Coupled strain and temperature gradient analysis in curved surfaces

A common method in experimental mechanics is 3D digital image correlation (DIC) for determining deformations of curved surfaces, whereby the motion of material points is measured. The temperatures induced by the deformation of surfaces or by other heat sources are of particular interest as well. In this context, surface temperatures can be measured using infrared thermography systems. However, the disadvantage of this approach is that a thermographic camera only provides the two‐dimensional temperature distribution (plane images) of three‐dimensionally curved surfaces. Thus, the temperature and temperature gradients on the curved surface assigned to material points cannot be specified. However, for particular problems the temperature and the motion of material points are of particular interest. In the literature, planar problems are studied and simplifying assumptions are made so that a simple coupling of thermography and DIC is possible. In contrast, this contribution presents a method where 3D DIC is coupled with thermography in such a way that the temperature, temperature gradient, as well as displacement and strain distributions at material points of curved surfaces can be determined over temporal processes. The method also allows a continuous strain and temperature gradient determination over the entire domain, since a special class of functions named radial basis functions is used for interpolating the measured data.


INTRODUCTION
In recent years, full-field measurement techniques, such as digital image correlation (DIC) and infrared thermography (TG), have risen in research interest since they allow the measurement of a comprehensive amount of data on the surface of a specimen.However, the question of how to evaluate the measured data arises.When considering the information of the surface deformation, which is obtained from DIC in a Lagrangian setting, convenient interpolation approaches are required to reconstruct the surface under investigation and to compute deformation related quantities.In contrast, the temperature data, which can be extracted from TG, are measured in a Eulerian setting.However, TG provides only 2D temperature distributions but does not provide the physical coordinates of the spatial points, where the temperatures are measured.As a result, obtaining the temperature and deformation information at a material point is not a trivial task.
The coupling of both measurement systems to simultaneously measure the surface deformation as well as the temperature is already studied in recent literature, see, for example, [1][2][3] and the cited literature.There are different approaches for correlating the deformation and temperature information.It is suitable to employ particular marks on the specimen's surface [4] or to evaluate the local strain and temperature evolution along pre-defined lines [5].In recent work by [6], the suitability of such approaches is shown even for elevated temperatures and high strain rates.Instead of drawing on TG, [7] choose the thermographic phosphor technique and DIC to simultaneously measure temperature and strain information.
However, most of the aforementioned publications only deal with planar problems, which especially is sufficient for the investigation of plane specimens with simple geometries.In contrast, the investigation of additively manufactured components, which often possess wavy surfaces, requires the correlation of DIC-and TG-data in a three-dimensional setting.The interpolation of surface information from DIC can be done with so-called radial basis functions (RBFs), which are suitable especially for the analysis of curved surfaces [8,9].In [10] the coupling between both measurement techniques is described in detail and is applied for the determination of temperature gradients in curved surfaces.The novelty of the present work is that the surface interpolation with RBFs and the three-dimensional coupling between DIC and TG are combined to compute the principal strains, associated directions, as well as the temperature gradient at any point on curved surfaces, particularly, of an additively manufactured specimen.
In this contribution, Einstein summation convention is used -summation is applied to indices that occur twice in a term.Latin letters used as indices indicate a summation from 1 to 3 and Greek letters indicate a summation from 1 to 2. An index that occurs once in a term means that it is held constant for the expression.

DESCRIPTION OF SURFACES AND DEFORMATIONS
This section summarizes the basic procedure for strain analysis.The surface description is explained first, followed by the strain analysis.Assuming a surface, it can be described by two parameters, allowing the Cartesian coordinates of the material points that comprise the surface to be formulated as a function of these surface parameters.The use of convective coordinates leads to ⃗ (, ) =   (, )⃗   ,  = (Ψ 1 , Ψ 2 ).Due to the motion of the material points, the surface deforms.The tangent vectors are obtained with ⃗   =  ⃗ (, )∕Ψ  and the gradient vectors are determined with the metric coefficients, ⃗   =    With the principal stretches, the principal strains are calculated.In this contribution, the Green-Lagrange strains are considered,   = ( 2  − 1)∕2.

DEFORMATION ANALYSIS USING DIC DATA AND RBFs
Since DIC only provides the coordinates of specific material points on the surface at specific times   ,  = 1, 2, … , the coordinates have to be interpolated at each time.According to [8], the coordinates are interpolated using RBFs, Each RBF m(⋅) depends on a normalized radius   = ρ (, Ψ ) = ‖ − Ψ ‖ 2 ∕ 0 , which is evaluated with the surface parameters and the center point Ψ belonging to the RBF concerned.Each RBF has a center point that is distinguishable from the center points of the other RBFs.In this contribution, the number of RBFs or center points  cp corresponds to the number of material points tracked by DIC.Furthermore, linear monomials are considered, { n1 (), n2 (), n3 ()} = {1, Ψ 1 , Ψ 2 } to account for rigid body motions.To define the surface parameters, an approach is used that provides a simple relationship between the surface parameters and the Cartesian coordinates of the material points.In the investigations presented here, the material points of the surface in the reference configuration can be uniquely distinguished by their  and  coordinates.Therefore, these coordinates are used as surface parameters (Ψ 1 , Ψ 2 ) = (, ), which correspond to  are determined with the measured Cartesian coordinates of the material points at time   via a linear system of equations.The determined interpolation function is subsequently used to determine the principal strains and corresponding principal directions according to Section 2, see, for details, [8].

COORDINATE TRANSFORMATION BETWEEN DIC AND TG
In order to project the temperature data onto the surface reconstructed by RBFs, the procedure described in [10] is applied.
The most important aspects are briefly described below.To project the temperature data, the translation and the rotation between the two measuring systems DIC and TG must be determined, see Figure 1.The DIC system already defines a coordinate system with respect to which the Cartesian coordinates of the tracked material points are specified.Since the TG system does not provide the physical coordinates of the spatial points, where the temperature data are measured, and only a 2D temperature distribution is obtained, a coordinate system for the TG system must be defined first.In addition, a camera model must be introduced for the TG camera that describes the imaging process of the material points on the image plane of the TG camera (gray area in Figure 1).The simplest camera model, and the one used here, is parallel projection.For this camera model, the length-to-pixel ratio is needed to convert pixel coordinates to physical lengths.In order to determine the translation and rotation between the two measurement systems, a tripod is used, which also defines an oblique coordinate system.The dimensions of the tripod are known.The curvilinear coordinate system introduced by the tripod can be formulated in terms of the coordinate system from DIC or the defined coordinate system from TG The appearing unknowns Δ 0 are determined by the covariant metric coefficients,

(A) (B)
F I G U R E 2 Distance measurement between TG camera and tripod (left), coupling of DIC and TG (right).
In order to determine the rotation, the gradient vectors of the curvilinear coordinate system of the tripod are formulated with the coordinate system of the system, Afterwards, the rotation between the coordinate systems of DIC and TG is determined using the determined gradient vectors and the tangent vectors of the tripod expressed in the coordinate system of the DIC system, The coefficients   describe the rotation.To determine the translation between the DIC and TG coordinate system, the position vector of a tripod point is considered, In this case, the translation ⃗  Θ and the vector ⃗   are unknown.Since only the distances between the tripod points in the ⃗  Θ 3 direction are known, it is necessary to measure the distance of a tripod point in one direction.The distance measurement setup is shown in Figure 2A.Finally, the measured distance of the tripod point is used to determine the translation between the DIC and TG coordinate system.Uncertainty studies were performed in [10] to quantify the influence of the tripod, distance measurement, and length-to-pixel ratio on the temperature analysis results.

DETERMINATION OF TEMPERATURE GRADIENT
Based on the surface interpolation from Section 3 and the coordinate transformation from Section 4, the main aspects of the temperature gradient determination are discussed below.A detailed description of the procedure is given in [10].Since the two-dimensional thermography images only provide temperatures at equidistant points (pixels), the temperature data are also interpolated with RBFs.However, in contrast to Section 3, a scalar quantity, namely the temperature, is interpolated and the parameters of the interpolation function are defined in a different way.For this purpose, the pixel coordinates are used, which are converted into physical lengths in the thermography image  Θ = ( Θ 1 ,  Θ 2 ) using the length-to-pixel ratio.These are subsequently used as parameters for the temperature interpolation, F I G U R E 3 Experimental setup (left) and analyzed specimen (right).
Moreover, linear monomials are also considered.Now, the surface interpolation can be used to determine the Cartesian coordinates  = ( 1 ,  2 ,  3 ) of a material point from its surface parameters,  → , the coordinate transformation can be used to determine the physical lengths in the thermography image from the Cartesian coordinates,  →  Θ , and the temperature interpolation can be used to determine the temperature from the physical lengths in the thermography image,  Θ → Θ. Figure 2B shows the coupled temperature and position determination of a material point.This allows the temperature to be formulated as a function of the different coordinates, Θ = Θ( Θ ) = Θ( Θ ()) = Θ( Θ (())), and the in-plane temperature gradient to be determined, Hence, the chain rule must be applied to obtain the derivative of the temperature with respect to the surface parameters, (12)

APPLICATION TO EXPERIMENT
In this section, the data measured in [10,Sec. 4.3] are used to simultaneously determine the principal strains as well as the associated principal directions on the surface, in addition to the temperature gradient.An enlarged detail of the experimental setup used in [10] is shown in Figure 3A, and the specifications of the experimental setup and measurement systems can be found in [10,Sec. 4.1].The specimen is a cylindrical shell (length 50 mm, 30 mm, height 12 mm, thickness 7 mm) made using wire arc additive manufacturing, see Figure 3B.To enable both DIC and TG, a speckle pattern of known emissivity is applied on the surface of the specimen before the experiment is performed.In the experiment, the heating plate, which is coated with thermal paste, is heated up to 100 • C before the specimen is placed on it.Measurement starts when the specimen is placed on the heating plate.The DIC system provides a trigger signal to the TG system during the measurement so that both systems are coupled and the data are recorded at the same time.
In the following, using the measured data at time  = 0 s (reference configuration) and  = 15 s (current configuration), the procedure presented in this contribution is used to determine the temperature gradient as well as the principal strains and associated principal directions caused by the inhomogeneous temperature distribution at time  = 15 s.DIC and TG data are interpolated using the inverse multiquadric function, m(  ) =   ∕ √  2  +  2  , as RBF and  0 = 1mm to normalize the radius.Due to the different densities of the center points, the interpolation of the DIC data is done with   = 0.9 and the interpolation of the TG data is done with   = 1.3.
Figure 4A shows the three-dimensional surface of the specimen reconstructed from the DIC data, on which the TG data are plotted.At certain points on the surface, arrows indicate the temperature gradient.The temperature gradient is larger in the direction of the contact surfaces of the specimen, while it is small in the upper region.Figure 4B shows the top view of the surface with the temperature field and gradient from Figure 4A.
Figure 5A shows the absolute value of the maximum principal strain on the curved surface and Figure 5B shows the absolute value of the minimum principal strain on the curved surface.In both figures, the associated principal directions are indicated by arrows at the points where the temperature gradient has already been shown.The principal directions are always perpendicular to each other, which is not necessarily apparent due to the curved specimen surface and the top view (projection) in Figures 5A,B , respectively.Furthermore, due to the curved surface, it can be seen that there are locally very fluctuating principal strain directions.

CONCLUSIONS
The approach presented in this contribution allows the coupling of strain and temperature gradient determination on curved surfaces using digital image correlation (DIC) and infrared thermography (TG).This requires the reconstruction of the surface from the DIC data, which involves the definition of surface parameters, the definition of a coordinate system, and camera model for the TG system, and the determination of translation and rotation between the two measurement systems, which is done here using an experimental tripod.This allows TG data to be projected onto curved surfaces.Since both DIC and TG only provide data at discrete points, interpolation is necessary, which is done in this contribution using RBFs.The use of RBF interpolation leads to continuously differentiable coordinates and temperature fields.This allows evaluation of principal strains and associated principal directions, as well as the temperature gradient at any point on curved surfaces.

A C K N O W L E D G M E N T S
Open access funding enabled and organized by Projekt DEAL.

R E F E R E N C E S
⃗   , [  ] = [  ] −1 ,   = ⃗   ⋅ ⃗   .With the tangent vectors in the current configuration ⃗   and the gradient vectors in the reference configuration ⃗   , the deformation of the surface is described by the in-plane deformation gradient, F = ⃗   ⊗ ⃗   .To obtain the principal strains and the associated principal directions in the surface, the in-plane right Cauchy-Green tensor Ĉ, which is related to the in-plane right stretch tensor Û, is determined first, Ĉ = Û2 = F F   .Afterwards, the principal stretches   and the associated principal directions ⃗   are obtained by solving the eigenvalue problem, ( Ĉ −  2 ) ⃗  = ⃗ 0.

F I G U R E 1
Determination of the translation and rotation between the coordinate systems of DIC and TG by means of a tripod.a projection of the surface onto a plane.The projected material points tracked by DIC are used as the center points.The unknowns  ()  and  ()

F I G U R E 4
Temperature distribution and gradient on the curved surface of the specimen (left), top view (right).F I G U R E 5 Absolute value of maximum principal strain (left) and minimum principal strain (right) with associated directions on the curved surface of the specimen.