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The purpose of this paper is to present an applied method to design the low-speed contact between a mass and surface of a beam using an analytical solution based on the first-order shear deformation beam theory. Also, a simulation of impact process is carried out by ABAQUS finite element (FE) code.

In theoretical formulation, first strains and stresses are obtained, then kinetic and potential energies are written, and using a combination of Ritz and Lagrange methods, a set of system of motion equations in the form of mass, stiffness and force matrices is obtained. Finally, the motion equations are solved using Runge–Kutta fourth order method.

The von Mises stress contours at the impact point and contact force from the ABAQUS simulation are illustrated and it is revealed that the theoretical solution is in good agreement with the FE code. The effect of changes in projectile speed, projectile diameter and projectile mass on the results is carefully examined with particular attention to evaluate histories of the impact force and beam recess. One of the important results is that changes in projectile speed have a greater effect on the results than changes in projectile diameter, and also changes in projectile mass have the least effect.

This paper presents a combination of methods of energy, Ritz and Lagrange and also FE code to simulate the problem of sandwich beams under low velocity impact.

The purpose of this paper is to investigate the nonsimultaneous impact of three impactors with spherical tip on the response of a low-velocity impact on a beam.

In this research, the third-order shear deformation theory of the beam with hyperbolic shear-strain function is used. Hamilton’s principle is applied to derive the motion equations. To simulate nonsimultaneous impacts, by using the Hertz nonlinear contact law, the contact of the impactors with different times is simulated. Comparisons with other articles are carried out in the one impactor form.

In the parametric study, the histories of the contact force and displacement of the beam are investigated in the presence of only one impactor in the center of the beam and also in the presence of three impactors, one in the center of the beam and the other two around the first impactor with a delay. One of the important and noteworthy points is that the presence of two impactors with a delay causes the maximum contact force and contact time to decrease and the maximum displacement of the beam center to increase.

The original point of this paper is what is the difference between the impact response of one projectile and three nonsimultaneous projectiles on the beam.

The purpose of this study is a numerical simulation and an analytical analysis about the low-velocity impact on a functionally graded porous plate with porosity distribution in the thickness direction. In this article, polymethyl methacrylate is used for matrix, and single-walled carbon nanotube (CNTs) (10,10) with consideration agglomeration sizes and lumping of CNT inside the agglomerations is applied for reinforcement.

In analytical formulation, the non-linear Hertz contact law is applied for interaction between projectile and plate surface. High-order shear deformation plate theory is developed, and energy of the system for impactor and plate is written. The governing equations are derived using Ritz method and Lagrange equations and are solved using the fourth-order Runge–Kutta method. Also, ABAQUS finite element model of functionally graded porous plate with all edges simply supported and reinforced by CNT under low-velocity impact is simulated and is compared with those is achieved in the present analytical approach.

In parametric studies, the influence of porosity distribution patterns include uniform, non-uniform symmetric and non-uniform asymmetric on the histories of contact force and impactor displacement of simply supported plate reinforced by CNT are presented. Eventually, the effects of porosity coefficient, impactor initial velocity, impactor radius and CNTs lumping inside agglomerations for non-uniform symmetric distribution patterns are discussed in impact event in detail.

In this paper, the effect of combination of polymethyl methacrylate and CNTs with consideration agglomeration sizes and lumping of CNTs inside the agglomerations in the form of a functionally graded porous plate is studied in the problem of low-velocity impact analysis.

The purpose of this study is to investigate the strain rate effect on the problem of low-velocity impact (LVI) on a beam, including silicon nitride and stainless steel materials.

Based on the nonlinear Hertz impact mechanism, the energies related to the impactor and the beam are written, and motion equations are derived using the Lagrangian mechanics and Ritz method. The strain rate term is represented as a damping matrix in the equations of motion. In the issue of LVI on the silicon nitride and stainless steel beam, the effect of internal viscous damping coefficient in simply–simply and clamped–free boundary conditions are studied. Also, the influence of the volume fraction index in the range between zero and one and greater than one on the impact response is investigated.

The results make it clear that the strain rate parameter had little effect on the response in LVI. Also, an increase in the volume fraction index has led to a decrease in the contact force and an increase in the rebound velocity of the impactor.

The effect of strain rate on LVI is theoretically studied in this paper, while in most of the papers, this effect is investigated experimentally and numerically.

Engineering composite laminates/structures are usually subjected to complex and variable loads, which result in interlayer delamination damage. However, damaged laminate may cause the whole structure to fail before reaching the design level. Therefore, the purpose of this paper is to develop an equivalent model to effectively evaluate compressive residual strength.

In this paper, taking carbon fiber reinforced composite T300/69 specimens as the study object, first, the compressive residual strength under different impact energy is obtained. Then, zero-thickness cohesive elements, Hashin failure criteria and Camanho nonlinear degradation scheme are used to simulate the full-process simulation for compression after edge impact (CAEI). Lastly, based on an improved Whitney–Nuismer criterion, the equation of edge hole stress distribution, characteristic length and compressive residual strength is used to verify the correctness of the equivalent model.

An equivalent relationship between the compressive residual strength of damaged laminates and laminates with edge hole is established. For T300/69 laminates with a thickness of 2.4 mm, the compressive residual strength after damage under an impact energy of 3 J is equivalent to that when the hole aperture R = 2.25 mm and the hole aperture R = 9.18 mm when impact energy is 6 J. Besides, the relationship under the same size and different thickness is obtained.

The value of this study is to provide a reference for the equivalent behavior of damaged laminates. An equivalent model proposed in this paper will contribute to the research of compressive residual strength and provide a theoretical basis for practical engineering application.

In this paper, the standard Peridynamic Timoshenko beam model accounting for the shear deformation is chosen to describe the thick beam kinematics. Unfortunately, when applied to very thin beam structures, the standard Peridynamics (PD) encounters the shear locking phenomenon, leading to incorrect solutions.

PD differs from classical continuum mechanics and other nonlocal theories that do not involve spatial derivatives of the displacement field. PD is based on the integral equation instead of differential equations to handle discontinuities and other singularities.

The shear locking can be successfully alleviated using the developed selective integration method. In particular, this technique has been implemented in the standard PD, which allows an accurate result for a wide range of slenderness from very thin to thick (10 < L/t < 10³) structures. It can also accelerate the computational time for particular dynamic problems using fewer neighboring integration particles. Several numerical examples are solved to demonstrate the effectiveness of the proposed method for modeling beam structures.

The paper highlights the severe shear locking phenomenon in the Peridynamic Timoshenko beam available in the literature, especially for very thin structures. A new alternative for the alleviation of shear locking in the Peridynamic Timoshenko beam, using selective integration. Hence the developed Peridynamic Timoshenko beam model is effective for thin and thick structures. A new peridynamic formulation for the low-velocity impact beam models is presented and validated.

1. The paper highlights the severe shear locking phenomenon in the Peridynamic Timoshenko beam proposed in the literature, especially for very thin structures.

2. The developed Peridynamic Timoshenko beam model based on selective integration is effective for thin and thick structures.

3. A new peridynamic formulation for the low-velocity impact beam models is presented and validated.

The paper highlights the severe shear locking phenomenon in the Peridynamic Timoshenko beam proposed in the literature, especially for very thin structures.

The developed Peridynamic Timoshenko beam model based on selective integration is effective for thin and thick structures.

A new peridynamic formulation for the low-velocity impact beam models is presented and validated.

This article focuses on the low-velocity impact (LVI) output of carbon nanotubes (CNTs)’ reinforcement circular plates, considering agglomeration size effect and clumping of CNTs’ inner side of the agglomerations.

A representative volume element (RVE) is used to determine the nanocomposite properties reinforced with agglomerated CNTs with random orientation. First-order shear deformation theory (FSDT) is used to obtain the motion equations of LVI analysis. These equations are handled by developing a Ritz method and Lagrangian mechanics. To extract the mass and stiffness matrices, terms with second and higher degrees are ignored.

Formulation validation is performed by providing various examples, including comparisons with other research and ABAQUS FE code. The effects of agglomeration size, clumping of CNTs’ inner side of the agglomerations, CNT volume fraction and impact location on the responses of impact load, projectile displacement and plate deflection are analytically studied. These achievements illuminate how the influence of agglomeration size is very small on the impact response. Also, the influence of clumping of CNTs’ inner side of the agglomerations is significant, and as it increases, the displacement values and impact time increase, and the impact force decreases.

In this article, to avoid additional calculations, the parameters of the mass matrix and the stiffness coefficients are linearized to obtain the equations of motion of the impact on the circular plate.

The present study assesses the impact resistance of the shear thickening fluids-filled (STFs-filled) foam through drop-hammer impact tests.

The maximum residual impact load and specific impact energy absorption rate of STF-filled foam are studied with varying thickness (4–14 mm), densities (0.35–0.6 g/cm3) and hardness (40–50 Rockwell Hardness C Scale (HRC)) under different ambient temperatures (−20−20 °C) and impact energies (25–75 J).

The following conclusions are obtained from this study: (1) the higher the impact energy, the greater the maximum residual impact force and energy absorption efficiency of the material; (2) the impact resistance of STF-filled foam can be improved with the decrease of ambient temperature, achieving the highest energy absorption rate at −10?. (3) STF-filled foam substrate has the highest impact resistance, the lowest maximum residual impact force and the highest energy absorption coefficient when the density is 0.35  g/cm3, the hardness is 45HC and the thickness is 10 mm.

This is the first paper to analyze the impact of both environmental factors and material properties on the impact resistance of STF-filled foam. The results show that the decrease in temperature and the increase in hardness can enhance the impact resistance of STF-filled foam.

The purpose of this study is to model the theory of the low-velocity impact (LVI) process on sandwich beams consisting of flexible cores and face sheets reinforced with functionally graded carbon nanotubes (CNTs).

A series of parameters derived from molecular dynamics are used to consider the size scale in the mixture rule for the combination of CNTs and resin. A procedure involving the use of the first-order shear deformation theory of the beam is used to provide the displacement field of the sandwich beam. The energy method and subsequently the generalized Lagrange method are used to derive the motion equations. Due to the use of Hertz’s nonlinear theory to calculate the contact force, the equations of motion are nonlinear. Validation of the problem is carried out by comparing natural frequencies with other papers.

The influence of a series of parameters such as CNTs distributions pattern in the face sheets, the influence of the CNTs volume fraction and the influence of the core thickness to the face sheets thickness ratio in the issue of LVI on sandwich beams with clamped-clamped boundary conditions is investigated. The result shows that the type of CNTs pattern in the face sheet and the CNTs volume fraction have a very important effect on the answer to the problem, which is caused by the change in the value of the Young’s modulus of the beam at the contact surface. Changes in the core thickness to the face sheets thickness ratio has little effect on the impact response.

Considering the important application of sandwich structures in vehicles, aviation and ships, in this research, sandwich beams consisting of flexible core and CNTs-reinforced face sheets are investigated under LVI.

The purpose of this study is to extract the response of the simultaneous low-velocity impact of multiple impactors on a porous functionally graded (FG) aluminum plate.

To design a porous FG structure, a series of functions are applied using the porosity coefficient, and mechanical properties including Young’s modulus, shear modulus and the density of the porous structure are presented as a function of the axis placed in the direction of the plate thickness. The first-order shear deformation theory of the plate is used. To simulate the contact process between each impactor and the plate, a nonlinear Hertz contact force is considered for that impactor independently.

ABAQUS finite element software is used for the verification process of the theorical equations. The effects of porous function type, radius and initial velocity of impactor are investigated for the simultaneous impact of five impactors on porous FG aluminum plate with a simply supported boundary condition. Histories of contact force and displacement of the impactor placed in the center of the beam are analyzed in detail with the changes of the mentioned parameters.

Due to the advantages of porous aluminum plate such as high energy absorption and low weight, such structures may be subjected to the simultaneous impact of multiple impactors, which is studied in this research.