Characteristic Material Length Scale Research Articles

Size-dependent steady creeping microfluid flow based on the boundary element method

In order to investigate size-dependent creeping plane microfluidic flow, a boundary element method is implemented that involves the calculation of interior quantities and multi-domain problems. The governing equations are formulated using the skew-symmetric character of the couple-stress tensor and its energy conjugate mean-curvature tensor, as established in consistent couple stress theory. This theoretical formulation includes one characteristic material length scale parameter l that represents the size-dependency of the problem. Here, we present the boundary integral representation and numerical implementation for two-dimensional size-dependent steady state creeping incompressible flow, in which velocities, angular velocities, force- and couple-tractions are the primary variables. Beyond the previously known singular fundamental solution kernels for point force and point couple in the velocity and angular velocity integral equations, here we present the integral equations for calculating internal mean curvatures and couple-stresses, along with their corresponding kernels. The formulation is then applied to solve some computational problems both to investigate the size effects on the response resulting from the theory, and to validate the strength of the numerical method. For this purpose, we study the important effect of different boundary conditions on the flow pattern and consider a multi-fluid domain Couette flow problem.

Boundary element formulation for steady state plane problems in size-dependent thermoelasticity

A boundary element method is developed to examine two-dimensional size-dependent thermoelastic response in isotropic solids. The formulation is founded on the recently established consistent couple stress theory, in which both the couple-stress tensor and its energy conjugate mean curvature tensor are skew-symmetric. For isotropic materials, there is no thermal mean curvature deformation, and the thermoelastic effect is solely the result of thermal strain deformation. As a result, size-dependency is quantified by one characteristic material length scale parameter l, while the thermal coupling is activated through the classical thermal expansion coefficient α. Interestingly, in this size-dependent multi-physics model, the thermal governing equation is independent of the deformation. However, the mechanical governing equations depend on the temperature field. Here, we develop the boundary integral representation and numerical implementation for this size-dependent thermoelastic boundary element method (BEM) for plane problems, which involves temperatures, displacements, rotations, normal heat fluxes, force-tractions and couple-tractions as primary variables within a boundary-only formulation. Then, we apply this new BEM formulation to several basic computational problems in an effort to validate the robustness of the numerical implementation and to examine size-dependent response.

Unsteady Helical Flows of a Size-Dependent Couple-Stress Fluid

The helical flows of couple-stress fluids in a straight circular cylinder are studied in the framework of the newly developed, fully determinate linear couple-stress theory. The fluid flow is generated by the helical motion of the cylinder with time-dependent velocity. Also, the couple-stress vector is given on the cylindrical surface and the nonslip condition is considered. Using the integral transform method, analytical solutions to the axial velocity, azimuthal velocity, nonsymmetric force-stress tensor, and couple-stress vector are obtained. The obtained solutions incorporate the characteristic material length scale, which is essential to understand the fluid behavior at microscales. If characteristic length of the couple-stress fluid is zero, the results to the classical fluid are recovered. The influence of the scale parameter on the fluid velocity, axial flow rate, force-stress tensor, and couple-stress vector is analyzed by numerical calculus and graphical illustrations. It is found that the small values of the scale parameter have a significant influence on the flow parameters.

Cryogenic nanoindentation size effect in [0 0 1]-oriented face-centered cubic and body-centered cubic single crystals

Cryogenic nanoindentation experiments performed on [0 0 1]-oriented single crystalline Nb, W, Al, and Au in an in situ nanomechanical instrument with customized cryogenic testing capability revealed temperature dependence on nanoindentation size effect. The Nix-Gao model, commonly used to capture indentation size effect at room temperature, does not take into account thermal effects and hence is not able to explain these experimental results where both hardness at infinite indentation depth and characteristic material length scale were found to be strong functions of temperature. Physical attributes are critically examined in the framework of intrinsic lattice resistance and dislocation cross-slip probability.

Consistent skew-symmetric couple stress theory for size-dependent creeping flow

Within the framework of continuum fluid dynamics, couple-stresses appear as an inevitable consequence of non-central forces and the discrete character of matter at the finest scales. As a result, the force-stress tensor becomes non-symmetric and classical theory may not accurately predict the behavior. Recent theoretical work has shown that the couple-stress tensor must be skew-symmetric and that mean curvature rate is the energy-conjugate kinematical measure. The resulting fully consistent couple stress theory incorporates a characteristic material length scale for the fluid that becomes increasingly important, as the characteristic geometric dimension of the problem approaches that level. This size-dependent non-Newtonian theory is essential to understand a range of behavior at micro-scales with potential applications to blood flow and lubrication, among others. Here we concentrate on steady creeping flow within this newly developed fully determinate linear couple stress theory and formulate a boundary integral representation for two-dimensional problems. These boundary integral equations are written in terms of velocities, angular velocities, force-tractions and moment-tractions as primary variables. Details are provided on the derivation of fundamental solutions and on the corresponding boundary element implementation. Afterwards, the new boundary element method is applied for the solution of two basic problems to explore some consequences of this size-dependent couple stress fluid mechanics.

Interaction Between Dislocations in a Couple Stress Medium

Taylor’s theory of crystal plasticity is reformulated for dislocations in a couple stress medium. The divergence between Taylor’s approach and an approach that includes the effects of couple stresses on dislocation interactions is demonstrated. It is shown that dislocations separated by a distance that is comparable to a characteristic material length scale, have mutual interaction somewhat weaker than that predicted by classical elasticity.

Local and non-local Gurson-based ductile damage and failure modelling at large deformation

Abstract The purpose of this work is the formulation, numerical implementation and initial application of a non-local extension of existing Gurson-based modelling for isotropic ductile damage and attendant crack growth. It is being carried out under the premise that void coalescence results not only in accelerated damage development (e.g., Needleman and Tvergaard, 1984), but also in damage delocalisation (i.e., via interaction between neighbouring Gurson RVE's). To this end, we proceed by analogy with the approach of Needleman and Tvergaard (1984) who replaced the Gurson void volume fraction f with a (local) effective damage parameter f ∗ in the Gurson yield condition to account for the effect of void coalescence on the material behaviour. In the current case, the role of f ∗ is taken over and generalised by an effective continuum damage field ν. A field relation for ν is formulated here in the framework of continuum thermodynamics. In the simplest case, the resulting relation is formally analogous to the inhomogeneous temperature equation in which void nucleation and growth represent (local) sources for ν and in which void coalescence takes place in a process zone whose dimension is determined by a characteristic material lengthscale. Analogous to temperature, then, ν represents an additional continuum degree-of-freedom here, resulting in a coupled deformation-damage field model. In the last part of the work, the complete model for coupled damage-deformation is implemented numerically using the finite-element method on the basis of backward-Euler integration and consistent linearisation. Using this implementation, the behaviour of the current extended Gurson-based damage model is investigated for the case of simple tension of an inhomogeneous steel block. In particular, the corresponding simulation results document quantitatively the dependence of the delocalisation of the model damage process and minimisation of mesh-dependence on the characteristic dimension of the damage process zone.

Material-length-scale-controlled nanoindentation size effects due to strain-gradient plasticity

This paper proposes a concept of strain-gradient plasticity that is based on characteristic material length scale. Experimental results for indentation size effect are correlated with the predictions from a dislocation-based strain-gradient plasticity model. Nanoindentations on single crystals of Al, Ag, Ni, polycrystalline Cu and poly-synthetically twinned (PST) lamellar α2- and γ-TiAl are conducted with an atomic force microscope with an add-on force transducer from Hysitron, Inc. It is found that the indentation size effect is controlled by a characteristic material length scale l, which is a function of the Burger’s vector, the shear modulus, and a material reference stress. The material length scales are found to be in the order lAg>lNi>lAl>lCu>lα2-TiAl>lγ-TiAl, which corresponds to the indentation size effect.

Cosserat modeling of cellular solids

Cellular solids inherit their macroscopic mechanical properties directly from the cellular microstructure. However, the characteristic material length scale is often not small compared to macroscopic dimensions, which limits the applicability of classical continuum-type constitutive models. Cosserat theory, however, offers a continuum framework that naturally features a length scale related to rotation gradients. In this paper a homogenization procedure is proposed that enables the derivation of macroscopic Cosserat constitutive equations based on the underlying microstructural morphology and material behavior. To cite this article: P.R. Onck, C. R. Mecanique 330 (2002) 717–722.

Length scale effect on mechanical behavior due to strain gradient plasticity

The characteristic material length scale l as introduced in the phenomenological theory of strain gradient plasticity by Fleck and Hutchinson [1] is crucial for the application of the theory. Three dislocation models are proposed in this paper for the derivation of the material length scales as there is a rational connection between dislocation theory and phenomenological theory. The length scale is determined as a function of material fundamental parameters and is independent of how the ‘overall effective strain’ is measured, or how the dislocations interact. The length scale is generally about 1.5 μm or less, agreeing with the observation that the size effect appears approximately at 1 μm and sub-micrometer levels for indentation deformations. The dislocation interaction is also discussed in the framework of strain gradient plasticity.

Indentation size effects in crystalline materials: A law for strain gradient plasticity

We show that the indentation size effect for crystalline materials can be accurately modeled using the concept of geometrically necessary dislocations. The model leads to the following characteristic form for the depth dependence of the hardness: H H 0 1+ h ∗ h where H is the hardness for a given depth of indentation, h, H 0 is the hardness in the limit of infinite depth and h ∗ is a characteristic length that depends on the shape of the indenter, the shear modulus and H 0. Indentation experiments on annealed (111) copper single crystals and cold worked polycrystalline copper show that this relation is well-obeyed. We also show that this relation describes the indentation size effect observed for single crystals of silver. We use this model to derive the following law for strain gradient plasticity: ( σ σ 0 ) 2 = 1 + l ̂ χ , where σ is the effective flow stress in the presence of a gradient, σ 0 is the flow stress in the absence of a gradient, χ is the effective strain gradient and l ̂ a characteristic material length scale, which is, in turn, related to the flow stress of the material in the absence of a strain gradient, l ̂ ≈ b( μ σ 0 ) 2 . For materials characterized by the power law σ 0 = σ ref ε 1 n , the above law can be recast in a form with a strain-independent material length scale l. ( builtσ σ ref ) 2 = ε 2 n + l χ l = b( μ σ ref ) 2 = l ̂ ( σ 0 σ ref ) 2 . This law resembles the phenomenological law developed by Fleck and Hutchinson, with their phenomenological length scale interpreted in terms of measurable material parametersbl].

The influence of an interface reaction on the creep response of damaged materials

In this paper we examine how the operation of an interface reaction influences the response of a body which deforms as the result of the transportation of material around the grains by grain-boundary diffusion. A characteristic material length scale λ is identified. If λ is much less than a suitably chosen physical length scale for a problem then the rate of diffusion controls the overall rate of the process. If, however, λ is comparable with the physical length scale then the interface reaction dominates the response. A range of problems are examined which provide a number of different possible physical length scale.

Lubrication of a roller on a plane in combined rolling, sliding and normal motion with additives

The micropolar fluid theory-a possible non-Newtonian model for fluids with additives in which the average molecular size may be comparable to a characteristic material length scale—is applied to the problem of lubrication of a lightly loaded roller on a plane in combined sliding, rolling and normal motion. Three new parameters N, L and q are defined. Various bearing characteristics are presented graphically. It is shown that these three dimensionless parameters are of great significance in discussing the flow behavior of a lubricant with additives. The results of Dowson et al.[2] can be obtained as a particular case of the present study. The overall conclusion of the study, in conformity with the experimental results, is an increase in effective viscosity, resulting in an increase in the load carrying capacity and a decrease in coefficient of friction when additives are used.