Nonlinear compressive stiffness in impacted composite laminates determined by an inverse method

This paper presents use of an inverse method and non-contact optical measurements for determining the apparent compressive stiffness reduction in impact damage zones in composite laminates. The tensile stiffness distribution and nonlinearity is also briefly covered. The method is based on iterative updating of the material properties in a finite element model with the objective to match the predicted displacement fields to those measured optically in impacted specimens under load. To examine the effect of the damage on initial imperfections, strain and buckling, the displacement fields obtained experimentally by digital image correlation are demonstrated and discussed. Finally, the method is applied to the obtained full-field measurements and the influence of applied strain on the nonlinear tensile stiffness and apparent compressive stiffness of real impact damage zones is evaluated. Material nonlinearity in tension is found to increase towards the damage centre where fibre damage is more severe. Stiffness in compression can only be represented by a uniform apparent material nonlinearity, which is strongly linked to local buckling.     

A Self-Learning Data-Driven Development of Failure Criteria of Unknown Anisotropic Ductile Materials with Deep Learning Neural Network

This paper first proposes a new self-learning data-driven methodology that can develop the failure criteria of unknown anisotropic ductile materials from the minimal number of experimental tests. Establishing failure criteria of anisotropic ductile materials requires time-consuming tests and manual data evaluation. The proposed method can overcome such practical challenges. The methodology is formalized by combining four ideas: 1) The deep learning neural network (DLNN)-based material constitutive model, 2) Self-learning inverse finite element (SELIFE) simulation, 3) Algorithmic identification of failure points from the self-learned stress-strain curves and 4) Derivation of the failure criteria through symbolic regression of the genetic programming. Stress update and the algorithmic tangent operator were formulated in terms of DLNN parameters for nonlinear finite element analysis. Then, the SELIFE simulation algorithm gradually makes the DLNN model learn highly complex multi-axial stress and strain relationships, being guided by the experimental boundary measurements. Following the failure point identification, a self-learning data-driven failure criteria are eventually developed with the help of a reliable symbolic regression algorithm. The methodology and the self-learning data-driven failure criteria were verified by comparing with a reference failure criteria and simulating with different materials orientations, respectively.     

Adaptation of the virtual fields method for the identification of biphasic hyperelastic model parameters in soft biological tissues with osmotic swelling

Biphasic hyperelastic models have become popular for soft hydrated tissues, and there is a pressing need for appropriate identification methods using full-field measurement techniques such as digital volume correlation. This paper proposes to address this need with the virtual fields method (VFM). The main asset of the proposed approach is that it avoids the repeated resolution of complex nonlinear finite element models. By choosing special virtual fields, the VFM approach can extract hyperelastic parameters of the solid part of the biphasic medium without resorting to identifying the model parameters driving the osmotic effects in the interstitial fluid. The proposed approach is verified and validated through three different examples: the first and second using simulated data and then the third using experimental data obtained from porcine descending thoracic aortas samples in osmotically active solution.     

Variability response functions for beams with nonlinear constitutive laws

The ability to determine probabilistic information of response quantities in structural mechanics (e.g. displacements, stresses) is restricted due to lack of information on the probabilistic characteristics of uncertain system parameters. The concept of the Variability Response Function (VRF) has been proposed as a means to systematically capture the effect of the stochastic spectral characteristics of uncertain system parameters modeled by homogeneous stochastic fields on the uncertain structural response. The key property of the VRF in its classical sense is its independence from the marginal probability distribution function (PDF) and the spectral density function (SDF) of the uncertain system parameters (it depends only on the deterministic structural configuration and boundary conditions). In this paper, the existence, the uniqueness, and the SDF- and PDF-independence of a variability response function is formally proven for the first time for statically determinate beam structures following a specific class of nonlinear constitutive laws (power laws). For statically indeterminate nonlinear structures, the generalized variability response function (GVRF) methodology is shown to produce GVRFs for statically indeterminate nonlinear beams with a square-root constitutive law that are almost SDF-independent and only mildly dependent on the marginal PDF. This PDF-dependence is not significant and all GVRFs computed in this study have very similar shapes. This is important as it implies that conclusions related to the effect of correlation length scales on the response uncertainty can be inferred in general. However, the GVRF methodology for nonlinear statically indeterminate structures is only suitable when a closed-form expression is known to exist for the VRF of statically determinate structures having the same constitutive law.     

An inverse method for determining elastic material properties and a material interface

A numerical procedure which integrates optimization, finite element analysis and automatic finite element mesh generation is developed for solving a two-dimensional inverse/parameter estimation problem in solid mechanics. The problem consists of determining the location and size of a circular inclusion in a finite matrix and the elastic material properties of the inclusion and the matrix. Traction and displacement boundary conditions sufficient for solving a direct problem are applied to the boundary of the domain. In addition, displacements are measured at discrete points on the part of the boundary where the tractions are prescribed. The inverse problem is solved using a modified Levenberg-Marquardt method to match the measured displacements to a finite element model solution which depends on the unknown parameters. Numerical experiments are presented to show how different factors in the problem and the solution procedure influence the accuracy of the estimated parameters.     

Analytical Full-field Solutions of a Piezoelectric Layered Half-plane Subjected to Generalized Loadings

The two-dimensional problem of a planar transversely isotropic piezoelectric layered half-plane subjected to generalized line forces and edge dislocations in the layer is analyzed by using the Fourier-transform method and the series expansion technique. The full-field solutions for displacements, stresses, electrical displacements and electric fields are expressed in explicit closed forms. The complete solutions consist only of the simplest solutions for an infinite piezoelectric medium with applied loadings. It is shown in this study that the physical meaning of this solution is the image method. The explicit solutions include Green's function for originally applied loadings in an infinite piezoelectric medium and the remaining terms are image singularities which are induced to satisfy free surface and interface continuity conditions. The mathematical method used in this study provides an automatic determination for the locations and magnitudes of all image singularities. The locations and magnitudes of image singularities are dependent on the piezoelectric material constants of the layered half-plane and the location of the applied loading. With the aid of the generalized Peach-Koehler formula, the image forces acting on dislocations are derived from the full-field solutions of the generalized stresses. Numerical results for the full-field distributions of stresses and electric fields in the piezoelectric layered half-plane and image forces for edge dislocations are presented based on the available analytical solutions.     

Identification of Constitutive Material Model Parameters for High-Strain Rate Metal Cutting Conditions Using Evolutionary Computational Algorithms

Advances in plasticity-based analytical modeling and finite element methods (FEM) based numerical modeling of metal cutting have resulted in capabilities of predicting the physical phenomena in metal cutting such as forces, temperatures, and stresses generated. However, accuracy and reliability of these predictions rely on a work material constitutive model describing the flow stress, at which work material starts to plastically deform. This paper presents a methodology to determine deformation behavior of work materials in high-strain rate metal cutting conditions and utilizes evolutionary computational methods in identifying constitutive model parameters. The Johnson-Cook (JC) constitutive model and cooperative particle swarm optimization (CPSO) method are combined to investigate the effects of high-strain rate dependency, thermal softening and strain rate-temperature coupling on the material flow stress. The methodology is applied in predicting JC constitutive model parameters, and the results are compared with the other solutions. Evolutionary computational algorithms have outperformed the classical data fitting solutions. This methodology can also be extended to other constitutive material models.     

A scaled boundary finite element formulation for dynamic elastoplastic analysis

This study presents the development of the scaled boundary finite element method (SBFEM) to simulate elastoplastic stress wave propagation problems subjected to transient dynamic loadings. Material nonlinearity is considered by first reformulating the SBFEM to obtain an explicit form of shape functions for polygons with an arbitrary number of sides. The material constitutive matrix and the residual stress fields are then determined as analytical polynomial functions in the scaled boundary coordinates through a local least squares fit to evaluate the elastoplastic stiffness matrix and the residual load vector semianalytically. The treatment of the inertial force within the solution of the nonlinear system of equations is also presented within the SBFEM framework. The nonlinear equation system is solved using the unconditionally stable Newmark time integration algorithm. The proposed formulation is validated using several benchmark numerical examples.     

位移测量反演分析的正、逆定式化方法及其算法程序实现

位移测量反演分析已应用于岩石工程、结构检测、生物力学诊断等领域。介绍了位移反演分析的逆定式化方法与正定式化方法,以及基于这两种方法的初始地应力与岩层介质模量识别的两个算法程序,对比分析了两者在地应力模型、目标参数、反演算法等几个方面区别,重点说明了约束变尺度法优化算法能较好地解决非线性规划问题。并应用于岩石力学施工预报,获得了不同合理性和精度的反分析结果,以及区间最优的反演参数。     

Inverse design techniques for improving composite prosthetic devices

The design and performance of composite prosthetic devices can be improved by tailoring the material properties to achieve a prescribed response. An example of such a response would be displacements and stresses exhibited by healthy, undisturbed femoral bone. In this paper, an inverse design methodology, used in the Volumetrically Controlled Manufacturing (VCM) process, is developed and tested for improving the design of orthopedic prosthetic devices. First, a three-dimensional finite element (FE) model is developed based on available Computed Tomography (CT) data. The FE model is used to evaluate the response of the model subjected to a typical load. Second, as a part of the VCM process, the inverse design process is used to formulate a design problem that is in the form of a constrained least-squares problem. The intent is to find the material properties of the FE model to obtain a known displacement field on the stem-cancellous interface. Third, a solution methodology is developed to solve this constrained least squares problem using the finite element analysis for function evaluations and a gradient-based nonlinear programming (NLP) method to solve the design problem. Two test problems are solved to illustrate the developed methodology. The results indicate that material properties can be tailored to meet specific response requirements.     

Identification of Gurson's material constants by using Kalman filter

A new approach based on the inverse analysis is proposed for estimating material parameters of nonlinear constitutive equations. Using the measurable response of experimental specimens, an inverse analysis is carried out to predict most suitable values of unknown material constants. In general, the accuracy of prediction depends on geometries of specimens and types of measurements. In order to identify optimal experimental procedure, the Kalman filter technique is employed. We have chosen the Gurson model for porous elastic-plastic materials as the material model and its two parameters as the unknown constants. Gurson's constitutive model has been widely used for studying ductile fracture as well as shear localization of various metals. Detailed finite element simulations are performed to demonstrate the effectiveness of the proposed method in determination of the two parameters relating to void nucleation. In the Kalman filter procedure, it is found that the rate of convergence to the correct solutions depends on shapes of test specimens, initial estimates of the unknown parameters, and accuracy of measured data as well as computed reference data. Our analysis predicts that when two differently shaped specimens under tension are used (i.e., a plate with a center hole and another with double side notches), a significant improvement occurs in the rate of convergence.     

基于纤维模型的钢管混凝土组合框架连续倒塌非线性动力分析

基于非线性纤维梁-柱单元理论建立了钢管混凝土空间框架有限元模型,分析其在非正常荷载作用下(火灾、爆炸、撞击等)的抗连续性倒塌性能。以ABAQUS软件为求解平台进行了钢材和混凝土材料模型的二次开发,并采用已有研究者的试验结果进行了模型验证,在此基础上采用抽柱法进行了一典型的12层钢管混凝土空间框架体系在不同初始损伤模型下的连续性倒塌非线性动力分析。分析结果表明,不同抽柱工况下,框架梁柱内力由于失效柱破坏将发生内力重分布,相邻构件弯矩和轴力变化较大,其卸载后传力路径遵循就近原则;底层中柱失效后,其上部节点竖向位移最大,底层角柱失效后,上部节点竖向位移最小。总体上钢管混凝土空间框架在竖向荷载作用下具有较好的抗连续倒塌性能。     

Explicit analysis technique for nonlinear inverse mechanics problems

Nonlinear material models are conventionally used in forward analysis to predict the global mechanical response of boundary value problems. Such models are not expected to exactly reproduce global experimental response in all cases. Accordingly, the measured global response at specific domain or surface points can guide an inverse nonlinear structural analysis to successively recover a representative material model. By assuming an initial set of stress–strain data points, the load–displacement response at the control points is computed in a forward incremental analysis without iterations. This analysis retains the stresses at the integration points. The corresponding strains are not expected to be accurate since the computed displacements are not anticipated to match the measured displacements at the control points. Therefore, a conjugate incremental displacement analysis is performed at the same load steps to correct for displacements and strains everywhere by matching the measured displacements at the control points. It is found that the predicted stress–strain data set at the most highly stressed integration point provides the most accurate representation of an improved material model. This data set is used in the next two-phase incremental analysis pass as the material model. The process is repeated until the forward analysis phase reproduces the measured displacements at the control points requiring no corrections. Accordingly, the selection of a single stress–strain data set yields an explicit recovery of the nonlinear elasticity material model without any interpolation or averaging schemes, which significantly reduces data storage and computational effort. The applicability of the present explicit approach is demonstrated on simple mechanical models, a skeletal structural system and a 2D finite element mesh.     

Thermo-elastic material parameters identification using modified error in constitutive equation approach

This paper presents an identification procedure for anisotropic thermo-elastic heterogeneous material profile based on modified error in constitutive equation (MECE) approach. The inverse problem is posed as an optimization problem where the objective functional evaluates the difference in constitutive relation that associates kinematically admissible strain field to the statically admissible stress field. An additional term due to corruption in measurement data is included in the cost functional as a penalty form. While following standard MECE-based identification procedure, we have proposed a trace norm of the constitutive discrepancy functional that arises due to two dissimilar fields for material parameter update. In the process, we obtain explicit parameter update formula for general anisotropic thermo-elastic material. However, unlike elastic case, parameter update equations are nonlinear due to thermo-elastic constitutive relation. Finally, the potential of the proposed procedure in estimating anisotropic material parameters is illustrated through some large-scale parameter estimation problems.     

Identification of Gurson–Tvergaard material model parameters via Kalman filtering technique. I. Theory

In this paper quasi-static ductile fracture processes are simulated within the framework of the finite element method by means of the Gurson–Tvergaard isotropic constitutive model for progressively cavitating elastoplastic solids. The progressive degradation of the material strength properties in the fracture process zone due to micro-void growth to coalescence is modeled through the computational cell concept. Among the several model parameters to be calibrated in the computations, attention is restricted to the Tvergaard coefficients q ₁ and q ₂ and to the initial porosity f ₀ in the unstressed configuration. To identify these model parameters the inverse problem is solved via the extended Kalman filter for nonlinear systems coupled to a numerical methodology for the sensitivity analysis. In part I of this work the theory of Kalman filtering and sensitivity analysis is presented. First results concerning the identification of the Tvergaard parameters for a whole crack growth in single edge notched bend specimens made of a pressure vessel steel are presented. In order to enhance the convergence towards the final solution of the identification procedure, during the tests measurements are made of the displacements of points located in the central portion of the notched specimens, where model parameters highly affect the system state variables. In part II of this work a numerical validation of the proposed procedure in terms of uniqueness of the final identified solution, requirements of accuracy for the Bayesian initialization of the model parameters and sensitivity to the experimental measurement errors will be presented and discussed.     

Inverse estimation of variable thermal parameters in a functionally graded annular fin using dragon fly optimization

This paper presents an inverse study of heat transfer of a conductive, convective and radiative annular fin made of a functionally graded material. Three major parameters such as conductive–convective parameter, conductive–radiative parameter and the parameter describing the variation of thermal conductivity are inversely estimated from a specified temperature field. The forward solution of temperature field is obtained from the closed form solution of nonlinear heat transfer equation using Homotopy perturbation method (HPM). A dragonfly algorithm that simulates the swarming behaviour of dragonflies, as analogous, is employed in finding out the inverse parameters. The temperature values of the forward solution are used as input data for the inverse analysis. The inverse parameters are then estimated iteratively by minimizing the objective function until the guessed temperature field approximately satisfies the preassigned temperature field of the forward solution. The inverse simulation following HPM-based forward solution converges faster than ordinary differential equation-based forward solution. The reconstructed temperature fields obtained from the various combination of inverse parameters give good agreement (～1% error) with the desired temperature field. Thus, the presented inverse model provides an opportunity to the fin designer for selecting the several feasible combinations of thermal parameters suggesting the material design that result in a prescribed temperature field.     

Analysis on the fatigue damage evolution of notched specimens with consideration of cyclic plasticity

This paper presents a damage mechanics method applied successfully to assess fatigue life of notched specimens with plastic deformation at the notch tip. A damage‐coupled elasto‐plastic constitutive model is employed in which nonlinear kinematic hardening is considered. The accumulated damage is described by a stress‐based damage model and a plastic strain‐based damage model, which depend on the cyclic stress and accumulated plastic strain, respectively. A three‐dimensional finite element implementation of these models is developed to predict the crack initiation life of notched specimens. Two cases, a notched plate under tension‐compression loadings and an SAE notched shaft under bending‐torsion loadings including non‐proportional loadings, are studied and the predicted results are compared with experimental data.     

Three-dimensional nonlinear thermo-elastic analysis of functionally graded cylindrical shells with piezoelectric layers by differential quadrature method

Three-dimensional nonlinear thermo-elastic analysis of a functionally graded cylindrical shell with piezoelectric layers under the effect of asymmetric thermo-electro-mechanical loads is carried out. The strain–displacement relations are based on the nonlinear Lagrangian strain–displacement relations; that is, nonlinear terms containing derivatives of the displacement in the radial direction are included. Material properties of the shell are assumed to be graded in the radial direction according to a power law but the Poisson’s ratio is assumed to be constant. Cylindrical shells are assumed to be under the effect of pressure loading in cosine form, ring pressure loads, electric and temperature fields. Numerical results of stress, displacement, electric and thermal fields are obtained by using two versions of the differential quadrature methods, namely polynomial and Fourier quadrature methods. The convergence of the solution is studied, and results of the axisymmetric loadings are verified with reported results for a cylindrical shell with material properties obeying a power law. Effects of the grading index of material properties, the temperature difference, the ratio of the mean radius to the thickness of the shell, boundary conditions, the thickness of piezoelectric layers and electric excitation on stress, displacement, electric and temperature fields are presented.     

Towards Eliminating the Displacement Bias Due to Out‐Of‐Plane Motion in 2D Inverse Problems: A Case of General Rigid‐Body Motion

This article reports an important development related to the inverse characterization of material constitutive parameters using 2D optical displacement field measurements. The out‐of‐plane motion of the specimen, which has traditionally been considered detrimental to the accuracy of these experiments, is generally of two types: (a) a global out‐of‐plane rigid‐body motion of the specimen relative to the camera and (b) out‐of‐plane deformations resulting from material heterogeneity or out‐of‐plane loads. In an earlier article, we proposed to partially relax the condition of no out‐of‐plane motion by allowing for (b) in 2D inverse procedures, in the context of finite element update method, and introduced a compensation strategy by redefining the cost function on the object plane of the acquisition system. The experimental errors due to (a) were assumed negligible. Here, we propose that the global rigid‐body motion (a) may also be recovered within the inverse procedures, hence completely waiving the condition of strictly in‐plane displacements for inverse problems. The recovery is achieved by identifying and including the possible modes of global rigid‐body motion within the cost function together with careful selection of test configuration. The effects of individual rigid‐body modes on the computed displacement fields are studied in detail and utilized as a guideline for selection of test configuration. The approach is fully demonstrated and validated by simulated as well as real experiments for determining elastic constants of isotropic and orthotropic materials using different experimental setups. Effects of improving the optimization routine, for cost function minimization, and experimental noise are also presented.     

Calibrating constitutive models with full-field data via physics informed neural networks

The calibration of solid constitutive models with full-field experimental data is a long-standing challenge, especially in materials that undergo large deformations. In this paper, we propose a physics-informed deep-learning framework for the discovery of hyperelastic constitutive model parameterizations given full-field surface displacement data and global force-displacement data. Contrary to the majority of recent literature in this field, we work with the weak form of the governing equations rather than the strong form to impose physical constraints upon the neural network predictions. The approach presented in this paper is computationally efficient, suitable for irregular geometric domains, and readily ingests displacement data without the need for interpolation onto a computational grid. A selection of canonical hyperelastic material models suitable for different material classes is considered including the Neo–Hookean, Gent, and Blatz–Ko constitutive models as exemplars for general non-linear elastic behaviour, elastomer behaviour with finite strain lock-up, and compressible foam behaviour, respectively. We demonstrate that physics informed machine learning is an enabling technology and may shift the paradigm of how full-field experimental data are utilized to calibrate constitutive models under finite deformations.