Nonlinear Strain Research Articles

A Review of Sheet Metal Forming Evaluation of Advanced High-Strength Steels (AHSS)

This paper presents a review on the formability evaluation of AHSS, enhancing necking-based failure criteria limitations. Complementary fracture/damage constitutive modeling approaches specifically tailored to formability evaluation, validated through numerical and experimental methods, are also subjects of research. AHSS are widely processed through sheet metal forming processes. Although an excellent choice when lightweight, high-strength, and ductility are critical factors, their multi-phase microstructure accentuates forming challenges. To accurately model forming behavior, necking-based failure criteria as well as direct fracture models require improvements. As a necking-based failure model, the conventional forming limit diagram/curve (FLD/FLC) presents limitations in estimating direct fracture (surface cracks, edge cracks, shear cracks), as well as deformation histories under non-linear strain paths. Thus, significant research efforts are being made towards the development of advanced fracture constitutive models capable of predicting fracture scenarios without necking, which are more frequently observed in the realm of AHSS. Scientific community research is divided into several directions aiming at improving the forming and fracture behavior accuracy of parts subjected to sheet metal forming operations. In this review paper, a comprehensive overview of ductile fracture modeling is presented. Firstly, the limitations of FLD/FLC in modeling fracture behavior in sheet metal forming operations are studied, followed by recent trends in constitutive material modeling. Afterwards, advancements in material characterization methods to cover a broad range of stress states are discussed. Finally, damage and fracture models predicting failure in AHSS are investigated. This review paper supplies relevant information on the current issues the sheet metal forming community is challenged with due to the trend towards AHSS employment in the automotive industry.

Buckling and post-buckling of anisogrid lattice-core sandwich plates with nanocomposite skins

This research offers an in-depth examination of the buckling and post-buckling responses observed in sandwich plates distinguished by the presence of an anisogrid lattice core and face layers composed of functionally graded carbon nanotube-reinforced composites (FG-CNTRC). To model this structure, a global continuous model for the lattice core is employed, while the displacement field is approximated using the First-Order Shear Deformation Theory (FSDT). Geometrically nonlinear strain relations based on the von-Kármán assumptions are incorporated to accurately capture the structural response. The homogenization of the FG-CNTRC skins is achieved through the modified rule of mixture, enabling estimations of effective material properties. To separate pre-buckling and post-buckling phases, we implement the adjacent-equilibrium criterion. Solving the linear (pertaining to buckling) and nonlinear (related to post-buckling) equations is based on the Generalized Differential Quadrature (GDQ) technique alongside an iterative method grounded in the displacement control strategy. Our analysis explores the influence of core and skin characteristics on the stability of these sandwich plates. By examining critical buckling thresholds, post-buckling equilibrium paths, and the underlying mechanisms driving these responses, we contribute insights into the structural stability and performance of such composite structures. This study not only enhances our understanding of buckling behavior in anisogrid lattice core sandwich plates but also holds promise for diverse engineering applications where structural stability is paramount.

Numerically assisted calibration procedure of nonlinear in-plane shear properties of unidirectional composite laminae based on off-axis tensile experiments

In this paper, a novel methodology is presented to evaluate the true nonlinear shear response of continuous fiber-reinforced plastic (CFRP) unidirectional laminae. It requires simple off-axis tensile experiments to be conducted and a finite element (FE) representation of them with a material constitutive law capable of handling nonlinearity in shear. Upon successful evaluation of the normal stiffness properties, the shear stress-strain response is derived via numerical calibration of the underlying FE models. The presented approach is also demonstrated by processing the raw data of an extensive material characterization test campaign conducted on a thermoplastic matrix CFRP. The outcome is compared to the conventional methods for shear response derivation using relevant standards such as ASTM D3518 and ASTM D5379. It has been successfully demonstrated that the pseudo-hardening phenomenon obtained from standard shear experiments as a result of fibers aligning with load direction can be eliminated with off-axis specimen tension experiments, thus, the true shear stress versus deformation response can be extracted up to failure. The main purpose of current work is to demonstrate the inconsistency in the available standard methods related to mechanical testing-based derivation of nonlinear in-plane shear behavior of UD plies. In addition, a novel technique is presented to achieve a more accurate prediction of nonlinear shear stress and strain along the entire representative loading range that contributes to more accurate simulations of composite parts up to failure and thus, better strength predictions.

Investigation on the size effect of compacted clay mode I fracture based on NDB specimens

The size effect is closely related to the cracking resistance of compacted clay, which is crucial in engineering applications, but little attention has been given to it in previous studies. This study investigated the size effect on mode I fracture behaviors of compacted clay using single-edge notched deep beam (NDB) specimens. Through the digital image correlation (DIC) method, the crack initiation mechanism of compacted clay was analyzed. The results showed that the specimen size and dimensionless crack length did not change the failure pattern, while peak load, peak displacement, and fracture energy gradually increased with the increase of specimen size. Based on the load–displacement curves and the horizontal strain evolution laws, the three horizontal strain stages at the crack tip crack initiation point were defined, i.e., the steady strain growth stage, the nonlinear strain growth stage, and the nonlinear strain intensification stage. Through the Bažant size effect model (SEM), the fracture process zone length Cf and mode I fracture toughness KIC of compacted clay without the size effect were inferred. The separated form of SEM was applied, and the contribution ratios of the strength criterion and linear elastic fracture mechanics were quantitatively analyzed. Combing the characteristics of the NDB specimens, a prediction model considering the support spacing, specimen structure parameters, and fracture toughness was established.

A physically-based mixed hardening model for the prediction of the ductility limits of thin metal sheets using a CPFE approach

An advanced Crystal Plasticity Finite Element (CPFE) approach is developed to accurately predict the ductility limit strains of thin metal sheets. This method uses polycrystalline unit cells to represent the metal sheets at the macroscopic level. The macroscopic behavior of these unit cells is determined based on that of the constituent single crystals using the periodic homogenization multiscale scheme. At the single crystal scale, the constitutive framework follows a finite strain rate-independent formulation, with the flow rule governed by the Schmid law. The evolution of the single crystal yield surface is described through a physically-based mixed hardening model, where isotropic hardening is characterized by a dislocation density-based formulation, while kinematic hardening is described by the nonlinear Armstrong–Frederick model. The unit cell ductility limit strains are predicted by the Rice bifurcation criterion. The reliability of the mixed hardening model in accurately reproducing mechanical behavior is confirmed through simulations of uniaxial tension/compression loading. Then, the developed computational strategy is used to investigate the impact of key microstructural hardening parameters on the initiation of localized necking under linear strain paths. The numerical predictions reveal the significant influence of these parameters on the formability of thin metal sheets. Additionally, the analysis of ductility limits under non-linear strain paths demonstrates a strong dependency of the numerical predictions on strain path changes. The numerical predictions obtained by the developed CPFE multiscale strategy are compared with experimental results from the literature. In summary, the proposed approach provides a reliable tool for accurately predicting the ductility limits of thin metal sheets, offering valuable insights for engineering applications.

Experimental investigation of steel-reinforced precast shear wall with replaceable energy dissipators

To improve the bearing and energy dissipation of the precast concrete (PC) shear wall, this paper proposes an angle-section and C-section steel embedded PC wall with external low-yield-point steel (LYP) energy dissipators (EDs). The wall is an energy-dissipation and load-bearing bi-functional system during the seismic and a design method for this wall is also presented in this paper. Two full-size precast shear walls with and without external EDs (PWE and PW, respectively), were designed based on the proposed method and tested under an axial compression ratio of 0.2. Seismic performance in terms of failure modes, hysteretic behavior, strength envelope, stiffness degradation, energy dissipation, and steel strains was investigated and discussed. According to the crack pattern observed during the test, the PW and PWE were found to be in flexural–shear failure mode, and the PWE exhibited better seismic behavior compared with the PW, The yield and peak loads of the PWE compared with those of the PW increased by 23% and 22%, respectively. The equivalent viscous damping ratios of the two walls exceeded 0.05. When the walls were under a large displacement, the equivalent viscous damping coefficient of the PWE was higher than that of the PW. Moreover, the cumulative energy dissipation of the PWE exceeded that of the PW by 15–55%. The strain in the embedded steel in the PWE was less than that in the PW, indicating that the PW suffered more nonlinear strain and concrete damage. Test results indicate that the supplementation of replaceable external EDs can improve lateral bearing and energy dissipation capacities as well as decrease the nonlinear strain to reduce the extent of damage to wall panels. Additionally, the analytical model of PWE based on the fiber model was proposed and the rationality of the model was verified by comparison of the numerical and tested results. Furthermore, a parameter analysis was conducted to investigate the seismic behavior of PWE considering various structural properties.

A nonlocal higher-order shear deformation approach for nonline ar static analysis of magneto-electro-elastic sandwich Micro/Nano-plates with FG-CNT core in hygrothermal environment

In this study, the nonlocal static analysis of Magneto-Electro-Elastic (MEE) sandwich micro/nano-plates with Functionally Graded Carbon Nanotubes (FG-CNTs) core in a hygrothermal environment is studied. The nonlocal static formulation is developed based on Reddy's High-order Shear Deformation theory (HSDT). The von Kármán nonlinear strains are used and the governing equations of the HSDT are derived accounting for Eringen's nonlocal stress-gradient model. The governing equations are solved using the Galerkin method, which involves expanding the displacement field based on orthogonal functions. The variation of the volume fractions through the thickness of FG-CNTs core layer has been computed using two different homogenization techniques: the rule of mixtures and the Mori–Tanaka scheme. A detailed parametric study to show the characteristics of the FG-CNTs core, nonlocal parameters, thermal and moisture changes, types of CNT reinforcement and volume fraction, electric and magnetic potentials, and geometric parameters. The results are compared with the first-order shear deformation theory (FSDT) and classical plate theory to show the accuracy of nonlocal High-order Shear Deformation theory(NHSDT) formulation.

Installation of PHDs accelerates the self-weight consolidation of dredged sludge yards by considering the influence of the stacking process

In dredging projects, dredged sludge is usually stacked in yards and treated by installing prefabricated horizontal drains (PHDs). A research gap exists in the large-strain self-weight consolidation of dredged sludge stacked in yards treated with PHDs. In this study, a nonlinear large strain consolidation model for dredged sludge treated with PHDs was established, considering its nonlinear compressibility and permeability, the change in stacking thickness with time, and varying PHD layers. Besides, the solution of this model was obtained using the finite difference method, and its reliability was verified by comparing it with that of one-dimensional consolidation and plane strain consolidation under special conditions. On this basis, the effects of different factors on the large strain consolidation of dredged sludge yards were analyzed. The results show that the consolidation rates of dredged sludge are affected by the stacking time as well as the number and laying ratio of PHDs. The design of dredged sludge yards can be optimized in terms of the laying position and ratio of PHD. The sludge treatment capacity is associated with the stacking time.

A new size-dependent nonlinear model for piezoelectric semiconductor nanofibers by considering the effects of strain gradient and flexoelectricity

Flexotronics induced by the strain gradient and flexoelectricity in a semiconductor with symmetric or asymmetric crystal structures have attracted considerable attention as a promising approach for tunable electronic processes. However, the electric nonlinear properties between the carriers and electric field are ignored in most of related studies. In this study, a new size-dependent nonlinear model is developed to investigate the interaction among the size effects (caused by strain gradient and flexoelectricity), piezo-effects and semiconducting properties in a piezoelectric semiconductor (PS) nanofiber with asymmetric crystal structures. In addition, the Von-Kármán's nonlinear strain is also considered to check the effect of geometric nonlinearity. It is found that the piezo-effect can be enhanced (reduced) by the effect of flexoelectricity (strain gradient) due to a new positive (negative) superimposed increment of electric field under a pair of tensile stress. Besides, the size effect will be weakened by the semiconducting property. The studies will provide guidance for theoretical analysis of size effect in nano PSs and design of flexotronic devices.

The displacement mechanism of the cracked rock – a seismic design and prediction study using XFEM and ANNs

Materials with sufficient strength and stiffness can transfer nonlinear design loads without damage. The present study compares crack propagation speed and shape in rock-like material and sandstone when subjected to seismic acceleration. The nonlinear extended finite element method (NXFEM) has been used in numerical simulation. It assumes the model has a pre-existing crack at 0° from the horizontal. The mechanical properties of the model, crack propagation shape, and crack speed were selected as the main parameters. The nonlinear stress and strain along the crack have been compared in two simulated models. NXFEM and Artificial Neural Networks (ANNs) were used to predict the displacement. The simulation results illustrate that the materials’ crack propagation mechanism and mechanical properties control the stress, strain, and displacement at the selected points in the model. In addition, crack propagation in materials is related to elastic-plastic stresses and strains along the crack path. The speed and shape of the crack are associated with the mechanical properties of the materials. The prediction of crack paths helps to understand failure patterns. Comparison of the seismic response of the rock-like material with sandstone helps to assess the stress, strain, and displacement levels during cracking. This study’s findings agree with the literature report and field observations.

Driving torque model of the bionic soft arm’s hyperelastic bellows

ABSTRACT The technology of soft continuum robots represents an advance in the field of robotics to benefit a wide range of industries such as healthcare, manufacturing or environmental exploration. Soft continuum robots can be driven pneumatically by bellows as the soft continuum manipulator presented in this article. A driving torque model of the bellows considering hyperelastic material properties, friction and restoring torques is derived, whose purpose is suitability for model-based control design and subsequent trajectory generation. Hence, the driving torque model must be real-time capable. This is realized by an iterative algorithm calculating the bellows’ torque transmission by assuming a two-dimensional no-slip membrane contact. Nonlinear strain behaviours and hysteresis effects of the bellows are considered by the Ogden material model. The driving torque model’s performance is validated experimentally by measuring the external torques of the bellows.

Nonlinear wave propagation in graphene incorporating second strain gradient theory

The exploration of graphene has attracted extensive interest owing to its significant structural and mechanical properties. In this research, we numerically investigate wave propagation in defect-free single-layer graphene, considering its geometrically nonlinear behavior through second-strain gradient elasticity. To capture the geometric nonlinearity, firstly, the nonlinear strain–displacement relations are introduced. The governing equation and associated boundary conditions are derived using Hamilton’s principle. Then, the weak form, including the element matrices, is established. The eigenvalue problem is solved for 2D wave propagation through periodic structures theory. Finally, the dynamical properties such as band structures, mode shapes, energy flow, and wave beaming effects are analyzed. The numerical results reveal that the geometric nonlinearity through the second strain gradient influences the wave propagation characteristics in graphene. The findings are significant and contribute to the understanding of graphene’s dynamic response, with implications for the engineering applications of graphene-based nanostructures.

Nonlinear Vibration of Truncated Open Conical Nanoshells Under Harmonic Excitation

The nonlinear forced vibration behavior of truncated open conical nanoshells is investigated in this paper. Nonlocal elasticity theory has been employed to modify the size-dependent effect of micro/nanostructure. Von Karman nonlinear strains are used to consider the geometric nonlinearity of the nanoshells. Using Hamilton’s principle, the equations of motion and boundary conditions of truncated open conical nanoshells are derived. Galerkin method is used to solve the governing equations. Then, to determine the nonlinear forced vibration behavior of truncated open conical nanoshells, complex averaging and arc-length continuation methods are employed. Finally, the effect of nonlocal parameters, cone apex angle, the amplitude of harmonic excitation, structural damping coefficient, and geometrical parameters on the nonlinear frequency response of the truncated open conical nanoshells are discussed.

Nonlinear strain energy formulation of spatially deflected strip flexures

Modeling spatial deflections of strip flexures in compliant mechanisms has been one of the most challenging problems due to the nonlinear coupling effects between bending, torsion and stretching. Energy-based approaches can mitigate the modeling complexity by treating a compliant mechanism in its entirety and eliminating the analysis of load equilibriums between the adjacent elements. To leverage energy-based approaches, this work formulates the nonlinear strain energy stored in spatially deflected strip flexures. The strain energy, expressed as the integration of the product of stress and strain over the entire volume, is simplified using Taylor series expansion followed by truncation of higher order terms, leading to a closed-form expression with relevant nonlinearities retained. The coefficient matrices in the strain expression are validated by the load–deflection relations obtained in our previous work. The effectiveness of the energy expression combined with Castigliano’s first theorem is demonstrated by two compliant mechanisms containing multiple strip flexures, and the results are validated by those of nonlinear finite element models. The energy expression offers a convenient and mathematically simple way for modeling compliant mechanisms containing spatially deflected strip flexures.

A user material approach for the solution of multi-field problems in Abaqus: Theoretical foundations, gradient-enhanced damage mechanics and thermo-mechanical coupling

The solution of multi-field problems and the numerical implementation by means of the finite element method constitute a sophisticated part of the characterisation of industrial processes. A comprehensive implementation framework for such a system of coupled field equations into a non-linear large strain finite element formulation is provided. The procedure is derived for a micromorphic approach in a thermo-mechanical setting. Although the provided framework contributes to a particular three-field problem it is not limited to a specific application or a specific number of coupled field equations from a conceptual point of view. The solution of the considered system of equations is separated into two coupled domains, with the balance of linear momentum and a balance equation of heat equation-type being solved on each of them. Since both, the balance of micromorphic momentum and the heat balance equation, are partial differential equations of Laplace-type, the resulting two-instance problem can be solved in the framework of commercial finite element software, such as Abaqus, based on a thermo-mechanical user material. To assess the framework for a particular constitutive model, a gradient-enhanced damage model in a thermo-mechanical setting is applied and representative simulation results are discussed. The Abaqus framework is made available as an open-source code on GitHub (https://github.com/InstituteOfMechanics/Thermomechanical_Gradient_Enhanced_Damage_UMAT).

Tension buckling and postbuckling of nanocomposite laminated plates with in-plane negative Poisson’s ratio

Abstract Mechanical metamaterials with negative Poisson’s ratio (NPR) have emerged as a novel class of engineering material, and have attracted increasing attention in various engineering sectors. Most studies available on the buckling problem of laminated plates with positive or NPR are those under uniaxial compression. Here, we report that the buckling phenomenon may occur for auxetic nanocomposite laminated plates under uniaxial tension when the unloaded edges of the plates are immovable. Two types of nanocomposites are considered, including graphene/Cu and carbon nanotube/Cu composites. Governing equations of the auxetic nanocomposite laminated plates are formulated based on the framework of Reddy’s higher-order shear deformation theory. In modeling, the von Kármán nonlinear strain–displacement relationship, temperature-dependent material properties, thermal effects, and the plate–substrate interaction are considered. The explicit analytical solutions for postbuckling of auxetic nanocomposite laminated plates subjected to uniaxial tension are obtained for the first time by employing a two-step perturbation approach. Numerical investigations are performed for tension buckling and postbuckling behaviors of auxetic nanocomposite laminated rectangular plates with in-plane NPR rested on an elastic substrate under temperature environments.

Geometrically nonlinear bending of stiffened composite skewed cylindrical shells under transverse pressure

The literature review on relative performances of laminated composite skewed shells confirms that research reports on bending performances of moderately thin, stiffened, laminated composite skewed cylindrical panels, using the geometrically nonlinear approach, are not available. This paper aims to fill that deficiency and proposes a finite element code combining eight-noded, doubly curved elements with modified Sanders’ first approximation theory for thin shells and von Kármán-type nonlinear strains. Correctness of the proposed geometrically nonlinear bending formulation for skewed shells are verified through solutions of benchmark problems. The deflections, force, and moment resultants are reported for different skew angles, laminations, stacking sequences, radius of curvature, plan dimension ratios, and stiffener properties like orientations, numbers, and eccentric positions. The results are discussed critically which reveals that shells having curved edges free and straight edges clamped fabricated using 0°/90°/0° laminate offer the best performances. The biaxial stiffeners, nx = 7, ny = 7, show the minimum deflections and stress resultants. The skewed shells offer greater deflections and hence, must be avoided in industrial practices.

A comprehensive formulation for determining static characteristics of mosaic multi-stable composite laminates under large deformation and large rotation

Multi-stable composite laminates are composite materials that exhibit multi-stable states, making them highly suitable for use in morphing structures. These materials are capable of maintaining each stable state without expending any energy. As a result, they are used extensively in numerous applications and garnered the interest of scholars and aerospace organizations. In the context of practical applications, such as morphing structures, it is insufficient for designers to rely solely on common bi-stable composite laminates that exhibit large deformations and medium rotations to achieve their desired objectives. Consequently, based on the objectives of the design, there are two potential resolutions to address this limitation. A designer may utilize mosaic multi-stable composite laminates to achieve a morphing structure that exhibits high flexibility, significant deformation, and substantial rotation. The utilization of a series connection between a bi-stable composite laminate and a symmetric composite laminate results in the formation of a mosaic bi-stable composite laminate with variable stiffness. Furthermore, the amalgamation of two asymmetric composite laminates with inverted orientations engenders a mosaic tri-stable composite laminate. The present research examines the static characteristics of mosaic bi-stable and tri-stable composite laminates. It also seeks to analyze the factors affecting the behavior of these types of laminates. A geometrically exact model was formulated for this objective. Apart from the geometrically exact model, a widely used and uncomplicated model relying on the conventional Classical Laminated-Plate Theory (CLPT) and Von-Karman nonlinear strains was employed. The proposed models were validated through finite element simulations. The system's static equations were derived using the virtual work principle and the Rayleigh-Ritz method. The present study examines and explores quasi-static snap-through behavior between stable states through the application of concentrated forces. The findings indicate a high level of concurrence between the outcomes derived from the geometrically exact model and the finite element analyses, particularly in composite laminates exhibiting significant deformations and rotations.

Solitary deformation waves in two coaxial shells made of material with combined nonlinearity and forming the walls of annular and circular cross-section channels filled with viscous fluid

The aim of the paper is to obtain a system of nonlinear evolution equations for two coaxial cylindrical shells containing viscous fluid between them and in the inner shell, as well as numerical modeling of the propagation processes for nonlinear solitary longitudinal strain waves in these shells. The case when the stress-strain coupling law for the shell material has a hardening combined nonlinearity in the form of a function with fractional exponent and a quadratic function is considered. Methods. To formulate the problem of shell hydroelasticity, the Lagrangian–Eulerian approach for recording the equations of dynamics and boundary conditions is used. The multiscale perturbation method is applied to analyze the formulated problem. As a result of asymptotic analysis, a system of two evolution equations, which are generalized Schamel– Korteweg– de Vries equations, is obtained, and it is shown that, in general, the system requires numerical investigation. The new difference scheme obtained using the Grobner basis technique is proposed to discretize the system of evolution equations. Results. The exact solution of the system of evolution equations for the special case of no fluid in the inner shell is found. Numerical modeling has shown that in the absence of fluid in the inner shell, the solitary deformation waves have supersonic velocity. In addition, for the above case, it was found that the strain waves in the shells retain their velocity and amplitude after interaction, i.e., they are solitons. On the other hand, calculations have shown that in the presence of a viscous fluid in the inner shell, attenuation of strain solitons is observed, and their propagation velocity becomes subsonic.

Effect of non-linear strain stiffening in eDAH and unjamming.

In cell clusters, the prominent factors at play encompass contractility-based enhanced tissue surface tension and cell unjamming transition. The former effect pertains to the boundary effect, while the latter constitutes a bulk effect. Both effects share outcomes of inducing significant elongation in cells. This elongation is so substantial that it surpasses the limits of linear elasticity, thereby giving rise to additional effects. To investigate these effects, we employ atomic force microscopy (AFM) to analyze how the mechanical properties of individual cells change under such considerable elongation. Our selection of cell lines includes MCF-10A, chosen for its pronounced demonstration of the extended differential adhesion hypothesis (eDAH), and MDA-MB-436, selected due to its manifestation of cell unjamming behavior. In the AFM analyses, we observe a common trend in both cases: as elongation increases, both cell lines exhibit strain stiffening. Notably, this effect is more prominent in MCF-10A compared to MDA-MB-436. Subsequently, we employ AFM on a dynamic range of 1-200 Hz to probe the mechanical characteristics of cell spheroids, focusing on both surface and bulk mechanics. Our findings align with the results from single cell investigations. Specifically, MCF-10A cells, characterized by strong contractile tissue tension, exhibit the greatest stiffness on their surface. Conversely, MDA-MB-436 cells, which experience significant elongation, showcase their highest stiffness within the bulk region. Consequently, the concept of single cell strain stiffening emerges as a crucial element in understanding the mechanics of multicellular spheroids (MCSs), even in the case of MDA-MB-436 cells, which are comparatively softer in nature.