Modelling the constitutive behaviour of thin metal sheet using strain gradient theory

Two mesh refinement indicators based on the gradients of effective stress (GSIG) and effective plastic strain (GEPS), respectively, are proposed for adaptive finite element analysis of the large deformation, quasi‐static or dynamic response of shell structures. The mesh refinement indicators are based on equi‐distributing the variation of stresses or plastic strains over the elements of the mesh. A program module is developed and implemented in the non‐linear explicit finite element code LS‐DYNA. This module provides element‐wise refinement evaluations so that selective mesh refinements are carried out in regions of the mesh where the values of local indicators exceed a user‐specified tolerance. The FE model of a conventional deep drawing process is used as a numerical model, including both material and geometrical non‐linearities, in order to demonstrate the versatility of the two refinement indicators. Four different refinement indicators, based on angle change, thickness change, GSIG and GEPS, are applied in this investigation. The numerical results are compared with experimental results regarding the thickness distribution versus cup height, cup height variation versus circumference angle, effective plastic strain in the deformed sheet and punch force. It is shown that the proposed indicators can identify finite elements which have high gradients of effective stress or effective plastic strain so that the mesh is refined in the regions undergoing the most severe deformations and the numerical results are improved.

Micromechanical modeling of damage evolution and fracture behavior in particle reinforced metal matrix composites based on the conventional theory of mechanism-based strain gradient plasticity

Strain gradient plasticity with nonlinear evolutionary energetic higher order stresses

The effective threshold for fatigue crack propagation: a plastic size effect?

Lightweight sheet metals are attractive for aerospace and automotive applications due to their exceptional properties, such as low density and high strength. Sheet metal forming (SMF) is a key technology to manufacturing lightweight thin-walled complex-shaped components. With the development of SMF, numerical simulation and theoretical modelling are promoted to enhance the performance of new SMF technologies. Thus, it is extraordinarily valuable to present a comprehensive review of historical development in SMF followed by state-of-the-art advanced characterization and modelling approaches for lightweight metallic materials. First, the importance of lightweight materials and their relationship with SMF followed by the historical development of SMF are reviewed. Then, the progress of advanced finite element technologies for simulating metal forming with lightweight alloys is covered. The constitutive modelling of lightweight alloys with an explanation of state-of-the-art advanced characterization to identify the constitutive parameters are presented. Then, the formability of sheet metals with major influencing factors, the techniques for measuring surface strains in SMF and the experimental and modelling approaches for determining the formability limits are clarified. Finally, the review is concluded by affording discussion of the present and future trends which may be used in SMF for lightweight metallic materials.

A new approach on the influence of sheet thickness and material plastic properties on the limit strains in thin sheet metal forming is developed, using the strain gradient model that predict the local necking onset from initial thickness imperfections. As a result of the analysis, the definition of a roughness concept parameter is presented: the initial roughness profile inclination to thickness ratio parameter. Also, the critical normalized strain gradient at the onset of local necking can be calculated from the initial assumed roughness profile. A brief review of stretch forming of sheet metal, the diagrams and the limiting strain curves for local necking, FLC, and the limit strain theories are presented. The main characteristics of the sheet metal forming processes are also identified and are based in the press shop practice. The limit strains for different thickness sizes are obtained using the present model and a numerical code developed by the author. Present theoretical model produced reasonable predictions about the influence of thickness size on the FLC.

Investigation of a gradient enriched Gurson-Tvergaard model for porous strain hardening materials

Solution for cross- and angle-ply laminated Kirchhoff nano plates in bending using strain gradient theory

The main requirement of large deformation problems such as high-speed machining, impact, and various primarily metal forming, is to develop constitutive relations which are widely applicable and capable of accounting for complex paths of deformation. Achieving such desirable goals for material like metals and steel alloys involves a comprehensive study of their microstructures and experimental observations under different loading conditions. In general, metal structures display a strong rate- and temperature-dependence when deformed non-uniformly into the inelastic range. This effect has important implications for an increasing number of applications in structural and engineering mechanics. The mechanical behavior of these applications cannot be characterized by classical (rate-independent) continuum theories because they incorporate no ‘material length scales’. It is therefore necessary to develop a rate-dependent (viscoplasticity) continuum theory bridging the gap between the classical continuum theories and the microstructure simulations. Physically based vicoplasticity models for different types of metals (body centered cubic, face centered cubic and hexagonal close-packed) and steel alloys are derived in this work for this purpose. We adopt a multi-scale, hierarchical thermodynamic consistent framework to construct the material constitutive relations for the rate-dependent behavior. The concept of thermal activation energy, dislocations interactions mechanisms and the role of dislocations dynamics in crystals are used in the derivation process taking into consideration the contribution of the plastic strain evolution of dislocation density to the flow stress of polycrystalline metals. Material length scales are implicitly introduced into the governing equations through material rate-dependency (viscosity). The proposed framework is implemented into the commercially well-known finite element software ABAQUS. The finite element simulations of material instability problems converge to meaningful results upon further refinement of the finite element mesh due to the successful incorporation of the material length scale in the model formulations. It is shown that the model predicted results compare very well with different experimental data over a wide range of temperatures (77K°-1000K°) and strain rates (10-3-104s-1). It is also concluded from this dissertation that the width of localization zone (shear band) exhibits tremendous changes with different initial temperatures (i.e., different initial viscosities and accordingly different length scales).

In this paper, formulation of elastic-plastic analysis of functionally graded (FG) spherical pressure vessels under internal pressure based on strain gradient plasticity is presented. The material properties are assumed to vary in a power law manner in the radial direction. A linear hardening rule for the material behavior in the plastic region is assumed. After deriving the governing differential equations, a closed form solution is obtained. At the first step, the obtained results were validated against other available results in the literature. Then the effects of changing the inner radius from a few micro-meters to one meter, FG power index and strain gradient coefficient on stress and plastic region size are studied based on classical and strain gradient theories. Also, the effect of internal pressure on the size of plastic region is studied.

The governing equations of motion, together with the associated boundary conditions, are derived for the second strain gradient Timoshenko micro- and nano-beams. The second strain gradient theory is a highly powerful nonclassical continuum theory, capable of capturing the size effects in micro- and nano-scale structures. In case studies, the static and free-vibration behaviors of a hinged–hinged beam are investigated utilizing the presented second strain gradient theory-based Timoshenko beam model. The obtained results are compared with those of the available models in the literature, which are based on the (first) strain gradient theory, the modified couple stress theory, and the classical continuum theory.

The classical continuum theory not only underestimates the stiffness of microscale structures such as microbeams but is also unable to capture the size dependency, a phenomenon observed in these structures. Hence, the non-classical continuum theories such as the strain gradient elasticity have been developed. In this paper, a Timoshenko beam finite element is developed based on the strain gradient theory and employed to evaluate the mechanical behavior of microbeams used in microelectromechanical systems. The new beam element is a comprehensive beam element that recovers the formulations of strain gradient Euler–Bernoulli beam element, modified couple stress (another non-classical theory) Timoshenko and Euler–Bernoulli beam elements, and also classical Timoshenko and Euler–Bernoulli beam elements; note that the shear-locking phenomenon will not happen for the new Timoshenko beam element. The stiffness and mass matrices of the new element are derived in closed forms by following an energy-based approach and using Hamilton’s principle. It is noted that unlike the classical beam elements, the stiffness matrix of the new element has a size-dependent nature that can capture the size-dependent behavior of microbeams. The shape functions of the newly developed beam element are determined by solving the equilibrium equations of strain gradient Timoshenko beams, which brings about a size-dependent characteristic for them. The new beam element is employed to evaluate the static deflection of a microcantilever, and the results are compared to the experimental data as well as the results obtained by using the classical beam element and the couple stress plane element. The new beam element is also implemented to calculate the static deflection, vibration frequency, and pull-in voltage of electrostatically actuated microbeams. The current results are compared to the experimental data as well as the classical FEM outcomes. It is observed that the results of the new element are in excellent agreement with the experimental data while the gap between the experimental and classical FEM results is significant.

Surface treatment processes, such as shot peening, are routinely used by several manufacturing industries (automotive, aerospace, medical, etc.) to impart compressive residual stresses onto the surface of metal components to increase damage tolerance and fatigue life. These compressive stresses are engineered into components to delay initiation of surface cracks and retard crack growth rate thereby improving fatigue life. In addition, the plastic deformation resulting from the shot peening process hardens the material increasing the yield strength and elastic stress range. For elastic material response, accurate fatigue life estimates that include credit for residual stresses are possible for complex geometries with complicated load histories. However, when a shot peened component is subjected to elevated temperatures or cyclic loading, thermal relaxation and cyclic plasticity, respectively, produce inelastic material response from which residual stresses may change continuously. When the material response is inelastic, taking full credit for compressive residual stresses in fatigue models would result in a nonconservative prediction. As a result, designers are reluctant to incorporate any compressive residual stresses into fatigue life predictions of turbine engine components, subject to elevated temperatures and inelastic loading conditions. This paper investigates relaxation of shot peened residual stresses in IN100 when subjected to applied stress profiles that produce large plastic strains and strain gradients in laboratory test samples. Model predictions supported by experiments results are presented to explain that redistribution of residual stress depth profiles are a function of geometry and applied stress/strain profile.

The stability problem of cylindrical shells is addressed using higher-order continuum theories in a generalized framework. The length-scale effect which becomes prominent at microscale can be included in the continuum theory using gradient-based nonlocal theories such as the strain gradient elasticity theories. In this work, expressions for critical buckling stress under uniaxial compression are derived using an energy approach. The results are compared with the classical continuum theory, which can be obtained by setting the length-scale parameters to zero. A special case is obtained by setting two length scale parameters to zero. Thus, it is shown that both the couple stress theory and classical continuum theory forms a special case of the strain gradient theory. The effect of various parameters such as the shell-radius, shell-length, and length-scale parameters on the buckling stress are investigated. The dimensions and constants corresponding to that of a carbon nanotube, where the length-scale effect becomes prominent, is considered for this investigation.

Using differential quadrature method (DQM), this study investigated pull-in instability of beam-type nano-switches under the effects of small-scale and intermolecular forces including the van der Waals (vdW) and the Casimir forces. In these nano-switches, electrostatic forces served as the driving force, and von-Karman type nonlinear strain was used to examine nonlinear geometric effects. To derive nonlinear governing equations as well as the related boundary conditions for the nano-beam, variation method was used. Besides, to study the influence of size effect, the nonlocal elasticity theory was employed and the resulting governing equations were solved using DQM. Finally, the pull-in parameters were studied using the nonlocal theory and the results were compared with the numerical results of the classical continuum theory as well as experimental results contained in the references. Results demonstrated that taking into consideration the von-Karman type nonlinear strain increases the beam stiffness and hence, the pull-in voltage. Besides, use of the small scale, compared with the classical theory of elasticity, yields results much closer to experimental results.