Residual Stress Modeling

The elastic stress-strain relations for an orthotropic lamina under plane stress conditions are 

The final transverse chemical shrinkage strain is and ach is the degree of cure when chemical shrinkage is complete. The parameters c, and c2 are empirical constants obtained from chemical shrinkage characterization tests. In most cases occh can be approximated as the degree of cure at gelation. 

The thermal strains can be modeled using the longitudinal and transverse thermal expansion coefficients. From experimental testing of IM6/3100 [5] these coefficients were not found to be significantly dependent on degree of cure. In this case the thermal strains are 

The longitudinal and transverse stresses in the individual plies are given by the laminated plate theory as [19] 

The major Poisson s ratio is vl2 and E = E2/El is the ratio of the transverse to longitudinal modulus. Equation 8.24 gives the induced curvature for anticlastic deformation of an unsymmetric cross-ply laminate. The curvature is dependent on the thermal and chemical strain mismatch (e, — e2), lamina mechanical properties (v12, E) and the half-thickness, h. 

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Arzhaev A.I., Bougaenko S.E., Smirnov Yu.I., Aladinsky V V, Makhanev V.O., Saburov Yu. Residual stress modelling and analysis for INPP primary circuit pipeline welds. Transactions of the 14th Int. Conf. on Structural Mechanics in Reactor Technology (SMiRT 14), Lyon, 1997, Vol. 1, Div, B, pp. 345-352. 

The quasielastic method as developed by Schapery [26] is used in the development of the viscoelastic residual stress model. The use of the quasielastic method is motivated by the fact that the relaxation moduli are required in the viscoelastic analysis of residual stresses, whereas the experimental characterization of composite materials is usually in terms of the creep compliances. An excellent account of the development of the quasielastic method is given in [27]. The underlying restriction in the application of the quasielastic method is that the compliance response of the material shows little curvature when plotted versus log time [28]. Harper [27] shows excellent agreement between the quasielastic method and direct inversion for AS4/3510-6 graphite/epoxy composite. For most graphite/thermoset systems, the restrictions imposed by the quasielastic method are satisfied. 

The present LES concept has in most cases been used as a research tool to study isotropic and homogeneous turbulence within the more theoretical fields of science. Note that the residual-stress models for homogeneous turbulence are not adequate describing industrial non-isotropic inhomogeneous turbulent flows ([137] [121], chap 13). 

The opposite viewpoint, advocated by Boris et al. [16], is that no explicit filtering should be performed and no explicit residual stress model should be used (Oj = 0). Instead, an appropriate numerical method is used to attempt to solve the Navier-Stokes equation for v(r, f). Because the grid is not fine enough to resolve the solution to the Navier-Stokes equation, significant numerical stresses arise. Thus, filtering and residual-stress modeling are performed implicitly by the numerical method. 

Robust validation data for residual-stress models require experimentally intensive and costly diffraction testing, using neutrons or x-rays. The particular value of synchrotron x-ray techniques has been illustrated for several aluminum welding studies, including FSW applied to dissimilar alloys (Ref 88-90). Bringing together the finite element analysis of residual stress and the extensive synchrotron data is a matter of current research. 

Using flaw visuahzation system data the strength and fracture mechanics estimations are carried out in accordance with defect assessment regulatory procedure M-02-91 [5]. Recently, the additions had been included in the procedure, concerning interpretation of expert flaw visualization sysf em data, computer modelling, residual stresses, in-site properties of metal, methods of fracture analysis. 

In order to describe inherited stress state of weldment the finite element modelling results are used. A series of finite element calculations were conducted to model step-by-step residual stresses as well as its redistribution due to heat treatment and operation [3]. The solutions for the reference weldment geometries are collected in the data base. If necessary (some variants of repair) the modelling is executed for this specific case. 

Applications The general applications of XRD comprise routine phase identification, quantitative analysis, compositional studies of crystalline solid compounds, texture and residual stress analysis, high-and low-temperature studies, low-angle analysis, films, etc. Single-crystal X-ray diffraction has been used for detailed structural analysis of many pure polymer additives (antioxidants, flame retardants, plasticisers, fillers, pigments and dyes, etc.) and for conformational analysis. A variety of analytical techniques are used to identify and classify different crystal polymorphs, notably XRD, microscopy, DSC, FTIR and NIRS. A comprehensive review of the analytical techniques employed for the analysis of polymorphs has been compiled [324]. The Rietveld method has been used to model a mineral-filled PPS compound [325]. 

The divergence of (4.21) yields a Poisson equation for p. However, the residual stress tensor r6 is unknown because it involves unresolved SGS terms (i.e., UfiJfi). Closure of the residual stress tensor is thus a major challenge in LES modeling of turbulent flows. 

The form of (4.21) is very similar to that of (2.93), p. 47, for the mean velocity (U) found by Reynolds averaging. However, unlike the Reynolds stresses, the residual stresses depend on how the filter function G is defined (Pope 2000). Perhaps the simplest model for the residual stress tensor was proposed by Smagorinsky (1963) ... 

Composite materials inherently develop residual stresses during processing. This happens because the two (or more) phases that constitute the composite behave differently when subjected to nonmechanical loading. For example, consider a reinforcing phase that has low thermal expansion characteristics embedded in a matrix phase with high thermal expansion characteristics. If the material is initially stress free and the temperature is decreased, then the matrix will try to shrink more than the reinforcement. This places the reinforcement in a state of compression (i.e. a compressive residual stress). If the phases are well bonded, then models can be developed to predict the residual stress field that is induced during processing. 

The modeling of residual stress development during cure can be used to optimize the processing conditions to reduce or control residual stresses. The current process model is used next to assess the effects of several processing conditions on residual stresses. Reduced cure temperature, longer dwell times, slower cool down rate, and the use of novel cure cycles are all feasible for the reduction of residual stresses. 

The MRC cycle calls for a 182°C cure temperature. The effect of cure temperature on residual stress was investigated by curing specimens at four other cure temperatures (171, 165, 160, and 149°C) while holding the dwell time (4 hours) constant. In Figure 8.18 the dimensionless curvature for these specimens is plotted versus the cure temperature. The curvature is reduced as the cure temperature is decreased with significant reduction in curvature obtained for dwell temperatures of 165°C or less. The final curvature as predicted by the viscoelastic process model is overlaid with the experimental data in Figure 8.18 and is shown to capture the trend. 

On the epoxy side of the interface, high fracture toughness and low residual stresses 72,73) are a requirement for optimum transverse strength in graphite and glass-epoxy 1A) composites. Since the adsorption of epoxy components has been shown to be probable, the local structure of the epoxy at the interphase will most likely not be the same as in the bulk. This local anisotropy caused by the interphase is a limitation in the predictive capability of micromechanical models which do not include the interphase as a component. 

For stage VI, the analysis of inherent (or residual) stresses resulting from nonuniform cooling and heat treatment of final articles appears to be critical.Thus, stages IV to VI should be the subject of mathematical modelling specific to reactive processing (chemical molding processes). 

The calculation of residual stresses in the polymerization process during the formation of an amorphous material was formulated earlier.12 The theory was based on a model of a linear viscoelastic material with properties dependent on temperature T and the degree of conversion p. In this model the effect of the degree of conversion was treated by a new "polymerization-time" superposition method, which is analogous to the temperature-time superposition discussed earlier. 

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