Eigenstrain Distribution Research Articles

An inverse method for curing process-induced eigenstrain reconstruction of laminated composites

Accurate acquisition of the curing process-induced eigenstrain distribution is vital for predicting residual stress of laminated composites. In this work we develop a novel inverse method for full-field PIE reconstruction of laminated composites. We design standard laminates to obtain the PIE of unidirectional and woven lamina which are key ingredients of full-field PIE. Then an inverse model is established to accurately calculate lamina’s PIE from the measured residual middle-plane strain and curvature of standard laminates. Full-field PIE of validation laminates are then reconstructed by the obtained lamina’s PIE. Experimental results are in good agreement with the reconstructed results, verifying the accuracy and effectiveness of our proposed method. Finally, we implement the inverse method in accurately reconstructing full-field PIE of a hybrid laminated composite I-beam. We also observe that the PIE is process dependent, indicating that PIE can be used to evaluate the effect of curing process on residual stress.

Investigating the applicability of the layered average eigenstrain method in laser shock peening

The eigenstrain model is widely used to predict residual stress and deformation resulting from laser shock peening (LSP). However, most existing eigenstrain models rely on the input of layered average eigenstrain (LAE), which has not been fully verified, particularly considering the non-uniform eigenstrain distribution in the workpiece. To address this issue, a modified eigenstrain model is proposed that uses periodic array eigenstrain (PAE) as input to verify the accuracy of the LAE method. First, a dynamic LSP model is developed to obtain the initial eigenstrain, which is then used in both the LAE and PAE models. The effects of overlapping rate and laser energy of LSP are analyzed by comparing the results of residual stress and displacement predictions. The results show that although the LAE model accurately predicts the displacement induced by LSP, it has errors in predicting residual stress. Although the LAE model has errors in predicting residual stress, its results can still be used to predict the stress intensity factor (SIF) of laser shock peening with some fluctuations. Therefore, we conclude that the LAE method is applicable under high overlapping conditions in LSP, while the proposed PAE-based eigenstrain model provides a more accurate approach for predicting residual stress and deformation during LSP. This has practical implications for the design and optimization of LSP processes.

Nonlinear eigenstrain analysis for compressible Blatz–Ko solids

The objective of this research is to present eigenstrain analysis for a certain class of compressible, isotropic nonlinear elastic solids known as Blatz–Ko materials. The deformations and stress fields caused by eigenstrain distributions in (1) a cylindrical wedge with axisymmetric cylindrical eigenstrains, (2) a cylindrical bar with axisymmetric torsional and cylindrical eigenstrains, and (3) a spherical ball with spherically-symmetric eigenstrain distribution, which are supposed to be composed of the special Blatz–Ko material, are studied. In each problem, the form of deformation is first presumed to put the associated material manifold with Riemannian geometry in Euclidean space, and then the governing equilibrium equations are derived. The obtained nonlinear equations are solved and assessed when the structure has an inclusion with uniform eigenstrains. Moreover, this paper investigates the special solutions in which the stress becomes uniform or hydrostatic inside and outside the inclusion, or singular at the sphere center or cylinder axis. To have a better understanding of the analysis, an example is evaluated numerically in each section.

Elastic–plastic analysis of high load ratio fatigue tests on a shot-peened quenched and tempered steel, combining the Chaboche model and the Theory of Critical Distances

This study aimed to predict the uniaxial fatigue strength of shot peened notched specimens made of 42CrMo4 quenched and tempered steel. The Chaboche kinematic hardening rule was employed to model the cyclic plastic behaviour of the tested samples and the evolution of the residual stresses, which were measured with X-ray diffraction and then reproduced through a finite element software with a novel proposed procedure. This latter introduced a distribution of eigenstrains to reproduce the initial residual stresses and a hardening of the material to replicate the residual stresses measured after the loading for a run-out specimen. The Theory of Critical Distances combined with the Smith-Watson and Topper multiaxial fatigue criterion were then employed to predict the fatigue strength resulting in a good level of accuracy.

Hindered thermal warping triggers tensile cracking in the cores of compressed columns of a fire-loaded tunnel segment structure: Efficiency and accuracy of beam theory prediction, compared to FEM

The nonlinear Finite Element Method (FEM) is the current gold standard for the thermo-mechanical analysis of reinforced concrete structures. As an alternative, this paper is devoted to a model reduction strategy which reduces the CPU time by a factor of 500. This strategy combines Fourier series-based solutions for the thermal conduction problem, and thermo-elastic Timoshenko beam theory. Temperature histories known to be relevant for fire accidents enter series solutions quantifying the conduction of heat into a closed cell frame consisting of slabs, walls, and columns. Corresponding temperature profiles are translated into thermal eigenstrains. The latter are represented as the sum of three portions: (i) their cross-sectional averages (called thermal eigenstretches); (ii) their cross-sectional moments (called thermal eigencurvatures); and (iii) the remaining eigenstrain distributions (called eigenwarping). The latter portion is hindered at the cross-sectional scale, giving rise to non-linearly distributed self-equilibrated thermal stresses. The eigenstretches and eigencurvatures, in turn, are constrained at the scale of the frame structure. Together with external mechanical loads, they enter the exact solutions of thermo-elastic Timoshenko beam theory with equivalent cross-sections accounting for the different material properties of concrete and steel. Axial normal stresses, quantified from beam-theory-related normal forces and bending moments, are superimposed with the hindered-warping-induced stresses. These stresses agree well with corresponding results obtained by the nonlinear FEM. As regards the load carrying behavior of the columns, excessive thermal tensile strains at the periphery of the columns trigger, in the core of the columns, large tensile stresses which even exceed the strength of concrete. Respective cracking events are considered through reduced effective columnar cross-sections. Right after initiation of cracking, around 12min after the start of the heating process, the cracks propagate for some 30sec quite rapidly, and very much slower thereafter. If the initial cross-sections of the columns are increased, more pronounced hindered thermal warping, together with less quickly evolving compressive forces, results in earlier cracking. Overall, it is concluded that tensile cracking is the key material non-linearity, at least during the first 30min of the fire test, with maximum temperatures up to 300°C.

Characterization of glass fiber reinforced polymer via digital volume correlation: Quantification of strain activity and damage growth

Strain and damage distributions within a dogbone specimen containing a rectangular notch were assessed in-situ via X-ray computed tomography coupled with digital volume correlation. The specimen cut from a continuous glass fiber mat reinforced polyester resin composite plate was subjected to cyclic tensile loading. The strain–damage interplay was evaluated by analyzing major eigen strain and correlation residual fields. The horizontal strained band emanating from the notch root was present from the beginning of loading. The final failure proceeded along this strained band. The first cracks did not initiate in the notched region, despite high strain gradients. Strain activity and damage growth were quantified by analyzing the cumulative probabilities of major eigen strains. The comparison was made between the notched region and the remaining part of the inspected specimen. It is shown that, although the mean levels were higher in the notched region, the standard deviations of the major eigen strain distributions were higher in the remaining part of the specimen.

A fuzzy finite element model based on the eigenstrain method to evaluate the welding distortion of T-joint fillet welded structures

T-joint fillet welded structure is one of the most commonly welded structures in the assembly of vessels, so it is crucial to accurately predict the welding distortion of T-joint fillet as fast as possible for in-process quality controls in ship building industry. Eigenstrain distribution which forms after welding accounts for the appearance of welding distortion, the results obtained from experiment and thermal elastoplastic finite element (FE) analysis indicate that it is closely related with the formation of heat-affected zone (HAZ), however, a quantitative description needs to be determined based on limited experimental data. To deal with this problem, a fuzzy finite element model (fFEM) is proposed to evaluate welding distortion in T-joint fillet welding using the eigenstrain method. A set of coefficients which determines the eigenstrain distribution is introduced as the optimization parameters in fFEM. The optimized coefficients of eigenstrains can be found by introducing the genetic algorithm (GA) and combining it with the commercial FE software ANSYS. The eigenstrain distribution calculated by the optimized coefficients can predict the welding distortion with high accuracy and efficiency, the same optimized coefficients and corresponding eigenstrain distribution can be applied to various dimensions of T-joint welded structures if the welding parameters remain unchanged.

The Fundamental Formulation for Inhomogeneous Inclusion Problems with the Equivalent Eigenstrain Principle

In this paper, and on the basis of the equivalent eigenstrain principle, a fundamental formulation for inhomogeneous inclusion problems is proposed, which is to transform the inhomogeneous inclusion problems into auxiliary equivalent homogenous inclusion problems. Then, the analysis, which is based on the equivalent homogenous inclusions, would significantly reduce the workload and would enable the analytical solutions that are possible for a series of inhomogeneous inclusion problems. It also provides a feasible way to evaluate the effective properties of composite materials in terms of their equivalent homogenous materials. This formulation allows for solving the problems: (i) With an arbitrarily connected and shaped inhomogeneous inclusion; (ii) Under an arbitrary internal load by means of the nonuniform eigenstrain distribution; and (iii) With any kind of external load, such as singularity, uniform far field, and so on. To demonstrate the implementation of the formulation, an oblate inclusion that interacts with a dilatational eigenstrain nucleus is analyzed, and an explicit solution is obtained. The fundamental formulation that is introduced here will find application in the mechanics of composites, inclusions, phase transformation, plasticity, fractures, etc.

Behavior of predried mature concrete beams subject to partial wetting and drying cycles

The presented research focuses on the behavior of predried concrete beams with 2.5 m span subjected to cycles of nonsymmetric wetting and drying. Wetting is induced by partially immersing the specimens in a water basin. The submerged portion of the specimens was relatively low (1/5 to 1/10 of their height) leading to a highly nonuniform and nonsymmetric distribution of eigenstrains due to concrete swelling. Unlike conventional experiments on the volume changes of concrete, the measured quantity is not the axial deformation but the vertical displacement instead. This paper presents the experimental data obtained within two wetting and drying cycles, running over 1 year.

Analytical solutions for coated circular inhomogeneity with non-uniform axisymmetric eigenstrain distribution

In this paper, a general model for coated circular inhomogeneous inclusion problems is proposed and studied, where non-uniform axisymmetric eigenstrains are independently distributed within the circular zone and annular zone (coating layer). The main difficulties are from the dissimilarity of three-phase materials, the non-uniform axisymmetric eigenstrain distribution, and the presence of intermediate annularity, which are hardly treated within the classical Eshelby inclusion mechanics. By employing the equivalent eigenstrain principle, these challenges can be completely tackled, and a general explicitly analytical solution is obtained. From this solution, a couple of solutions for typical inclusion problems are readily degenerated, which have been derived with great efforts in literature. With the general model and its solution, the analytical solution for the coated hollow-cored inhomogeneous inclusion problem with non-uniform axisymmetric eigenstrain is provided, and an eigenstrain-based nano-inclusion model is subsequently proposed. The applications of coated circular inhomogeneity with eigenstrains can be found in coated fiber composites, thermoelastic problems, and micro- & nano-mechanics.

An iterative approach combined with multi-dimensional fitting of limited measured stress points to reconstruct residual stress field generated by laser shock peening

In this paper, an iterative approach combined with least squares fitting of limited measured points is introduced to reconstruct residual stress distribution after laser shock peening (LSP) treatment. The iterative approach is started by introducing prescribed stress components in the data measurement region (the mapped zone) in the finite element (FE) model, then a static equilibrium analysis is conducted, which will cause the stress distribution in the mapped zone and unmapped zone both redistribute if they don't converge to the target stresses. By repeatedly introducing the input stress in the mapped zone to converge to the prescribed stress components, the complete residual stress distribution in the whole region can be obtained. Firstly, to validate the viability of this method, based on the full stress components information of ‘measured’ sample points extracted from FE models, residual stress reconstructions of single-pass welding of a thick plate and a single shot of LSP have been carried out, respectively. After that, to examine the robustness of the method, the iterative approach combined with multi-dimensional least squares fitting is tested using a limited set of ‘measured’ sample points extracted from the FE model of single shot LSP treatment. It is found that besides the commonly studied depth direction, the in-plane stress components are also varied with its in-plane locations, which are due to “edge effects”. The residual stress fields for the whole region reconstructed by the proposed approach agree well with the actual stress fields even if some of the non-critical stress information are missing. Finally, the results are compared with those reconstructed by the eigenstrain method. It is shown that the proposed iteration approach can reconstruct the residual stress distribution in the whole region at a satisfactory level by solving a direct linear elastic static equilibrium problem, while the eigenstrain method needs to solve an inverse problem to obtain the appropriate eigenstrain distribution.

Inclusion-based boundary element method for virtual experiments of particulate composites containing arbitrarily shaped inhomogeneities

This paper extends our recent work [1] to evaluate the elastic fields and effective modulus of a composite containing arbitrarily shaped inhomogeneities for both two-dimensional (2D) and three-dimensional (3D) problems. Based on Eshelby’s equivalent inclusion method (EIM), the material mismatch between inhomogeneities and matrix phases is represented with a continuously distributed eigenstrain on inclusions. Since there exists singularities at the vertices of polyhedral inhomogeneities, domain discretization of inhomogeneities is used to interpolate the eigenstrain distribution, and high accuracy is obtained by using the closed-form integrals of the source field. The influence of the singularity decays in 1r2 and 1r3 for 2D and 3D problems respectively. Because Eshelby’s tensors depend on the shape instead of the size, the iBEM is particularly suitable for cross scale modeling of composites with a wide range of particle sizes. Although a number of elements are required to provide high fidelity results of the local field around the vertices, in general, very few elements or a single element are enough for each inhomogeneity to obtain the convergent solutions of the effective material properties, which enables virtual experiments of a composite containing many inhomogeneities. This novel numerical method has been verified with the finite element method (FEM) with much more elements. Virtual experiments of composites with many particles demonstrate its versatile capability and great potentials.

Elastic Solution of a Polyhedral Particle With a Polynomial Eigenstrain and Particle Discretization

AbstractThe paper extends the recent work (Wu, C., and Yin, H., 2021, “Elastic Solution of a Polygon-Shaped Inclusion With a Polynomial Eigenstrain,” ASME J. Appl. Mech., 88(6), p. 061002) of Eshelby’s tensors for polynomial eigenstrains from a two-dimensional (2D) to three-dimensional (3D) domain, which provides the solution to the elastic field with continuously distributed eigenstrain on a polyhedral inclusion approximated by the Taylor series of polynomials. Similarly, the polynomial eigenstrain is expanded at the centroid of the polyhedral inclusion with uniform, linear, and quadratic order terms, which provides tailorable accuracy of the elastic solutions of polyhedral inhomogeneity using Eshelby’s equivalent inclusion method. However, for both 2D and 3D cases, the stress distribution in the inhomogeneity exhibits a certain discrepancy from the finite element results at the neighborhood of the vertices due to the singularity of Eshelby’s tensors, which makes it inaccurate to use the Taylor series of polynomials at the centroid to catch the eigenstrain at the vertices. This paper formulates the domain discretization with tetrahedral elements to accurately solve for eigenstrain distribution and predict the stress field. With the eigenstrain determined at each node, the elastic field can be predicted with the closed-form domain integral of Green’s function. The parametric analysis shows the performance difference between the polynomial eigenstrain by the Taylor expansion at the centroid and the C0 continuous eigenstrain by particle discretization. Because the stress singularity is evaluated by the analytical form of Eshelby’s tensor, the elastic analysis is robust, stable, and efficient.

Elastic Solution of a Polygon-Shaped Inclusion With a Polynomial Eigenstrain

Abstract This paper presents the Eshelby’s tensor of a polygonal inclusion with a polynomial eigenstrain, which can provide an elastic solution to an arbitrary, convex inclusion with a continuously distributed eigenstrain by the Taylor series approximation. The Eshelby’s tensor for plane strain problem is derived from the fundamental solution of isotropic Green’s function with the Hadmard regularization, which is composed of the integrals of the derivatives of the harmonic and biharmonic potentials over the source domain. Using the Green’s theorem, they are converted to two line (contour) integrals over the polygonal cross section. This paper evaluates them by direct analytical integrals. Following Mura’s work, this paper formulates the method to derive linear, quadratic, and higher order of the Eshelby’s tensor in the polynomial form for arbitrary, convex polygonal shapes of inclusions. Numerical case studies were performed to verify the analytic results with the original Eshelby’s solution for a uniform eigenstrain in an ellipsoidal domain. It is of significance to consider higher order terms of eigenstrain for the polygon-shape inclusion problem because the eigenstrain distribution is generally non-uniform when Eshelby’s equivalent inclusion method is used. The stress disturbance due to a triangle particle in an infinite domain is demonstrated by comparison with the results of the finite element method (FEM). The present solution paves the way to accurately simulate the particle-particle, partial-boundary interactions of polygon-shape particles.

The effects of axisymmetric radial, circumferential and longitudinal eigenstrains on the traveling wave solution in a neo-Hookean cylindrical rod

This study deals with the impact of cylindrical eigenstrains on the traveling wave solutions of a neo-Hookean cylindrical rod. For this purpose, we consider an isotropic, incompressible neo-Hookean rod with the symmetrical distribution of radial, circumferential and longitudinal eigenstrains. To establish the momentum balance equations, we construct a Riemannian manifold as the reference configuration, and then place it in Euclidean space. Assuming that the rod has an axisymmetric region with uniform eigenstrains, we extend and simplify the governing equations to come up with the final nonlinear differential equation. After thorough analysis of this equation, we introduce a traveling wave which is not observed in an eigenstrain-free rod and also explain how the cylindrical eigenstrains affect the velocities. In addition, we propose an important solution, stating that with special quantities of the eigenstrains, any arbitrary function can be a traveling wave in the rod (provided that it is physically acceptable). To substantiate this claim, we find those eigenstrain parameters by which the equilibrium equation is satisfied automatically. Proving that these waves are of equal velocities, we can say that this solution is similar to d’Alembert’s solution in linear approaches.

Elasticity of a cylinder with axially varying dilatational eigenstrain

The paper presents a general solution of the isotropic elasticity problem for the sandwiched inclusion in an elastic cylinder and having axially varying eigenstrain. This problem is relevant in considerations of a cylinder with axially varying thermal expansion coefficient or a nanowire having an axial inhomogeneity of the crystal lattice parameter due to growth and processing conditions.The technique of the solution and the results for the elastic fields and energies of dilatational inclusions with different distribution of eigenstrain, namely, constant, trapezoidal, and diffusion-like, along the axis of the cylinder, are given in full details. The technique explores the axial superpositions of infinitely thin dilatation disks inserted in the cylinder, the elastic fields of which are found in analytical form. In addition, the energies of interfaces separated domains with constant eigenstrains in the cylinder are given. It is demonstrated that the blurring of the eigenstrain in the transition region leads to a decrease in interface energy.

Analytical solution for circular inhomogeneous inclusion problems with non-uniform axisymmetric eigenstrain distribution

This paper presents the analytical solution of a class of plane elasticity problems for circular inhomogeneous inclusions with non-uniform axisymmetric eigenstrain distribution, which includes both radial and hoop eigenstrains. The complex variable potential solution for a point-wise eigenstrain in an infinite plane solid is presented first, in which two principal strains and their directions are taken into account. Directly employing it as influence function, the complex variable potential for the circular homogeneous inclusion problem is formulated with Green's function method. The novelty of this approach is that it is able to take intrinsic advantage of complex variable approach and effectively tackle the mathematical difficulties encountered during formulation. Next, by using the principle of equivalent eigenstrain, the main challenge in solving inhomogeneous inclusion problems is overcome, allowing the general explicit analytical solution to be derived. Based on these results, three illustrative examples of practical significance are given: (i) dissimilar cylinder interference-fits within an infinite body, (ii) a pure dilatational eigenstrain problem within a circular inclusion, and (iii) a circular inclusion problem with a wedge disclination eigenstrain distribution. The fundamental formulation introduced here will find application in other aspects in the mechanics of fiber composites, thermoelasticity, and nano-mechanics of defects in solids.

An efficient method for the elastic field in a transversely isotropic full space due to arbitrary inclusions

The present study is on the analytical solution for the elastic field due to a cuboidal inclusion of uniform eigenstrain within a transversely isotropic full-space material, and a numerical method to model inclusions of any arbitrary shapes and with any eigenstrain distributions as the integration of a set of such cuboidal inclusions. The fast Fourier transform (FFT) is applied for efficient computation. The developed method and results are implemented to analyze the elastic field in a transversely isotropic full-space material containing inclusions of different shapes, different eigenstrain distributions, and multiple cuboids of different densities. Furthermore, the effect of material anisotropy on the stress field subjected to a spherical inclusion with pure dilatant eigenstrains is explored by comparing the behavior of a transversely isotropic material with that of a corresponding isotropic one. The numerical results show that the induced stresses are drastically influenced by the Young’s moduli of transversely isotropic materials, and that material constant C33 has a large influence on normal stress σ33.

Eshelby problem in continuous shape transition of helical inclusion

Change in Eshelby tensor associated with continuous shape transition from cylinder to helix was numerically computed by means of Green's function method, where helical shape transition is demonstrated by horizontal and vertical shape transitions parameters, δ=(1−d/D) and ρ=(1−d/p), respectively (d: helical core diameter, D: helical diameter, p: helical pitch). Regardless of the number of helical turns, significant change in Eshelby tensor was observed under the continuous shape transition from straight cylinder to mild twisted helix. In the meanwhile, the Eshelby tensor associated with the shape transition from the hollow-tube cylinder to the strong twisted helix is greatly influenced by the number of helical turns. By assuming spatial distribution of eigenstrain for the helix embedded in the cylinder, average internal stress inside and outside of helix was predicted by means of Tanaka-Mori theorem.

Nonlinear traveling wave propagation in a Neo-Hookean cylindrical rod with torsional eigenstrain

This paper investigates the torsional and longitudinal traveling wave propagation through a cylindrical rod with torsional eigenstrain distribution. We derive three dynamic equilibrium equations of axisymmetric motion of a rod which is assumed to be composed of an incompressible, isotropic neo-Hookean solid and then reduce them to one nonlinear differential equation. Employing phase plane analysis, we solve and assess this equation, and accordingly determine all smooth and non-smooth waves. We also examine the effects of torsional eigenstrain on the shapes and velocities of traveling waves. We show that a new type of traveling wave, which is not seen in a normal rod, arises due to the eigentwist. Moreover, we demonstrate that in a cylindrical rod with eigentwist, a torsional traveling wave is always associated with a longitudinal one while it can propagate alone in a normal rod.