Residual stresses in thin film systems: Effects of lattice mismatch,thermal mismatch and interface dislocations

This paper explores the mechanisms of the residual stress generation in thin film systems with large lattice mismatch strain, aiming to underpin the key mechanism for the observed variation of residual stress with the film thickness. Thermal mismatch, lattice mismatch and interface misfit dislocations caused by the disparity of the material layers were investigated in detail. The study revealed that the thickness-dependence of the residual stresses found in experiments cannot be elucidated by thermal mismatch, lattice mismatch, or their coupled effect. Instead, the interface misfit dislocations play the key role, leading to the variation of residual stresses in the films of thickness ranging from 100 nm to 500 nm. The agreement between the theoretical analysis and experimental results indicates that the effect of misfit dislocation is far-reaching and that the elastic analysis of dislocation, resolved by the finite element method, is sensible in predicting the residual stress distribution. It was quantitatively confirmed that dislocation density has a significant effect on the overall film stresses, but dislocation distribution has a negligible influence. Since the lattice mismatch strain varies with temperature, it was finally confirmed that the critical dislocation density that leads to the measured residual stress variation with film thickness should be determined from the lattice mismatch strain at the deposition temperature.     

Residual stresses in roller-expanded thin tubes

The residual-stress distribution of through-the-wall thickness in the roller-expanded region of thin-walled incoloy-800 Fe–Ni–Cr alloy tubing was determined. Such tubes are commonly used in the fabrication of steam generators for nuclear-power stations. For the present study, the test specimens consisted of short lengths of tubing which were roller-expanded into tubesheet simulation blocks. Some of the specimens were then heat treated. The measurement method involved the installation of strain-gage rosette strips on the inner tube wall. Strain measurements were first taken after the removal of the tubesheet simulation block. Residual stresses were then released by progressive chemical etching of the outer tube wall. In some cases the inner tube wall was etched instead and this required the removal of the inner strain-gage strip and its replacement by one attached to the outer wall. A calculation procedure based on the Sachs approach, first proposed for straight unrolled tubes, was used for determining the residual-stress distribution in the vicinity of the roller-expanded zone and through-the-wall thickness. Surface residual stresses of the order of 250 MPa were determined in the as-received specimens. Residual stresses in the stress-relieved heat-treated specimens were generally lower by about 40 percent. Contact stresses were nearly eliminated by the heat-treating process. The residual stresses in the various specimens of the same type compared to within a standard deviation of 35 MPa.     

Modeling of progressive delamination in a thin film driven by diffusion-induced stresses

A semi-analytical method based on the cohesive model has been developed to investigate the progressive growth of interface delamination in an axisymmetric thin film electrode driven by diffusion-induced stresses under the assumption that the electrode remains elastic during the Li-ion diffusion process. The evolutions of the cohesive zone and debonding zone with respect to charging time have been predicted. The cohesive zone propagates in an accelerating manner and the debonding zone advances in a slowing down manner. The key parameters that control the interfacial stresses and delamination have been identified from the obtained governing equations. And according to the discussions on the key parameters, design insights into the geometry, charging velocity and material properties of the electrode have been provided.     

Residual stresses in gas-assisted injection molding

Residual stresses are a major issue in the mechanical and optical behavior of injection-molded parts. In this study, we analyze their development in the case of gas-assisted injection molding (GAIM) of amorphous polymers. Flow-induced residual stresses are computed within a decoupled approach, in which elastic effects are neglected in the momentum balance, assuming a generalized Newtonian material behavior. In a staggered procedure, the computed viscous flow kinematics are used to calculate normal stresses employing a compressible version of the Rolie-Poly model. For the computation of thermally and pressure-induced residual stresses, a linear thermo-viscoelastic model is used. A 3-D finite element model for GAIM is employed, which is able to capture the kinematics of the flow front and whose capabilities to predict the thickness of the residual material layer have been validated by Haagh and Van de Vosse (Int J Numer Methods Fluids 28:1355–1369, 1998). In order to establish a clear comparison, the development of residual stresses is analyzed using standard injection molding and GAIM for a test geometry.     

Residual stresses in squeeze-cast aluminum rods

The longitudinal slitting technique has been applied to determining and comparing the residual stresses in as-cast and squeeze-cast aluminum rods. Residual stresses in the squeeze-cast aluminum alloy rods are found to increase with applied punch pressures under a constant die-base thermocouple reference temperature. For the variations of residual stresses with varying die-base thermocouple reference temperatures, a peak residual stress is found to occur at a die-base thermocouple reference temperature of 100° C. A semi-empirical formula is derived for the determination of the maximum longitudinal residual stress in the tapered cylindrical as-cast aluminum alloy, from which the maximum longitudinal residual stresses for squeeze cast can be determined, using the residual-stress ratios obtained experimentally     

Residual stresses in evaporated AI-Si films

Residual stresses in vacuum evaporated thin Al-Si films were evaluated by measuring the deflection of a thin cantilevered substrate during removal of the film. Post-thermal treatment and thermal cyclings at temperatures between ?269° C and 560° C were also introduced to determine their effects on the residual stresses of the films.     

Residual stresses in cyclically loaded two-phase solid

A plane-stress finite element model of a two-phase solid is analyzed for residual stresses that develop under cyclic loading. One phase is assumed to be dilute and to yield at stresses below those of the matrix. Simple triangular constant strain finite elements are used and the soft material is assumed to be a linear work hardening one. The load is applied incrementally and the stiffness matrix is updated at each step to its tangential values. The emphasis is on the residual stresses which develop within the soft element and in the neighborhood. Several distributions of soft material were examined. In the cases we consider, the residual stresses change sign on each load cycle, that is, they are not constant. They are also an appreciable percentage of the yield stress of the soft phase. The residual stresses primarily develop in the first cycle and settle down in just a few cycles. The location of a soft element is somewhat of a factor in how far away its effect is felt by other elements. In addition to the calculation of residual stresses, the temperature changes are also calculated. The plastic work is converted into heat in the soft elements and this in turn is conducted to the cooler elastic matrix. It is found that adiabatic conditions can be assumed for steel specimens for strain rates above .033 sec⁻¹ and at rates above .23 sec⁻¹ for aluminum.     

Residual stresses in turned AISI 4340 steel

The residual-stress distribution in the surface region of workpieces of annealed AISI 4340 steel that is turned under unlubricated conditions is determined using a deflection etching technique. The absolute value of the residual stresses at the machined surface are low and increase with an increase in depth beneath the machined surface to a maximum. They then decrease with a further increase in depth eventually becoming vanishingly small. Peak residual stresses are tensile at cutting speeds of 0.5 and 1.0 ms⁻¹ and are compressive at a cutting speed of 1.5 ms⁻¹ for all feed rates and depths of cut. Peak residual stresses and depth of the stressed region increase with an increase in feed rate and depth of cut, but decrease with an increase in cutting speed. The results of this investigation can be interpreted in terms of the variation of tool forces with cutting conditions.     

Residual stresses in plastically encapsulated microelectronic devices

The residual stresses in a model of a plastically encapsulated microelectronic device are determined photoelastically using birefringent transfer-molding compounds. The measured stresses are compared with a laminated-platetheory analysis and then used to guide a two-dimensional finite-element analysis of the device. Photoelasticity was also used to determine the effects of postcure on the residualstress distribution.     

Residual stresses near a rail end

Stress fields near a cut end of a rail containing longitudinal residual stress typical of roller-straightened rail were studied using analysis and a finite element model. For a self-equilibrating residual stress distribution with equal maximum and minimum stresses, the distance to reach 95% of the mid-rail residual stress field is from 0.7 to 1.8 times the rail height, with the finite element model predicting a length of 1.1 times the rail height. This gives a measure of the accuracy of the simpler analytical models. At the rail end, the longitudinal residual stress goes to zero, and the vertical residual stress near mid-web reaches a maximum of approximately 27 ksi (186 MPa) (1.35 times the maximum mid-rail longitudinal residual stress of 20 ksi, or 138 MPa). The maximum shear stresses are 6 ksi (41 MPa) and −8 ksi (−55 MPa) near the head-web and web-base intersections, respectively, approximately 2 in. (51 mm) from the end of a 7.3 in. (185 mm) high rail. The shear stress is zero at the cut end and in mid-rail. The worst possible end-crack is a horizontal web crack in the vertical residual stress field at the rail end. The stress intensity K_I on such a crack is estimated to reach 20 ksi√in. (22 MPa√m) for cracks 0.5 in. (13mm) long. This is already 0.4 to 0.8 times K_I for carbon and alloy rails, and about 0.5 times K_Ic for a long crack.     

Non-uniform, axisymmetric misfit strain： in thin films bonded on plate substrates/substrate systems： the relation between non-uniform film stresses and system curvatures

Current methodologies used for the inference of thin film stress through curvature measurements are strictly restricted to stress and curvature states which are assumed to remain uniform over the entire film/substrate system. By considering a circular thin film/substrate system subject to non-uniform, but axisymmetric misfit strain distributions in the thin film, we derived relations between the film stresses and the misfit strain, and between the plate system’s curvatures and the misfit strain. These relations feature a ‘‘local’’ part which involves a direct dependence of the stress or curvature components on the misfit strain at the same point, and a ‘‘non-local’’ part which reflects the effect of misfit strain of other points on the location of scrutiny. Most notably, we also derived relations between the polar components of the film stress and those of system curvatures which allow for the experimental inference of such stresses from full-field curvature measurements in the presence of arbitrary radial non-uniformities. These relations also feature a ‘‘non-local’’ dependence on curvatures making a full-field measurement a necessity. Finally, it is shown that the interfacial shear tractions between the film and the substrate are proportional to the radial gradients of the first curvature invariant and can also be inferred experimentally.     

Residual stresses in thick electrodeposits of a nickel-cobalt alloy

Some electroplated metals contain residual stresses which can cause warpage or premature failure of parts plated or electrofomed with these materials. Noticeably absent from the literature are residual-stress data for finished parts. Typically for plated or electroformed parts, residual stresses are determined independently on thin strips and then piece parts are plated. This research describes a technique which can be used to measure stress on finished parts. The method involves drilling a hole in the part and measuring the resulting change of strain in the vicinity of the hole. Viability of this technique was demonstrated by measuring the stress in a nickel-cobalt deposit plated on an aluminum cylinder. Two separate runs, one 50 deg removed from the other, provided almost identical results; stress was 160 MN/m² (23,200 psi). Two other runs in a region where plating was somewhat thinner provided slightly lower results probably because all boundary-condition requirements were not met. The computed residual-stress values compared quite favorably with independent rigid-strip measurements of 131 MN/m² (19,000 psi) obtained for the solution before and after plating of the cylinder.     

Residual stresses in copper-2% beryllium alloy strips

Residual-stress profiles were determined along the rolling direction in cold-rolled, annealed, and roller-leveled copper-2% beryllium alloy strips using the X-ray-diffraction technique. The data were corrected for the effects of beam penetration and subsurface-stress relaxation due to material removal. The type, magnitude and distribution of residual stresses were found to depend on the amount of cold reduction of the strips. Surface residual stresses as high as 40 percent of the material yield strength were observed, with the stress values decreasing rapidly through the depth. Corrections for the effects of both beam penetration and stress relaxation were found to be very important at higher percentages of strip cold reduction. Thermal treatments were found to significantly reduce the level of residual stress. However, mechanical treatments (roller leveling) produced no change in the magnitude of stress at the strip surface, but resulted in an irregular pattern of stress.     

Residual surface stresses in laminated cross-ply fiber-epoxy composite materials

Residual (curing) stresses in a cross-ply laminated plate are related to the strains released when individual plies are separated. Released displacements are determined using high-sensitivity moiré interferometry and linearized strain-displacement equations are used to determine residual strains. Elastic orthotropic stress-strain relations are used to calculate residual stresses remote from free-edges of a [90₂₀/0₂₀/90₂₀] graphite-epoxy cross-ply panel. The measured strains compare favorably with those predicted by laminated plate theory. In a second example, the circumferential and radial residual strains and stresses at the end-section of a thick-walled cross-ply graphite-epoxy cylinder are determined. Paper was presented at the 1992 SEM Spring Conference on Experimental Mechanics held in Las Vegas NV on June 8–11.     

Residual stresses in reverse-bend test samples for stress-corrosion testing

Cracking of Inconel Alloy 600 (registered trademark) u-bend tubes used in pressure-water-reactor (PWR) steam generators has been a major concern in the nuclear-power industry over the past several years. The mechanism of cracking has been determined to be intergranular stress-corrosion cracking with residual stresses a major contributor. A simple specimen known as a reverse u-bend (RUB) has been used by a number of laboratories to simulate the high stresses and plastic strain extant in the most susceptible regions of the u-bend tubes. This paper presents the results of residual-stress measurements on four RUB samples, each from a different laboratory.The results indicate that the individual RUB fabrication procedures used by different laboratories tend to produce different residual-stress patterns in the highly strained regions over 700 tensile to nearly 700-MPa compressive on different samples. Stress gradients on the order of 140 MPa/mm were found on some samples. The residual-stress patterns were seen to qualitatively predict the stress-corrosion-cracking pattern experienced on similar samples.Electric Power Research Institute.