Adiabatic shear localization in α-titanium: experiments, modeling and microstructural evolution

The localization of dynamic shearing deformations in α-titanium is examined using a novel experimental technique (the compression-torsion Kolsky bar) that allows the recovery of specimens within which an adiabatic shear band has been grown as the result of a single torsional pulse. The specimens are circumferentially notched thick-walled cylinders that are subjected to simultaneous, independently controlled, dynamic compression and torsion. Explicit finite element computations are performed to obtain the stress, strain, temperature and pressure distributions within the specimens under the measured boundary conditions. The constitutive behavior input to the computational simulations is obtained from independent high-strain-rate experiments (involving only homogeneous deformations) on the same material. Shear band growth and microstructural evolution in the specimens are investigated by sectioning the specimens at different depths from the outer radial surface. TEM observations across the shear bands reveal the following microstructural evolution: (a) planar dislocation motion and twinning; (b) grouping of dislocations into cells; (c) formation of elongated subgrains along the shear direction, and (d) development of equiaxed nanocrystalline grains 50- in diameter. The microstructures observed are analogous to those reported for severely cold-rolled metals.     

Combined deformation- and temperature-induced scission in a rubber cylinder in torsion

When an elastomeric material is deformed and subjected to temperatures above some characteristic value T_cr (near for natural rubber), it undergoes time and temperature dependent chemical changes consisting of scission and crosslinking of its macromolecular structure. The process continues until the temperature decreases below T_cr. Experiments carried out in uniaxial extension have shown that the chemical changes are independent of stretch ratio within moderate stretches. It is reasonable to expect that the chemical changes would be affected by sufficiently large deformations, an interaction referred to as ‘mechanochemistry’. A kinetic theory of the breakdown of solids by Zhurkov [Kinetic concept of strength of solids, Int. J. Fract. Mech. 1 (1965) 311-323. [15]] attributes this interaction to the lowering of activation energy by mechanical work.In a recent constitutive theory, an expression was developed that relates the chemical kinetics of scission of the original elastomeric network to time, temperature and activation energy. The kinetic theory of Zhurkov suggests a method for modifying this expression to account for the influence of deformation. This is explored in the case of simple shear deformations, such as those occurring during torsion of elastomeric cylinders held at fixed length. Using the approach of Penn and Kearsley [The scaling law for finite torsion of elastic cylinders, Trans. Soc. Rheology 20 (1976) 227-238. [16]], it is shown that experiments in torsion can be used to determine the influence of shear deformations on the chemical kinetics of scission.     

Surface stress effects on the resonant properties of metal nanowires: The importance of finite deformation kinematics and the impact of the residual surface stress

We utilize the recently developed surface Cauchy-Born model, which extends the standard Cauchy-Born theory to account for surface stresses due to undercoordinated surface atoms, to study the coupled influence of boundary conditions and surface stresses on the resonant properties of gold nanowires with surfaces. There are two major purposes to the present work. First, we quantify, for the first time, variations in the nanowire resonant frequencies due to surface stresses as compared to the corresponding bulk material which does not observe surface effects within a finite deformation framework depending on whether fixed/free or fixed/fixed boundary conditions are utilized. We find that while the resonant frequencies of fixed/fixed nanowires are elevated as compared to the corresponding bulk material, the resonant frequencies of fixed/free nanowires are reduced as a result of compressive strain caused by the surface stresses. Furthermore, we find that for a diverse range of nanowire geometries, the variation in resonant frequencies for both boundary conditions due to surface stresses is a geometric effect that is characterized by the nanowire aspect ratio. The present results are found to agree well with existing experimental data for both types of boundary conditions.The second major goal of this work is to quantify, for the first time, how both the residual (strain-independent) and surface elastic (strain-dependent) parts of the surface stress impact the resonant frequencies of metal nanowires within the framework of nonlinear, finite deformation kinematics. We find that if finite deformation kinematics are considered, the strain-independent surface stress substantially alters the resonant frequencies of the nanowires; however, we also find that the strain-dependent surface stress has a significant effect, one that can be comparable to or even larger than the effect of the strain-independent surface stress depending on the boundary condition, in shifting the resonant frequencies of the nanowires as compared to the bulk material.     

A gradient theory of small-deformation isotropic plasticity that accounts for the Burgers vector and for dissipation due to plastic spin

This study develops a gradient theory of small-deformation viscoplasticity based on: a system of microforces consistent with its peculiar balance; a mechanical version of the second law that includes, via the microforces, work performed during viscoplastic flow; a constitutive theory that accounts for the Burgers vector through a free energy dependent on , with H^p the plastic part of the elastic-plastic decomposition of the displacement gradient. The microforce balance and the constitutive equations, restricted by the second law, are shown to be together equivalent to a nonlocal flow rule in the form of a coupled pair of second-order partial differential equations. The first of these is an equation for the plastic strain-rate in which the stress T plays a basic role; the second, which is independent of T, is an equation for the plastic spin. A consequence of this second equation is that the plastic spin vanishes identically when the free energy is independent of, but not generally otherwise. A formal discussion based on experience with other gradient theories suggests that sufficiently far from boundaries solutions should not differ appreciably from classical solutions, but close to microscopically hard boundaries, boundary layers characterized by a large Burgers vector and large plastic spin should form.Because of the nonlocal nature of the flow rule, the classical macroscopic boundary conditions need be supplemented by nonstandard boundary conditions associated with viscoplastic flow. As an aid to solution, a variational formulation of the flow rule is derived.Finally, we sketch a generalization of the theory that allows for isotropic hardening resulting from dissipative constitutive dependences on .     

Inflation and eversion of functionally graded non-linear elastic incompressible circular cylinders

We study axisymmetric radial deformations of a circular cylinder composed of an inhomogeneous Mooney-Rivlin material with the two material parameters varying continuously through the cylinder thickness either by a power law or an affine relation. It is found that for the exponent of the power law function equal to 1, the hoop stress for an internally pressurized cylinder is uniform in the cylinder. One can tailor the gradation of these two material parameters to make the maximum tensile hoop stress occur either on the inner surface or on the outer surface. Also, the stress concentration in a pressurized thick cylinder strongly depends upon the value of the exponent of the power law variation of the two material parameters. For an affine through-the-thickness variation of the two elastic moduli the hoop stress at the point is nearly the same as that in a cylinder composed of a homogeneous material. Here R_in and R_ou equal, respectively, the inner and the outer radii of the cylinder in the unstressed reference configuration, and R is the radial coordinate of a point in the reference configuration. The stress distribution in an everted cylinder strongly depends upon its thickness in the reference configuration.     

A theory of strain-gradient plasticity for isotropic, plastically irrotational materials. Part I: Small deformations

This study develops a small-deformation theory of strain-gradient plasticity for isotropic materials in the absence of plastic rotation. The theory is based on a system of microstresses consistent with a microforce balance; a mechanical version of the second law that includes, via microstresses, work performed during viscoplastic flow; a constitutive theory that allows:

•

the microstresses to depend on , the gradient of the plastic strain-rate, and

•

the free energy ψ to depend on the Burgers tensor .

The microforce balance when augmented by constitutive relations for the microstresses results in a nonlocal flow rule in the form of a tensorial second-order partial differential equation for the plastic strain. The microstresses are strictly dissipative when ψ is independent of the Burgers tensor, but when ψ depends on G the gradient microstress is partially energetic, and this, in turn, leads to a back stress and (hence) to Bauschinger-effects in the flow rule. It is further shown that dependencies of the microstresses on lead to strengthening and weakening effects in the flow rule.Typical macroscopic boundary conditions are supplemented by nonstandard microscopic boundary conditions associated with flow, and, as an aid to numerical solutions, a weak (virtual power) formulation of the nonlocal flow rule is derived.     

Exact solutions for radial deformations of a functionally graded isotropic and incompressible second-order elastic cylinder

We find closed-form solutions for axisymmetric plane strain deformations of a functionally graded circular cylinder comprised of an isotropic and incompressible second-order elastic material with moduli varying only in the radial direction. Cylinder's inner and outer surfaces are loaded by hydrostatic pressures. These solutions are specialized to cases where only one of the two surfaces is loaded. It is found that for a linear through-the-thickness variation of the elastic moduli, the hoop stress for the first-order solution (or in a cylinder comprised of a linear elastic material) is a constant but that for the second-order solution varies through the thickness. The radial displacement, the radial stress and the hoop stress do not depend upon the second-order elastic constant but the hydrostatic pressure and hence the axial stress depends upon it. When the two elastic moduli vary as the radius raised to the power two or four, the radial and the hoop stresses in an infinite space with a pressurized cylindrical cavity equal the pressure in the cavity. For an affine variation of the elastic moduli, the hoop stress in an internally loaded cylinder made of a linear elastic isotropic and incompressible material at the point is the same as that in a homogeneous cylinder. Here R_in and R_ou equal, respectively, the inner and the outer radius of the undeformed cylinder and R the radial coordinate of a point in the unstressed reference configuration.     

Deformation of FCC nanowires by twinning and slip

We present atomistic simulations of the tensile and compressive loading of single crystal face-centered cubic (FCC) nanowires with and orientations to study the propensity of the nanowires to deform via twinning or slip. By studying the deformation characteristics of three FCC materials with disparate stacking fault energies (gold, copper and nickel), we find that the deformation mechanisms in the nanowires are a function of the intrinsic material properties, applied stress state, axial crystallographic orientation and exposed transverse surfaces. The key finding of this work is the first order effect that side surface orientation has on the operant mode of inelastic deformation in both and nanowires. Comparisons to expected deformation modes, as calculated using crystallographic Schmid factors for tension and compression, are provided to illustrate how transverse surface orientations can directly alter the deformation mechanisms in materials with nanometer scale dimensions.     

Plasticity size effects in free-standing submicron polycrystalline FCC films subjected to pure tension

The membrane deflection experiment developed by Espinosa and co-workers was used to examine size effects on mechanical properties of free-standing polycrystalline FCC thin films. We present stress-strain curves obtained on films 0.2, 0.3, 0.5 and thick including specimen widths of 2.5, 5.0, 10.0 and for each thickness. Elastic modulus was consistently measured in the range of 53- for Au, 125- for Cu and 65- for Al. Several size effects were observed including yield stress variations with membrane width and film thickness in pure tension. The yield stress of the membranes was found to increase as membrane width and thickness decreased. It was also observed that thickness plays a major role in deformation behavior and fracture of polycrystalline FCC metals. A strengthening size scale of one over film thickness was identified. In the case of Au free-standing films, a major transition in the material inelastic response occurs when thickness is changed from 1 to . In this transition, the yield stress more than doubled when film thickness was decreased, with the thick specimen exhibiting a more brittle-like failure and the thick specimen exhibiting a strain softening behavior. Similar plasticity size effects were observed in Cu and Al. Scanning electron microscopy performed on Au films revealed that the number of grains through the thickness essentially halved, from approximately 5 to 2, as thickness decreased. It is postulated that this feature affects the number of dislocations sources, active slip systems, and dislocation motion paths leading to the observed strengthening. This statistical effect is corroborated by the stress-strain data in the sense that data scatter increases with increase in thickness, i.e., plasticity activity.The size effects here reported are the first of their kind in the sense that the measurements were performed on free-standing polycrystalline FCC thin films subjected to macroscopic homogeneous axial deformation, i.e., in the absence of deformation gradients, in contrast to nanoindentation, beam deflection, and torsion, where deformation gradients occur. To the best of our understanding, continuum plasticity models in their current form cannot capture the observed size scale effects.     

On the effective stress in unsaturated porous continua with double porosity

Using mixture theory we formulate the balance laws for unsaturated porous media composed of a double-porosity solid matrix infiltrated by liquid and gas. In this context, the term ‘double porosity’ pertains to the microstructural characteristic that allows the pore spaces in a continuum to be classified into two pore subspaces. We use the first law of thermodynamics to identify energy-conjugate variables and derive an expression for the ‘effective’, or constitutive, stress that is energy-conjugate to the rate of deformation of the solid matrix. The effective stress has the form , where σ is the total Cauchy stress tensor, B is the Biot coefficient, and is the mean fluid pressure weighted according to the local degrees of saturation and pore fractions. We identify other emerging energy-conjugate pairs relevant for constitutive modeling of double-porosity unsaturated continua, including the local suction versus degree of saturation pair and the pore volume fraction versus weighted pore pressure difference pair. Finally, we use the second law of thermodynamics to determine conditions for maximum plastic dissipation in the regime of inelastic deformation for the unsaturated two-porosity mixture.     

On the statistical analysis of local fracture stresses in notched bars

Test results for critical local fracture stresses are analysed statistically for both “as-received” and “degraded” pressure-vessel weld metal. The values were determined from the fracture loads of blunt-notch four-point-bend specimens fractured over a range of low test temperatures, making use of results from a finite-element stress analysis of the stress-strain distributions ahead of the notch root. The “degraded” material tested in this work has been austenitized at a high temperature, followed by both prestraining and temper embrittlement. This has led to a situation in which the fracture stress for the “degraded” material is reduced significantly below that for the “as-received” material. The fracture mechanisms are different in that the “degraded” material shows evidence of intergranular fracture as well as cleavage fracture (in coarse grain size) whereas the “as-received” material shows only cleavage fracture (in fine grain size). The critical stress (σ_F) distributions plotted on normal probability paper show that the experimental cumulative distribution function (CDF) is linear for each condition with different mean values: for “as-received” material and for “degraded” material. The values of standard deviation are small and almost identical (33-). The decrease of the local fracture stress after degradation is related to the local fracture micro-mechanisms. Statistical analysis of the results for the two conditions supports the hypothesis that the values of σ_F are essentially single valued, within random experimental errors. A similar analysis of the data treating both conditions as a single population reveals some interesting points relating to statistical modelling and lower-bound estimation for mechanical properties. Comparisons are made with Weibull analysis of the data. A further conclusion is that it is extremely important to base any statistical model on inferences drawn from micro-mechanical modelling of processes, and that examination of “normal” CDFs can often provide good indications of when it is necessary to subject data to further statistical and physical analysis.     

Geometrical properties of the vorticity vector derived using large-eddy simulation

In this paper, the geometrical properties of the resolved vorticity vector derived from large-eddy simulation are investigated using a statistical method. Numerical tests have been performed based on a turbulent Couette channel flow using three different dynamic linear and nonlinear subgrid-scale stress models. The geometrical properties of have a significant impact on various physical quantities and processes of the flow. To demonstrate, we examined helicity and helical structure, the attitude of with respect to the eigenframes of the resolved strain rate tensor and negative subgrid-scale stress tensor -τ_ij, enstrophy generation, and local vortex stretching and compression. It is observed that the presence of the wall has a strong anisotropic influence on the alignment patterns between and the eigenvectors of , and between and the resolved vortex stretching vector. Some interesting wall-limiting geometrical alignment patterns and probability density distributions in the form of Dirac delta functions associated with these alignment patterns are reported. To quantify the subgrid-scale modelling effects, the attitude of with respect to the eigenframe of -τ_ij is studied, and the geometrical alignment between and the Euler axis is also investigated. The Euler axis and angle for describing the relative rotation between the eigenframes of -τ_ij and are natural invariants of the rotation matrix, and are found to be effective for characterizing a subgrid-scale stress model and for quantifying the associated subgrid-scale modelling effects on the geometrical properties of .     

Boundary conditions in small-deformation, single-crystal plasticity that account for the Burgers vector

This paper discusses boundary conditions appropriate to a theory of single-crystal plasticity (Gurtin, J. Mech. Phys. Solids 50 (2002) 5) that includes an accounting for the Burgers vector through energetic and dissipative dependences on the tensor G=curlH^p, with H^p the plastic part in the additive decomposition of the displacement gradient into elastic and plastic parts. This theory results in a flow rule in the form of N coupled second-order partial differential equations for the slip-rates , and, consequently, requires higher-order boundary conditions. Motivated by the virtual-power principle in which the external power contains a boundary-integral linear in the slip-rates, hard-slip conditions in which

(A)

on a subsurface S_hard of the boundary

for all slip systems α are proposed. In this paper we develop a theory that is consistent with that of (Gurtin, 2002), but that leads to an external power containing a boundary-integral linear in the tensor , a result that motivates replacing (A) with the microhard condition

(B)

on the subsurface S_hard.

We show that, interestingly, (B) may be interpreted as the requirement that there be no flow of the Burgers vector across S_hard.What is most important, we establish uniqueness for the underlying initial/boundary-value problem associated with (B); since the conditions (A) are generally stronger than the conditions (B), this result indicates lack of existence for problems based on (A). For that reason, the hard-slip conditions (A) would seem inappropriate as boundary conditions.Finally, we discuss conditions at a grain boundary based on the flow of the Burgers vector at and across the boundary surface.     

Limit analysis of multi-layered plates. Part I: The homogenized Love-Kirchhoff model

The purpose of this paper is to determine , the overall homogenized Love-Kirchhoff strength domain of a rigid perfectly plastic multi-layered plate, and to study the relationship between the 3D and the homogenized Love-Kirchhoff plate limit analysis problems. In the Love-Kirchhoff model, the generalized stresses are the in-plane (membrane) and the out-of-plane (flexural) stress field resultants. The homogenization method proposed by Bourgeois [1997. Modélisation numérique des panneaux structuraux légers. Ph.D. Thesis, University Aix-Marseille] and Sab [2003. Yield design of thin periodic plates by a homogenization technique and an application to masonry wall. C. R. Méc. 331, 641-646] for in-plane periodic rigid perfectly plastic plates is justified using the asymptotic expansion method. For laminated plates, an explicit parametric representation of the yield surface is given thanks to the π-function (the plastic dissipation power density function) that describes the local strength domain at each point of the plate. This representation also provides a localization method for the determination of the 3D stress components corresponding to every generalized stress belonging to . For a laminated plate described with a yield function of the form , where σ^u is a positive even function of the out-of-plane coordinate x₃ and is a convex function of the local stress σ, two effective constants and a normalization procedure are introduced. A symmetric sandwich plate consisting of two Von-Mises materials ( in the skins and in the core) is studied. It is found that, for small enough contrast ratios (), the normalized strength domain is close to the one corresponding to a homogeneous Von-Mises plate [Ilyushin, A.-A., 1956. Plasticité. Eyrolles, Paris].