Suppressed plastic deformation at blunt crack-tips due to strain gradient effects

Large deformation gradients occur near a crack-tip and strain gradient dependent crack-tip deformation and stress fields are expected. Nevertheless, for material length scales much smaller than the scale of the deformation gradients, a conventional elastic–plastic solution is obtained. On the other hand, for significant large material length scales, a conventional elastic solution is obtained. This transition in behaviour is investigated based on a finite strain version of the Fleck–Hutchinson strain gradient plasticity model from 2001. The predictions show that for a wide range of material parameters, the transition from the conventional elastic–plastic to the elastic solution occurs for length scales ranging from 0.001 times the size of the plastic zone to a length scale of the same order of magnitude as the plastic zone.     

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.     

Strain gradient crystal plasticity effects on flow localization

In metal grains one of the most important failure mechanisms involves shear band localization. As the band width is small, the deformations are affected by material length scales. To study localization in single grains a rate-dependent crystal plasticity formulation for finite strains is presented for metals described by the reformulated Fleck–Hutchinson strain gradient plasticity theory. The theory is implemented numerically within a finite element framework using slip rate increments and displacement increments as state variables. The formulation reduces to the classical crystal plasticity theory in the absence of strain gradients. The model is used to study the effect of an internal material length scale on the localization of plastic flow in shear bands in a single crystal under plane strain tension. It is shown that the mesh sensitivity is removed when using the nonlocal material model considered. Furthermore, it is illustrated how different hardening functions affect the formation of shear bands.     

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.     

A study of microbend test by strain gradient plasticity

Metallic materials display strong size effect when the characteristic length associated with plastic deformation is on the order of microns. This size effect cannot be explained by classical plasticity theories since their constitutive relations do not have an intrinsic material length. Strain gradient plasticity has been developed to extend continuum plasticity to the micron or submicron regime. One major issue in strain gradient plasticity is the determination of the intrinsic material length that scales with strain gradients, and several microbend test specimens have been designed for this purpose. We have studied different microbend test specimens using the theory of strain gradient plasticity. The pure bending specimen, cantilever beam, and the microbend test specimen developed by Stolken and Evans (Stolken, J.S., Evans, A.G., 1998. A microbend test method for measuring the plasticity length scale Acta Mater. 46, 5109–5115) are found suitable for the determination of intrinsic material length in strain gradient plasticity. However, the double cantilever beam (both ends clamped) is unsuitable since its deformation is dominated by axial stretching. The strain gradient effects significantly increase the bending stiffness of a microbend test specimen. The deflection of a 10-μm thick beam is only a few percent of that estimated by classical plasticity.     

Perfect elastic-viscoplastic field at mode I dynamic propagating crack-tip

The viscosity of material is considered at propagating crack-tip. Under the assumption that the artificial viscosity coefficient is in inverse proportion to power law of the plastic strain rate, an elastic-viscoplastic asymptotic analysis is carried out for moving crack-tip fields in power-hardening materials under plane-strain condition. A continuous solution is obtained containing no discontinuities. The variations of numerical solution are discussed for mode I crack according to each parameter. It is shown that stress and strain both possess exponential singularity. The elasticity, plasticity and viscosity of material at crack-tip only can be matched reasonably under linear-hardening condition. And the tip field contains no elastic unloading zone for mode I crack. It approaches the limiting case, crack-tip is under ultra-viscose situation and energy accumulates, crack-tip begins to propagate under different compression situations.     

Mode I and mode II crack tip asymptotic fields with strain gradient effects

The strain gradient effect becomes significant when the size of fracture process zone around a crack tip is comparable to the intrinsic material lengthl, typically of the order of microns. Using the new strain gradient deformation theory given by Chen and Wang, the asymptotic fields near a crack tip in an elastic-plastic material with strain gradient effects are investigated. It is established that the dominant strain field is irrotational. For mode I plane stress crack tip asymptotic field, the stress asymptotic field and the couple stress asymptotic field can not exist simultaneously. In the stress dominated asymptotic field, the angular distributions of stresses are consistent with the classical plane stress HRR field; In the couple stress dominated asymptotic field, the angular distributions of couple stresses are consistent with that obtained by Huang et al. For mode II plane stress and plane strain crack tip asymptotic fields, only the stress-dominated asymptotic fields exist. The couple stress asymptotic field is less singular than the stress asymptotic fields. The stress asymptotic fields are the same as mode II plane stress and plane strain HRR fields, respectively. The increase in stresses is not observed in strain gradient plasticity for mode I and mode II, because the present theory is based only on the rotational gradient of deformation and the crack tip asymptotic fields are irrotational and dominated by the stretching gradient. The project supported by the National Natural Science Foundation of China (19704100), National Natural Science Foundation of Chinese Academy of Sciences (KJ951-1-20), CAS K.C. Wong Post-doctoral Research Award Fund and Post-doctoral Science Fund of China     

Size-dependent yield strength of thin films

Biaxial strain and pure shear of a thin film are analysed using a strain gradient plasticity theory presented by Gudmundson [Gudmundson, P., 2004. A unified treatment of strain gradient plasticity. Journal of the Mechanics and Physics of Solids 52, 1379–1406]. Constitutive equations are formulated based on the assumption that the free energy only depends on the elastic strain and that the dissipation is influenced by the plastic strain gradients. The three material length scale parameters controlling the gradient effects in a general case are here represented by a single one. Boundary conditions for plastic strains are formulated in terms of a surface energy that represents dislocation buildup at an elastic/plastic interface. This implies constrained plastic flow at the interface and it enables the simulation of interfaces with different constitutive properties. The surface energy is also controlled by a single length scale parameter, which together with the material length scale defines a particular material.Numerical results reveal that a boundary layer is developed in the film for both biaxial and shear loading, giving rise to size effects. The size effects are strongly connected to the buildup of surface energy at the interface. If the interface length scale is small, the size effect vanishes. For a stiffer interface, corresponding to a non-vanishing surface energy at the interface, the yield strength is found to scale with the inverse of film thickness.Numerical predictions by the theory are compared to different experimental data and to dislocation dynamics simulations. Estimates of material length scale parameters are presented.     

Effects of characteristic material lengths on mode III crack propagation in couple stress elastic–plastic materials

The asymptotic fields near the tip of a crack steadily propagating in a ductile material under Mode III loading conditions are investigated by adopting an incremental version of the indeterminate theory of couple stress plasticity displaying linear and isotropic strain hardening. The adopted constitutive model is able to account for the microstructure of the material by incorporating two distinct material characteristic lengths. It can also capture the strong size effects arising at small scales, which results from the underlying microstructures. According to the asymptotic crack tip fields for a stationary crack provided by the indeterminate theory of couple stress elasticity, the effects of microstructure mainly consist in a switch in the sign of tractions and displacement and in a substantial increase in the singularity of tractions ahead of the crack-tip, with respect to the classical solution of LEFM and EPFM. The increase in the stress singularity also occurs for small values of the strain hardening coefficient and is essentially due to the skew-symmetric stress field, since the symmetric stress field turns out to be non-singular. Moreover, the obtained results show that the ratio η introduced by Koiter has a limited effect on the strength of the stress singularity. However, it displays a strong influence on the angular distribution of the asymptotic crack tip fields.     

Plane-Strain Fracture with Curvature-Dependent Surface Tension: Mixed-Mode Loading

There have been a number of recent papers by various authors addressing static fracture in the setting of the linearized theory of elasticity in the bulk augmented by a model for surface mechanics on fracture surfaces with the goal of developing a fracture theory in which stresses and strains remain bounded at crack-tips without recourse to the introduction of a crack-tip cohesive-zone or process-zone. In this context, surface mechanics refers to viewing interfaces separating distinct material phases as dividing surfaces, in the sense of Gibbs, endowed with excess physical properties such as internal energy, entropy and stress. One model for the mechanics of fracture surfaces that has received much recent attention is based upon the Gurtin-Murdoch surface elasticity model. However, it has been shown recently that while this model removes the strong (square-root) crack-tip stress/strain singularity, it replaces it with a weak (logarithmic) one. A simpler model for surface stress assumes that the surface stress tensor is Eulerian, consisting only of surface tension. If surface tension is assumed to be a material constant and the classical fracture boundary condition is replaced by the jump momentum balance relations on crack surfaces, it has been shown that the classical strong (square-root) crack-tip stress/strain singularity is removed and replaced by a weak, logarithmic singularity. If, in addition, surface tension is assumed to have a (linearized) dependence upon the crack-surface mean-curvature, it has been shown for pure mode I (opening mode), the logarithmic stress/strain singularity is removed leaving bounded crack-tip stresses and strains. However, it has been shown that curvature-dependent surface tension is insufficient for removing the logarithmic singularity for mixed mode (mode I, mode II) cracks. The purpose of this note is to demonstrate that a simple modification of the curvature-dependent surface tension model leads to bounded crack-tip stresses and strains under mixed mode I and mode II loading.     

Mechanism-based strain gradient plasticity in C0 axisymmetric element

Non-uniform plastic deformation of materials exhibits a strong size dependence when the material and deformation length scales are of the same order at micro- and nano-metre levels. Recent progresses in testing equipment and computational facilities enhancing further the study on material characterization at these levels confirmed the size effect phenomenon. It has been shown that at this length scale, the material constitutive condition involves not only the state of strain but also the strain gradient plasticity. In this study, C⁰ axisymmetric element incorporating the mechanism-based strain gradient plasticity is developed. Classical continuum plasticity approach taking into consideration Taylor dislocation model is adopted. As the length scale and strain gradient affect only the constitutive relation, it is unnecessary to introduce either additional model variables or higher order stress components. This results in the ease and convenience in the implementation. Additional computational efforts and resources required of the proposed approach as compared with conventional finite element analyses are minimal. Numerical results on indentation tests at micron and submicron levels confirm the necessity of including the mechanism-based strain gradient plasticity with appropriate inherent material length scale. It is also interesting to note that the material is hardened under Berkovich compared to conical indenters when plastic strain gradient is considered but softened otherwise.     

I型定常扩展裂纹尖端的弹黏塑性场

考虑材料在扩展裂纹尖端的黏性效应，假设黏性系数与塑性应变率的幂次成反比，对幂硬化材料中平面应变扩展裂纹尖端场进行了弹黏塑性渐近分析，得到了不含间断的连续解，并讨论了I型裂纹数值解的性质随各参数的变化规律. 分析表明应力和应变均具有幂奇异性，并且只有在线性硬化时，尖端场的弹、黏、塑性才可以合理匹配. 对于I型裂纹，裂尖场不含弹性卸载区. 当裂纹扩展速度趋于零时，动态解趋于准静态解，表明准静态解是动态解的特殊形式；如果进一步考虑硬化系数为零的极限情况，便可退化为Hui和Riedel的非线性黏弹性解.     

Defects in flexoelectric solids

A solid is said to be flexoelectric when it polarizes in proportion to strain gradients. Since strain gradients are large near defects, we expect the flexoelectric effect to be prominent there and decay away at distances much larger than a flexoelectric length scale. Here, we quantify this expectation by computing displacement, stress and polarization fields near defects in flexoelectric solids. For point defects we recover some well known results from strain gradient elasticity and non-local piezoelectric theories, but with different length scales in the final expressions. For edge dislocations we show that the electric potential is a maximum in the vicinity of the dislocation core. We also estimate the polarized line charge density of an edge dislocation in an isotropic flexoelectric solid which is in agreement with some measurements in ice. We perform an asymptotic analysis of the crack tip fields in flexoelectric solids and show that our results share some features from solutions in strain gradient elasticity and piezoelectricity. We also compute the energy release rate for cracks using simple crack face boundary conditions and use them in classical criteria for crack growth to make predictions. Our analysis can serve as a starting point for more sophisticated analytic and computational treatments of defects in flexoelectric solids which are gaining increasing prominence in the field of nanoscience and nanotechnology.     

Mode I crack tip field with strain gradient effects

A plane strain mode I 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^[19]; the angular distributions of couple stresses are the same as the couple stress results^[19]. 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^[19], but for the stress dominated field, the angular distributions of stresses are consistent with the HRR field; for the couple stress dominated field, the angular distributions of couple stresses are consistent with those in Ref. [19]. 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 I is dominated by the tension gradient, which will be shown in another paper. Supported by the National Science Foundation of China (No. 19704100), Science Foundation of Chinese Academy of Sciences (Project KJ951-1-20), CAS K. C. Wong Post-doctoral Research Award Fund and the Post Doctoral Science Fund of China.     

Wedge indentation of thin films modelled by strain gradient plasticity

A plane strain study of wedge indentation of a thin film on a substrate is performed. The film is modelled with the strain gradient plasticity theory by Gudmundson [Gudmundson, P., 2004. A unified treatment of strain gradient plasticity. Journal of the Mechanics and Physics of Solids 52, 1379–1406] and analysed using finite element simulations. Several trends that have been experimentally observed elsewhere are captured in the predictions of the mechanical behaviour of the thin film. Such trends include increased hardness at shallow depths due to gradient effects as well as increased hardness at larger depths due to the influence of the substrate. In between, a plateau is found which is observed to scale linearly with the material length scale parameter. It is shown that the degree of hardening of the material has a strong influence on the substrate effect, where a high hardening modulus gives a larger impact on this effect. Furthermore, pile-up deformation dominated by plasticity at small values of the internal length scale parameter is turned into sink-in deformation where plasticity is suppressed for larger values of the length scale parameter. Finally, it is demonstrated that the effect of substrate compliance has a significant effect on the hardness predictions if the effective stiffness of the substrate is of the same order as the stiffness of the film.     

Perfect elastic-viscoplastic field at mode Ⅰ dynamic propagating crack-tip

The viscosity of material is considered at propagating crack-tip. Under the assumption that the artificial viscosity coefficient is in inverse proportion to power law of the plastic strain rate, an elastic-viscoplastic asymptotic analysis is carried out for moving crack-tip fields in power-hardening materials under plane-strain condition. A continuous solution is obtained containing no discontinuities. The variations of numerical solution are discussed for mode Ⅰ crack according to each parameter. It is shown that stress and strain both possess exponential singularity. The elasticity, plasticity and viscosity of material at crack-tip only can be matched reasonably under linear-hardening condition. And the tip field contains no elastic unloading zone for mode Ⅰ crack. It approaches the limiting case, crack-tip is under ultra-viscose situation and energy accumulates, crack-tip begins to propagate under different compression situations.     

Debonding failure and size effects in micro-reinforced composites

Failure in micro-reinforced composites is investigated numerically using the strain-gradient plasticity theory of Gudmundson [Gudmundson, P., 2004. A unified treatment of strain gradient plasticity. Journal of the Mechanics and Physics of Solids 52 (6) 1379–1406] in a plane strain visco-plastic formulation. Bi-axially loaded unit cells are used and failure is modeled using a cohesive zone at the reinforcement interface. During debonding a sudden stress drop in the overall average stress–strain response is observed. Adaptive higher-order boundary conditions are imposed at the reinforcement interface for realistically modeling the restrictions on moving dislocations as debonding occurs. It is found that the influence of the imposed higher-order boundary conditions at the interface is minor. If strain-gradient effects are accounted for a void with a smooth shape develops at the reinforcement interface while a smaller void having a sharp tip nucleates if strain-gradient effects are excluded. Using orthogonalization of the plastic strain gradient with three corresponding material length scales it is found that, the first length scale dominates the evaluated overall average stress–strain response, the second one only has a small effect and the third one has an intermediate effect. Finally, studies of reinforcement having elliptical cross-sections show rather significant gradients of stress which is not seen for the corresponding circular cross-sections. Also, an increased drop in the overall load carrying capacity is observed for cross-sections elongated perpendicular to the principal tensile direction compared to the corresponding circular cross-sections.     

Effect of yield surface vertex on crack-tip fields in mode III

An asymptotic crack-tip analysis of stress and strain fields is carried out for an antiplane shear crack (Mode III) based on a corner theory of plasticity. Because of the nonproportional loading history experienced by a material element near the crack tip in stable crack growth, classical flow theory may predict an overly stiff response of the elastic plastic solid, as is the case in plastic buckling problems. The corner theory used here accounts for this anomalous behavior. The results are compared with those of a similar analysis based on the J₂ flow theory of plasticity.     

Influence of material parameters and crystallography on the size effects describable by means of strain gradient plasticity

In the context of single-crystal strain gradient plasticity, we focus on the simple shear of a constrained strip in order to study the effects of the material parameters possibly involved in the modelling. The model consists of a deformation theory suggested and left undeveloped by Bardella [(2007). Some remarks on the strain gradient crystal plasticity modelling, with particular reference to the material length scales involved. Int. J. Plasticity 23, 296–322] in which, for each glide, three dissipative length scales are considered; they enter the model through the definition of an effective slip which brings into the isotropic hardening function the relevant plastic strain gradients, averaged by means of a p-norm. By means of the defect energy (i.e., a function of Nye's dislocation density tensor added to the free energy; see, e.g., Gurtin [2002. A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessary dislocations. J. Mech. Phys. Solids 50, 5–32]), the model further involves an energetic material length scale. The application suggests that two dissipative length scales may be enough to qualitatively describe the size effect of metals at the microscale, and they are chosen in such a way that the higher-order state variables of the model be the dislocation densities. Moreover, we show that, depending on the crystallography, the size effect governed by the defect energy may be different from what expected (based on the findings of [Bardella, L., 2006. A deformation theory of strain gradient crystal plasticity that accounts for geometrically necessary dislocations. J. Mech. Phys. Solids 54, 128–160] and [Gurtin et al. 2007. Gradient single-crystal plasticity with free energy dependent on dislocation densities. J. Mech. Phys. Solids 55, 1853–1878]), leading mostly to some strengthening. In order to investigate the model capability, we also exploit a Γ-convergence technique to find closed-form solutions in the “isotropic limit”. Finally, we analytically show that in the “perfect plasticity” case, should the dissipative length scales be set to zero, the presence of the sole energetic length scale may lead, as in standard plasticity, to non-uniqueness of solutions.