A Generalized Stress Inversion Approach With Application to Residual Stress Estimation

The process of manufacturing often introduces residual stresses within a structure. Some processes include nonuniform plastic deformation (such as through forging and bending), surface modification (such as machining), and material phase changes (such as through welding and quenching) [1]. If not properly accounted for, these stresses can lead to unexpected performance including failure. This phenomenon is even more prominent in the additive manufacturing (AM) process, which relies on the layer-by-layer deposition of material to build up a structure of interest. As each layer is deposited, thermal gradients and solidification result in residual stresses within internal volumes, some of which may exceed the yield strength of the material. Recent experiences in the AM community [2,3] have demonstrated the detrimental effects that residual stress have on AM parts.

We present a generalized relaxation-based stress recovery technique, capable of being applied to an arbitrary geometry. In this approach, a specimen is physically cut, and digital image correlation (DIC) [4] is used to measure the displacements due to the relaxation of the now-released stresses. Then, we solve the following inverse problem. Given some measured displacements, determine the tractions on the cut surfaces that caused the observed deformation. We formulate this inverse problem as a quadratic partial differential equation (PDE)-constrained optimization problem. Once tractions are obtained, we can apply Cauchy’s Stress principle to recover stresses that are released by a related cut. The traction identification problem falls within the category of source or force inversion problems, which have been treated in various applications across science and engineering. A few examples can be found in Refs. [5–7].

Our proposed method has several advantages. First, the generalized inversion method can be used on arbitrary geometries, provided that the geometry is sufficiently observable. By sufficiently observable, we mean two things. First, the geometry must have an accurate finite element (FE) model and is conducive for DIC measurements [4,8]. The other caveat is that the measured DIC data must actually contain some notion of the released residual stress; for example, the data must record some sufficient deformation. The second advantage is that using DIC offers flexibility in choosing cut and measurement locations. Data can be gathered on any surface as long as the surface is observable. In addition, we note that DIC obtains high-resolution and dense data over a relatively large surface area, thus capable of providing the generalized inversion method with as much information as possible and making the method more robust. Finally, in our proposed method, we can estimate all traction components and hence all components of stress. Because DIC is able to capture both in-plane and out-of-plane displacements, this information is used by the proposed method to invert for all traction components.

Our proposed approach encompasses a large family of problems, including the estimation of residual stresses through relaxation techniques. The particular situation in the latter class of problems is the presence of eigenstresses, which exist in the absence of any external loading. As shown in Sec. 2, these stress fields represent a particular case addressed by our approach. Residual stress identification is our primary motivation for the development of the proposed approach.

Relaxation-based methods for determining residual stress have a rich and well-developed history. These methods include layer removal, sectioning [1], hole drilling [1,9], slitting, and most recently, the contour method [10]. For many of these cases, the specimen geometry and the type of cut made correspond with the particular method, and they each have varying degrees of specimen damage and amount of stress information recovered. For example, although the layer removal and sectioning methods can determine stresses throughout the body, these methods may lead to challenging experimental situations related to cutting and measuring complex specimens. The hole-drilling method is semidestructive and makes minor perturbations on a larger structure, making it more attractive, although it can only determine stress fields close to the specimen surface. Digital image correlation has been combined with some of these existing methods such as hole drilling [11,12]. We want to emphasize that our goal is to develop a new methodology for measuring residual stress rather than augment previous ones.

The two methods that are the most similar to the generalized inversion technique proposed herein are the slitting method and the contour method. In the slitting method, only the normal traction is recovered on the surface of a cut [9,13–15]; this approach does not provide full-field stress information since it ignores shear tractions. Moreover, in the slitting method, it is assumed that the stress profile is constant through the thickness of the part. The slitting method is best suited for prismatic-shaped specimens. Finally, only one strain gauge is typically used in the slitting method, limiting the amount of information recorded by each cut.

The contour method is the most general of the previously developed approaches for measuring residual stress [10,16–18]. In the contour method, a finite element model is used to estimate residual stress throughout a body by imposing measured displacements normal to the surface as essential or Dirichlet boundary conditions. The measured contours are averaged to remove the effect of shear stresses in the cut plane. The stress field that results from imposing these averaged contours as boundary conditions is solely related to the normal components of traction on the surface.

There are a couple of limitations with the contour method that we will try to overcome with our proposed method. First, in the countour method, the body has to be symmetric with respect to the cut plane to suppress the influence of shear stresses through the averaging of the measured contours. Second, because measured displacements are enforced as boundary conditions, measurement noise can propagate, in a potentially amplified fashion, to the residual stress solution in a neighborhood of the cut surface. To see this, notice that stresses are proportional to strains, which in turn are computed by differentiating displacements. Hence, noise in the measurements is also differentiated leading to an amplification of errors. However, noise effects may be mitigated through averaging and filtering. In summary, the main limiting factor of the contour method is the need for symmetry about the cut plane or, equivalently, the fact that only stresses associated with normal components of tractions can be identified. Our proposed approach aims at identifying residual stresses in arbitrary geometries with arbitrary cuts, while formally treating noise within the framework of regularized inverse problems.

It is important to recognize that the mentioned limitations of the contour method may not be an issue depending on the problem at hand, and we acknowledge that this is one of the most general methods for recovering residual stresses currently available.

There exist nondestructive methods for quantifying residual stress [1,19–21]. For instance, in x-ray diffraction and neutron diffraction approaches, changes in the atomic lattice spacing of the sample are measured by using electromagnetic scattering and penetrating radiation, respectively [20,21]. However, these methods have trade-offs between the amount of energy used and the specimen penetration depth. For example, x-ray diffraction is traditionally limited to near-surface measurements although increasing the amount of radiation, as in the synchrotron x-ray method, directly corresponds to the measurable depth [1]. One drawback of these methods is the requirement of prohibitively expensive radiation sources. We will not further address nondestructive approaches in our work.

We note that other methods make use of finite element (FE) simulations. Specifically, some of these methods use the concept of eigenstrain [22], which arises from the fact that the total strain within a body is the sum of the elastic and inelastic (or eigenstrain) parts. Then, by determining the eigenstrain, the residual stress can be found. This inverse problem, known as the inverse eigenstrain technique as developed in Refs. [23–25], consists of determining the eigenstrain distribution by minimizing the error between the forward stress calculations and the measured stress data (found from elastic strain measurements). While similar in concept, this method is implemented using a least-squares solution and may not be scalable to large-scale problems. Other approaches such as Ref. [26] focus on using measurements localized in the incompatible region, reducing the need to determine the full eigenstrain distribution. Eigenstrain-free methodologies have also been developed. Using equilibrium-satisfying and compatible stress basis functions (such as Airy Stress), the residual stress field can be reconstructed from limited measurements [27]. In addition, data-driven approaches [28] have shown good residual stress reconstructions although they rely on a priori model of the residual stress distribution. Overall, our goal differs from the aforementioned approaches in that we are aiming to generalize current relaxation-based methods.

The proposed generalized inversion method mitigates many of the drawbacks of the current methods. The use of a finite element model allows for the recovery of stress on an arbitrary body, and the use of DIC data captures all three displacement components on an arbitrary surface. In addition, residual stresses are estimated using a regularized inversion approach tailored to deal with measurement noise. The generalized inversion method then combines these approaches to efficiently determine the full-field residual within a body with reasonable computational and experimental costs.

The remainder of this article is organized as follows. First, we present a formulation for a body undergoing deformation due to residual stress in Sec. 2. We show how a mechanically induced traction can be used to obtain stress throughout the body. Then, given measurements obtained by DIC, we setup the inverse problem and use PDE-constrained optimization to invert for tractions. We also provide an explanation of how the derivatives are calculated in the optimization framework. Then, we validate the method by comparing it with the slitting method on simplified numerical examples in Sec. 3. We also demonstrate its capability by inverting for stresses on a more complex AM part in Sec. 4. Finally, we provide closing remarks in Sec. 5.

In this section, we first define the forward problem in Sec. 2.1 by using the elastostatics formulation to solve for displacements (and then postprocess for stresses) in a body due to applied tractions. Then, we define the corresponding inverse problem in Sec. 2.2 where we find tractions given some observed displacement fields, and we convert the inverse problem to the corresponding PDE-constrained optimization problem to solve for tractions (and hence stress). We finally include some remarks on the observability of relaxed stress and its implication in our formulation in Sec. 2.3.

Consider a body Ω with an existing stress field present as shown in Fig. 1(a). Dirichlet boundary conditions are prescribed on Γ_D, and Γ_N denotes the Neumann boundary. We denote the existing stress field in the reference domain as $σ$⁠.

To identify an existing stress field, we appeal to the relaxation principle: by physically cutting or removing material from a body, the existing stress redistributes and causes (typically) elastic deformation. The ensuing deformation can be uniquely related to the relaxed stress using the governing equations of elasticity, which in turn we can use in our inversion approach.

We now present an inverse problem formulation to estimate the traction $T^$ from displacements observed after cutting the body and allowing it to relax, denoted as measured displacements $um$⁠. Notice that once $T^$ is identified, we can recover the released stress field through the boundary value problems in Eq. (3).

In this study, we used the L-curve criterion for selecting the Tikhonov parameter α that appears in Eq. (13). We selected this approach as there is no assumption made on the noise level. The method strives for a balance between the magnitude of the regularization term and the misfit error (i.e., error between predicted and measured quantities). More details on the L-curve approach can be found in Ref. [31].

As mentioned earlier, a residual stress field may be composed of multiple self-equilibrating fields. In this case, multiple cuts are needed to recover the entire residual stress since we can only observe or identify a field relaxed by a given cut. To address this problem, we can define a set of surface cuts ${ΓCj}j=1Nc$ with corresponding measured displacements ${umj}j=1Nc$⁠, where N_c is the number of cuts. Assuming that the body can be completely relaxed by a finite number of cuts, we can solve a sequence of optimization problems as shown in Eq. (17) (one for each cut). The number of cuts is selected as to render the body completely relaxed. Letting $σj$ be the identified stress for a given $ΓCj$⁠, the total residual stress field would be $σ=∑jσj$ due to the principle of superposition.

We would like to point out that the actual number of cuts that renders a body fully relaxed cannot be determined, in general, a priori. Moreover, it may be impossible to render a body completely relaxed by a finite number of cuts. We will limit the scope of this study to the case of identifying the released stress associated with a single cut. We will pursue the problem of identifying multiple self-equilibrating fields in the future work.

We demonstrate our generalized inversion method on three synthetic experiments: a thin-wall structure in Fig. 3(a), a beam on a baseplate structure in Fig. 3(b), and a bridge-like arch in Fig. 3(c).

By using the wall model in Sec. 3.1, we study two different induced stress distributions. In the first case in Sec. 3.1.1, we induce stresses by applying normal tractions on the exterior lateral surfaces. We then synthetically cut the wall down its center, allowing the body to relax and deform. The observed displacements is used to recover tractions along the cut plane and the associated released stress. In the second case in Sec. 3.1.2, we use temperature differences to induce a complex stress field in the structure. One of the goals of using a thin-wall structure is to provide a side-by-side comparison of our generalized inversion approach with the well-known slitting method [14]. As such, we restrict the generalized inversion method to recover only the normal traction components to provide the same set of solutions as that of the slitting method.

We also apply the generalized method to recover stresses induced by eigenstrains [37] in Sec. 3.2. In this example, we apply a spatially varying eigenstrain to a beam on a baseplate with a rectangular hole without any Dirichlet boundary conditions to generate a known residual stress profile. The baseplate rectangular hole is directly underneath the parts of the beam with the induced eigenstrain to isolate the eigenstrain effects. This example allows a direct comparison between the residual stresses released by the cut and the inverted-for stresses.

A potential application of our proposed method is to recover residual stresses in additively manufactured components as the process of layer deposition induces high stresses [38]. Hence, we present a third example that demonstrates how our generalized inversion approach can be applied to AM parts. We consider an arch structure in Sec. 3.3 and induce a stress field by applying a traction field over one of its leg surfaces. Then, using synthetic displacement data, we invert for all traction components on that cut plane. A physical experiment will also be performed on this arch model in Sec. 4.

In this example, we consider a cut down the wall model discretized into 50 incremental cuts. The thin-wall model has dimensions 30 mm × 20 mm × 2 mm, which is divided into 19,000 eight-noded hexahedral elements. Figure 4 shows in more detail the geometry and mesh used. We use displacements on the entire front surface of the wall for the inversion. To generate data for the slitting method comparison, the average strain at the bottom of the wall is recorded, as indicated by the strain gauge placement shown in Fig. 4.

We contrast results from the slitting and the proposed generalized method by comparing the normal tractions recovered on the cut plane. From Cauchy’s stress principle (see Eq. (3d)), the normal traction on the cut plane corresponds to the released stress component along the X-direction (σ_xx).

In this example, we induced a stress field by applying a parabolic pressure over the outer lateral surfaces of the thin wall as observed in Fig. 5(a). The applied pressure is used to induce a known stress field, which can then be used to judge the accuracy of reconstructions obtained using our proposed generalized approach and the slitting method. Figure 5(b) shows the true induced stress component along the X-direction (normal to the slit). The displacements (Fig. 5(a)) needed for the generalized inversion method are obtained by solving the forward problem with the slit present, and the strains needed for the slitting method are found by solving forward problems sequentially with an increasing slit depth. This slit runs through 96% of the wall, and the bottom four corners of the thin wall are constrained.

The normal traction and the corresponding stress field along the slit were recovered using the proposed generalized inversion approach and the slitting method. For each case, we present the inferred normal stress along the cut (Figs. 6(a)–6(c)). We can see in Fig. 6(b) that the solution recovered using the slitting method becomes oscillatory near the tip of the slit. This behavior reflects both the conditioning of the system matrix in Eq. (25) and the choice of the regularization operator. To illustrate this point, in Fig. 6(b), we used an ℓ₂-norm as the regularization operator, which keeps the magnitude of the solution bounded, but it is insensitive to oscillations. On the other hand, when we used a first-derivative smoothing operator, which penalizes oscillations, we obtained a smoother solution as shown in Fig. 6(c). We did not observe these oscillations in our proposed generalized inversion method even when an ℓ₂ norm was used for regularization. This behavior could be explained by the fact that displacements are measured instead of strains, and observations are taken over many points on a surface. Away from the tip region, both methods recover the normal traction (normal stress) profiles accurately for different levels of noise.

We would like to point out that the support conditions in this example render the body statically indeterminate. This fact does not limit the applicability of either method, despite the slitting method being used with simple supports in many practical cases. Also, the oscillatory solutions depicted in Fig. 6(b) for the slitting method are particular to the current example due to the choice of regularization parameter and the conditioning of $G$⁠. The latter is influenced by the number of slits used (i.e., the more the slits are used, the worse the condition number of $G$⁠). As mentioned earlier, improved results can be obtained by using derivative-based regularization operators and using a small number of slits, which translates into an improved conditioning of $G$⁠.

The relative error and the chosen Tikhonov value are presented in Table 1. We observe that the error in the solution increases with noise for both methods, as expected. High errors are observed for the ℓ₂ regularization case, which reflects the poor quality of the reconstructions near the end of the cut. On the other hand, notice that a derivative-based regularization operator leads to improved solutions and ameliorates the oscillations by injecting a priori knowledge about the released stress profile [13,14].

In the second case, the thin wall is subjected to a temperature gradient with its outer walls constrained. As shown in Fig. 7, the top half of the wall is subjected to a positive temperature difference of +50 °C, and the bottom half of the wall is subjected to a negative temperature difference of −50°C. In this example, the normal stress over a centerline is discontinuous at the interface between the heated and cooled blocks as shown in Fig. 8(b).

The inversion results for both the generalized method and the slitting method are shown in Fig. 9 and tabulated in Table 2. Similar to the previous example, the generalized and the slitting methods with noise-free measurements recovered the stress profile very accurately, while the error increased in both cases with increasing noise. However, we notice that the solution error for the generalized inversion method was lower for all cases with noise. From this result, we make two main observations. The first is that in this example using the first-derivative penalty operator in Fig. 9(c) does not yield a significant improvement over the ℓ₂ operator shown in Fig. 9(b). This indicates that the choice of optimal regularization operator is problem dependent. We refer the reader to Refs. [15,36] for additional choices of regularization. The second observation is that the generalized inversion method accurately captures the discontinuity in the stress profile. This improved accuracy in the generalized inversion method can be attributed to the fact that the entire displacement field over the front surface is used in the reconstruction. On the other hand, the conventional slitting method only uses one strain measurement near the bottom of the wall. We investigated the effect of adding more strain measurements on the reconstructions obtained with the slitting method. As expected, as the number of measurements increased as shown in Fig. 10(a), the solution error decreased as shown in Figs. 10(b) and 10(c).

It is possible to obtain comparable errors with both approaches (generalized and slitting) by refining the traction discretization and increasing the number of measurements. However, these actions have very different consequences on the methods in question. For instance, finer discretizations of the traction profile in the slitting method requires physically inducing more slits. Also, using just one strain gauge as the number of slits is increased can worsen the condition number of the system matrix in Eq. (25). Conversely, traction discretizations can be easily changed in the generalized method by refining the finite element mesh. The latter would yield a larger number of unknowns, which is compensated by the large amount of data in high-resolution DIC. In addition, we note that the generalized method allows for stress inversion on complex geometries, as we will show in the next example.

The imposed eigenstrain is shown in Fig. 12(a). A cut is made along the middle of the beam, and the body is allowed to deform. The displacement field released by the cut is shown in Fig. 12(b). A 5% Gaussian noise is added to the displacement fields taken along the top, side, and cut surfaces; this noisy data are considered as our synthetic measured data.

In addition, we compare stress components along the X-, Y-, and Z-axes of the beam. Figure 13 shows the true and recovered normal stress components, and Fig. 14 shows the true and recovered shear stress components. We note that given our observed surfaces, the generalized inversion method recovers the relaxed stress field with enough accuracy given the presence of 5% random noise.

In this section, we consider a small bridge-type arch model with dimensions of 20 mm × 4 mm × 8 mm that has a semi-circle of radius 4 mm cut as shown in Fig. 15. The arch sits upon a baseplate of 30 mm × 14 mm × 7 mm. The entire assembly is isotropic with steel parameters as listed in Fig. 15. The aim is to study the performance of our proposed inversion approach using a realistic geometry that will be studied using the experimental data in Sec. 4. The arch structure rests upon a much larger build plate, as indicated by the large flat footprint below the base of the arch. We define the cut surface to be the bottom of the right leg of the arch. To generate synthetic displacement data, two different sets of forces are applied at the foot of the arch. The first case in Sec. 3.3.1 considers linearly varying pressures, while the second case in Sec. 3.3.2 uses a uniform traction at the base of the arch. Note that we are performing two different inversions. The first considers (normal) pressures as the design variables, while the second considers all three traction components. Displacements are measured on the top and side of the arch (see Fig. 15) and are herein referred to as the data set.

The first example considers a linearly varying pressure at the bottom of the arch, which induces bending about the left foot as shown in Fig. 16. The pressures linearly vary along the X-direction and are constant along the Y-direction. We used our proposed generalized inversion approach to recover pressure on the cut plane using the measured displacements. Figure 17 presents the recovered pressure along the X-axis of the arch foot for a case with 0%, 1%, and 5% noise added to the measured displacements. For the purpose of illustration, we computed the mean and standard deviation of the recovered pressure coefficients along the Y-direction. Each point in Fig. 17 represents the mean pressure at a particular X point, and the error bars represent the standard deviation of the pressures along the Y-axis. Notice that a perfectly recovered pressure profile should only vary linearly along the X-axis and be constant along the Y-axis.

We can see that the reconstruction is accurate. Table 4 summarizes the chosen Tikhonov parameter and the δ error between the applied and recovered pressures. For this case, we observe the following expected behavior: for a noise-free data set, we are able to invert very closely to the true tractions for the applied pressure. As the noise level increases, the regularization parameter and the solution error increase.

For the second arch example, all three traction components on the cut plane are now considered. To that end, a uniform traction of 3 MPa in all three components is applied to the base of the arch to generate the displacement data. Figure 18 shows the applied traction profile along with the resulting displacements.

We studied two different discretizations at the base of the arch. First, we assume a spatially constant traction, resulting in a total of three design variables (one for each traction component). Second, we consider a base surface discretization as in the previous (linearly varying pressure) case, where we have 96 × 3 = 288 design variables. For the coarse discretization, no regularization is needed; for the finer discretization, a Tikhonov parameter is selected using the L-curve approach.

We present the box-and-whiskers plot for the signed relative error δ_d in Figs. 19(a) and 19(b) for the single and multiple side set cases, respectively. In the single side set case (Fig. 19(a)), we notice that the error is centered around zero, which is expected in linear inverse problems with no regularization and zero-mean Gaussian noise. Furthermore, we can observe that the variance of the y-component of traction is an order of magnitude lower than that of the x- and z-components. This result can be explained as follows. First, we know that displacements in the Y direction are primarily induced by the y-component of traction. Second, they are much larger than those in the X and Z directions since the overall structural stiffness for the arch is larger in the latter two directions. Hence, large displacements in the Y direction dominate the objective function (12), leading to the observed results. An example of the recovered traction using one side set is shown in Fig. 20(a).

We observe a similar trend when we consider the finer discretization case in Fig. 19(b); the variance of the y-component is much smaller than that of the other components. The same reason that was explained for the single side set case applies here. However, we also observe the effect of using regularization. The expected values of the traction components are biased below zero as their magnitude is penalized by the regularization term. The better accuracy in the z-component than in the x-component can be attributed to measured displacements being more sensitive to the former component than the latter due to the mechanical behavior of the arch. Figure 20(b) shows the recovered tractions for a particular noise realization for the multiple side set case. Table 5 summarizes the average and variance of the signed error (δ_d) of each component across all of the simulations.

In this example, we employed the generalized inversion method to determine residual stresses within an additively manufactured part. We printed an arch model with dimensions 20 mm × 8 mm × 4 mm using 316L stainless steel. Two DIC stereo-pairs were used to record images before and after the bottom foot was separated from the plate. One stereo-pair was used to calculate displacements at the top surface, while the other stereo-pair calculated displacements on the side view of the arch. Notice that a relatively large subset size of 49 × 49 pixels² was used to minimize the displacement noise and is acceptable due to the smooth gradients across the part. However, the large subset size does create a region around the edge where data are unavailable as shown in Fig. 21. To further reduce the DIC displacement noise on a particular surface, we averaged 25 images together to create an averaged noise-free image. We then combined these measurements from both surfaces to form one data set for the inversion process. These displacement data were mapped onto the finite element mesh used for the synthetic data example in Sec. 3.3. Specifically, the coordinate systems from the DIC system and the FE model were aligned, and the DIC data were interpolated onto the mesh. Figure 21 shows the interpolated data on the arch surface. Additional DIC settings are presented in Table 6 [8].

It is important to note that unlike the synthetic data experiments, there is no ground truth. Therefore, we explore two different representations of the unknown loads. In the first case, we use a distributed pressure, which limits the number of design variables that the data inform, hence regularizing the inverse problem to some extent. The second case corresponds to the use of three-dimensional tractions, which has the highest power of approximation, but may worsen the ill-conditioning of the inverse problem if some of the components are insensitive to the measured displacements. In the end, we are interested in the residual stresses in the body released by the cut. Therefore, if both representations lead to similar solutions (i.e., same approximation accuracy on the data), we would prefer the one with fewer design variables.

Figure 22 shows (a) recovered pressure and (b) recovered traction profiles. We tabulate the inverse problem parameters used and the obtained maximum von Mises in Table 7. Notice that the maximum stress is very similar in both cases. In the traction representation case, the normal components are one order of magnitude larger than the in-plane components, indicating that the normal component dominates the response. Hence, using a pressure representation is sufficient and a preferable option in this problem due to better sensitivity (improved conditioning) and fewer unknowns.

Figures 23(a) and 23(b) demonstrate the residual stress distribution; the stresses are similar to each other. In addition, we compute the relative error between the stress fields between these two models using Eq. (32), where $σa$ corresponds to the stress field from the pressures case, and $σb$ corresponds to the stress field from the tractions case. We report that the relative difference between these two stress fields is 5.44%. Finally, we compute the von Mises stress, and note that the relative difference between the two cases is 3.72%. The maximum Von Mises calculations from the pressures and tractions inversions are 398.6 and 394.0 MPa, respectively, and are within 1.5% of one another. Current material characterization of the utilized machine suggests that the yield strength for the printed parts is in the range of 300–350 MPa [39], and the yield stress of wrought stainless steel is approximately 220–270 MPa [40]. Thus, we have determined that the arch is in the plastic regime.

In this paper, we present a general PDE-constrained optimization approach coupled with digital image correlation to estimate stresses released in complex structures subjected to arbitrary cuts. Our results indicate that our proposed method can be successfully used to estimate residual stresses. However, more validation is required. The method should be validated against other stress recovery methods, such as neutron diffraction. In addition, as with all relaxation methods, the generalized inversion method can only invert for stresses that were released in the cutting process. Determining where to cut or even what surfaces to measure plays a crucial role in recovering residual stress. Insufficient observed information will yield a poorer residual stress profile. Similarly, given the complexities of AM parts, the assumption that all residual stress is released at the given cut may not hold; the process of iterative cuts including the decision to optimize potential cut locations must also be explored.

The authors thank Paul Farias for running the experiments and the sierra-sd team for their support. Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA-0003525.