Rigid Plastic Hardening Research Articles

Crashworthiness design of graded cellular materials: Experimental verification of the backward design strategy

Graded cellular materials with a suitable design may improve the crashworthiness of protective structures. A backward design strategy considering the loading-rate sensitivity of cellular material has been proposed in the companion paper to guide the crashworthiness design of graded cellular material for protecting a moving object. Detailed experimental results are presented in this study to verify the feasibility and estimate the accuracy of this design strategy. A rate-dependent, rigid–plastic hardening (D-R–PH) idealization was employed to model cellular materials. A two-specimen test method was developed to determine the dynamic material parameters and their relationship with the relative density for the crashworthiness design. A power-law relationship between the dynamic material parameters and the relative density for the closed-cell foam with acrylonitrile butadiene styrene (ABS) plastic as the matrix material was obtained and employed in the crashworthiness design. Additive manufacturing was employed to fabricate the designed graded cellular specimens with ABS plastic, and the Taylor–Hopkinson pressure bar experimental technique together with high-speed photography was applied to perform impact tests. Graded ABS closed-cell foam specimens for different design conditions, e.g., the initial impact velocity, impact mass, and design target of impact force, were designed by the D-R–PH backward design strategy and prepared, and the corresponding impact tests were conducted. The impact force applied on the mass remains relatively stable and meets the crashworthiness requirement in all test cases. Therefore, the feasibility and reliability of the crashworthiness backward design strategy are verified, and this design strategy can be used to guide the crashworthiness design in engineering practice.

Enhancement in the mechanical behaviour of a Schwarz Primitive periodic minimal surface lattice structure design

Triply periodic minimal surface (TPMS) sheet lattice structures are composed of continuous and smooth shells, enabling the achievement of a high surface-to-volume ratio and pore interconnectivity, which represent an emerging solution for lightweight applications. In this study, an improved Schwarz primitive lattice (P-lattice) structure was proposed by redefining the original opening diameter with a shape parameter. Prototypes of different configurations, such as the original P-lattice (OP) structure, modified P-lattice structure with a small opening diameter (SP), and modified P-lattice structure with a big opening diameter (BP) were fabricated via micro-selective laser melting using 316 L stainless steel. Quasi-static compression tests were performed on the fabricated samples. The experimental results indicated that the Young's modulus, compressive strength, and energy absorption of the SP lattice were increased by 25.84%, 15.63%, and 33.02%, respectively, compared with those of the OP structure. A finite element model was established to investigate the mechanical properties and energy absorption of all the designed configurations, and the results showed good agreement with the experimental observations. A rigid–plastic hardening model was also introduced to macroscopically predict the mechanical response and energy absorption of the as-designed lattice structures. The mechanical properties and energy absorption of the SP structure outperformed those of the OP and BP structures.

Crushing behavior and deformation mechanism of randomly honeycomb cylindrical shell structure

Abstract Honeycomb structure, which presents potential ability in energy absorption and impact resistance, has advantages of simplicity, lightweight, easy manufacturing and designability. In this study, honeycomb structure panel is expanded to cylindrical shell by adopting the rolled-up algorithm and the crushing behaviors of the randomly honeycomb cylindrical shell (RHCS) structures are investigated under axial loading. The influence of geometrical and topological parameters, i.e. the thickness-to-diameter ratio, cell irregularity, relative density on the deformation evolution and energy absorption performance is systematically performed. It is found that the deformation modes of RHCS structures are significantly affected by thickness-to-diameter ratio and cell irregularity. Moreover, the rate-independent, rigid–plastic hardening (R–PH) idealisation with two parameters is adopted to characterize the crushing responses of RHCS structures and the parameters are quantified as power-law relations with respect to relative density. Further, the crushing behaviors of density-graded RHCS structures with continuously varying in density are carried out under different impact velocities. Results indicated that the introduction of density gradients can strongly affect the deformation mode, crushing force and energy absorption performance of density-graded RHCS structures. Combining with analysis of crushing front propagation, the theoretical determination of crushing responses of the density-graded RHCS under different loading rates is presented based on R–PH model. The results of predicted crushing force-time histories of density-graded RHCS structures show versatile agreement with that of finite element values at both impact and support ends. Additionally, the energy absorption capacity that is related to deformation mechanisms of density-graded RHCS structures is investigated.

Impact Resistance and Design of Graded Cellular Cladding

One-dimensional impact response of graded cellular rods with monotonously increasing density distribution, which benefits in buffering the impact object in proximal end, is investigated by using shock models and cell-based finite element models. Based on the rigid–plastic hardening (R-PH) idealization, a theoretical model of the shock front propagation behavior in graded cellular rods subjected by impact is developed and solved numerically with a fourth-order Runge–Kutta scheme. Finite element analysis is performed using 2D random Voronoi technique, and the comparisons illustrate that the theoretical predictions and finite element results are in good agreement. The anti-impact responses of graded cellular materials with different gradient parameters are carried out, and it is demonstrated that the design indicators, including support stress, critical length and maximum deceleration of the object, are highly related to the gradient parameters of graded cellular materials. A design strategy for the impact alleviation of graded cellular materials is proposed. Thus, the gradient parameters can be finally determined to meet the requirement of the actual design indicators and it can also offer an optimal critical length for final graded cellular materials.

Modification and verification of the Deshpande–Fleck foam model: A variable ellipticity

The ellipticity involved in the Deshpande–Fleck foam model describing the constitutive behaviour of cellular materials was usually considered to be constant, but some very different values were suggested in the literature. A cell-based finite element model of closed-cell foam under uni-/multi-axial compression is employed to verify the Deshpande–Fleck foam model. The ellipticity is determined by applying uniaxial and hydrostatic compression tests and it is found to vary with the equivalent plastic strain, i.e., the ellipticity decreases with the equivalent plastic strain and then increases sustainably before full densification. According to the understandings from the numerical results, a fitting relation between the ellipticity and the equivalent plastic strain is suggested. The ellipticities of an open-cell foam and a closed-cell foam studied experimentally in the literature are fitted well with this relation. A modification to the Deshpande–Fleck foam model with a variable ellipticity is thus proposed. The modified Deshpande–Fleck foam model brings much accurate predictions with using a rigid–plastic hardening (R-PH) idealisation model, which describes the stress–strain relation of cellular material under uniaxial compression. Good agreement is also observed between the experimentally measured stress–strain responses and the predictions of the modified Deshpande–Fleck foam model, especially when considering the effect of the plastic Poisson's ratio with a non-associated flow rule. The findings herein are helpful to improve the prediction accuracy of the Deshpande–Fleck foam model.

Blast response of continuous-density graded cellular material based on the 3D Voronoi model

Abstract One-dimensional blast response of continuous-density graded cellular rods was investigated theoretically and numerically. Analytical model based on the rigid-plastic hardening (R-PH) model was used to predict the blast response of density-graded cellular rods. Finite element (FE) analysis was performed using a new model based on the 3D Voronoi technique. The FE results have a good agreement with the analytical predictions. The blast response and energy absorption of cellular rods with the same mass but different density distributions were examined under different blast loading. As a blast resistance structure, cellular materials with high energy absorption and low impulse transmit is attractive. However, high energy absorption and low impulse transmit cannot be achieved at the same time by changing the density distribution. The energy absorption capacity increases with the initial blast pressure and characteristic time of the exponentially decaying blast loading. By contract, when the blast loading exceeds the resistance capacity of cellular material, the transmitted stress will be enhanced which is detrimental to the structure being protected.

Stress Distribution in Graded Cellular Materials Under Dynamic Compression

Dynamic compression behaviors of density-homogeneous and density-graded irregular honeycombs are investigated using cell-based finite element models under a constant-velocity impact scenario. A method based on the cross-sectional engineering stress is developed to obtain the one-dimensional stress distribution along the loading direction in a cellular specimen. The cross-sectional engineering stress is contributed by two parts: the node-transitive stress and the contact-induced stress, which are caused by the nodal force and the contact of cell walls, respectively. It is found that the contact-induced stress is dominant for the significantly enhanced stress behind the shock front. The stress enhancement and the compaction wave propagation can be observed through the stress distributions in honeycombs under high-velocity compression. The single and double compaction wave modes are observed directly from the stress distributions. Theoretical analysis of the compaction wave propagation in the density-graded honeycombs based on the R-PH (rigid-plastic hardening) idealization is carried out and verified by the numerical simulations. It is found that stress distribution in cellular materials and the compaction wave propagation characteristics under dynamic compression can be approximately predicted by the R-PH shock model.

Blast Alleviation of Cellular Sacrificial Cladding: A Nonlinear Plastic Shock Model

The anti-blast behavior of cellular sacrificial cladding is investigated based on a continuum-based nonlinear plastic shock model. A rate-independent, rigid–plastic hardening (R-PH) model with two material parameters, namely the initial crushing stress and the strain hardening parameter, is employed to idealize the cellular material. The governing equation of the motion of cover plate is obtained and solved numerically with a fourth-order Runge–Kutta scheme. A comparison of the crushing percentage contours of sacrificial cladding based on the R-PH model and the rigid–perfectly plastic–locking (R-PP-L) model is carried out. Results transpire that the R-PP-L model is not accurate enough to evaluate the energy absorption. Dimensional analysis is employed to study the critical length of cellular sacrificial cladding and an empirical expression is determined by the controlling valuable method. An asymptotic solution is also obtained by applying the regular perturbation theory. Finally, the design criteria of cellular sacrificial cladding based on the R-PH shock model is verified by a cell-based finite element model.

Flow of viscoplastic materials through converging channels

AbstractMany classical rigid perfectly/plastic solutions cannot be extended to more complicated rigid plastic materials, such as rigid plastic hardening materials and rigid viscoplastic materials. The present paper reveals sources of this difficulty in the case of flow of rigid viscoplastic material through infinite converging rough wedge‐shaped and axisymmetric channels. Two types of viscoplastic models without strain hardening are considered, with and with no saturation stress. The maximum friction law is adopted at the friction surface. Qualitative features of the solutions are compared to those occurring in compression of a rigid plastic layer between rough, parallel plates. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Application of corotational rates of the logarithmic strain in constitutive modeling of hardening materials at finite deformations

In this paper a finite deformation constitutive model for rigid plastic hardening materials based on the logarithmic strain tensor is introduced. The flow rule of this constitutive model relates the corotational rate of the logarithmic strain tensor to the difference of the deviatoric Cauchy stress and the back stress tensors. The evolution equation for the kinematic hardening of this model relates the corotational rate of the back stress tensor to the corotational rate of the logarithmic strain tensor. Using Jaumann, Green–Naghdi, Eulerian and logarithmic corotational rates in the proposed constitutive model, stress–strain responses and subsequent yield surfaces are determined for rigid plastic kinematic and isotropic hardening materials in the simple shear problem at finite deformations.