Torsion Of Thin Wires Research Articles

A modified conventional theory of mechanism-based strain gradient plasticity considering both size and damage effects

The in-service deformation damage cannot be avoided, even for micro-scale metallic materials. To characterize the coupling effect of size and damage in micro-metallic materials, a modified incremental constitutive model is proposed on the basis of the conventional theory of mechanism-based strain gradient plasticity (CMSG), in which the assumptions of both small strain and isotropic damage are adopted. Other than retaining the essential features of the CMSG theory, the in-service deformation damage effects on the elastic moduli, plastic yielding criterion and intrinsic length scale, are well included. Typical micro-scale experimental results, including those from the torsion of thin wires, bending of ultra-thin beams, and tension and compression of micro-particle-reinforced metal matrix composites (MPMMCs), are theoretically analyzed using the new modified theory. Good agreements between the theoretical predictions and the experimental measurements are achieved, especially for the mechanical behaviors of MPMMCs subjected to relatively large deformation. Furthermore, it is interesting to find, upon increasing the uniaxial tension, the dominant damage mode in MPMMCs will transit from interface debonding to matrix damage, while matrix damage is always dominated in MPMMCs under uniaxial compression. In this study, we provide a convenient and precise way to characterize the in-service mechanical behaviors of micro-scale material systems, especially the commonly used composites with a characteristic length scale of micrometers.

The coupling effect of size and damage in micro-scale metallic materials

In order to characterize the deformation behavior accompanying damage of microstructures in micro-scale metallic materials, a new theoretical model is developed based on a low order strain gradient plasticity theory. Not only the size effect induced by strain gradient plasticity but also the one of microstructure damage induced by deformation is considered. The feature of the new theory includes two aspects: the strain gradient is taken as an internal variable to affect the tangential hardening modulus without the introduction of high-order stress or high-order boundary condition; both the elastic modulus and the involved intrinsic length are influenced by the microstructural damage. Two commonly used samples with size effect in micro-scales, i.e., the thin wire torsion and the ultra-thin beam bending, are re-analyzed with the new model. It is found that stiffness of the micro-scale material is gradually reduced along with the increasing deformation and the theoretical prediction is consistent well with the existing experimental data. All the results demonstrate that the present theory should be a promising way for predicting the mechanical behavior of a more complex system, for example, the micro-particle reinforced metal matrix composite and the recent-hot-studied nano-crystallized gradient materials.

Gradient plasticity used for modeling extrinsic and intrinsic size effects in the torsion of Au microwires

AbstractThe gradient plasticity theory proposed by Aifantis and coworkers has been successfully used to model size effect phenomena at the microscale and nanoscale, by introducing into the formulation an internal length scale associated with the phenomenological coefficients of the gradient plasticity model. In this paper, Aifantis’ gradient plasticity theory is applied to model the sample size-dependent torsion of thin wires, with a strain-dependent internal length scale as well as grain size dependence based on the Hall-Petch relationship. This study reveals that internal length scale is related with sample size and grain size, with such a connection determined by the ductility of the material.

Correlating the internal length in strain gradient plasticity theory with the microstructure of material

The internal length is the governing parameter in strain gradient theories which among other things have been used successfully to interpret size effects at the microscale. Physically, the internal length is supposed to be related with the microstructure of the material and evolves during the deformation. Based on Taylor hardening law, we propose a power-law relationship to describe the evolution of the variable internal length with strain. Then, the classical Fleck–Hutchinson strain gradient theory is extended with a strain-dependent internal length, and the generalized Fleck–Hutchinson theory is confirmed here, by comparing our model predictions to recent experimental data on tension and torsion of thin wires with varying diameter and grain size. Our work suggests that the internal length is a configuration-dependent parameter, closely related to dislocation characteristics and grain size, as well as sample geometry when this affects either the underlying microstructure or the ductility of the material.

Formulation of strain gradient plasticity with interface energy in a consistent thermodynamic framework

In this work, the strain gradient formulation is used within the context of the thermodynamic principle, internal state variables, and thermodynamic and dissipation potentials. These in turn provide balance of momentum, boundary conditions, yield condition and flow rule, and free energy and dissipative energies. This new formulation contributes to the following important related issues: (i) the effects of interface energy that are incorporated into the formulation to address various boundary conditions, strengthening and formation of the boundary layers, (ii) nonlocal temperature effects that are crucial, for instance, for simulating the behavior of high speed machining for metallic materials using the strain gradient plasticity models, (iii) a new form of the nonlocal flow rule, (iv) physical bases of the length scale parameter and its identification using nano-indentation experiments and (v) a wide range of applications of the theory. Applications to thin films on thick substrates for various loading conditions and torsion of thin wires are investigated here along with the appropriate length scale effect on the behavior of these structures. Numerical issues of the theory are discussed and results are obtained using Matlab and Mathematica for the nonlinear ordinary differential equations (NODE) which constitute the boundary value problem. This study reveals that the micro-stress term has an important effect on the development of the boundary layers and hardening of the material at both hard and soft interface boundary conditions in thin films. The interface boundary conditions are described by the interfacial length scale and interfacial strength parameters. These parameters are important to control the size effect and hardening of the material. For more complex geometries the generalized form of the boundary value problem using the nonlocal finite element formulation is required to address the problems involved.

A reformulation of mechanism-based strain gradient plasticity

We reformulate the theory of mechanism-based strain gradient plasticity (MSG) [J. Mech. Phys. Solids 47 1239 (1999); ibid 48 99 (2000); J. Mater. Res. 15 1786 (2000)] that involves the third-order tensor of higher-order stress to a much simpler version within Fleck and Hutchinson's theoretical framework of strain gradient plasticity theory [J. Mech. Phys. Solids 49 2245 (2001)]. Similar to MSG, the new theory is also based on the Taylor dislocation model, but the higher-order stress is a vector (rather than a third-order tensor) and is the work conjugate of the gradient of plastic strain. The intrinsic material length in the new theory is still related to the shear modulus, yield stress and Burgers vector in the same way as in MSG, but the simplicity of the new theory makes it much easier to be implemented in the finite element analysis. We present a few examples that display strong size effects at the micron and submicron scales, including the shear of an infinite layer, torsion of thin wires, bending of thin beams and growth of microvoids. These examples show that this new strain gradient plasticity theory based on the Taylor dislocation model captures the strong size effect.

Theory of deformation in small volumes of material

It has long been controversial as to whether small volumes of material are, or should be, more resistant to plastic deformation than is implied by the yield strength of bulk material. We generalize the established theory of critical thickness for strained layers to show that there is an increase in the initial yield stress for geometrical reasons wherever there is a strain gradient. The theory is in quantitative agreement, without free fitting parameters, with the classic experimental data on the torsion of thin wires and on the bending of thin beams.

Incompatibility and a simple gradient theory of plasticity

In the continuum theory, at finite strains the crystal lattice is assumed to distort only elastically during plastic flow, while generally the elastic distortion itself is not compatible with a single-valued displacement field. Lattice incompatibility is characterized by a certain skew-symmetry property of the gradient of the elastic deformation field, and this measure can play a natural role in nonlocal theories of plasticity. A simple constitutive proposal is discussed where incompatibility only enters the instantaneous hardening relations. As a result, the incremental boundary value problem for rate-independent and rate-dependent behaviors has a classical structure and rather straightforward modifications of standard finite element programs can be utilized. Two examples are presented in this paper: one for size-scale effects in the torsion of thin wires in the setting of an isotropic J 2 flow theory and the other for hardening of microstructures containing small particles embedded in a single crystal matrix.

A new hardening law for strain gradient plasticity

A new hardening law of the strain gradient theory is proposed in this paper, which retains the essential structure of the incremental version of conventional J 2 deformation theory and obeys thermodynamic restrictions. The key feature of the new proposal is that the term of strain gradient plasticity is represented as an internal variable to increase the tangent modulus. This feature which is in contrast to several proposed theories, allows the problem of incremental equilibrium equations to be stated without higher-order stress, higher-order strain rates or extra boundary conditions. The general idea is presented and compared with the theory given by Fleck and Hutchinson (Adv. in Appl. Mech. (1997) 295). The new hardening law is demonstrated by two experimental tests i.e. thin wire torsion and ultra-thin beam bending tests. The present theoretical results agree well with the experiment results.

Mechanism-based strain gradient plasticity—II. Analysis

A mechanism-based theory of strain gradient (MSG) plasticity has been proposed in Part I of this paper. The theory is based on a multiscale framework linking the microscale notion of statistically stored and geometrically necessary dislocations to the mesoscale notion of plastic strain and strain gradient. This theory is motivated by our recent analysis of indentation experiments which strongly suggest a linear dependence of the square of plastic flow stress on strain gradient. Such a linear dependence is consistent with the Taylor plastic work hardening model relating the flow stress to dislocation density. This part of this paper provides a detailed analysis of the new theory, including equilibrium equations and boundary conditions, constitutive equations for the mechanism-based strain gradient plasticity, and kinematic relations among strains, strain gradients and displacements. The theory is used to investigate several phenomena that are influenced by plastic strain gradients. In bending of thin beams and torsion of thin wires, mechanism-based strain gradient plasticity gives a significant increase in scaled bending moment and scaled torque due to strain gradient effects. For the growth of microvoids and cavitation instabilities, however, it is found that strain gradients have little effect on micron-sized voids, but submicron-sized voids can have a larger resistance against void growth. Finally, it is shown from the study of bimaterials in shear that the mesoscale cell size has little effect on global physical quantities (e.g. applied stresses), but may affect the local deformation field significantly.