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《高速刚体对复合材料冲击的影响》在现代科技领域，复合材料因其独特的力学性能，如高强度、轻量化和耐腐蚀性，被广泛应用于航空航天、汽车制造和体育器材等行业。然而，这些材料在受到高速冲击时的行为是研究的焦点，因为这种冲击可能导致结构损伤甚至失效。"high-velocity-impact-of-rigid-body-on-composites.zip"这个压缩包文件可能包含了一系列关于高速刚体对复合材料冲击的研究资料，包括实验数据、分析模型和仿真结果。高速冲击通常涉及物体以极高的速度相互碰撞，这在军事、交通和工程事故中都可能发生。在复合材料与刚体的碰撞过程中，由于材料内部的层间剪切、纤维断裂以及基体的塑性变形，复合材料可能会出现复杂多样的损伤模式。理解这些现象对于设计更耐冲击的复合材料结构至关重要。我们需要了解速度对冲击力的影响。根据动量守恒定律，碰撞的动能会转化为材料的变形能和声能。高速冲击下，能量的转换非常迅速，可能导致局部温度升高，引发热效应。此外，高冲击速度还会增加应力波在材料中的传播速度，可能导致瞬时的应力集中，从而造成严重的损伤。复合材料的特性使其在高速冲击下的响应与其他材料显著不同。其多层次的结构（如纤维增强树脂基体）意味着损伤可能沿着不同的方向发展。例如，纤维断裂可能导致材料强度下降，而层间剪切可能导致分层，这两者都会影响材料的整体性能。在研究中，研究人员可能会使用实验方法，如高速摄影技术来捕捉冲击过程，通过破坏表面的显微观察来评估损伤程度。同时，也会运用数值模拟工具，如有限元分析（FEA），来预测冲击响应并优化设计。这些工具可以帮助我们理解材料在不同冲击条件下的行为，并为设计抗冲击复合材料提供理论支持。此外，标签"velocity"暗示了文件内容可能专注于速度对冲击效果的影响。通过分析不同速度下的冲击结果，可以揭示速度与损伤程度之间的关系，这对于制定安全标准和防护措施具有重要意义。 "high-velocity-impact-of-rigid-body-on-composites.zip"这个压缩包可能包含了对高速冲击条件下复合材料性能的深入研究，涵盖了实验数据、理论分析和模拟计算等方面，对于提升复合材料在高风险环境下的应用安全性具有重要价值。无论是工程师设计更坚固的结构，还是科研人员探索新的材料强化途径，这些研究成果都将提供宝贵的参考。

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Contents lists available at ScienceDirect

Thin-Walled Structures

journal homepage: www.elsevier.com/locate/tws

Full length article

Deformation behavior of single and multi-layered materials under impact

Asheesh Sharma, Rohan Mishra, Sanyam Jain, Srikant S. Padhee, Prabhat K. Agnihotri

⁎

Department of Mechanical Engineering, Indian Institute of Technology Ropar, Rupnagar, 140001 Punjab, India

ARTICLE INFO

Keywords:

Abaqus explicit

Impact analysis

Critical failure velocity

Multilayer materials

VUMAT

ABSTRACT

Finite element (FE) simulations using Abaqus/Explicit are performed to study the deformation behavior of

materials under impact loading. Various conﬁgurations including monolithic and multi-layered plate having

combinations of ceramic, metal and composite material layers are investigated to determine the critical failure

velocity V

as a function of layer thickness and stacking. While cylindrical impactor is assumed to be rigid,

Johnson-Cook (JC), Johnson-Holmquist (JH2) and Hashin 3D and Puck criteria is used to characterize damage/

failure in metal (Al and steel), ceramic (SiC) and composite (carbon ﬁber/epoxy) layer respectively. Constitutive

equations for composite material are supplied via user subroutine VUMAT. The results of FEM simulations reveal

that Ceramic-Al-Carbon ﬁber/epoxy multilayer plate provides most desirable combination with higher critical

failure velocity, lower average density, lower pressure and displacement at the back plate as compared to other

material combinations considered in this work. Moreover, the analysis presented shows that the numerical

approach developed can be used as a tool to predict the geometry and material combinations of a multilayer

system to improve its resistance against impact loading.

1. Introduction

The study of interaction between two impacting bodies, known as

impact dynamics or terminal ballistics has many crucial applications.

Bullet impact on armor [1], occupant and pedestrian safety during

automobile accidents [2], tool drop on aircraft wing [3], are few ex-

amples where impact dynamics plays an important role. An in-depth

understanding of deformation behavior of materials under impact

loading helps not only in designing better products but more im-

portantly saving human life. Knowledge of material response under

impact loading will help in estimating, enhancing and extending life

and performance of any structure. Impact dynamics broadly depends

upon two primary variables, namely the geometry and material of

impacting bodies. Most often geometry of the impacting bodies gets

dictated by the design parameters other than impact requirements: an

aircraft wing has to be of an aero foil conﬁguration, an armor has to be

the shape of human body with limited thickness to enable mobility in

combat zone, and a bullet has to be of conical shape for eﬀective pe-

netration and so on. Ultimately, it boils down to the response of ma-

terial(s), which dictates the performance of the structure under impact

loading. Various theories have been proposed to model the average

behavior of metals [4,5], composites [6] and ceramics [7] under impact

loading. Depending upon their nature, diﬀerent materials have found

their niche domain of application. While ceramic materials are most

commonly used as front layer in armor for protection against bullet and

shrapnel due to their high penetration resistance capability, epoxy and

epoxy-based composites are ubiquitous in automobile industry for en-

ergy absorption due to high strain to failure and hyper-elastic nature

[8]. Domain dependent application of materials is most common solu-

tion strategy for any design engineer and some kind of saturation in

material choice has been reached recently in this regard [9]. None-

theless it is well understood that these solutions are not extremal and

further performance enhancement in the form of weight reduction or/

and higher level of energy absorption is possible with intelligent design

approach.

One of the possible ways in which the performance can be enhanced

is to use a multi-material solution preferably through tuned geometry

and stacking sequence. The basic idea is to enhance/amplify the per-

formance of material(s) through structural tailoring. It is reported that

the response of material to impact load changes with the change in

target plate conﬁguration [10] as well as shape of the impactor [11].

For example, the impact resistance of double layer steel plate found to

be better in comparison to monolithic steel plate having same thickness

[10]. Similarly, it is shown that by changing the ductility of steel in

lower layer there is a considerable increased in ballistic resistance of

double layered steel shields [12].

http://dx.doi.org/10.1016/j.tws.2017.08.021

Received 29 December 2016; Received in revised form 15 August 2017; Accepted 16 August 2017

⁎

Corresponding author.

E-mail address: prabhat@iitrpr.ac.in (P.K. Agnihotri).

Thin-Walled Structures 126 (2018) 193–204

Available online 31 August 2017

Motivated by the above observations and to further explore the

feasibility of multi-material solution to impact problems, Finite Element

(FE) based simulation has been performed in this study. Simulation of a

rigid cylindrical impactor on a deformable plate has been carried out in

Abaqus/Explicit. In the ﬁrst set of studies, single material is used for the

plate to check the accuracy, convergence and meshing requirement for

the simulation. In the second set of studies, multilayer material com-

binations are used through the thickness of the plate. In all the simu-

lations, the plate thickness is kept constant while the impactor velocity

gradually increased to record the minimum velocity at which impactor

will penetrate through the plate made of a given material or materials

combination.

2. Computational model

For the simulations, Finite Element models are developed in

Abaqus/Explicit, incorporating various material and damage models.

Computational cost and accuracy are two conﬂicting requirements in

these type of simulations and to strike a balance between the two,

special modeling techniques have been adopted which are explained in

the relevant sections.

2.1. Geometry, boundary conditions and convergence

2.1.1. Geometry

The simulation involves impact of an impactor on a plate of chosen

material at a certain velocity as shown in Fig. 1(a). The plate is 10 mm

thick with rest of the two dimensions being 200 mm × 200 mm and

entire boundary being constrained. The impactor is a rigid cylindrical

rod of radius 5 mm with a weight of 28 g striking at the centre of the

plate. As the entire system is symmetric about AA′ and BB′, only one

quarter of the plate and striker is modeled to reduce the computational

cost in FEM simulations as shown in Fig. 1(b).

2.1.2. Boundary conditions

Symmetric boundary conditions are imposed along the face OB and

OA′.

•

X-Symmetric boundary conditions are imposed on OB face as

=U 0

•

Y-Symmetry boundary conditions are imposed on OA` face as

=U 0

•

ENCASTRE boundary conditions are imposed for remaining sides of

the plate as

===UUU

123

Since the zone of interest is near the impact point, the quarter plate

is partitioned into two domains through parting line CDE. A ﬁner mesh

is used in the domain OCDE with relatively coarser mesh used in the

rest of the domain (see Fig. 1(c)) to reduce the number of elements and

hence computational cost of FEM simulations. Convergence study is

carried out to determine the minimum element size.

2.1.3. Meshing

All the parts are meshed using C3D8R (Cubic, 8-node, 3D element

with reduced hourglass control) elements, which are mostly used in

impact studies [13], in all the FEM calculations. The size of the element

is decided through convergence analysis of the single plate (aluminium)

FEM model. To this end, the FEM simulations are performed with dif-

ferent mesh sizes and the residual K.E. is compared as a function of

element size (see Fig. 2). It is observed from this analysis that the

Nomenclature

A Yield stress

B Hardening constant

C Strain rate constant

n Hardening exponent

Eﬀective plastic strain

To Reference temperature

Tm Melting temperature

P Pressure

q Mises stress

, failure parameters

Normalized intact stress

Fractured equivalent stress

ἐ Actual strain

∆ε

Equivalent plastic strain increment during a cycle of in-

tegration

Plastic strain to fracture under constant pressure

Minimal distance between a point where d = 0 and a fully

damaged point, d = 1

T Tensile strength

Equation of state constant

Fs Failure strain

β Fraction of damage energy to convert to internal pressure

Initial density

Fig. 1. (a) Full geometry (b) computational model after consideration of symmetry and (c) meshed geometry used in FEM simulations.

A. Sharma et al.

Thin-Walled Structures 126 (2018) 193–204

194

residual K.E. increases with an increase in number of elements through

the thickness of plate. However, the change in residual K.E. is marginal

(~ 13%) when the number of elements across the thickness is increased

from 50 to 100. Since the computational cost associated with 50

number of elements is signiﬁcantly lower as compared to 100 elements,

it is desirable to use 50 elements through the thickness of plate. Based

on this analysis, a ﬁner mesh with element size of 0.2 mm (thickness

(10 mm)/number of elements (50)) is chosen near the impact region

(20 mm × 20 mm) while a coarser mesh having element size of 5 mm is

used away from the impact region to make a tradeoﬀ between accuracy

and computational time. Fig. 3 shows the meshing of full computational

model.

2.2. Material models

Terminal ballistics between two solid bodies induces extremely high

pressure and temperatures around the impact interface for a very short

duration. These extreme conditions cause shock waves which even-

tually decay to elastic waves at large distances. In this short duration,

the material may undergo physical changes such as melting or eva-

poration as well as mechanical changes which leads to loss of strength,

local fracture and total shattering. Even though this state of material is

irreversible one, the state of thermodynamic equilibrium is maintained.

The material properties in this terminal ballistic belong either to

equation of state: which accounts for material compression at high

pressure and temperature or to the constitutive relations: which follow

the strength and failure characteristics of solids under high pressure,

temperature and strain rates even with the possibility of large de-

formations. In the present study, most commonly analyzed/explored

materials in impact dynamics such as metals (aluminium [13,14] and

steel [4,5]), ceramic (silicon carbide [15,16]) and composites (carbon

ﬁber/epoxy [17]) are chosen as plate material whereas the penetrator is

modeled to be rigid. The constitutive behavior of metals is character-

ized by Johnson Cook plasticity and failure model [18] along with the

Mie-Grũneisen equation of state [19]. Johnson-Holmquist (JH-2) model

[20] is used to study the deformation behavior of ceramic material.

While ﬁber failure is modeled using Hashin-3D failure [21] criteria, the

Puck [22] criteria is used to account for matrix failure for composites

material. Constitutive model for 3D composite is not available in

Abaqus/Explicit hence VUMAT is used to deﬁne the constitutive model

for composites. Even though the materials models used in this work are

well established, a brief description is given below for the sake of

completeness.

2.2.1. Material model for metals

The response of metals to applied loading not only depends on strain

but also on other parameters such as temperature, pressure, strain rate

etc. The deformation behavior of metals is characterized using fol-

lowing materials model.

2.2.1.1. Johnson Cook plasticity model. The Johnson Cook (JC) model is

a phenomenological model commonly used to predict the material

response of metals subjected to high strain rate and impact loading. The

JC constitutive model explicitly accounts for the eﬀects of strain, strain

rate and temperature as given in Eq. (1).

=+ + −

εεΤ Α Βε C ε T(,



,) [ ()][1 ln(



)][1 (

)]

yp p p

(1)

The ﬁrst term in square bracket accounts for the eﬀect of strain at

the point of deformation. In this

is the cumulative plastic strain,

equal to the initial yield stress at the reference strain rate at the re-

ference temperature,

is the hardening term and

is exponent on

cumulative equivalent plastic strain. The second term characterizes the

eﬀect of strain rate on the deformation behavior with

as strain rate

constant and



is ratio of instantaneous strain rate

(



)

to reference

strain rate

(



)

i.e.

=ε



. The eﬀect of temperature is captured by third

term. Here

is deﬁned as

−

()

, where

is melting tem-

perature of metal,

is the reference temperature and

is the in-

stantaneous temperature. Constants

, B, n, m are evaluated experi-

mentally.

2.2.1.2. Mie–Grũneisen equation of state model. Mie–Grũneisen equation

of state is used in conjunction with Johnson Cook material model. It

deﬁnes the pressure volume relationship of a solid at a given

temperature (Eqs. (2) and (3)) in two ways depending on whether the

material is compressed or expanded. The Grũneisen coeﬃcient (

)is

⎛

⎝

⎞

⎠

ΓV

(2)

−= −

ee(

)

(3)

Where p

,Vand e

are pressure, volume and internal energy at

reference state usually the state at which temperature is assumed to

be 0 K.

2.2.1.3. Johnson Cook dynamic failure model. Johnson Cook dynamic

failure model characterizes the damage in metals under high strain-rate

loading. It is based on the value of the equivalent plastic strain at

element integration points. Damage parameter

(

)

is deﬁned by the

relation,

Fig. 2. Convergence test: residual kinetic energy vs. number of elements along the

thickness of impacted plate.

Fig. 3. Finite element mesh of the computational model on the left and exploded view of

ﬁner meshing around the point of impact.

A. Sharma et al.

Thin-Walled Structures 126 (2018) 193–204

195

∑

⎛

⎝

⎜

∆

⎞

⎠

⎟

(4)

Where the plastic failure strain

(

)

is given by

⎜⎟

⎡

⎣

⎢

⎛

⎝

⎞

⎠

⎤

⎦

⎥

⎡

⎣

⎢

⎛

⎝

⎞

⎠

⎤

⎦

⎥

⎡

⎣

⎢

−

⎤

⎦

⎥

εddexpd

1ln



()

m room

12 3 4

room

(5)

In Eq. (5), the ﬁrst term takes care of pressure dependence and the

second term accounts for strain rate dependence. The third term is used

to accommodate the dependence on temperature. Constants d

, d

and d

are the failure parameters which are determined experi-

mentally. When damage parameter

(

)

is 1 for an element that parti-

cular element is deleted in the simulation. The Johnson-Cook model

used in this work is currently available in the Abaqus/Explicit material

model library for both shell and solid elements.

Al-6061 (aluminium) and St-37(steel) are the metallic materials

used for the material plates. Material properties of Johnson Cook model

for Al and steel is given in appendices in Tables A.1 and A.2, at the end

of the paper.

2.2.2. Johnson-Holmquist model for ceramics

Generally, ceramic materials are brittle in nature. They have high

compressive strength but low tensile strength and tend to exhibit pro-

gressive damage under compressive load due to the growth of micro

fractures. The Johnson-Holmquist material model (JH-2) [20] with

damage evolution is useful for modeling failure in brittle materials

subjected to large pressures and high strain rates and has been used in

this work. As per JH-2 model, the normalized equivalent stress is

evaluated as,

=− −

σDσσ

(

)

iif

(6)

The stresses in Eq. (6) are normalized to equivalent stress at Hu-

goniot Elastic Limit (HEL) i.e.

HEL

and

are normalized in-

tact and fractured equivalent stress respectively and D is damage

(0 < D < 1). Values of both these stresses are given as

=+ +

AP T C ε

(

)(1 ln



)

(7)

BP C ε

(

)(1 ln



)

(8)

is normalized dimensionless pressure through Hugoniot Elastic

Limit and is deﬁned as

HEL

. Similarly

HEL

is the normalized

maximum tensile fracture strength.

tends to zero as

approaches 1.

Additional material constants

ABNC

,, ,,

are determined through

experiments. The damage for fracture is accumulated in a manner si-

milar to that used in Johnson Cook fracture model and it is expressed as

∑

=∆Dεε/

(9)

∆ε

is the equivalent plastic strain increment during a cycle of in-

tegration and

=f(P) is the plastic strain to fracture under constant

pressure. When D is 1 for an element that particular element is deleted

in the simulation.

Silicon carbide is used as the ceramic material. All the essential

parameter values of silicon carbide are mentioned in the input ﬁle. The

properties which are used for silicon carbide (ceramic) are given in

Table A.3 in appendices.

2.2.3. Hashin 3-D and Puck criterion for composite materials

Composite materials are processed by stacking layers of lamina

which are made of ﬁber and matrix on top of each other to get desired

thickness. Fiber acts as a strength bearing member while matrix keep

them together. Therefore failure of composite structures is dictated by

the failure behavior of ﬁber and matrix. Various theories have been

proposed to accurately predict the failure of composite structures [23].

Among these, the Hashin 3D and Puck criterion which predict the

failure of unidirectional, transversely isotropic composite material are

phenomenologically accurate and the predicted values are close to ex-

perimental observations [24]. Each mode of failure is treated in-

dependently in these theories. In the FEM simulations performed here,

Hashin 3D failure model is used for ﬁber and Puck failure criterion is

used for matrix. The failure of composite materials is governed by the

following four failure criterion,

Equation for Tensile ﬁber failure if

> 1 where

⎜⎟⎜⎟⎜⎟

⎛

⎝

⎞

⎠

⎛

⎝

⎞

⎠

⎛

⎝

⎞

⎠

rft

(10)

Equation for Compressive ﬁber failure if

> 1 where

=rfc

(11)

Equation for Tensile matrix failure if

rmt

> 1 where

⎜⎟⎜ ⎟⎜⎟ ⎜

⎟

⎛

⎝

⎞

⎠

⎛

⎝

⎞

⎠

⎛

⎝

⎞

⎠

⎛

⎝

⎞

⎠

rmt

ft fc

(12)

Equation for Compressive matrix failure

rmc

> 1 where

⎜⎟⎜ ⎟⎜⎟ ⎜

⎟

⎛

⎝

⎞

⎠

⎛

⎝

⎞

⎠

⎛

⎝

⎞

⎠

⎛

⎝

⎞

⎠

rmc

ft fc

(13)

Where, f

denotes the allowable tensile strength in respective directions

(i = 1, 2, 3) and f

denotes allowable compressive strength in respective

direction (i = 1, 2, 3). The values of diﬀerent material parameter used

here are given in Table A.4 in appendices.

The above mentioned failure crit eria also govern the damage in-

itiation in composite layer and are available in Abaqus/Explicit.

However, these can be use d only with p lane formulation i.e. plane

stress, s hell, continuum shell and membrane elements. Even though, it

is possible to perform impact simu lations with thes e elements b ut to

have a standard comparison, the composite plate is modeled with 3D

elements by us ing user deﬁn ed material model VUMAT in Abaqus si-

mulations. Initially in the impact simulat ions at t =0,thecomposite

is assumed to be linear elastic and its deformation behavior is gov-

erned by Eqs. (A .1) and (A.2).Oncethedamageinitiates(asperEqs.

(10)–(13)), the stiﬀness degrades and the el astic constants are updated

as per Eqs. (A.5)– (A.15). The modeling strategy used here is in line

with [28].

2.3. Post processing

The simulation time in all the FEM calculations kept ﬁxed at 0.1 ms.

As the impactor is rigid, initial energy imparted to the impactor in the

form of kinetic energy (initial velocity of the impactor) is utilized in

deformation and fracture of the deformable plate. So, as impactor pe-

netrates through the thickness of plate, deﬁned as depth of penetration

(DOP), its kinetic energy will keep on reducing and will approach to

zero. The initial velocity of impactor is increased gradually and the

response of various plates with diﬀerent material combinations is stu-

died. To have a standard comparison, we deﬁne the critical failure

velocity V

, the velocity at which the impactor is just able to penetrate

through the thickness of single/multilayer plate i.e. at

≤

, the DOP

will be either less than or equal to the total thickness of the plate. For

velocities higher than V

, the impactor will pass through the plate and

as a result the calculated DOP will be greater than total plate thickness

which has no physical signiﬁcance.

2.4. Validation

In order to check the accuracy and predicting capability of proposed

FE modeling approach, the results of FEM simulations is confronted

with experimental data available in the literature. Fig. 4 compares the

residual velocity as a function of impact velocity obtained from current

FEM simulations and experimental data reported in [26] for the plate

A. Sharma et al.

Thin-Walled Structures 126 (2018) 193–204

196

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leaflet-velocity 风场流线效果原生js版本 .zip