WO2006008554A1

WO2006008554A1 - Method of determining residual stress within an object

Info

Publication number: WO2006008554A1
Application number: PCT/GB2005/002920
Authority: WO
Inventors: Alexander Korsunsky
Original assignee: Isis Innovation Limited
Priority date: 2004-07-23
Filing date: 2005-07-25
Publication date: 2006-01-26
Also published as: GB0416517D0

Abstract

A method of determining residual stress within an object, comprising: defining a model for calculating the residual stress state within the object given an inelastic strain distribution within the object; representing the inelastic strain distribution as a combination of basis functions depending on a set of parameters; inputting values of at least one of the residual stress, the change of stress, strain, deformation, and distortion of the object obtained from measurement; defining a cost functional for evaluating the magnitude of the difference between said at least one of the residual stress, change of stress, strain, deformation , end distortion within the object, predicted by the model based on the strain basis functions and set of parameters, and the input values of the same obtained from measurement; adjusting the parameters to minimise the cost functional; and reconstructing the residual stress state within the object using the model and the strain basis functions together with the parameter values obtained in the adjusting process.

Description

METHOD OF DETERMINING RESIDUAL STRESS WITHIN AN OBJECT

Mechanical stress is defined as the average internal force acting across cross- sectional area. This definition has several important consequences. Firstly, since both the force and (normal to) area are vectors, then stress is a second rank tensor, i.e. a construct that depends on orientation in a way similar to an outer product of two vectors. Secondly, the size of the area over which averaging is performed is not specified, and hence stress can be perceived at different 'magnification'.

Stresses are related to deformation (strain) through material constitutive law. Most materials display a range of deformations over which the loading is reversible, i.e. when the external load is removed the object returns precisely to its original shape and dimension. The relation between stresses and strains for this type of behaviour, termed elastic, may be linear or non-linear.

If the external load exceeds the elastic limit then some permanent deformation persists after load removal. In many structural materials such as metallic alloys this type of behaviour is primarily associated with plasticity. The strain experienced by the material under loading beyond the maximum elastic strain is called plastic, or more generally, inelastic.

If inelastic strain introduced into the material by loading or some other mechanism is uniform across the object, then all stresses disappear upon load removal, because no misfit arises between different parts of the object. If, on the other hand, the distribution of inelastic strains induced in the object is not uniform, then different parts of the object no longer fit together in the same way as they did prior to the introduction of inelastic strain. This means that different parts of the object must deform elastically to accommodate the misfit. This elastic accommodation of misfit gives rise to residual stresses.

The term 'residual stresses' refers to any self-equilibrating system of internal stresses that is present inside an object in the absence (or e.g. after the removal) of all external loading. For example, in metals residual stresses can be very low after extended stress-relieving treatment at high temperature, or very significant following e.g. cold working by forging or rolling, i.e. comparable in magnitude with the material yield stress.

Residual stresses exist on a variety of lengths and are associated with different structural scales. In conventional engineering components the macroscopic scale corresponds to the range from fractions of a millimetre to many centimetres. Residual stresses that exist at this length scale are referred to as Type I, or macroscopic, and correspond to the interaction between different parts of an object, e.g. surface and bulk regions.

Type II residual stresses describe force interaction between different grains in a polycrystalline aggregate. These stresses are considered at the length scale of several to hundreds of micrometres. They often arise due to the inhomogeneity of properties between grains, e.g. elastic stiffness, thermal expansion, plastic strain, etc. These properties may differ very markedly if grains of different types are present in the same material, e.g. ceramic in a metallic matrix; different crystallographic phases such as martensite and austenite co-existing in a steel, etc. Even in an alloy that is nominally macroscopically homogeneous, i.e. consists of grains of only one type of material, Type II stresses arise due to anisotropy, i.e. the dependence of thermomechanical properties of these individual grains on orientation.

Type III stresses arise at the length scale of crystal lattice defects or their arrangements, e.g. vacancies, dislocations, dislocation pile-ups, etc., and are associated with the lengths from the sub-nanometre and into hundreds of nanometres scale. They arise because crystal lattice defects represent elementary instances of misfit at length scales comparable with lattice spacing, and induce local distortion and elastic deformation. In many instances of engineering analysis the length scale of consideration is in the millimetre range, the material is considered as a continuum, and Type I macroscopic stresses are of primary interest. Under service loading conditions these stresses are thought to be superimposed on the applied stresses, and thus enter such considerations as those needed for the prediction of component durability, e.g. fatigue or creep life.

Experimental methods of residual stress evaluation can be classified. Material removal techniques rely on the principle of registering additional deformation induced by sectioning, machining or dissolving away parts of the object. If the object was residual stress free, then material removal does not induce any deformation. Furthermore, once the object is sectioned into very small pieces, then macroscopic stresses in these pieces are progressively reduced to zero with diminishing piece size. It follows that by tracing the deformation of the pieces during this progressive diminution it is possible to extract information about the initial residually stress state of the object. Various implementations exist of this destructive method of residual stress evaluation. For example, if the object has simple plate geometry and residual stress distribution that only depends on the out-of-plane coordinate, then simple analysis of plate distortion during progressive layer removal allows the residual stress to be reconstructed (Moore and Evans). Further techniques of note include slotting and hole drilling.

One particular method should be noted here, known as the contour method (M B Prime). The principle of the method consists of performing a minimally disturbing cut, e.g. by electric discharge machining; measuring to high accuracy the deviation of the cut surface topology from flat; and applying the measured normal displacements, in the opposite sense, to a finite element model of the object. The solution then gives residual stress distribution inside the object. Note that no information about the in-plane displacements is obtained or used.

Non-destructive techniques rely on measuring some physical property of the material that exhibits a correlation with residual stress, and deducing residual stress state. For example, acoustic wave speed and magnetic properties have been shown to correlate with some aspects of the residual stress state.

The above techniques are all capable of providing information about Type I macroscopic residual stress only. Diffraction of penetrating radiation on the crystal lattice is a powerful method of measuring stress-induced changes in the lattice spacing. The method has the crucial advantage of providing information about Type I, II and III stresses. However, deducing stress from the measurement of lattice spacing is not straightforward. Firstly, in order to determine strain the measured spacing must be compared to some stress-free value, which may be difficult to determine. Secondly, evaluation of the stress tensor involves the application of elastic constitutive law to the strain tensor; a process associated with uncertainty either in some strain components, or in the material's elastic properties.

Determining the residual stress within an object, such as a turbine blade or aircraft wing, is clearly a technologically important problem because it has implications regarding the object durability and potential failure in service. There exist a number of problems with the above discussed techniques. It is often the case that experimental measurements needed for stress evaluation are approximate, laborious, expensive, and limited in number. So there are the problems of how to make best use of a limited number of measurements, how to combine measurements obtained by different techniques, and what to do about incompatible or inconsistent measurements. There is also the problem that some measurements only obtain information locally or at the surface, so there are the problems of how to calculate stresses in the bulk and how to obtain information on inaccessible parts which cannot be measured directly. Finally, there is no general reconstruction for obtaining residual stress distribution based on measurements conducted on objects, particularly when the object has been sectioned or machined. It is an object of the invention to alleviate any of the above problems.

Accordingly, the present invention provides a method of determining residual stress within an object, comprising: defining a model for calculating the residual stress state within the object given an inelastic strain distribution within the object; representing the inelastic strain distribution as a combination of basis functions depending on a set of parameters; inputting values of at least one of the residual stress, the change of stress, strain, deformation, and distortion within the object obtained from measurement; defining a cost functional for evaluating the magnitude of the difference between said at least one of the residual stress, change of stress, strain, deformation , and distortion within the object, predicted by the model based on the strain basis functions and set of parameters, and the input measured values of the same; adjusting the parameters to minimise the cost functional; and reconstructing the residual stress state within the object using the model and the strain basis functions together with the coefficient values obtained in the adjusting process.

The process of adjusting the parameters to minimise the cost functional can effectively convert input values of stress into a distribution of inelastic strains (hereinafter referred to as the eigenstrain distribution), represented by basis functions. In other words, the problem of determining residual stresses within the object is best posed in terms of the search for a stress state that provides the best agreement with the information available from experimental measurements. Determining the eigenstrain distribution has previously been difficult, but the advantage of expending the effort to find it, according to the invention, is that it is invariant (i.e. transferable) between subsets of the original object obtained by sectioning. The method of the invention will also always find a solution to the eigenstrain which fits with the available information, even where there are inconsistencies in the data. In the calculation, different data can be weighted differently in the cost functional. The approach can be applied to any geometry, any material, any mechanism or origin of eigenstrain, and can incorporate any number and type of relevant experimental measurement.

Embodiments of the invention will now be described, by way of example only, with reference to the accompanying drawing in which:

Fig. 1 is a schematic flow chart of the method of the invention.

Referring to Fig. 1, in box 10, a model for calculating the residual stress state within the object given an inelastic (eigenstrain) strain distribution within the object is defined. Suitable models are known and include analytical methods, such as direct numerical integration of convolution between the eigenstrain distribution and the eigenstrain influence function (Korsunsky, 2001), or by numerical methods, such as the finite element method (FEM), or boundary integral or element method, or finite difference method, or Rayleigh-Ritz or Galerkin method. The FEM can be implemented in the form of commercially available finite element package, such as ABAQUS, ANSYS, DYNA, etc. The eigenstrain distribution can be introduced into the model in the form of initial stress distribution or by simulating it as thermally- induced strain. The finite element package can then perform a single solution of the elastic problem and output the results for the points or sampling regions (appropriate averages over lines, areas or volumes) at positions relevant to the experimentally measured data, e.g. distortion, residual strain or stress, etc. The model incorporates suitable boundary conditions, such as defined by the shape of the object, or portion thereof. In box 20, the inelastic strain distribution is represented as a combination of basis functions depending on a set of parameters. The basis functions are suitable for the representation of eigenstrain distributions. For each of the N basis functions a corresponding solution of the residual stress is obtained using the above model.

Basis functions can be selected from any suitable system, e.g. power series, polynomial or trigonometric orthogonal sets, etc. A set of parameters is associated with each basis function. For example, each basis function might have a parameter which represents a coefficient (linear or non-linear) applied to the basis function. However, according to a preferred embodiment of the invention, it is often the case that eigenstrain distributions are confined in space to a particular region, e.g. plastic deformation zone, weld line and the surrounding heat affected zone, etc. In order to represent this property of eigenstrain distribution, the above basis functions may be convolved with a smooth bounded support function, i.e. a function that is different from zero only within a bounded domain, and decays to zero outside the domain in a suitable smooth way. In this case, the parameters describing the eigenstrain distribution are therefore the coefficients of the basis functions, as well as the domain and transition characteristics of the bounded support function defining the region where eigenstrains are non-zero. Although eigenstrain at a given point is a multi-component (six component) object, in many practical situations fixed relationships exist between eigenstrain components. For example, if eigenstrain distribution is produced by uniaxial plastic deformation in the z-direction, then shear eigenstrains vanish due to symmetry, while direct eigenstrains can be postulated to obey the relationship ε^* _^ = ε_w ^* - -0.5£^* _z , so that the number of unknown distributions is reduced to one.

In box 30, values of at least one of the residual stress, the change of stress, strain, deformation, and distortion of the object obtained from measurement are inputted. The input values of the residual stress can be obtained by at least one technique selected from the group comprising material removal, sectioning, machining, etching, hole drilling, slotting, trepanning, together with an attendant method of measuring at least one of deformation, strain, distortion and residual stress. The input values of the residual stress can also be obtained by measurement of diffraction of radiation on coupons, sections or slices of the object. The input values of the residual stress can be obtained by measurement of bulk residual strains by diffraction of penetrating radiation, such as X-rays and neutrons. The input values of the residual stress, or change of stress, or strain, or deformation, or distortion can be obtained using a technique comprising material removal and precise distortion or deformation measurement, and can also further comprise determination of the shape and position of the object surface. The measurement can comprise one or more selected from the group comprising: strain gauging, measurement using a coordinate measurement machine, optical methods, photogrammetry methods, and profϊlometry. Values from several different measurement techniques can be combined as input data. In box 40, a cost functional is defined for evaluating the magnitude of the difference between said at least one of the residual stress, change of stress, strain, deformation , and distortion within the object, predicted by the model based on the strain basis functions and set of parameters, and the input values of the same obtained from measurement; and in box 50 a minimisation is performed by adjusting the parameters to minimise the cost functional.

In more detail, a niinimization problem of cost functional at M collocation (measurement) positions with respect to the combination of basis function eigenstrain models is formulated and reduced. For example, the functional can be chosen in the form of weighted sum over all collocation (measurement) points z-1 , ... ,M of squares of the difference between the measurement (e) and prediction Ce ) at X_J: J = ∑w,.(_e(x,.)-£(_Xi))²

1=1 M

The weightings are w,- . The minimisation of cost functional J is performed with respect to the eigenstrain distribution parameters. If the parameters enter the combination in a non-linear way, e.g. if the extent of the distribution is varied, then the numerical solution of the direct problem (basis eigenstrain model) has to be repeated at each step of the iteration. Minimisation can be accomplished by using a suitable method, e.g by formulating and solving an adjoint problem to determine the gradient with respect to parameter c, dJ/dc, and applying the steepest descent or other non-linear minimisation algorithm, or by any other specially designed algorithm, or a standard algorithm available from numerical libraries. If the linear combinations of basis eigenstrain models is considered, then the cost functional /assumes the form

This form is quadratic in terms of the unknown coefficients c_j of the basis eigenstrain functions. Therefore the minimum can be found directly by differentiating the above equation with respect to c_\ and deriving a linear system of equations. For M>N the minimization problem is well-posed and is readily solved.

Finally, at box 60, the residual stress state everywhere within the object is reconstructed using the model and the strain basis functions together with the parameter values obtained in the adjusting (i.e. minimisation) process.

Embodiments of the invention are performed by a computer program executed on a computer system. The computer system may be any type of computer system, but is typically a conventional personal computer executing a computer program written in any suitable language. The computer program may be stored on a computer-readable medium, which may be of any type, for example: a recording medium, such as a disc-shaped medium insertable into a drive of the computer system, and which may store information magnetically, optically or magneto- optically; a fixed recording medium of the computer system such as a hard drive; or a solid-state computer memory. The measurement values may be input into the computer system directly, for example from a coordinate measurement machine or strain gauges, or the computer system may read information representing the values from a store of previously obtained measurement values.

Claims

1. A method of determining residual stress within an object, comprising: defining a model for calculating the residual stress state within the object given an inelastic strain distribution within the object; representing the inelastic strain distribution as a combination of basis functions depending on a set of parameters; inputting values of at least one of the residual stress, the change of stress, strain, deformation, and distortion of the object obtained from measurement; defining a cost functional for evaluating the magnitude of the difference between said at least one of the residual stress, change of stress, strain, deformation , and distortion within the object, predicted by the model based on the strain basis functions and set of parameters, and the input values of the same obtained from measurement; adjusting the parameters to minimise the cost functional; and reconstructing the residual stress state within the object using the model and the strain basis functions together with the parameter values obtained in the adjusting process.

2. A method according to claim 1, further comprising obtaining the input values of the residual stress by at least one technique selected from the group comprising material removal, sectioning, machining, etching, hole drilling, slotting, trepanning, together with an attendant method of measuring at least one of deformation, strain, distortion and residual stress.

3. A method according to claim 1 or 2, further comprising obtaining the input values of the residual stress by measurement of diffraction of radiation on coupons, sections or slices of the object.

4. A method according to claim 1, 2 or 3, further comprising obtaining the input values of the residual stress by measurement of bulk residual strains by diffraction of penetrating radiation.

5. A method according to claim 4, wherein the penetrating radiation comprises one of X-rays and neutrons.

6. A method according to any one of the preceding claims further comprising obtaining the input values of the residual stress, or change of stress, or strain, or deformation, or distortion using a technique comprising material removal and precise distortion or deformation measurement.

7. A method according to claim 6, wherein the measurement comprises determination of the shape and position of the object surface.

8. A method according to claim 6 or 7, wherein the measurement comprises one or more selected from the group comprising: strain gauging, measurement using coordinate measurement machine, optical methods, photogrammetry methods, and profilometry.

9. A method according to any one of the preceding claims, wherein the model for calculating the residual stress state is based on a finite element method, or boundary integral or element method, or finite difference method, or Rayleigh-Ritz or Galerkin method, or direct numerical integration of convolution between the inelastic strain distribution and an inelastic strain influence function.

10. A method according to any one of the preceding claims, wherein the cost functional is in the form of a weighted sum of squares of the difference between model prediction and input value obtained from measurement.

11. A method according to any one of the preceding claims, wherein said set of parameters comprise a coefficient for each basis function and at least one value for defining the spatial extent of the associated eigenstrain distribution.

12. A computer program capable of being executed on a computer system and capable when so executed of causing the computer system to perform a method according to any one of the preceding claims.

13. A computer readable medium storing a computer program according to claim 12.