Strain gradient near interface of coaxial cylinders in torsion

Classical elastoplastic theory predicts that the rotation angle near an interface between two mismatched materials is discontinuous under shear. The strain gradient effects, however, can be significant within a narrow region near the interface. This can be shown by application of the strain gradient plasticity. The matching expansion method was used to obtain asymptotic results. Comparison is then made with those found numerically for the interface torsion problem of a two-layered cylindrical tube. The strain gradient plasticity theory solution differs from that of the classical elastoplastic theory solution, depending on the properties aside from the interface behavior and the loading mode. A failure criterion is also proposed that accounts for the strain gradients.     

Finite deformation analysis of mechanism-based strain gradient plasticity: torsion and crack tip field

A finite deformation theory of mechanism-based strain gradient (MSG) plasticity is developed in this paper based on the Taylor dislocation model. The theory ensures the proper decomposition of deformation in order to exclude the volumetric deformation from the strain gradient tensor since the latter represents the density of geometrically necessary dislocations. The solution for a thin cylinder under large torsion is obtained. The numerical method is used to investigate the finite deformation crack tip field in MSG plasticity. It is established that the stress level around a crack tip in MSG plasticity is significantly higher than its counterpart (i.e. HRR field) in classical plasticity.     

Stress gradient versus strain gradient constitutive models within elasticity

A stress gradient elasticity theory is developed which is based on the Eringen method to address nonlocal elasticity by means of differential equations. By suitable thermodynamics arguments (involving the free enthalpy instead of the free internal energy), the restrictions on the related constitutive equations are determined, which include the well-known Eringen stress gradient constitutive equations, as well as the associated (so far uncertain) boundary conditions. The proposed theory exhibits complementary characters with respect to the analogous strain gradient elasticity theory. The associated boundary-value problem is shown to admit a unique solution characterized by a Hellinger–Reissner type variational principle. The main differences between the Eringen stress gradient model and the concomitant Aifantis strain gradient model are pointed out. A rigorous formulation of the stress gradient Euler–Bernoulli beam is provided; the response of this beam model is discussed as for its sensitivity to the stress gradient effects and compared with the analogous strain gradient beam model.     

Stress gradient continuum theory

A stress gradient continuum theory is presented that fundamentally differs from the well-established strain gradient model. It is based on the assumption that the deviatoric part of the gradient of the Cauchy stress tensor can contribute to the free energy density of solid materials. It requires the introduction of so-called micro-displacement degrees of freedom in addition to the usual displacement components. An isotropic linear elasticity theory is worked out for two-dimensional stress gradient media. The analytical solution of a simple boundary value problem illustrates the essential differences between stress and strain gradient models.     

On the torsion problem of strain gradient elastic bars

The torsion problem of elastic bars of any cross-sections is discussed, into the context of strain gradient elasticity. It is proven that torsion problem is feasible only for the bars with circular cross-sections. For the other bars (with non-circular cross sections), the non-classical boundary conditions are not satisfied.     

Photovisco-elasto-plastic analysis tested on polyester by the scattered-light method

A three-dimensional photovisco-elasto-plastic model considering the strain rate effect was investigated by the scattered-light method using polyester as a model material. To examine the mechanical and optical properties of the material, tension and torsion tests were carried out on cylindrical specimens under various strain rates at 30°C. The effects of strain rate on the stress-strain relation and scattered-light fringe appearance were evaluated. The equivalent shearing stress-strain relation can be approximated by the Ramberg-Osgood equation with rate-dependent modulus and yield stress. The fringe gradient, when normalized by a rate-dependent yield gradient, can be related to an equivalent strain in the same form regardless of the strain rate. The strain rate can be evaluated from the measurement of the rate of increase of the fringe gradient. Hence, the relation between the fringe gradient and its rate of increase was derived as a function of strain rate. Finally, a method is proposed for the estimation of the visco-elasto-plastic stress and strain in a three-dimensional specimen from the measurement of only the fringe gradient and its rate of increase. The method was successfully applied not only to uniaxial tension but also to pure torsion.     

A finite element implementation of the stress gradient theory

In this contribution, a finite element implementation of the stress gradient theory is proposed. The implementation relies on a reformulation of the governing set of partial differential equations in terms of one primary tensor-valued field variable of third order, the so-called generalised displacement field. Whereas the volumetric part of the generalised displacement field is closely related to the classic displacement field, the deviatoric part can be interpreted in terms of micro-displacements. The associated weak formulation moreover stipulates boundary conditions in terms of the normal projection of the generalised displacement field or of the (complete) stress tensor. A detailed study of representative boundary value problems of stress gradient elasticity shows the applicability of the proposed formulation. In particular, the finite element implementation is validated based on the analytical solutions for a cylindrical bar under tension and torsion derived by means of Bessel functions. In both tension and torsion cases, a smaller is softer size effect is evidenced in striking contrast to the corresponding strain gradient elasticity solutions.

     

Variational formulation of a simplified strain gradient elasticity theory and its application to a pressurized thick-walled cylinder problem

A detailed variational formulation is provided for a simplified strain gradient elasticity theory by using the principle of minimum total potential energy. This leads to the simultaneous determination of the equilibrium equations and the complete boundary conditions of the theory for the first time. To supplement the stress-based formulation, the coordinate-invariant displacement form of the simplified strain gradient elasticity theory is also derived anew. In view of the lack of a consistent and complete formulation, derivation details are included for the tutorial purpose. It is shown that both the stress and displacement forms of the simplified strain gradient elasticity theory obtained reduce to their counterparts in classical elasticity when the strain gradient effect (a measure of the underlying material microstructure) is not considered. As a direct application of the newly obtained displacement form of the theory, the problem of a pressurized thick-walled cylinder is analytically solved. The solution contains a material length scale parameter and can account for microstructural effects, which is qualitatively different from Lamé’s solution in classical elasticity. In the absence of the strain gradient effect, this strain gradient elasticity solution reduces to Lamé’s solution. The numerical results reveal that microstructural effects can be large and Lamé’s solution may not be accurate for materials exhibiting significant microstructure dependence.     

A structural gradient theory of torsion,the effects of pretwist,and the tension of pre-twisted DNA

A structural gradient theory of torsion of thin-walled beams is developed. A non-local estimate of the mean value of the angle of twist of the beam leads to a shear gradient that is energetically consistent with a bi-moment, in the spirit of the averaging theory of Vardoulakis and Giannakopoulos (2006). The geometric details of the cross section play the role of the microstructure of the beam, introducing a size effect in the torsion problem. The appropriate boundary conditions are derived from the variational formulation of the problem. The proposed gradient elasticity theory is identical to Vlasov’s torsion theory of thin walled elastic beams. The tension of pre-twisted DNA is analyzed at high axial loads, where enthalpic elasticity prevails. A size effect is naturally introduced, indicating that shorter DNA lengths lead to stiffer response in torsion. It is shown also that the complete unwinding of DNA triggers the debonding of its strands.     

Characterizing Torsional Properties of Microwires Using an Automated Torsion Balance

In order to characterize the torsional behavior of microwires, an automated torsion tester is established based on the principle of torsion balance. The main challenges in developing a torsion tester at small scales are addressed. An in-situ torsional vibration method for precisely calibrating the torque meter is developed. The torsion tester permits the measurement of torque to nN m, as a function of surface shear strain to a sensitivity of sub-microstrain. Using this technique, we performed (monotonic and/or cyclic) torsion tests on polycrystalline copper and gold wires. It is found that (i) a size effect appears in both the initial yielding and the plastic flow of torsional response; (ii) a reverse plasticity occurs upon unloading in cyclic torsion response; and (iii) the Hall-Petch effect and the strain gradient effect are synergistic. We also performed cyclic torsion tests on human hairs and spider silk which are natural protein fibers with a different morphological structure to metallic wires. It is shown that the single hair exhibits torsional recovery, and that the spider silk displays torsionally superelastic behavior whereby it is able to withstand great shear strain.     

NEW STRAIN GRADIENT THEORY AND ANALYSIS

A new strain gradient theory which is based on energy nonlocal model is proposed in this paper, and the theory is applied to investigate the size effects in thin metallic wire torsion, ultra-thin beam bending and micro-indentation of polycrystalline copper. First, an energy nonlocal model is suggested. Second, based on the model, a new strain gradient theory is derived. Third, the new theory is applied to analyze three representative experiments.     

A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation

In recent years there have been many papers that considered the effects of material length scales in the study of mechanics of solids at micro- and/or nano-scales. There are a number of approaches and, among them, one set of papers deals with Eringen's differential nonlocal model and another deals with the strain gradient theories. The modified couple stress theory, which also accounts for a material length scale, is a form of a strain gradient theory. The large body of literature that has come into existence in the last several years has created significant confusion among researchers about the length scales that these various theories contain. The present paper has the objective of establishing the fact that the length scales present in nonlocal elasticity and strain gradient theory describe two entirely different physical characteristics of materials and structures at nanoscale. By using two principle kernel functions, the paper further presents a theory with application examples which relates the classical nonlocal elasticity and strain gradient theory and it results in a higher-order nonlocal strain gradient theory. In this theory, a higher-order nonlocal strain gradient elasticity system which considers higher-order stress gradients and strain gradient nonlocality is proposed. It is based on the nonlocal effects of the strain field and first gradient strain field. This theory intends to generalize the classical nonlocal elasticity theory by introducing a higher-order strain tensor with nonlocality into the stored energy function. The theory is distinctive because the classical nonlocal stress theory does not include nonlocality of higher-order stresses while the common strain gradient theory only considers local higher-order strain gradients without nonlocal effects in a global sense. By establishing the constitutive relation within the thermodynamic framework, the governing equations of equilibrium and all boundary conditions are derived via the variational approach. Two additional kinds of parameters, the higher-order nonlocal parameters and the nonlocal gradient length coefficients are introduced to account for the size-dependent characteristics of nonlocal gradient materials at nanoscale. To illustrate its application values, the theory is applied for wave propagation in a nonlocal strain gradient system and the new dispersion relations derived are presented through examples for wave propagating in Euler–Bernoulli and Timoshenko nanobeams. The numerical results based on the new nonlocal strain gradient theory reveal some new findings with respect to lattice dynamics and wave propagation experiment that could not be matched by both the classical nonlocal stress model and the contemporary strain gradient theory. Thus, this higher-order nonlocal strain gradient model provides an explanation to some observations in the classical and nonlocal stress theories as well as the strain gradient theory in these aspects.     

Crack tip field andJ-integral with strain gradient effect

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 conventionalJ 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-IK-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, theJ-integral is no longer path independent and increases monotonically as the radius of the calculated circular contour decreases. 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)     

On finite plastic deformation of anisotropic metallic materials

The classical flow theory of plasticity has been extended to the large strain range for anisotropic metallic materials. The following concepts have been incorporated into the constitutive framework: (1) the convected coordinates and the contravariant true stress, (2) an observer independent yield function, (3) the convected rate for general kinematics of deformation, and (4) the rotation of material texture expressed by a constitutive spin. The theory has been applied to the problem of free-end torsion of a thin-walled tube. The predicted results of shear stress-strain curve, axial strain versus shear strain curve, back stress versus shear strain curve, and initial and subsequent yield surfaces compare favorably with experimental data obtained by the author and his co-workers. It has been shown that the yield function defined by the contravariant true stress can account for the distortion of the yield loci.     

基于Hellinger-Reissner变分原理的应变梯度杂交元设计

从一般的偶应力理论出发,基于Hellinger-Reissner变分原理,通过对有限元 离散体系的位移试解引入非协调位移函数,得到了偶应力理论下有限元离散系统的能量相容 条件,并由此建立了应变梯度杂交元的应力函数优化条件. 根据该优化条件,构造了一 个C0类的平面4节点梯度杂交元,数值结果表明,该单元对可压缩和不可压缩状态的 梯度材料均可给出合理的数值结果,再现材料的尺度效应.     

Measurement of elastic-plastic stresses by scattered-light photomechanics

Several birefringent materials were studied for their suitability for use in three-dimensional photoplasticity. This study resulted in the selection of cellulose propionate as model material. Its close match in index of refraction with ordinary mineral oil makes cellulose propionate suitable for scattered-light photomechanics. Viscoelastic behavior of the material is used to simulate elastic-plastic behavior of metals. The stress, strain and optical behavior of the material has been studied under slow loading in finite steps. A successful solution of an elastic-plastic torsion problem was obtained, demonstrating the applicability of the techniques of scattered-light photoplasticity to three-dimensional problems. This experimental method does not require the unloading and slicing of the model, avoiding any errors that might be introduced by residual stresses due to unloading. A major advantage of the method is the use of live loading, which allows the investigation of several load levels with the same model. Stress- and strain-concentration factors for the grooved shaft in torsion showed excellent agreement with Neuber's analytical results. Distributions of shear stress and shear strain across the minimum section of the model were compared to elastic theory. Integration of the shear-stress distributions showed good agreement with the measured values of applied torque.     

On the Torsion of a Class of Nonlinear Fiber Composite Cylinders

This paper presents an analysis of the torsion of a solid or annular circular cylinder consisting of nonlinear material in the form of an elastic matrix with embedded unidirectional elastic fibers parallel to the cylinder axis. The specific class of composite considered is one for which nonlinear fiber-matrix interface slip is captured by uniform cohesive zones of vanishing thickness. Previous work on the effective antiplane shear response of this material leads to a stress–strain relation depending on the interface slip together with an integral equation governing its evolution. Here, we obtain an approximate single mode solution to the integral equation and utilize it to solve the torsion problem. Equations governing the radial distributions of shear stress and interface slip are obtained and formulae for torque–twist rate are presented. The existence of singular surfaces, i.e., surfaces across which the slip and the shear stress experience jump discontinuities are analyzed in detail. Specific results are presented for an interface force law that allows for interface failure in shear.     

Buckling analysis of cylindrical thin-shells using strain gradient elasticity theory

The stability problem of cylindrical shells is addressed using higher-order continuum theories in a generalized framework. The length-scale effect which becomes prominent at microscale can be included in the continuum theory using gradient-based nonlocal theories such as the strain gradient elasticity theories. In this work, expressions for critical buckling stress under uniaxial compression are derived using an energy approach. The results are compared with the classical continuum theory, which can be obtained by setting the length-scale parameters to zero. A special case is obtained by setting two length scale parameters to zero. Thus, it is shown that both the couple stress theory and classical continuum theory forms a special case of the strain gradient theory. The effect of various parameters such as the shell-radius, shell-length, and length-scale parameters on the buckling stress are investigated. The dimensions and constants corresponding to that of a carbon nanotube, where the length-scale effect becomes prominent, is considered for this investigation.     

Wedge and twist disclinations in second strain gradient elasticity

The aim of this paper is to study disclinations in the framework of a second strain gradient elasticity theory. This second strain gradient elasticity has been proposed based on the first and second gradients of the strain tensor by Lazar et al. [Lazar, M., Maugin, G.A., Aifantis, E.C., 2006. Dislocations in second strain gradient elasticity. Int. J. Solids Struct. 43, 1787–1817]. Such a theory is an extension of the first strain gradient elasticity [Lazar, M., Maugin, G.A., 2005. Nonsingular stress and strain fields of dislocations and disclinations in first strain gradient elasticity. Int. J. Eng. Sci. 43, 1157–1184] with triple stress. By means of the stress function method, the exact analytical solutions for stress and strain fields of straight disclinations in an infinitely extended linear isotropic medium have been found. An important result is that the force stress, double stress and triple stress produced by wedge and twist disclinations are nonsingular. Meanwhile, the corresponding elastic strain and its gradients are also nonsingular. Analytical results indicate that the second strain gradient theory has the capacity of eliminating all unphysical singularities of physical fields.     

Stress concentration and effective stiffness of aligned fiber reinforced composite with anisotropic constituents

The paper addresses the problem of calculation of the local stress field and effective elastic properties of a unidirectional fiber reinforced composite with anisotropic constituents. For this aim, the representative unit cell approach has been utilized. The micro geometry of the composite is modeled by a periodic structure with a unit cell containing multiple circular fibers. The number of fibers is sufficient to account for the micro structure statistics of composite. A new method based on the multipole expansion technique is developed to obtain the exact series solution for the micro stress field. The method combines the principle of superposition, technique of complex potentials and some new results in the theory of special functions. A proper choice of potentials and new results for their series expansions allow one to reduce the boundary-value problem for the multiple-connected domain to an ordinary, well-posed set of linear algebraic equations. This reduction provides high numerical efficiency of the developed method. Exact expressions for the components of the effective stiffness tensor have been obtained by analytical averaging of the strain and stress fields.