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.     

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.     

Onset of failure in finitely strained layered composites subjected to combined normal and shear loading

A limiting factor in the design of fiber-reinforced composites is their failure under axial compression along the fiber direction. These critical axial stresses are significantly reduced in the presence of shear stresses. This investigation is motivated by the desire to study the onset of failure in fiber-reinforced composites under arbitrary multi-axial loading and in the absence of the experimentally inevitable imperfections and finite boundaries.By using a finite strain continuum mechanics formulation for the bifurcation (buckling) problem of a rate-independent, perfectly periodic (layered) solid of infinite extent, we are able to study the influence of load orientation, material properties and fiber volume fraction on the onset of instability in fiber-reinforced composites. Two applications of the general theory are presented in detail, one for a finitely strained elastic rubber composite and another for a graphite-epoxy composite, whose constitutive properties have been determined experimentally. For the latter case, extensive comparisons are made between the predictions of our general theory and the available experimental results as well as to the existing approximate structural theories. It is found that the load orientation, material properties and fiber volume fraction have substantial effects on the onset of failure stresses as well as on the type of the corresponding mode (local or global).     

Experiments and theory in strain gradient elasticity

Conventional strain-based mechanics theory does not account for contributions from strain gradients. Failure to include strain gradient contributions can lead to underestimates of stresses and size-dependent behaviors in small-scale structures. In this paper, a new set of higher-order metrics is developed to characterize strain gradient behaviors. This set enables the application of the higher-order equilibrium conditions to strain gradient elasticity theory and reduces the number of independent elastic length scale parameters from five to three. On the basis of this new strain gradient theory, a strain gradient elastic bending theory for plane-strain beams is developed. Solutions for cantilever bending with a moment and line force applied at the free end are constructed based on the new higher-order bending theory. In classical bending theory, the normalized bending rigidity is independent of the length and thickness of the beam. In the solutions developed from the higher-order bending theory, the normalized higher-order bending rigidity has a new dependence on the thickness of the beam and on a higher-order bending parameter, b_h. To determine the significance of the size dependence, we fabricated micron-sized beams and conducted bending tests using a nanoindenter. We found that the normalized beam rigidity exhibited an inverse squared dependence on the beam's thickness as predicted by the strain gradient elastic bending theory, and that the higher-order bending parameter, b_h, is on the micron-scale. Potential errors from the experiments, model and fabrication were estimated and determined to be small relative to the observed increase in beam's bending rigidity. The present results indicate that the elastic strain gradient effect is significant in elastic deformation of small-scale structures.     

A thermodynamic based higher-order gradient theory for size dependent plasticity

A physically motivated and thermodynamically consistent formulation of small strain higher-order gradient plasticity theory is presented. Based on dislocation mechanics interpretations, gradients of variables associated with kinematic and isotropic hardenings are introduced. This framework is a two non-local parameter framework that takes into consideration large variations in the plastic strain tensor and large variations in the plasticity history variable; the equivalent (effective) plastic strain. The presence of plastic strain gradients is motivated by the evolution of dislocation density tensor that results from non-vanishing net Burgers vector and, hence, incorporating additional kinematic hardening (anisotropy) effects through lattice incompatibility. The presence of gradients in the effective (scalar) plastic strain is motivated by the accumulation of geometrically necessary dislocations and, hence, incorporating additional isotropic hardening effects (i.e. strengthening). It is demonstrated that the non-local yield condition, flow rule, and non-zero microscopic boundary conditions can be derived directly from the principle of virtual power. It is also shown that the local Clausius–Duhem inequality does not hold for gradient-dependent material and, therefore, a non-local form should be adopted. The non-local Clausius–Duhem inequality has an additional term that results from microstructural long-range energy interchanges between the material points within the body. A detailed discussion on the physics and the application of proper microscopic boundary conditions, either on free surfaces, clamped surfaces, or intermediate constrained surfaces, is presented. It is shown that there is a close connection between interface/surface energy of an interface or free surface and the microscopic boundary conditions in terms of microtraction stresses. Some generalities and utility of this theory are discussed and comparisons with other gradient theories are given. Applications of the proposed theory for size effects in thin films are presented.     

Size-dependent vibration of functionally graded curved microbeams based on the modified strain gradient elasticity theory

On the basis of the modified strain gradient elasticity theory, the free vibration characteristics of curved microbeams made of functionally graded materials (FGMs) whose material properties vary in the thickness direction are investigated. A size-dependent first-order shear deformation beam model is developed containing three internal material length scale parameters to incorporate small-scale effect. Through Hamilton’s principle, the higher-order governing equations of motion and boundary conditions are derived. Natural frequencies of FGM curved microbeams corresponding to different mode numbers are evaluated for over a wide range of material property gradient index, dimensionless length scale parameter and aspect ratio. Moreover, the results obtained via the present non-classical first-order shear deformation beam model are compared with those of degenerated beam models based on the modified couple stress and the classical theories. It is found that the difference between the natural frequencies predicted by the various beam models is more significant for lower values of dimensionless length scale parameter and higher values of mode number.     

Interfacial gradient plasticity governs scale-dependent yield strength and strain hardening rates in micro/nano structured metals

The effect of the material microstructural interfaces increases as the surface-to-volume ratio increases. It is shown in this work that interfacial effects have a profound impact on the scale-dependent yield strength and strain hardening of micro/nano-systems even under uniform stressing. This is achieved by adopting a higher-order gradient-dependent plasticity theory [Abu Al-Rub, R.K., Voyiadjis, G.Z., Bammann, D.J., 2007. A thermodynamic based higher-order gradient theory for size dependent plasticity. Int. J. Solids Struct. 44, 2888–2923] that enforces microscopic boundary conditions at interfaces and free surfaces. Those nonstandard boundary conditions relate a microtraction stress to the interfacial energy at the interface. In addition to the nonlocal yield condition for the material’s bulk, a microscopic yield condition for the interface is presented, which determines the stress at which the interface begins to deform plastically and harden. Hence, two material length scales are incorporated: one for the bulk and the other for the interface. Different expressions for the interfacial energy are investigated. The effect of the interfacial yield strength and interfacial hardening are studied by analytically solving a one-dimensional Hall–Petch-type size effect problem. It is found that when assuming compliant interfaces the interface properties control both the material’s global yield strength and rates of strain hardening such that the interfacial strength controls the global yield strength whereas the interfacial hardening controls both the global yield strength and strain hardening rates. On the other hand, when assuming a stiff interface, the bulk length scale controls both the global yield strength and strain hardening rates. Moreover, it is found that in order to correctly predict the increase in the yield strength with decreasing size, the interfacial length scale should scale the magnitude of both the interfacial yield strength and interfacial hardening.     

Shear banding analysis of geomaterials by strain gradient enhanced damage model

Shear band localization is investigated by a strain-gradient-enhanced damage model for quasi-brittle geomaterials. This model introduces the strain gradients and their higher-order conjugate stresses into the framework of continuum damage mechanics. The influence of the strain gradients on the constitutive behaviour is taken into account through a generalized damage evolutionary law. A weak-form variational principle is employed to address the additional boundary conditions introduced by the incorporation of the strain gradients and the conjugate higher-order stresses. Damage localization under simple shear condition is analytically investigated by using the theory of discontinuous bifurcation and the concept of the second-order characteristic surface. Analytical solutions for the distributions of strain rates and strain gradient rates, as well as the band width of localised damage are found. Numerical analysis demonstrates the shear band width is proportionally related to the internal length scale through a coefficient function of Poisson’s ratio and a parameter representing the shape of uniaxial stress–strain curve. It is also shown that the obtained distributions of strains and strain gradients are well in accordance with the underlying assumptions for the second-order discontinuous shear band boundary and the weak discontinuous bifurcation theory.     

Analysis of micro-sized beams for various boundary conditions based on the strain gradient elasticity theory

Bending analysis of micro-sized beams based on the Bernoulli-Euler beam theory is presented within the modified strain gradient elasticity and modified couple stress theories. The governing equations and the related boundary conditions are derived from the variational principles. These equations are solved analytically for deflection, bending, and rotation responses of micro-sized beams. Propped cantilever, both ends clamped, both ends simply supported, and cantilever cases are taken into consideration as boundary conditions. The influence of size effect and additional material parameters on the static response of micro-sized beams in bending is examined. The effect of Poisson’s ratio is also investigated in detail. It is concluded from the results that the bending values obtained by these higher-order elasticity theories have a significant difference with those calculated by the classical elasticity theory.     

A unified treatment of strain gradient plasticity

A theoretical framework is presented that has potential to cover a large range of strain gradient plasticity effects in isotropic materials. Both incremental plasticity and viscoplasticity models are presented. Many of the alternative models that have been presented in the literature are included as special cases. Based on the expression for plastic dissipation, it is in accordance with Gurtin (J. Mech. Phys. Solids 48 (2000) 989; Int. J. Plast. 19 (2003) 47) argued that the plastic flow direction is governed by a microstress q_ij and not the deviatoric Cauchy stress σ_ij′ that has been assumed by many others. The structure of the governing equations is of second order in the displacements and the plastic strains which makes it comparatively easy to implement in a finite element programme. In addition, a framework for the formulation of consistent boundary conditions is presented. It is shown that there is a close connection between surface energy of an interface and boundary conditions in terms of plastic strains and moment stresses. This should make it possible to study boundary layer effects at the interface between grains or phases. Consistent boundary conditions for an expanding elastic-plastic boundary are as well formulated. As examples, biaxial tension of a thin film on a thick substrate, torsion of a thin wire and a spherical void under remote hydrostatic tension are investigated.     

Modeling the interfacial effect on the yield strength and flow stress of thin metal films on substrates

It is shown in this paper that interfacial effects have a profound impact on the scale-dependent yield strength and strain hardening rates (flow stress) of metallic thin films on elastic substrates. This is achieved by developing a higher-order strain gradient plasticity theory based on the principle of virtual power and the laws of thermodynamics. This theory enforces microscopic boundary conditions at interfaces which relate a microtraction stress to the interfacial energy at the interface. It is shown that the film bulk length scale controls the size effect if a rigid interface is assumed whereas the interfacial length scale dominates if a compliant interface is assumed.     

A conventional theory of mechanism-based strain gradient plasticity

There exist two frameworks of strain gradient plasticity theories to model size effects observed at the micron and sub-micron scales in experiments. The first framework involves the higher-order stress and therefore requires extra boundary conditions, such as the theory of mechanism-based strain gradient (MSG) plasticity [J Mech Phys Solids 47 (1999) 1239; J Mech Phys Solids 48 (2000) 99; J Mater Res 15 (2000) 1786] established from the Taylor dislocation model. The other framework does not involve the higher-order stress, and the strain gradient effect come into play via the incremental plastic moduli. A conventional theory of mechanism-based strain gradient plasticity is established in this paper. It is also based on the Taylor dislocation model, but it does not involve the higher-order stress and therefore falls into the second strain gradient plasticity framework that preserves the structure of conventional plasticity theories. The plastic strain gradient appears only in the constitutive model, and the equilibrium equations and boundary conditions are the same as the conventional continuum theories. It is shown that the difference between this theory and the higher-order MSG plasticity theory based on the same dislocation model is only significant within a thin boundary layer of the solid.     

Gradient elasticity in statics and dynamics: An overview of formulations,length scale identification procedures,finite element implementations and new results

In this paper, we discuss various formats of gradient elasticity and their performance in static and dynamic applications. Gradient elasticity theories provide extensions of the classical equations of elasticity with additional higher-order spatial derivatives of strains, stresses and/or accelerations. We focus on the versatile class of gradient elasticity theories whereby the higher-order terms are the Laplacian of the corresponding lower-order terms. One of the challenges of formulating gradient elasticity theories is to keep the number of additional constitutive parameters to a minimum. We start with discussing the general Mindlin theory, that in its most general form has 903 constitutive elastic parameters but which were reduced by Mindlin to three independent material length scales. Further simplifications are often possible. In particular, the Aifantis theory has only one additional parameter in statics and opens up a whole new field of analytical and numerical solution procedures. We also address how this can be extended to dynamics. An overview of length scale identification and quantification procedures is given. Finite element implementations of the most commonly used versions of gradient elasticity are discussed together with the variationally consistent boundary conditions. Details are provided for particular formats of gradient elasticity that can be implemented with simple, linear finite element shape functions. New numerical results show the removal of singularities in statics and dynamics, as well as the size-dependent mechanical response predicted by gradient elasticity.     

Size effects in generalised continuum crystal plasticity for two-phase laminates

The solutions of a boundary value problem are explored for various classes of generalised crystal plasticity models including Cosserat, strain gradient and micromorphic crystal plasticity. The considered microstructure consists of a two-phase laminate containing a purely elastic and an elasto-plastic phase undergoing single or double slip. The local distributions of plastic slip, lattice rotation and stresses are derived when the microstructure is subjected to simple shear. The arising size effects are characterised by the overall extra back stress component resulting from the action of higher order stresses, a characteristic length l_c describing the size-dependent domain of material response, and by the corresponding scaling law lⁿ as a function of microstructural length scale, l. Explicit relations for these quantities are derived and compared for the different models. The conditions at the interface between the elastic and elasto-plastic phases are shown to play a major role in the solution. A range of material parameters is shown to exist for which the Cosserat and micromorphic approaches exhibit the same behaviour. The models display in general significantly different asymptotic regimes for small microstructural length scales. Scaling power laws with the exponent continuously ranging from 0 to −2 are obtained depending on the values of the material parameters. The unusual exponent value −2 is obtained for the strain gradient plasticity model, denoted “curl H^p” in this work. These results provide guidelines for the identification of higher order material parameters of crystal plasticity models from experimental data, such as precipitate size effects in precipitate strengthened alloys.     

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.     

Dislocation-based micropolar single crystal plasticity: Comparison of multi- and single criterion theories

Two new formulations of micropolar single crystal plasticity are presented within a geometrically linear setting. The construction of yield criteria and flow rules for generalized continuum theories with higher-order stresses can be done in one of two ways: (i) a single criterion can be introduced in terms of a combined equivalent stress and inelastic rate or (ii) or individual criteria can be specified for each conjugate stress/inelastic kinematic rate pair, a so-called multi-criterion theory. Both single and multi-criterion theories are developed and discussed within the context of dislocation-based constitutive models. Parallels and distinctions are made between the proposed theories and some of the alternative generalized crystal plasticity models that can be found in the literature. Parametric numerical simulations of a constrained thin film subjected to simple shear are conducted via finite element analysis using a simplified 2-D version of the fully 3-D theory to highlight the influence of specific model components on the resulting deformation under both loading and unloading conditions. The deformation behavior is quantified in terms of the average stress-strain response and the local shear strain and geometrically necessary dislocation density distributions. It is demonstrated that micropolar single crystal plasticity can qualitatively capture the same range of behaviors as slip gradient-based models, while offering a simpler numerical implementation and without introducing plastic slip rates as generalized traction-conjugate velocities subject to an additional microforce balance.     

The Facet method: A hierarchical multilevel modelling scheme for anisotropic convex plastic potentials

An entirely new analytical expression describing plastic anisotropy is presented. It is designed to be used in combination with multilevel models. It makes use of the theory of dual plastic potentials, which is shortly revisited. An analysis of convexity is presented. Note that the new method is not optimal when not used in combination with a multilevel model; other methods are better suited for identification on the basis of mechanical test data. Compared with already existing methods which work with multilevel models, the new method has the following advantages: (i) it is automatically convex anywhere in the six-dimensional stress or strain rate space; (ii) it can be used for materials with a stress differential effect, such as hcp metals or pre-strained cubic materials; (iii) its identification procedure is such that not only the Taylor theory, but also more advanced theories, such as the Alamel-model or self-consistent models, can be used to identify the parameters; (iv) an analytical expression of the plastic potential can be obtained in both strain rate and stress space, which is an important advantage when implementing the model in finite element codes for metal forming. Equipotential surfaces in strain rate space and corresponding yield loci obtained by the new method for four materials (one ferrite single crystal, one aluminium alloy and two types of steel) are presented and discussed.     

Forming limit stresses predicted by phenomenological plasticity theories with anisotropic work-hardening behavior

Forming limit stresses of sheet metals subjected to linear and combined stress paths are analyzed using the M-K model in conjunction with two anisotropic work-hardening models: a work-hardening model which is capable of describing Bauschinger and cross-hardening effects, and a work-hardening model which cannot predict the cross-hardening effect. It is found that the forming limit stress is path-independent when the stress–strain curves for the linear and combined stress paths agree well with each other. On the other hand, the forming limit stress for the combined stress path depends on the strain path when the prestrain changes the subsequent stress–strain relation. We conclude that the stress-based forming limit criterion is efficient only for a material with a work-hardening behavior that is not affected by strain path change. The influence of the work-hardening behavior on the forming limit stress is discussed in detail.     

Modelling of the interface between a thin film and a substrate within a strain gradient plasticity framework

Interfaces play an important role for the plastic deformation at the micron scale. In this paper, two types of interface models for isotropic materials are developed and applied in a thin film analysis. The first type, which can also be motivated from dislocation theory, assumes that the plastic work at the interface is stored as a surface energy that is linear in plastic strain. In the second model, the plastic work is completely dissipated and there is no build-up of a surface energy. Both formulations introduce one length scale parameter for the bulk material and one for the interface, which together control the film behaviour. It is demonstrated that the two interface models give equivalent results for a monotonous, increasing load. The combined influence of bulk and interface is numerically studied and it is shown that size effects are obtained, which are controlled by the length scale parameters of bulk and interface.