small scale parameters which describe the stiffness-softening and stiffness-hardening size effects of nanomaterials, respectively. By applying Hamilton's principle, the motion equation and the associated boundary condition are derived. A two-step perturbation method is introduced to handle the deep postbuckling and nonlinear bending problems of nanobeams analytically. Afterwards, the influence of geometrical, material, and elastic foundation parameters on the nonlinear mechanical behaviors of nanobeams is discussed. Numerical results show that the stability and precision of the perturbation solutions can be guaranteed, and the two types of size effects become increasingly important as the slenderness ratio increases. Moreover, the in-plane conditions and the high-order nonlinear terms appearing in the bending curvature expression play an important role in the nonlinear behaviors of nanobeams as the maximum deflection increases.展开更多
In this research, vibration and wave propagation analysis of a twisted micro- beam on Pasternak foundation is investigated. The strain-displacement relations （kine-matic equations） are calculated by the displacement...In this research, vibration and wave propagation analysis of a twisted micro- beam on Pasternak foundation is investigated. The strain-displacement relations （kine-matic equations） are calculated by the displacement fields of the twisted micro-beam. The strain gradient theory （SGT） is used to implement the size dependent effect at micro-scale. Finally, using an energy method and Hamilton＇s principle, the governing equations of motion for the twisted micro-beam are derived. Natural frequencies and the wave prop- agation speed of the twisted micro-beam are calculated with an analytical method. Also, the natural frequency, the phase speed, the cut-off frequency, and the wave number of the twisted micro-beam are obtained by considering three material length scale parameters, the rate of twist angle, the thickness, the length of twisted micro-beam, and the elastic medium. The results of this work indicate that the phase speed in a twisted micro-beam increases with an increase in the rate of twist angle. Moreover, the wave number is in- versely related with the thickness of micro-beam. Meanwhile, it is directly related to the wave propagation frequency. Increasing the rate of twist angle causes the increase in the natural frequency especially with higher thickness. The effect of the twist angle rate on the group velocity is observed at a lower wave propagation frequency.展开更多
Plastic limit load of viscoplastic thick-walled cylinder and spherical shell subjected to internal pressure is investigated analytically using a strain gradient plasticity theory. As a result, the current solutions ca...Plastic limit load of viscoplastic thick-walled cylinder and spherical shell subjected to internal pressure is investigated analytically using a strain gradient plasticity theory. As a result, the current solutions can capture the size effect at the micron scale. Numerical results show that the smaller the inner radius of the cylinder or spherical shell, the more significant the scale effects. Results also show that the size effect is more evident with increasing strain or strain-rate sensitivity index. The classical plastic-based solutions of the same problems are shown to be a special case of the present solution.展开更多
A dynamic Timoshenko beam model is established based on the new nonlocal strain gradient theory and slip boundary theory to study the wave propagation behaviors of fluid-filled carbon nanotubes （CNTs） at nanoscale. ...A dynamic Timoshenko beam model is established based on the new nonlocal strain gradient theory and slip boundary theory to study the wave propagation behaviors of fluid-filled carbon nanotubes （CNTs） at nanoscale. The nanoscale effects caused by the CNTs and the inner fluid are simulated by the nonlocal strain gradient effect and the slip boundary effect, respectively. The governing equations of motion are derived and resolved to investigate the wave characteristics in detail. The numerical solution shows that the strain gradient effect leads to the stiffness enhancement of CNTs when the nonlocal stress effect causes the decrease in stiffness. The dynamic properties of CNTs are affected by the coupling of these two scale effects. The flow velocity of fluid inside the CNT is increased due to the slip boundary effect, resulting in the promotion of wave propagation in the dynamic system.展开更多
A plane strain mode 1 crack tip field with strain gradient effects is investigated.A new strain gradient theory is used.An elastic-power law hardening strain gradient material is considered and two hardening laws,i.e....A plane strain mode 1 crack tip field with strain gradient effects is investigated.A new strain gradient theory is used.An elastic-power law hardening strain gradient material is considered and two hardening laws,i.e.a separation law and an integration law are used respectively.As for the material with the separation law hardening,the angular distributions of stresses are consistent with the HRR field,which differs from the stress results;the angular distributions of couple stresses are the same as the couple stress results.For the material with the integration law hardening,the stress field and the couple stress field can not exist simultaneously,which is the same as the conclusion,but for the stress dominated field,the an- gular distributions of stresses are consistent with the HRR field;for the couple stress dominated field,the an- gular distributions of couple stresses are consistent with those in Ref.However,the increase in stresses is not observed in strain gradient plasticity because the present theory is based on the rotation gradient of the deformation only,while the crack tip field of mode 1 is dominated by the tension gradient,which will be shown in another paper.展开更多
This paper presents a separated law of hardening in plasticity with strain gradient effects. The value of the length parameter l contained in this model was estimated from the experimental data for copper.
In this work,the static and dynamic response of a piezoelectric semiconductor cantilever under the transverse end force with consideration of flexoelectricity and strain gradient elasticity is systematically investiga...In this work,the static and dynamic response of a piezoelectric semiconductor cantilever under the transverse end force with consideration of flexoelectricity and strain gradient elasticity is systematically investigated.The one-dimensional governing equations and the corresponding boundary conditions are derived based on Hamilton’s principle.After that,combining with the linearized equations for the conservation of charge,the effects of characteristic length and flexoelectric coefficient on the working performance of a ZnO nanowire are demonstrated as a numerical case,including the static mechanical and electric fields,natural frequencies,and the frequency–response characteristics at resonances.The results indicate that the flexoelectric effect has a great influence on the electric properties of the nanowire,while the strain gradient effect directly contributes to its mechanical properties.To some extent,the increase in characteristic length is equivalent to the stiffness strengthening.The qualitative results and quantitative data are beneficial for revealing the underlying physical mechanism and provide guidance for the design of piezoelectric semiconductor devices.展开更多
基金Supported by the National Natural Science Foundation of China under Grant No.51975138the High-Tech Ship Scientific Research Project from the Ministry of Industry and Information Technology under Grant No.CJ05N20the National Defense Basic Research Project under Grant No.JCKY2023604C006.
文摘Marine thin plates are susceptible to welding deformation owing to their low structural stiffness.Therefore,the efficient and accurate prediction of welding deformation is essential for improving welding quality.The traditional thermal elastic-plastic finite element method(TEP-FEM)can accurately predict welding deformation.However,its efficiency is low because of the complex nonlinear transient computation,making it difficult to meet the needs of rapid engineering evaluation.To address this challenge,this study proposes an efficient prediction method for welding deformation in marine thin plate butt welds.This method is based on the coupled temperature gradient-thermal strain method(TG-TSM)that integrates inherent strain theory with a shell element finite element model.The proposed method first extracts the distribution pattern and characteristic value of welding-induced inherent strain through TEP-FEM analysis.This strain is then converted into the equivalent thermal load applied to the shell element model for rapid computation.The proposed method-particularly,the gradual temperature gradient-thermal strain method(GTG-TSM)-achieved improved computational efficiency and consistent precision.Furthermore,the proposed method required much less computation time than the traditional TEP-FEM.Thus,this study lays the foundation for future prediction of welding deformation in more complex marine thin plates.
基金Supported by the Science and Technology Project of Guangxi(Guike AD23023002)。
文摘In this paper,we propose a three-term conjugate gradient method for solving unconstrained optimization problems based on the Hestenes-Stiefel(HS)conjugate gradient method and Polak-Ribiere-Polyak(PRP)conjugate gradient method.Under the condition of standard Wolfe line search,the proposed search direction is the descent direction.For general nonlinear functions,the method is globally convergent.Finally,numerical results show that the proposed method is efficient.
基金Project supported by the National Natural Science Foundation of China(Nos.11922206,11702089,12272132)the Postgraduate Scientific Research Innovation Project of Hunan Province(No.CX20240388)。
文摘The acuurate prediction of the time-dependent mechanical behavior and deformation mechanisms of second-phase reinforced alloys under size effects is critical for the development of high-strength ductile metals and alloys for dynamic applications.However,solving their responses using high-fidelity numerical methods is computationally expensive and,in many cases,impractical.To address this issue,a dual-scale incremental variational formulation is proposed that incorporates the influence of plastic gradients on plastic evolution characteristics,integrating a strain-rate-dependent strain gradient plasticity model and including plastic gradients in the inelastic dissipation potential.Subsequently,two minimization problems based on the energy dissipation mechanisms of strain gradient plasticity,corresponding to the macroscopic and microscopic structures,are solved,leading to the development of a homogenization-based dual-scale solution algorithm.Finally,the effectiveness of the variational model and tangent algorithm is validated through a series of numerical simulations.The contributions of this work are as follows:first,it advances the theory of self-consistent computational homogenization modeling based on the energy dissipation mechanisms of plastic strain rates and their gradients,along with the development of a rigorous multi-level finite element method(FE2)solution procedure;second,the proposed algorithm provides an efficient and accurate method for evaluating the time-dependent mechanical behavior of second-phase reinforced alloys under strain gradient effects,exploring how these effects vary with the strain rate,and investigating their potential interactions.
基金support from the Anusandhan National Research Foundation(ANRF),erstwhile Science and Engineering Research Board(SERB),India,under the startup research grant program(SRG/2022/000566).
文摘The increasing integration of small-scale structures in engineering,particularly in Micro-Electro-Mechanical Systems(MEMS),necessitates advanced modeling approaches to accurately capture their complex mechanical behavior.Classical continuum theories are inadequate at micro-and nanoscales,particularly concerning size effects,singularities,and phenomena like strain softening or phase transitions.This limitation follows from their lack of intrinsic length scale parameters,crucial for representingmicrostructural features.Theoretical and experimental findings emphasize the critical role of these parameters on small scales.This review thoroughly examines various strain gradient elasticity(SGE)theories commonly employed in literature to capture these size-dependent effects on the elastic response.Given the complexity arising from numerous SGE frameworks available in the literature,including first-and second-order gradient theories,we conduct a comprehensive and comparative analysis of common SGE models.This analysis highlights their unique physical interpretations and compares their effectiveness in modeling the size-dependent behavior of low-dimensional structures.A brief discussion on estimating additional material constants,such as intrinsic length scales,is also included to improve the practical relevance of SGE.Following this theoretical treatment,the review covers analytical and numerical methods for solving the associated higher-order governing differential equations.Finally,we present a detailed overview of strain gradient applications in multiscale andmultiphysics response of solids.Interesting research on exploring the relevance of SGE for reduced-order modeling of complex macrostructures,a universal multiphysics coupling in low-dimensional structures without being restricted to limited material symmetries(as in the case of microstructures),is also presented here for interested readers.Finally,we briefly discuss alternative nonlocal elasticity approaches(integral and integro-differential)for incorporating size effects,and conclude with some potential areas for future research on strain gradients.This review aims to provide a clear understanding of strain gradient theories and their broad applicability beyond classical elasticity.
基金supported by the Natural Science Foundation of Heilongjiang Province(JQ2022E004)。
文摘The changes in strain gradient induced by grain boundaries are crucial for enhancing the plasticity of gradient magnesium(Mg)alloys.The change of strain distribution influence by grain boundaries during plastic deformation of the gradient structure was examined.In this paper,the gradient structure AZ31 Mg-alloy plate with the surface fine grain(FG)to the center coarse grain(CG)was fabricated using hard plate rolling(HPR).The microstructure and strain distribution of Mg-alloy with a gradient structure were analyzed by electron backscatter diffraction(EBSD)and Digital image correlation(DIC)during uniaxial tensile.The findings indicate that the gradient structure sample(GS sample)displays a uniform strain distribution during the tensile process.Coarse-grain sample(CG sample)have obvious strain concentration,which leads to premature fracture.Based on EBSD characterization,low-angle grain boundaries(LAGBs)accumulates in the CG during plastic deformation.Orientation of CG tends to the(0001)basal.At the same time,the density of geometrically necessary dislocations(GNDs)inside CG has changed,which improves the Heterogeneous deformation induced(HDI)stress of gradient structure.During the uniaxial tensile,LAGBs accumulates in CG and changes the strain distribution of the gradient structure,which induces the accumulation of GNDs,and hence improving the properties of the GS Mg-alloy.These findings unveil the mechanism of strength-plasticity synergism of GS alloys from a new perspective and offer insights into the application of GS in Mg-alloys.
文摘This paper presents a new criterion for determining the unloading points quantitatively and consistently in a multi-stage triaxial test.The radial strain gradient(RSG)is first introduced as an arc tangent function of the rate of change of radial strain to time.RSG is observed to correlate closely with the stress state of a compressed sample,and reaches a horizontal asymptote as approaching failure.For a given rock type,RSG value at peak stress is almost the same,irrespective of the porosity and permeability.These findings lead to the development of RSG criterion:Unloading points can be precisely determined at the time when RSG reaches a pre-determined value that is a little smaller than or equal to the RSG at peak stress.The RSG criterion is validated against other criteria and the single-stage triaxial test on various types of rocks.Failure envelopes from the RSG criterion match well with those from single-stage tests.A practical procedure is recommended to use the RSG criterion:an unconfined compression or single-stage test is first conducted to determine the RSG at peak stress for one sample,the unloading point is then selected to be a value close to the RSG at peak stress,and the multi-stage test is finally performed on another sample using the pre-selected RSG unloading criterion.Generally,the RSG criterion is applicable for any type of rocks,especially brittle rocks,where other criteria are not suitable.Further,it can be practically implemented on the most available rock mechanical testing instruments.
基金Project supported by the National Natural Science Foundation of China(No.12172169)the Priority Academic Program Development of Jiangsu Higher Education Institutions。
文摘This paper extends the one-dimensional(1D)nonlocal strain gradient integral model(NStraGIM)to the two-dimensional(2D)Kirchhoff axisymmetric nanoplates,based on nonlocal strain gradient integral relations formulated along both the radial and circumferential directions.By transforming the proposed integral constitutive equations into the equivalent differential forms,complemented by the corresponding constitutive boundary conditions(CBCs),a well-posed mathematical formulation is established for analyzing the axisymmetric bending and buckling of annular/circular functionally graded(FG)sandwich nanoplates.The boundary conditions at the inner edge of a solid nanoplate are derived by L'H?spital's rule.The numerical solution is obtained by the generalized differential quadrature method(GDQM).The accuracy of the proposed model is validated through comparison with the data from the existing literature.A parameter study is conducted to demonstrate the effects of FG sandwich parameters,size parameters,and nonlocal gradient parameters.
文摘By means of a comprehensive theory of elasticity, namely, a nonlocal strain gradient continuum theory, size-dependent nonlinear axial instability characteristics of cylindrical nanoshells made of functionally graded material(FGM) are examined. To take small scale effects into consideration in a more accurate way, a nonlocal stress field parameter and an internal length scale parameter are incorporated simultaneously into an exponential shear deformation shell theory. The variation of material properties associated with FGM nanoshells is supposed along the shell thickness, and it is modeled based on the Mori-Tanaka homogenization scheme. With a boundary layer theory of shell buckling and a perturbation-based solving process, the nonlocal strain gradient load-deflection and load-shortening stability paths are derived explicitly. It is observed that the strain gradient size effect causes to the increases of both the critical axial buckling load and the width of snap-through phenomenon related to the postbuckling regime, while the nonlocal size dependency leads to the decreases of them. Moreover, the influence of the nonlocal type of small scale effect on the axial instability characteristics of FGM nanoshells is more than that of the strain gradient one.
文摘In this paper,we analytically study vibration of functionally graded piezoelectric(FGP)nanoplates based on the nonlocal strain gradient theory.The top and bottom surfaces of the nanoplate are made of PZT-5H and PZT-4,respectively.We employ Hamilton’s principle and derive the governing differential equations.Then,we use Navier’s solution to obtain the natural frequencies of the FGP nanoplate.In the first step,we compare our results with the obtained results for the piezoelectric nanoplates in the previous studies.In the second step,we neglect the piezoelectric effect and compare our results with those obtained for the functionally graded(FG)nanoplates.Finally,the effects of the FG power index,the nonlocal parameter,the aspect ratio,and the lengthto-thickness ratio,and the nanoplate shape on natural frequencies are investigated.
基金Project supported by the National Natural Science Foundation of China(Nos.11872233 and11472163)the China Scholarship Council(No.201706890041)the Innovation Program of Shanghai Municipal Education Commission(No.2017-01-07-00-09-E00019)
文摘In this pap er, a novel size-dep endent functionally graded (FG) cylindrical shell model is develop ed based on the nonlocal strain gradient theory in conjunction with the Gurtin-Murdoch surface elasticity theory . The new model containing a nonlocal parameter, a material length scale parameter, and several surface elastic constants can capture three typical typ es of size e ects simultaneously , which are the nonlocal stress ef- fect, the strain gradient e ect, and the surface energy e ects. With the help of Hamilton’s principle and rst-order shear deformation theory , the non-classical governing equations and related b oundary conditions are derived. By using the prop osed model, the free vibra- tion problem of FG cylindrical nanoshells with material prop erties varying continuously through the thickness according to a p ower-law distribution is analytically solved, and the closed-form solutions for natural frequencies under various b oundary conditions are obtained. After verifying the reliability of the prop osed model and analytical method by comparing the degenerated results with those available in the literature, the in uences of nonlocal parameter, material length scale parameter, p ower-law index, radius-to-thickness ratio, length-to-radius ratio, and surface e ects on the vibration characteristic of func- tionally graded cylindrical nanoshells are examined in detail.
基金The project supported by the National Natural Science Foundation of China (19704100 and 10202023) and the Natural Science Foundation of Chinese Academy of Sciences (KJ951-1-20)
文摘The mode I plane strain crack tip field with strain gradient effects is presented in this paper based on a simplified strain gradient theory within the framework proposed by Acharya and Bassani.The theory retains the essential structure of the incremental version of the conventional J_2 deformation theory.No higher-order stress is introduced and no extra boundary value conditions beyond the conventional ones are required.The strain gradient effects are considered in the constitutive relation only through the instantaneous tangent modulus.The strain gradient measures are included into the tangent modulus as internal parameters.Therefore the boundary value problem is the same as that in the conventional theory.Two typical crack problems are studied:(a)the crack tip field under the small scale yielding condition induced by a linear elastic mode-I K-field and(b)the complete field for a compact tension specimen.The calculated results clearly show that the stress level near the crack tip with strain gradient effects is considerable higher than that in the classical theory.The singularity of the strain field near the crack tip is nearly equal to the square-root singularity and the singularity of the stress field is slightly greater than it.Consequently,the J-integral is no longer path independent and increases monotonically as the radius of the calculated circular contour decreases.
文摘It has not been a simple matter to obtain a sound extension of the classical J2 flow theory of plasticity that incorporates a dependence on plastic strain gradients and that is capable of capturing size-dependent behaviour of metals at the micron scale. Two classes of basic extensions of classical J2 theory have been proposed： one with increments in higher order stresses related to increments of strain gradients and the other characterized by the higher order stresses themselves expressed in terms of increments of strain gradients. The theories proposed by Muhlhans and Aifantis in 1991 and Fleck and Hutchinson in 2001 are in the first class, and, as formulated, these do not always satisfy thermodynamic requirements on plastic dissipation. On the other hand, theories of the second class proposed by Gudmundson in 2004 and Gurtin and Anand in 2009 have the physical deficiency that the higher order stress quantities can change discontinuously for bodies subject to arbitrarily small load changes. The present paper lays out this background to the quest for a sound phenomenological extension of the rateindependent J2 flow theory of plasticity to include a de- pendence on gradients of plastic strain. A modification of the Fleck-Hutchinson formulation that ensures its thermo- dynamic integrity is presented and contrasted with a comparable formulation of the second class where in the higher or- der stresses are expressed in terms of the plastic strain rate. Both versions are constructed to reduce to the classical J2 flow theory of plasticity when the gradients can be neglected and to coincide with the simpler and more readily formulated J2 deformation theory of gradient plasticity for deformation histories characterized by proportional straining.
文摘Abstract For an infinite slab of strain gradient sensitive material subjected to plane-strain tensile loading, compu- tation established and analysis confirmed that passivation of the lateral boundaries at some stage of loading inhibits plastic deformation upon further loading. This result is not surprising in itself except that, remarkably, if the gradient terms contribute to the dissipation, the plastic deformation is switched off completely and only resumes at a clearly defined higher load, corresponding to a total strain ec, say. The analysis presented in this paper confirms the delay of plastic deformation following passivation and determines the exact manner in which the plastic flow resumes. The plastic strain rate is continuous at the exact point ec of resumption of plastic flow and, for the first small increment Ae = e - ec in the imposed total strain, the corresponding increment in plastic strain, AeP, is proportional to （Ae）2. The constant A in the relation AeP（0） = A（Ae）2, where AeP（0） denotes the plastic strain increment at the centre of the slab, has been determined explicitly; it depends on the hardening modulus of the material. The presence of energetic gradient terms has no effect on the value of ec unless the dissipative terms are absent, in which case passivation reduces the rate of plastic deformation but introduces no delay. This qualitative effect of dissipative gradient terms opens the possibility of experimen- tal discrimination of their presence or absence. The analysisemploys an incremental variational formulation that is likely to find use in other problems.
基金supported by the National Natural Science Foundation of China(Nos.11672252 and11602204)the Fundamental Research Funds for the Central Universities,Southwest Jiaotong University(No.2682016CX096)
文摘In this paper, multi-scale modeling for nanobeams with large deflection is conducted in the framework of the nonlocal strain gradient theory and the Euler-Bernoulli beam theory with exact bending curvature. The proposed size-dependent nonlinear beam model incorporates structure-foundation interaction along with two small scale parameters which describe the stiffness-softening and stiffness-hardening size effects of nanomaterials, respectively. By applying Hamilton's principle, the motion equation and the associated boundary condition are derived. A two-step perturbation method is introduced to handle the deep postbuckling and nonlinear bending problems of nanobeams analytically. Afterwards, the influence of geometrical, material, and elastic foundation parameters on the nonlinear mechanical behaviors of nanobeams is discussed. Numerical results show that the stability and precision of the perturbation solutions can be guaranteed, and the two types of size effects become increasingly important as the slenderness ratio increases. Moreover, the in-plane conditions and the high-order nonlinear terms appearing in the bending curvature expression play an important role in the nonlinear behaviors of nanobeams as the maximum deflection increases.
基金Project supported by the Iranian Nanotechnology Development Committee and the University of Kashan(No.463855/11)
文摘In this research, vibration and wave propagation analysis of a twisted micro- beam on Pasternak foundation is investigated. The strain-displacement relations （kine-matic equations） are calculated by the displacement fields of the twisted micro-beam. The strain gradient theory （SGT） is used to implement the size dependent effect at micro-scale. Finally, using an energy method and Hamilton＇s principle, the governing equations of motion for the twisted micro-beam are derived. Natural frequencies and the wave prop- agation speed of the twisted micro-beam are calculated with an analytical method. Also, the natural frequency, the phase speed, the cut-off frequency, and the wave number of the twisted micro-beam are obtained by considering three material length scale parameters, the rate of twist angle, the thickness, the length of twisted micro-beam, and the elastic medium. The results of this work indicate that the phase speed in a twisted micro-beam increases with an increase in the rate of twist angle. Moreover, the wave number is in- versely related with the thickness of micro-beam. Meanwhile, it is directly related to the wave propagation frequency. Increasing the rate of twist angle causes the increase in the natural frequency especially with higher thickness. The effect of the twist angle rate on the group velocity is observed at a lower wave propagation frequency.
基金supported by the Ph. D. Programs Foundation of Ministry of Education of China(No. 20050403002)
文摘Plastic limit load of viscoplastic thick-walled cylinder and spherical shell subjected to internal pressure is investigated analytically using a strain gradient plasticity theory. As a result, the current solutions can capture the size effect at the micron scale. Numerical results show that the smaller the inner radius of the cylinder or spherical shell, the more significant the scale effects. Results also show that the size effect is more evident with increasing strain or strain-rate sensitivity index. The classical plastic-based solutions of the same problems are shown to be a special case of the present solution.
基金This project is supported by the National Natural Science Foundation of China （Grant No. 11462010）.
文摘A dynamic Timoshenko beam model is established based on the new nonlocal strain gradient theory and slip boundary theory to study the wave propagation behaviors of fluid-filled carbon nanotubes （CNTs） at nanoscale. The nanoscale effects caused by the CNTs and the inner fluid are simulated by the nonlocal strain gradient effect and the slip boundary effect, respectively. The governing equations of motion are derived and resolved to investigate the wave characteristics in detail. The numerical solution shows that the strain gradient effect leads to the stiffness enhancement of CNTs when the nonlocal stress effect causes the decrease in stiffness. The dynamic properties of CNTs are affected by the coupling of these two scale effects. The flow velocity of fluid inside the CNT is increased due to the slip boundary effect, resulting in the promotion of wave propagation in the dynamic system.
基金the National Natural Science Foundation of China (No.19704100)Science Foundation of Chinese Academy of Sciences (Project KJ951-1-20)CASK.C.Wong Post-doctoral Research Award Fund and the Post Doctoral Science Fund of China.
文摘A plane strain mode 1 crack tip field with strain gradient effects is investigated.A new strain gradient theory is used.An elastic-power law hardening strain gradient material is considered and two hardening laws,i.e.a separation law and an integration law are used respectively.As for the material with the separation law hardening,the angular distributions of stresses are consistent with the HRR field,which differs from the stress results;the angular distributions of couple stresses are the same as the couple stress results.For the material with the integration law hardening,the stress field and the couple stress field can not exist simultaneously,which is the same as the conclusion,but for the stress dominated field,the an- gular distributions of stresses are consistent with the HRR field;for the couple stress dominated field,the an- gular distributions of couple stresses are consistent with those in Ref.However,the increase in stresses is not observed in strain gradient plasticity because the present theory is based on the rotation gradient of the deformation only,while the crack tip field of mode 1 is dominated by the tension gradient,which will be shown in another paper.
基金The project supported by the National Natural Science Foundation of China
文摘This paper presents a separated law of hardening in plasticity with strain gradient effects. The value of the length parameter l contained in this model was estimated from the experimental data for copper.
基金supported by the National Natural Science Foundation of China(12061131013 and 11972276)the State Key Laboratory of Mechanics and Control of Mechanical Structures at NUAA(No.MCMS-E-0520K02)+1 种基金the Fundamental Research Funds for the Central Universities(NE2020002 and NS2019007)the start-up fund supported by NUAA,and a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions(PAPD).
文摘In this work,the static and dynamic response of a piezoelectric semiconductor cantilever under the transverse end force with consideration of flexoelectricity and strain gradient elasticity is systematically investigated.The one-dimensional governing equations and the corresponding boundary conditions are derived based on Hamilton’s principle.After that,combining with the linearized equations for the conservation of charge,the effects of characteristic length and flexoelectric coefficient on the working performance of a ZnO nanowire are demonstrated as a numerical case,including the static mechanical and electric fields,natural frequencies,and the frequency–response characteristics at resonances.The results indicate that the flexoelectric effect has a great influence on the electric properties of the nanowire,while the strain gradient effect directly contributes to its mechanical properties.To some extent,the increase in characteristic length is equivalent to the stiffness strengthening.The qualitative results and quantitative data are beneficial for revealing the underlying physical mechanism and provide guidance for the design of piezoelectric semiconductor devices.
