Impact Analysis of Mechanical Systems
Using Stress Wave Propagation Methodology
Rasoul Moradi, Michael L. McCoy and Hamid M. Lankarani
INTRODUCTION
In an impact condition of a mechanical system, traditional static solid mechanics
approaches are invalid as they do not account for the dynamic response of the system.
Methodologies that account for the dynamic stress/strain effects are namely
stereomechanics, energy method, contact mechanics, and stress wave propagation. This
chapter presents the fundamental governing equations for mechanical stress wave
propagation within engineering solids due to an impact or sudden loading event.
The response of an impacted solid depends on the energy of the striking body, which
results in the initiation of elastic stress waves for a low-energy impact, plastic wave
propagation for high intensity impacts, and hydrodynamic behavior of solids for higher
intensity impacts. The basic wave equation is presented in this chapter for a onedimensional case along with the impacted medium’s properties which influence wave
propagation. The concept of wave impedance is discussed along with the continuity
equations of wave motion at boundary conditions for force, stress and velocities.
Developments of transmission and reflection coefficients are shown. Impact conditions
for elastic, plastic and shock waves are presented. Application of the stress wave theory
for impact analysis will be demonstrated on the use of the Split Hopkinson Bar (SHB)
test. From this experimental impact test, the dynamic stress-strain response of the
material can be determined. Stress, strain and strain rate equations for the tested medium
are developed as well as the transmission and reflection of the stress wave.
The advantage of stress wave method is an accurate stress analysis on the impacted
elastic solid. In addition, the variation of local strain/stress levels in the solid can be
identified as a function of time and space. On the other hand, stress wave propagation is
highly mathematical and requires a large amount of simplification of the impacted
mechanical system. In engineering applications where complex geometries are involved,
the stress wave method can be captured by the use of explicit finite element analysis
(FEA). The use of Finite Element Methods (FEM) will be discussed in this chapter
illustrating stress wave propagating in solids under impact conditions. FEM parameters
such as contact methodologies, element selection, material constitutional equations,
boundary conditions and proper incremental time step will be examined to generate
accurate engineering modeling of the strains and stresses in wave propagation. Both a
simple bar impact and the SHB test are examined using FEM techniques.
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BACKGROUND
According to Kozlov (1991), the first investigation of impact goes back to 1668, and was
carried out by Wallis, Wren and Huygens. Newton later referred to Wren's work in his
famous work, “Mathematical Foundations of Natural Philosophy”, published in 1687.
The subject of impact attracts the interest of scientists and engineers from different areas
of knowledge from astrophysics to robotics. The common goal is to develop theories
which can predict the behavior of colliding objects. The mechanical engineer's interest in
impact problems is motivated by the desire to develop valid models for mechanical
systems’ behavior where impact is inherent to their function in order to predict the after
impact configuration as well as the applied force and transmitted energy during the
impact to each impacting body. In the evolution of impact theory, several major
approaches are recognized. Depending on the parameters desired from the analysis such
as: velocities, stresses, deflections, plastic deformation or energy absorption, along with
the types of simplifying assumptions about the impact event, the appropriate approach
can be utilized. The duration of the contact period governs the choice of the method used
for analyzing the impact.
In the design of mechanical systems under static or quasi-static loading, one should
consider that the stress field satisfies the equilibrium equation together with the body
forces and the static boundary condition. In these cases, the effects of inertia are entirely
neglected. For a body made up of perfectly plastic material, the ultimate loading is the
greatest loading under which a solution to the static problem can be found so that the
yield criterion is not violated. If a loading beyond the limit load is applied, then the static
problem solution will not appropriate, and the inertia effects must be taken into account.
If the time of loading is short, most of the external work may be transformed into kinetic
energy so that excessive deformation is prevented. For example, when a nail is struck by
a hammer, it may experience a force which produces a stress wave in excess of its static
yield strength without permanent deformation.
Impact is accompanied by a stress wave that propagates in the impacting bodies away
from the region of impact. If the energy transformed into vibrations becomes an
important fraction of the total energy, the inertia forces in the material have to be
considered. The classical statics-based approach then becomes an insufficient method to
examine an impact problem. The wave propagation approach is covered extensively by
Goldsmith (1960) and Zukas et al. (1992) for a wide variety of problems. In this chapter,
first the fundamental theories of stress wave propagation are briefly explained and then
application methods of utilizing stress wave propagation approach using the finite
element method to model impact phenomena will be discussed.
Overall, the dynamics behavior of impacted solids may roughly be divided into three
classes (Lindholm, 1971). For loading conditions that result in stresses below the yield
point, materials behave elastically. For metals, the classical elastic Hook’s law is
applicable in this case. A number of detailed mathematical solutions have been developed
in the literature for different loading conditions in this class. For these problems, both the
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geometry of the entire structure as well as the material property play a major role in
resisting external forces. As the intensity of the applied loading is increased, the material
is driven into the plastic range. The response of the system tends to become highly
localized and is more affected by the constitutive properties of the material in the vicinity
of load application than the geometry of the total structure. The behavior here involves
large deformations, heating and often failure of the colliding solids through a variety of
mechanisms. Two stress waves now propagate through the solid, an “elastic wave”
traveling at the speed of sound in the solid followed by a much slower in velocity (plastic
wave speed) but more intense “plastic wave”. With still further increases in loading
intensity or impact velocity, the pressures are generated by the impacts that exceed the
strength of the colliding solids by several orders of magnitudes. These propagate as
“shock waves” and behave hydrodynamically like a fluid. Table 1 summarizes the
material behavior in different impact regimes.
Table 1. Dynamic Aspects of Mechanical Testing
Loading Regime
Creep
Static
Rapid
Impact
Ballistic
1k
1
20m
10m
10n
1m
1
50
1K
1M
Creep
Rate or
Stress
Strain
Vibration
Elastic and
Plastic
Shock
Wave
Ignored
Ignored
Considered
Considered
Considered
Thermal
Isothermal
Isothermal
Adiabatic
Adiabatic
Adiabatic
General Stress
Levels
Low
Moderate
Moderate
High
High
Typical Time
Characteristic
Typical Strain
Rate (Per Sec)
Method of
Engineering
Inertia Forces
The theories of elasticity and plasticity provide the basis for wave analysis in solids.
Depending on the type of data available and information required from the stress-wave
equation, relationship could be set up for the analysis. If the constitutive equations of
material as well as the conditions of impact are available, it is possible to describe the
stress history within the interior of the material. The inverse problem, in which the
material properties are to be found from the experimental impact test, is equally
important. In theory, the advantage of stress wave method is an accurate stress analysis
on the impacted elastic solid. Also, the variation of local strain/stress levels in the solid
can be identified as a function of time and space. On the other hand, stress wave
propagation is highly mathematical and requires a large amount of simplification of the
impacted mechanical system. This limits its application to mainly one-dimensional
problems such as a rigid body impacting the end of an elastic cantilever rod (Juvinall,
1967). In engineering applications where complex geometries are involved, the stress
wave method can be captured by the use of finite element analysis (FEA). A physical test
that uses stress wave mechanics to determine the dynamic properties of an engineering
material is the Split Hopkinson Bar (SHB) test. From this test, the dynamic stress-strain
response of the material can be determined.
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ELASTIC WAVE PROPAGATION
When an impact force or impulse is applied to an elastic body, the generated disturbance
travels through the solid as stress waves, which are analogous to “sound waves” traveling
through air. The particles in a thin layer of material at the contact region are set into
motion. The remainder of the body, remote from the loading, remains undisturbed for
some finite length of time. As time passes, the thin region of moving particles expands
and propagates into the body in the form of an elastic deformation wave. Behind the
wave front, the body is deformed and the particles are in motion. Ahead of the wave
front, the body remains undeformed and at rest. If the geometry of the body is simple and
uniform, and if the applied force is well defined and uniformly applied, equations for
wave propagation in an elastic media may be utilized to evaluate stresses and
deformations in the body. It has been found that the propagating waves reflect internally
from boundaries of the body and interfere with one other. Depending on the boundary
conditions, “standing” or “interfering strain waves may produce the local larger strains
and stresses associated with impact condition than that of ordinary static loading. If the
impact force or impulse exhibits a velocity of less than that of the speed of sound in the
impacted solid, the wave propagation will be elastic.
For a material subjected to stresses through some external dynamic loading or testing
apparatus, the traditional static solid mechanic approaches cannot be applied. For
loadings that cause stresses below the yield limit of the material, “elastic stress waves”
are generated. The theory of elasticity provides the basis for wave analysis in solids. The
structure experiences deformations that can be determined from the combinations of
equations of motion can be utilized to obtain an infinitesimal element. The inertia effects
of the elements, the material’s constitutive and compatibility relations can be utilized to
obtain the displacement equation of motion called Navier’s equations, which is the
equation of motion for the elastic waves in a solid. For impact, since the intensity or rate
of loading is high enough, the inertia forces in the material have to be considered.
The solution to problems in impact mechanics requires the application of the basic laws
of mechanics and physics, as well as a description of the behavior of the material being
considered. The system of equations governing the motion of a homogeneous, isotropic,
linearly elastic body consists of the stress equations of motion, the Hook’s law, and the
strain-displacement relationships given by,
1. Strain-displacement relations, (kinematics):
1
εij = (ui,j + uj,i )
2
(1)
2. Material compatibility conditions (constitutive equations):
σij = λδij εkk + 2µεij
(2)
3. Equations of motion applied to an infinitesimal element, (equilibrium):
σij,j + ρbi = ρai
(3)
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vE
in which scripts, i, j = 1,2, 3; and λ = (1+v)(1−2v) , Lame constant, and µ =
E
2(1+v)
, shear
modulus, are two independent elastic constants which define all elastic material
properties for an isotropic engineering material. Eqs. 1~3 are the vectorial equations
which can be written in three directions to obtain each scalar equation. For example, the
equation of equilibrium is derived from summation of forces in the x, y and z directions
and equating these forces to the change in momentum of the element yielding the
equations of motions, given by,
σij,j + ρbi = ρü i
∂
∂
∂
∂2
∂x
∂y
∂z
∂t
⎧ ∂x σx + ∂y τxy + ∂z τxz + ρbx = ρ ∂t2 ux
yields ⎪ ∂
∂
∂
∂2
�⎯⎯� ∂x τxy + ∂y σy + ∂z τyz + ρby = ρ ∂t2 uy
⎨
⎪ ∂ τ + ∂ τ + ∂ σ + ρb = ρ ∂2 u
xz
yz
z
z
2 z
⎩
(4)
Eqs. 1~3 may be combined to obtain the displacement equation of motion, called
Navier’s equations, which is the equation of motion for the elastic wave in a solid:
µui,jj + (λ + µ)uj,ji + ρbi =
ρü i
∂
∂2
2
⎧ (λ + µ) ∂x ∆ + µ∆ (ux ) + ρbx = ρ ∂t2 ux
yields ⎪
∂
∂2
�⎯⎯� (λ + µ) ∂y ∆ + µ∆2 (uy ) + ρby = ρ ∂t2 uy
⎨
∂
∂2
⎪ (λ
2
⎩ + µ) ∆ + µ∆ (uz ) + ρbz = ρ 2 uz
In which ∆= ∇. u = uj,j =
∂u
∂x
+
∂v
∂y
∂z
+
∂w
∂z
(5)
∂t
is divergence of displacement vector u. For an
impact, the influence of body force during the short period of impact can be neglected
against the high impulse load from the impact, and then the equations become:
µui,jj + (λ + µ)uj,ji = ρü i
(6)
Taking divergence (∇) of the equation above gives the longitudinal wave equation as:
where
(8)
µui,jji + (λ + µ)uj,jii = ρü i,i
or
2
2
∂ ∆
∆
= Mρ ∂x∂ ∂x
∂t2
i i
(7)
E(1−v)
M = (λ + 2µ) = (1+v)(1−2v)
“P-waves” are those in which the particle motion induced by the disturbance is normal to
the wave front, parallel to the pulse propagation direction, and the strain is pure
dilatation. P-waves are also nomenclature by the terms dilatational, longitudinal, primary,
or pressure waves. A P-wave is associated with normal stress and can propagate in all
types of media. In linear elasticity, the P-wave modulus M = ρcL2 , also known as the
“longitudinal wave modulus”, is one of the elastic moduli available to describe isotropic
homogeneous materials, where cL is the velocity of a P-wave in the infinite elastic solid.
Analogous to the method above, we can take the curl (∇ × ) of the Eq. 6 to obtain the
shear wave equation as:
µ∇ × ui,jj + (λ + µ)∇ × uj,ji = ρ∇ × ü i
But ∇ × uj,ji = 0 and ∇ × ui,jj = (∇ × ui ),jj. Defining ∇ × ui = ψ, then,
(9)
µψi,jj = ρψ̈i
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Eq. 9 is related to the propagation of shear-wave with the velocity of cS2 =
µ
ρ
, where μ is
the shear modulus and ρ is the density of the solid. The distortional or transverse or shear
or secondary, “S-waves” are those wherein material particles move in a plane at right
angles to that, normal to the pulse propagation direction in which the wave front
propagates at a velocity of cS . An S-wave is associated with shearing stress and can
propagate only in media with shear stiffness, that is, in solids and not liquids.
The statement of the “elasto-dynamic” problem will be completed with establishing the
second-order partial differential wave equations and initial and boundary conditions
which could be the essential (geometric or Dirichlet) and/or natural (dynamic or
Neumann) boundary conditions describing the primary or secondary variables of the
second order partial differential equation.
Using the original P-wave equation in 1-, 2-, and 3-dimensions yields:
⎧ 1 − D:
⎪
2 − D:
⎨
⎪ 3 − D:
⎩
�
∂2 u
∂2 x
�
∂2 u
∂2 x
∂2 u
+
∂2 y
∂2 u
+
+
cL 2 ∂t2
=0
�−
1 ∂2 u
cL 2 ∂t2
=0
∂2 u
∂2 y
∂2 u
∂2 z
1 ∂2 u
−
∂2 x
�−
1 ∂2 u
cL 2 ∂t2
=0
(10)
In general, solution for the equations above (Eqs. 7 and 9) in 3-D is highly complicated
and very few closed solutions exist using this method due to its complexity. Stress
propagation in slender bars is considered to be P-wave only if the ratio of the length to
the diameter of the bar is greater than 10, for which lateral considerations may be
ignored. Then for a bounded media, the equation of wave propagation yields:
∂2 u
1 ∂2 u
− 2 2
∂2 x
cL ∂t
= 0 , cL 2 =
E
(11)
ρ
If the bar is not slender, the inertia of lateral contraction into must also be taken into
consideration:
∂2 u
∂t2
= cL 2
∂2 u
∂2 x
+ v2k2
∂4 u
∂x2 ∂t2
, cL 2 =
E
ρ
(12)
In which ν is the Poisson’s ratio and k is the radius of gyration. A useful parameter is the
ratio, α, of the S-wave to P-wave speeds:
𝐶
(1−2𝑣)
2(1−𝑣)
𝛼 = 𝐶𝑠 = �
𝐿
(13)
The second-order hyperbolic partial differential equation, Eq. 12 can be solved in two
methods: separation of partial differential equations into two second-order ordinary
differential equations, and mapping the coordinates to the new coordinate system which
is called “d’Alembert’s method”.
For the separation of variable method, the displacement is defined by u(x, t) = F(x)G(t).
The solution of the problem then reduces to solving of two second order differential
equation as:
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F ′′ + k 2 F = 0
(14)
̈G + λn 2 G = 0
(15)
where, λn2=k2cL2. Using boundary conditions associated with the boundary value-SturmLioville problem, F(x) = Acos(kx) + Bsin(kx), the unknowns A and B can be
determined. Initial value problem of equation yields: Gn (t) = An cos(λn t) + Bn sin(λn t).
Then,
(16)
u(x, t) = ∑∞
n=1 un (x, t)
Applying the initial condition yields the unknowns An and Bn and the displacement
(strain) and stress can be evaluated at different locations and times.
Using d’Alembert’s solution, coordinate transformation from (x, t) to (ζ, η) such that
ζ = x − cL t, η = x + cL t in Eq. 7 yields the canonical form of hyperbolic equations,
∂2 u
∂ζ ∂η
= 0.
It means that u(ζ, η) can be written as u(x, t) = f(x − cL t) + g(x + cL t),
the wave would be divided by two left- and right-running waves with a constant velocity
(Wasely, 1973). In this method, the two wave shapes at any time are functions of x only
and move in the positive and negative directions of x with a constant velocity.
The only difference between the propagation of elastic stress disturbances in bounded and
unbounded media is geometrical. In theory, the transmission of such disturbances can be
treated by solving the equations of small motion with the appropriate boundary
conditions. In practice, however, addition of boundaries introduces immense complexities
into the mathematical formulation of the problem so that very few closed solutions exist.
In stress wave calculations, two different velocities must be considered. The velocity of
the stress wave traveling at wave speed cL and the particle’s motion velocity designated
with symbol v. The particle velocity is the velocity of the material as the stress wave
transmits energy through the medium. In the case of an impact, the particle velocity is
the striker’s initial impact speed, usually designated as vo. As the stress wave passes, the
particle’s velocity changes from zero to vo. All the material behind the stress wave is
now at vo. The relationship between these two velocities cL and vo is developed from the
impulse-momentum equation, where the impulse of force F changes the velocity vo of the
mass ρcL∆t resulting in the momentum change of 𝑝𝑓 − 𝑝𝑖 in the impacted rod in time
period ∆t:
𝑡
𝑡
𝑓
𝑓
∫𝑡 𝐹𝑑𝑡 = ∫𝑡 𝜎0 𝐴𝑑𝑡 = 𝜎0 𝐴∆t = 𝑝𝑓− 𝑝𝑖 = ρAcL ∆tvo
𝑖
𝑖
(17)
here again ρ is density of the material, and cL∆t is the length of material now at velocity
vo. The initial magnitude of the impact stress is given by 𝜎𝑜 = ρcL vo . In the same
manner for the transverse (shear) stress pulse; σ = ρcs vo . This analysis assumes the
mass M of the striker is significant enough to generate a change in velocity of vo in the
mass m of the rod.
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Boundary conditions of the solid body affect the response of the stress wave that
propagates into it. For the most general boundary condition, the incident wave will be
partially reflected and partially transmitted. The degree of reflection and transmission at
the boundary condition is dependent upon the mechanical impedance ρcL of the materials
at the boundary. The total deflection or stress at any point along the rod is the sum of the
nth propagating wave and (nth –1) reflecting wave. At a boundary condition, continuity
of displacements and forces (𝜎A) is required. Designating subscript i for incident, r for
reflected and t for transmitted, the continuity of displacements amplitude A of the waves
are related by A i + A r = A t . Solving the d’Alembert’s solution to the boundary
conditions, the displacements at stresses at boundary conditions are determined by their
“mechanical impedances” as:
Ar =
𝜌 𝑐
1− 2 𝐿2
𝜌1 𝑐𝐿1
𝜌 𝑐
1+ 2 𝐿2
𝜌1 𝑐𝐿1
Ai
At =
𝜌 𝑐
1+ 2 𝐿2
𝜌 𝑐
2 2 𝐿2
𝜌 𝑐
2
𝜌1 𝑐𝐿1
1 𝐿1
σi
σt = 𝜌2𝑐𝐿2
𝜌1 𝑐𝐿1
+1
Ai
(18)
σr =
𝜌2 𝑐𝐿2
−1
𝜌1 𝑐𝐿1
𝜌2 𝑐𝐿2
+1
𝜌1 𝑐𝐿1
σi
Here subscript 1 represents properties of first medium carrying the incident wave to the
boundary, while subscript 2 is the represents the properties of the second medium
transmitting the wave. The “mechanical impedance” is defined as the product of mass
density ρ and the P-wave propagation velocity cL. For a free-end boundary condition, the
mechanical impedance is 𝜌2 𝑐𝐿2 = 0, and it is observed at the free boundary condition that
the displacement is double that of the impacted end’s displacement. The stress at the end
of rod will become zero after the passing of the incident stress wave as the reflected stress
wave will be the negative of the incident stress wave, and when summed together, cancel
each other. For a fixed-end boundary condition, the mechanical impedance is 𝜌2 𝑐𝐿2 = ∞
and the fixed-end deflection is zero. The fixed-end stress will be twice the incident stress
wave as the reflected stress wave now equals the magnitude of the incident wave. The
fixed-end boundary condition transmits twice the incident wave also. Eq. 18 assumes
equal medium areas on either side of the boundary: if not, stresses need to be adjusted
accordingly to area.
An example of one-dimensional elastic impact is given in Fig. 1 in which Fig. 1(a)
diagrams an impact of a rigid body of mass M with a velocity v0 into an elastic rod with
of density ρ and elastic modulus E. Fig. 1(b) shows the compressive elastic stress wave
propagating at cL through the bar with a compressive initial impact stress of σ0 =
ρcL v0 = v0 �Eρ. As the initial stress σ0 is propagating towards the right side, the free-
end stress is diminishing as the impactor starts to rebound away from the target. The
diminishing stress is noted as σe . This diminishing stress is governed by the force
equilibrium on the impactor with mass M and velocity of ve reacted upon the bar stress of
σe and bar area A, such that,
M
218
dve
dt
+ σe A = 0, where σe = ρcL ve = ve �Eρ
(19)
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Substituting
dve
dt
e 1
= dσ
, equation (19) become a first order ordinary differential
dt �Eρ
equation yielding,
σe = σ0 e
−A�Eρ
t
M
,
0<𝑡<
2L
c
(20)
Fig. 1. One-Dimensional Stress Wave Propagation due to Impact Loading.
ρ
ρ
σο
ρ
σ
σο
ρ
σο
At time, t1 = L⁄cL , the indent compressive wave at compressive stress σ0 is reflected as
a compressive wave with stress σ0 accordingly to Eq. 18. At t1, the end stress is now
2σ0 , compressive stress as shown in Fig. 1(d) by the sum of the nth propagating wave
and (nth –1) reflecting wave. Timoshenko (1955) provides a solution for the maximum
superposed stress experienced by the rod with a total mass of m when in impacted with
mass M at some point and time history of the impact as:
σmax = σ0 (�M⁄m + 1)
(21)
PLASTIC WAVE PROPAGATION
For high energy impacts where plastic strains occur outside of the contact area, some
form of plastic analysis is required (McCoy et al., 2010). When plastic strains go beyond
the scale of contained deformation, the elastic wave propagation model can no longer be
applied to analyze impact problems. Elastic theory cannot model the conversion of
kinetic energy into heat energy during the plastic deformations. Impacts of this nature
must be described by relationships which account for large strains and plastic
deformations that occur in the process. The perfect elastic-plastic analysis exhibits the
most practical way in predicting large plastic strains due to impact loading. Bohnenblust
(1950), Conroy (1955, 1956) and Symonds (1953) developed methods to analyzed beams
undergoing plastic deformations using rigid perfect-plastic and elastic-perfect plastic
material constitutes.
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The plastic wave propagation method extends the elastic wave theory. This is the domain
of high velocity impact generally associated with explosives and projectiles. Goldsmith
presents an extended study of the subject using the theory of plastic wave propagation. In
the theory of plastic strain propagation, the material is considered to be incompressible in
the plastic domain. Also, the state equation relating stress, strain and strain rate is
assumed to be independent of temperature. Maugin (1992) and Lubliner (1990) postulate
that where ductile materials are used, the loading is applied over a long period of time,
high temperatures are involved or high strain rates occur, and rate dependence cannot be
ignored in describing the plastic behavior of materials. Zukas et al. (1992) present an
extensive treatment of plastic wave propagation using both rate-dependent and rateindependent theories.
As in the case of elastic wave theory, the plastic propagation wave method is too complex
to analyze an impact problem with great complexity. Review of literature has indicated
that the only impact problems found to be solved by the plastic wave propagation method
have been the tensile impact of a semi-infinite wire and the impact compression of
cylinders. Three-dimensional plastic wave study is out of the scope of this study and only
the brief explanation of the one-dimensional plastic wave is given here.
In the bar impact analysis, plastic flow near the impact end introduces three-dimensional
effects (radial, inertial, heating), so that one-dimensional theory can be applied only at
points far away from the point of application of the load. With increasing the striking
velocity, a three-dimensional theory is required for complete analysis of experimental
results (Zukas, 1982). Plate geometry offers the opportunity to study materials behavior
at higher load loads and shorter times while offering again the simplicity of onedimensional analysis, uniaxial strain. In fact, the uniaxial strain can be achieved when the
plane wave propagating through a material with dimensions and constraints are such that
the lateral strains are zero. Similar to one-dimensional stress wave in bar analysis, plate
impact analyses neglect effects of thermo-mechanical coupling, which can be significant
at strains exceeding 30% (Lee, 1971). Much of the work in literature has assumed
hydrodynamic behavior of the material. However, an elastic precursor can produce
significant volumetric strain. An elastic unloading wave can significantly change the
local state of the material before the arrival of the plastic wave so that finite elastic and
plastic effects may need to be accounted for.
Following the elastic stress-strain relationship and the fact the plastic strain is
incompressible, the stress at the direction of the strain can be calculated by examination
of the total principle strain, which is the sum of the elastic strain εe and the plastic strain
εp, namely εi = εi e + εi p , i = 1, 2, 3 where i is the principle strain direction:
ε1 =
σ1 (1−2υ)
E
+
2σ2 (1−2υ)
E
ε2 = ε3 = 0
(22)
Where E is again the elastic modulus, ν is the Poisson’s ratio and σ is the principle stress.
The plasticity condition according to either von-Mises or Tresca failure theory relates the
principle stress to the yield strength σy:
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σ1 − σ2 = σY
(23)
Using the definition of the bulk modulus K into the equations, the above equation
becomes.
σ1 = Kε1 +
2σY
3
(24)
, where K = E/3(1 − 2ν)
The most important difference between the uniaxial stress and uniaxial strain is the bulk
compressibility term. The stress continues to increase regardless of the yield strength or
strain hardening due to the plastic impact. For ballistic impact or other high-rate
phenomena where the material does not have enough time to deform laterally, a condition
of uniaxial strain is established.
The maximum stress for uniaxial strain in one-dimensional elastic wave propagation is
called “Hugoniot elastic limit” σHEL . This is also the dynamic yield stress for the impact.
If Hugoniot elastic limit is exceeded, a plastic stress wave will be developed and
propagate after the elastic stress wave. The elastic wave will move with speed of
cE followed by a plastic wave moving with speed of cp . As shown in Fig. 2, the speed cp
of the plastic wave is a function of the slope of the stress-strain curve at a given value of
strain,
cE2 =
E(1−v)
ρ0 (1−2v)(1+v)
1 dσ
cp2 (σ) = ρ
0
dε
(25)
Fig. 2. Uniaxial Stress-State Perfectly Elastic-Strain Hardening Material.
SHOCK WAVE PROPAGATION
For impact velocities that are much greater than the speed of sound or elastic propagation
velocity cE , “shock waves” form. In this situation, where cE < cp , the continuous plastic
wave front breaks down and a single discontinuous shock front is formed traveling at a
shock velocity U, as illustrated in Fig. 3 Across the shock front, there is a discontinuity in
stress, density, velocity, and internal energy. Shock waves will be formed under
conditions of extremely high impulsive stress and will propagate in a material in a
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manner similar to the fluid dynamics situation. Using simplified equation of state
analogous to the case of elastic wave, the shock velocity of will be obtained as,
Us2 =
1 P1 −P0
ρ20 v0 −v1
, v0 =
Fig. 3. Progress of Plane Shock Wave.
1
ρ0
and v1 =
1
ρ1
(26)
As mentioned before, to study the propagation of longitudinal stress and strain waves in a
thin bar, it is common to represent the problem by a one-dimensional approximation. The
wavelengths are assumed to be much longer than the transverse dimensions of the bar.
This approximation yields good results at points of the bar enough away from the bar
ends. Near the ends, three-dimensional corrections are necessary. Let x denote the
Lagrangian coordinate along the bar axis and u(x, t) the corresponding displacement, the
engineering strain ε(x, t), and velocity v(x, t) are then given by,
ε=
∂u
∂x
v=
The kinematic compatibility relation gives:
𝜕𝜀
𝜕𝑡
=
𝜕𝑣
𝜕𝑥
∂u
∂t
(27)
(28)
The equations of motion for zero-body-force condition would be reduced to:
𝜕𝜎
𝜕𝑥
=𝜌
𝜕𝑣
𝜕𝑡
(29)
A shock front is said to occur at a point x = α(t) of the bar if the velocity v is
discontinuous at that point. The shock front is moving at a finite speed c in the positive xdirection, that is, c = α̇ (t) > 0 in designating the values of v just to the right (in front) of
the shock and just to the left of (behind) the shock by v+ and v−, respectively. The jump
in v is defined as:
|v| = v − − v +
(30)
Lubliner (2006) derived the jump in velocity, stress and strain relations by treating the
shock front as a thin zone in which these quantities change very rapidly with constant
rates. If the shock thickness is h, then for a front moving to the left and to the right we
have:
|v| ≅ ±h
∂v
∂x
(31)
Since the duration of the shock passage at a given point is h/c, then,
|v| ≅
h ∂v
c ∂t
(32)
Applying these approximations to Eqs. 28 and 29, the shock relations are:
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1
𝑐
|ε| = ± |v|
|σ| = ±ρc|v|
(33)
where the + and − signs apply to fronts moving to the right and to the left, respectively.
Eliminating |v|, the shock-speed equation yields,
𝜎
ρc 2 = � � = E
𝜀
(34)
By means of some approximating assumptions, Taylor (1948) and Lubliner (2006)
derived a formula for the dynamic yield stress of a rigid-plastic bar impacted into a rigid
target in terms of the impact speed and the specimen dimensions before and after impact
utilizing different approaches. For a bar made of work-hardening material, the problem
was treated by Lee and Tupper (1954). If the conventional stress-strain relation is given
by σ = F(ε) and the initial yield stress is σE , then the material just ahead of the shock
front may be assumed to be about to yield, so that σ = σE there, while immediately
behind the front the stress is σ
� = F(ε�). The stress jump is therefore σ = σ
� − σE .
Assuming the elastic-plastic material property for the bar changes the nature of the
problem drastically. In an elastic solid, disturbances cannot be propagated at a speed
faster than the elastic wave speed. Assuming a bilinear stress-strain material as shown in
Fig.2 for a one-dimensional problem, the velocity of each wave front has its own
characteristic speed dependent on the respective moduli of the elastic and plastic regions,
E, and E1, resulting in the wave profile shown in Fig. 4. Karman (1942), Taylor (1942)
and Rakhmatulin (1945) extended the Donnell theory (1930) independently for a
uniaxially loading bar by considering the bilinear elastic-plastic material and independent
of strain rate. Using a Lagrangian coordinate system with the x axis parallel to the bar
axis, the equation of motion in the x direction is given as:
ρ
∂2 u
∂t2
=
dσ ∂ε
dε ∂x
(35)
Applying the boundary condition for bar impacted at the end, and letting ξ = x/t, the
three solutions were obtained as:
v
For |x| < c1 t: ε = constant = 1 = ε1
(36)
For c1 t < |x| < c0 t: E(ε) =
For c0 t < |x|: ε = 0
x2
t2
c1
(37)
(38)
The solution for strain as a function of ξ = x/t is presented in Fig. 5 which illustrates the
two wave fronts traveling with its own characteristic velocity dependent on the slope of
the tangent to the stress-strain curve at that point.
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Fig. 4. Elastic and Plastic Wave profile for Bilinear Material.
σ
σ𝑦
E
� 𝑡
ρ
�
E1
𝑡
ρ
x
Fig. 5. Strain Distributions in Rod Produced by Constant Velocity Impact at End.
A much more comprehensive treatment of shock wave can be found in Duvall (1972,
1971, 1961), Duvall and Fawles (1963), Murri et al. (1974), Rinehart (1975), and Seigel
(1977).
ENGINEERING APPLICATION OF STRESS WAVE THEORY
Engineers use stress strain curves to design machine components and structures.
Engineers compare their design loads to the strength of the components used in their
design. Under dynamic loading conditions, using static strengths is sometimes an
underestimation of the material strength. In typical engineering metals, such as
aluminums and steels, the material response of these engineering materials typically
increases the yield strength of the material due to rapid loading. Impact loading
increases strain rate on engineering components and structures beyond that of
static testing rates. The increased strength as the result of impact conditions is
referred to as the dynamic strength of the material. The declaration of the
dynamic strength must be paired with strain rate or impact velocity of the test and
heat treat or cold work condition of the material.
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Dynamic strengths have been studied by both uniaxial tension and compression
testing of bars. For example, in tension impacts at impact velocities of 25 fps,
increases in material strengths for mild steels were observed as that shown in Fig.
6 (Blake, 2009). As seen in this Fig., the lower strength materials exhibit the
greatest percent increase in dynamic strength, (Wiffen, 1950).
Also, the compressive yield strengths of engineering metals also increase due to
impact loading conditions. For mild steel, the compressive 0.2% yield strength
was observed to increased from a static value of 40.3 ksi to 105 ksi. Likewise, for
a heat-treated and tempered 4130 steel the increase in dynamic yield strength was
significant, from 118.5 ksi to 190.5 ksi, (Wiffen, 1950).
The testing of materials for strengths at high strain rates are conducted in an
appratus with long rods or bars exhibiting length to diameter ratios exceeding ten
to promote uniaxial stress wave propagion. A mechanical experimental machine
called the Split Hopkinson Bar (SHB) apparatus is a device which is capable of load
engineering materials with strain rates up to 102 to 104 in/in-sec. Fig. 7 provides a
schematic of a typical SHB apparatus.
Fig. 6. Increase in Material Strength Due to Impact Loading (High Strain Response)
Typical Ferrous Material.
300
Static Ultimate
Strength, ksi
200
100
0
Increase in Dyamic Strength as Result of Impact Loading at 25 fps
% Increase in Strength
0
10
20
30
40
50
Fig. 7. Schematic of Split Hopkinson Bar Apparatus.
Incident Bar
Striker Bar
vo
Input
εi
Specimen
εr
Incident Bar Strain Gage
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Transmit Bar
εt
Transmit Bar Strain Gage
225
The material under test for its response to high strain rates is sandwiched between
two long rods. The long rods are fabricated from high strength alloy steel such as
4130 or 300M. The input bar is termed the incident bar, while the output bar is
termed the transmit bar. Both the incident and transmit bars are sized in diameter
to produce elastic stress waves when impacted with the striker bar and usually
both bars have identical diameter and lengths.
The specimen is held in place in slight compression by the friction of the two bars
in their rail bearings and the specimen is free to fall out of the sandwich as the
stress wave propagates through the bar interfaces. As seen in Fig. 8, which is an
actual SHB apparatus at the National Institute of Aviation Research on the campus
of Wichita State University, the incident and transmission bars are supported on
system of rail bearings which allows only for an axial degee of freedom of
movement in the appraturs, the typical time duration of a SHB test is about at 500600μ seconds.
The dynamic test starts with the striker bar strikes the incident bar. The striker bar
obtains its velocity through compressed air source and a gas gun. The magitude of
the pressure in the gun determines the velocity of the striker bar. This velocity of
the striker sets the maximum strain rate which could be exhibited in the specimen.
The maximum limiting strain rate and maximum limiting compressive stess that
can be established in SHB test is given by (Remesh, 2008):
ε̇ =
v0
Lo
σ s = Eb
(39)
Ab v 0
Aso 2cb
(40)
Fig. 8 Split Hopkinson Bar Apparatus at Wichita State University (www.niar.twsu.edu)
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where the subscript s represents the specimen’s properties, while subscript b
represents the incident and transmit bar properties, E is the elastic modulus, v o is
the striker velocity, cb is the elastic propagation speed in the incident and transmit
bars, L so is the specimen’s original gage length and A represents the area of a bar.
Note, subscript (so) indicates pre-test a dimension of the specimen.
Strain gages are attached to the apparatus bars according to Fig. 7, one gage on the
incident bar and one gage on the transmit bar. When the elastic stress propagating
down the incident bar passes the incident strain gage, the strain gage measure the
elastic strain and at this point; the strain is termed the incident strain ε i. At the
specimen-incident bar interface, the mechanical impedance changes and a portion
of the incident strain wave is transmitted through to the specimen while the other
portion of the incident strain wave is reflected back into the incident bar
designated as ε r. The proportions of strain transmitted and reflected are that given
by Eq. 18, noting strain is defined as, ε=σ/E. At the specimen-transmit bar
interface, the transmitted portion of the initial incident strain is reflected back and
transmitted through this interface. The transmitted strain from the specimen to the
transmitter bar is termed ε t. These three important strains of ε i, ε r and εt used to
measure dynamic strength by the SHB test are diagrammed on Fig. 9.
From Eq. 18, the velocity at the specimen-incident bar interface is given by v 1 (t)=
c b(εi (t)-ε r(t)), while the velocity of the specimen-transmit bar interface is v2 (t)=
c b(εt (t)). These velocities can be arranged to define the strain rate on the
specimen under test:
εṡ (t) =
v1 −v2
Lso
=
cb
(ε (t) − εr (t) − εt (t))
Lso i
(41)
With the elastic strain known in the incident and transmitter bars, the forces P at
the two specimen interfaces may be calculated with P 1 at the specimen-incident
and P 2 at the specimen-transmit interface and which are given by:
P1 (t) = Eb (εi (t) + εr (t))AB
P2 (t) = Eb εt (t)AB
(42)
(43)
The mean normal stress in the specimen would be calculated as the average force
divided by the specimen’s original cross sectional area Aso:
𝜎𝑠 (𝑡) =
𝑃1 +𝑃2
2A𝑠𝑜
=
𝐸𝑏 𝐴𝑏
(ε (t) + εr (t) + εt (t))
2 𝐴𝑠𝑜 i
(44)
Examination of Fig. 9, which is the wave propagation diagram for the SHB test,
provides some insight of the behavior of the stress wave propagation and how high
strain rates and stress levels are developed in the specimen. The compressive
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227
strain incident wave ε i travels left and hits the specimen-incident interface where a
portion of this wave is transmitted into the specimen setting up a compressive
stress wave in the specimen and a portion of it is reflected in ε r. Due to the
mismatch in mechanical impedance as the intentional differences in the bar’s to
specimen’s areas, with the specimen exhibiting the lower impedance, a large
portion of the transmitted wave through the specimen-incident interface
reverberates within the specimen setting up compressive wave after compressive
wave within the specimen. The state of stress in the specimen is the nth incident
wave summed with the n th-1 reflected wave. Thus a high compressive state of
stress is set up in the specimen. At the point for which the compressive stress
exceeds the yield strength of the specimen, plastic flow occurs which essentially
damps out further wave propagation through the system of bars. At this portion of
the experiment, the specimen’s stress state is in near equilibrium and P 1 = P 2,
combining this with Eqs. (42) and (43) shows:
εi (t) + εr (t) = εt (t)
(45)
Combining Equation (45) into Equation (44), the stress in the specimen maybe
determined by the strain measured in the transmit bar εt:
𝜎𝑠 (𝑡) =
𝐴𝑏 𝐸𝐵
𝜖 (𝑡)
𝐴𝑠𝑜 𝑡
(46)
Fig. 9 Wave Propagation Diagram of Split Hopkinson Bar Apparatus.
Combining Eq. (45) into Eq. (41), the strain rate in the specimen maybe
determined by the strain measured in the incident bar for the reflected wave εr:
𝜀𝑠̇ (𝑡) = −2
𝑐𝑏
𝜖 (𝑡)
𝐿𝑠𝑜 𝑟
(47)
The negative sign in Eq. (47) is due to the fact that the reflected pulse is a tension
strain wave. In the SHB test, the compressive incident strain wave, a compressive
strain is defined as positive value.
Integration of Eq. (47) leads to the strain history of the specimen:
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𝑡
𝜖𝑠 (𝑡) = ∫0 −2
𝑐𝑏
𝜖 (𝑡)𝑑𝑡
𝐿𝑠𝑜 𝑟
(48)
A stress strain curve is computed at the strain rate of the SHB test, which is
defined at the average strain rate over its time history to reflect the effect of strain
rate on material behavior. This stress-strain curve should be computed in true
stress and true strain as at high strains to reflect plastic incompressibility effects.
FINITE ELEMENT EMULATION OF STRESS WAVE PROPAGATION
IN IMPACTS
Reasons for Using the FEA Method in Impact Analysis
As discussed, the stress wave method is quite limited to impacts of simple
mechanical systems. Thus, use of the FEA method becomes the engineering tool
for examining an impacted system. FEA has the capability of solving complex
systems for which the stress wave method becomes impractical. Prior to
sophisticated FEA programs, experimental analysis using strain gages and
accelerometers was the methodology used to examine impacts that exhibit any
degree of complexity. Experimental analysis is expensive requiring prototype
development, test equipment, and a test environment. In an impact test, damage or
destruction of the prototype is required before much meaningful quantitative
information can be obtained about the design. A significant advantage of FEA
over the experimental method is the engineering examination of virtual prototypes
in a virtual environment. This allows for a more concurrent analysis of the design
during the designing process reducing engineering costs and testing while
increasing product performance.
Inherent Difficulties of Using the FEA Method
The practicing engineer is not the developer of the FEA code, but a user. The user
has no control of the approximations that were incorporated into the software, but
ignorance of these approximations and how to deal with them will result in poor
analysis when using the software.
For impact problems involving large
displacements and strains, the choice of material constitutive is important. The
user should be aware of the types and assumptions used in the analysis package
concerning material constitutive and choose the proper one for their impact
conditions. Time increments for advancing the solution through time must be
chosen in a manner to produce an accurate engineering analysis. A too large of a
time step will produce erroneous results while a too short of time step will waste
resources. The familiarity of a code to produce reasonable solutions to practical
engineering problems requires three to six months of examination and trials by the
user, (Zukus, 2000).
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Practical Aspects in Modeling Impact Problems with FEA
A list of recommended practices as the result of literature review on the subject
when using FEA techniques for solving impact problems are tabulated below to
help provide generate consideration when using FEA:
1. Element Types. Lagrangian formulation produces unacceptably high errors
when excessive mesh distortion occurs. Quadrilateral elements can distort into
shapes that produce negative volumes intern generating serious computational
errors. Impact bodies should be meshed with low order elements such as 3 node 6DOF triangular or 4 node 12-DOF tetrahedrons. These lower ordered elements
were found be more resistant to distortion and reduce computational effort, (Zukus,
2000).
2. Element Aspect Ratio. In order to capture the wave propagation effects in 2 and
3-D problems, the element aspect ratio must be less than four, (Nagtegall, 1974).
3. Element Arrangement. When meshing with triangular elements use four
triangular elements per quad shape, instead of two. The two-element arrangement
introduces asymmetry to the analysis, (Nagtegall, 1974).
4. Mesh. Ideally the mesh density should be uniform, but for consideration of
computational effort, the mesh density should be kept below an element-toelement change in size of 10% for reasonable accuracy. A minimum of three
elements should span the smallest dimension of the impact region. Mesh densities
should be higher in plastic deformation zones of contact bodies, (Littlefield, 1996).
5. Time Step. Choose an appropriate time step for the integration of the motion
equation. Start at 5-10% of the fundamental period of the mechanical system. For
stress wave analysis, use time step equal to the smallest element size of the mesh
divided by elastic propagation speed, if the computational cost can be accepted.
6. Computational Effort. If possible, deploy 2-D models and take advantage of
symmetry to reduce computer time.
7. Analysis Formulation. The total Lagrangian formulation is appropriate for
elastic-plastic analysis involving large displacements, large rotations but small
strains. The updated Lagrangian formulation is appropriate for elastic-plastic
analysis involving large displacements, large rotations and small or large strains.
8. Contact Elements. A key for successful modeling is the correct usage of the
contact or gap elements to enforce the contact boundary conditions when they
occur. The contact element area should be a close approximation of the adjoining
FEA element area that is involved in the contact region. Thus, if a 1” diameter bar
was impacting a plate, the summation of the contact area for these gap elements
should be equal to 0.7845 inch 2 . The contact stiffness should be the Young’s
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modulus of the softer contacting body. As an example, if an aluminum body was
contacting a steel body the contact modulus should be that of the aluminum
Young’s modulus. The contact distance d should represent the actual physical
distance at which the contacts initiate. Case in point, a beam element represents
the neutral axis of real beam with extreme fiber distance c. A plate element
represents a plate with thickness t. The contact distance for this beam/plate impact
should be c + t/2. If automatic contact control is used in a FEA analysis package,
the user must a good understanding of how the contact parameters are generated.
Otherwise, erroneous results will occur.
9. Comparison of Code to Benchmarks. Impact benchmarks must be modeled and
analyzed on an FEA code to help verify that the code and analyst can correctly
model the impact scenario.
10. Understanding of the Problem. Develop a static model of the impact problem
using dynamic load factors calculated by the energy method in order to understand
the critical areas of stress. Refine the mesh where high stress levels are exhibited
in the static model. Likewise, increase the mesh size in low and constant stress
level regions.
11. Realization. The FEA method is an approximation whose accuracy is
dependent on the analyst skills, which improve with education and practice.
FEA Emulation of Fixed-end Rod Impact
The impact problem of Fig. 1 possesses a closed solution for its wave propagation
speed, initial impact stress and maximum stress in the bar as the result of wave
constructive interference by Eqs. 17~19. Both a 3-D and 2-D FEA analysis of this
impact problem was performed to demonstrate that the FEA method is capable of
producing engineering accuracy to that of the closed solution. A FEM model of the
problem was developed using both 3-D brick elements and 2-D axisymmetric
element to examine the robustness of the models to theory with using different
element types. The problem geometry and material properties for the 3-D and 2-D
models are specified in Table 2.
Table 2. Fixed-end Rod Impact Problem Parameters.
System
Element
Velocity,
ips
Diameter
, in
Lengt
h, in
Weight,
lbf
Elastic Modulus,
Mpsi
Poisson’s
Ratio
Striker
186.7
1
4.5
10
30
0.33
Bar
0
1
25
5.52
30
0.33
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FEA Emulation of Fixed-end Rod Impact using 3-D Brick Elements
A 3-D model of the impacted rod of Fig. 1 which was assumed to be fabricated
from alloy steel was meshed with a total of 8536, 8-node, 24-DOF brick elements.
Material constitute for the impactor and rod was elastic isotropic with the material
properties of that shown in Table 2. A total 29,044 DOF were exhibited in the
model after application of boundary conditions.
Boundary conditions of
prescribed zero displacement in the translation degrees of freedom were applied at
the end of the rod where it is fixed. The impactor was given an initial velocity of
186.7 inches per second. Fig. 10 diagrams this model at the impact interface of
the impactor and bar.
Accordingly to Eq. 17, the initial normal stress due to impact would be 27.68 ksi
in compression which would be well below the elastic limit of alloy steel, thus
setting up an elastic stress wave. In addition, according to theory, the propagation
speed for the elastic wave would be c L which for steel is 202,389 inches per
second. Table 3 compares the theoretical closed solutions to the impact to those of
the FEA model. As seen, the FEA model is capable of producing engineering
accuracy of the closed solution, all within 3%.
Table 3. Comparison of Theoretical and FEA End Impact Solutions.
FEA Model
Mod
el
DOF
Afte
r BC
Solid
Brick
Elements
29,0
44
Num
ber of
Elem
ents
FEA
Time
Increm
ent,
μsec
Theore
tical
Initial
Impact
Stress,
ksi
FEA
Initial
Impac
t
Stress
, ksi
Differe
nce
Initial
Impact
Stress,
ksi
Theoret
ical
Propag
ation
Speed,
IPS
FEA
Propag
ation
Speed,
IPS
Differen
ce
Propaga
tion
Speed
8563
2
27.68
27.58
-0.36%
202,387
208,333
2.94%
Report outputs from the 3-D FEA model are diagrammed in the following figures
highlighting the stress waves:
1.
2.
3.
4.
5.
Fig. 11, initial impact stress of 27.85 ksi reported by the FEA analysis.
Fig. 12, stress wave forming at impacted end, side view of impact.
Fig. 13, stress wave approximately half way propagated through the bar
in 60 μseconds from impact.
Fig. 14, stress history at bar impacted face.
Fig. 15, stress history at bar end.
From Fig. 15, the doubling of the stress level is seen at the fixed end. This stress
is double the incident stress wave as predicted by Eq. 18. No anomalies from
theory observed in these FEA outputs.
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Fig. 10. (3-D) Model of Fixed-end Bar Impact.
Fig. 11. Von Mises Stress of Frontal Node Initial Impact, 27.581 ksi (Impactor
Drawn Transparent).
FEA Emulation of Fixed Rod Impact using 2-D Asymmetrical Elements
A FEM model of the problem of Fig. 1 was developed using 2-D asymmetrical
elements to demonstrate the FEA solution is within engineering accuracy of the
closed solution using this simpler element and reduced DOF.
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The model assumed the bars fabricated from alloy steel and was meshed with
8102, 4-node, 2-DOF axisymmetric elements. Material constitute was elastic
isotropic for the impactor and bar with the material properties of that shown in
Table 2. A total 16,889 DOF were exhibited in the model after application of
boundary conditions. Boundary conditions of prescribed zero displacement in the
translation degrees of freedom were applied at the end of the rod where it is fixed
along with the appropriate axisymmetric boundary conditions of an axisymmetric
application of FEA. The impactor was given an initial velocity of 186.7 inches per
second.
Accordingly to Eq. 17, the initially normal stress due to impact would be 27.68 ksi
in compression which would be well below the elastic limit of alloy steel, thus
setting up an elastic stress wave. In addition, according to theory, the propagation
speed for the elastic wave would be c L which for steel is 202,389 inches per
second. Table 4 compares the theoretical closed solutions to the impact to those of
the FEA model. As seen, the FEA model is capable of producing engineering
accuracy of the closed solution, all within 3% using the simpler element type with
less DOF and computation time.
Table 4. Comparison of Theoretical and FEA End Impact Solutions.
FEA Model
Mod
el
DOF
Afte
r BC
Axisymmetric
Elements
16,8
89
234
Num
ber of
Elem
ents
FEA
Time
Increm
ent,
μsec
Theore
tical
Initial
Impact
Stress,
ksi
FEA
Initial
Impac
t
Stress
, ksi
Differe
nce
Initial
Impact
Stress,
ksi
Theoret
ical
Propag
ation
Speed,
IPS
FEA
Propag
ation
Speed,
IPS
Differen
ce
Propaga
tion
Speed
8102
2
27.68
27.42
-0.93%
202,387
208,333
2.94%
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Fig. 12. Elastic Stress Wave Forming At Impact, t=20μsec.
Fig. 13. Elastic Stress Wave Propagating Towards Fixed-end, t=80 μsec.
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235
Fig. 14. Time History of Von Mises Stress in PSI on Node 2106, Front Node of
Impacted Bar.
Stress von Mises (2106) (lbf/(in^2))
50000
45000
40000
35000
30000
25000
20000
15000
10000
5000
0
0
0.0001
0.0002
0.0003
Time (s)
0.0004
0.0005
0.0006
Fig. 15. Time History of Von Mises Stress in PSI on Node 2425, End Node of
Impacted Bar.
Stress von Mises (2425) (lbf/(in^2))
50000
45000
40000
35000
30000
25000
20000
15000
10000
5000
0
0.0000
236
0.0001
0.0002
0.0003
Time (s)
0.0004
0.0005
0.0006
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Report outputs from the 3-D FEA model are diagrammed in the following figures:
1.
2.
3.
4.
Fig. 16, stress wave forming at impacted end, initial impact stress of
27.42 ksi.
Fig. 17, stress wave approximately half way propagated through the bar
in 60 μseconds from impact.
Fig. 18, stress history at bar impacted face.
Fig. 19, stress history at bar end.
It is seen from Table 4, that the simpler axisymmetric element is capable of
representing the impact problem and stress wave propagation in 1-D. With no
anomalies were observed in the FEA outputs, the FEA method using axisymmetric
elements will be used to analyze a more complex mechanical system under impact
loading such as the SHB test.
FEA Emulation of SHB Apparatus
The goal of this FEA emulation is to verify if the FEA method solution will
replicate the stress and strain Eqs. (44) ~ (48) developed by stress wave
propagation theory for the SHB test. Strain time histories will be extracted from
the one node each in the incident and transmit bars to represent the strain gages
shown in Fig. 8.
In the SHB test, the strain gages measure the incident strain ε i and transmit strain
ε t. In the same function as the strain gages, two nodes will be used to extract the
strain time histories from the FEA model. These nodes will be designated as
pseudo-strain gages. From the strain history taken from these two nodes on the
apparatus bars, the strain, strain rate and stress will be calculated from Eqs. (44) ~
(48) for the specimen’s loading conditions as prescribed from stress wave
propagation theory.
For comparison to these equations, it is permissible to directly examine the time
histories of strain, strain rate and stress from one of the specimen’s nodes. This
direct examination of the dynamic loading on the specimen will then be compared
with the equation generated dynamic loading from Eqs. (44) ~ (48) to demonstrate
if the FEA method is capable of producing engineering accuracy of the SHB
impact problem; An impact problem which involves stress wave propagation with
complex mechanical impedance interfaces.
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237
Fig. 16. Elastic Stress Wave Forming At Impact.
Fig. 17. Elastic Stress Wave Propagating Towards Fixed-End.
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Fig. 18. Time History of Von Mises Stress in PSI, Front Node of Impacted Bar.
Stress von Mises (1920) (lbf/(in^2))
50000
45000
40000
35000
30000
25000
20000
15000
10000
5000
0
0
0.0001
0.0002
0.0003
Time (s)
0.0004
0.0005
0.0006
Fig. 19. Time History of Von Mises Stress in PSI, End Node of Impacted Bar.
Stress von Mises (1909) (lbf/(in^2))
45000
40000
35000
30000
25000
20000
15000
10000
5000
0
0
0.0001
0.0002
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0.0003
Time (s)
0.0004
0.0005
0.0006
239
The geometry and material construction of the SHB apparatus at Wichita State
University was modeled by FEA emulation. Materials and geometry follow in
Table 5. The model consisted of the striker, incident bar, specimen bar and
transmit bar, all meshed with 2-D axisymmetric elements. Boundary conditions
applied were that only required for the use of axisymmetric elements. Thus, the
striker, incident bar, specimen bar and transmit bar were free to move in the axial
direction. Contact boundary conditions were applied at the interfaces of the SHB
bars and the specimen under test to transmit loads from the stress wave, but allow
separation of the impacted objects when applicable due to conservation of
momentum. The impact is partial inelastic and conservation of mechanical energy
is not conserved.
Table 5. SHB Impact Problem Parameters.
System
Element
Velocity,
ips
Diameter
, in
Length
,in
Weight,
lbf
Elastic Modulus,
Mpsi
Poisson’s
Ratio
Striker
1500
1
4.5
0.89
30
0.33
Incident Bar
0
1
60
13.3
30
0.33
Specimen
0
0.5
0.5
0.019
10.3
0.33
Transmit
Bar
0
1
4
13.3
30
0.33
The material constitutes for the striker, incident and transmits bars were that of
elastic isotropic. These bars are sized and have the strength such that the striker
invokes an elastic stress wave in the bar components. The specimen was modeled
with material properties of 7075-O aluminum and at striker speed of 1500 IPS,
plastic deformation would be exhibited in the 7075-0 material of the specimen.
Thus, the material constitute for the aluminum specimen was elastic-plastic with
kinematic hardening with a yield stress of 14.939 ksi and a strain hardening
coefficient of 114.379 ksi. The total number of elements of the SHB model was
3512 using all axisymmetric elements with of 6270 DOF after boundary conditions
for axisymmetric problems. No axial constraints were enforced. Time increment
was 1 μsecond with a total simulation time of 600 μseconds. Fig. 20 diagrams the
FEA model of the SHB apparatus.
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Fig. 20. FEA Model of SHB Apparatus.
Outputs from the FEA emulation of the SHB test are described in the following.
Figs. 21~23 show the compressive stress wave from the striker impact propagate
through the incident bar, reflected at and transmit into the specimen and then
transmit through to the transmit bar as predicted by stress wave theory. The
propagation of the stress wave is fully captured and demonstrated by the FEA
method.
Fig. 24 is a time history of node 2728 on the incident bar representing at strain
gage. It can be seen by this plot the maximum compressive stress wave reaching
the node 2728 at a 160 μseconds and traveling to the left. The reflected wave from
the mechanical impedance mismatch at the specimen, a positive strain wave passes
at 480 μseconds and traveling to right. The indicated wave speed would be 64
inches traveled in the delta of 326 μseconds, or indication of wave speed at
200,000 ips, as predicted by wave theory. Fig. 24 also confirms the wave and
reflection sense at the interface of the specimen according to Eq. 18. Fig. 25 plots
the strain rate on the specimen as calculated per Eq. (47) using the time history of
the reflected wave from pseudo-gage node 2728. The peak strain rate is estimated
at 3600 in/in-second. For the SHB test, the average strain rate of Fig. 25 would be
used to report the dynamic strength as parameter of the strain rate, which due to
the shape of the strain rate pulse in the Fig. 25 would be 1800 in/in-second.
Fig. 26 shows time history plot of the transmit strain from pseudo-strain gage,
Node 731. This demonstrated the transmission of a portion of the incident wave as
predicted by stress wave theory. Using the transmit strain history of Fig. 26 and
Eq. (46), the specimen time history of stress is shown in Fig. 27. The peak stress
is approximately 25 ksi, which is approximately 10 ksi over the static yield
strength of 7075-0.
Figs. 28~30 are the direct examination of the strain, strain rate and stress from the
specimen at its node 280. When these direct extractions of dynamic loading are
compared to the theoretical calculations, they were found to be within 5%. For
example the direct peak stress was found to be 24 ksi as compared to 25 ksi with
the pseudo-strain gages.
In comparison of shapes and magnitudes of the direct examination of the loading
conditions on the specimen to the loading conditions derived from the pseudostrain gages, the shapes and magnitudes were found extremely similar. This
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241
demonstrates that the FEM method can be used to solve complex impact problems
involving stress wave propagation.
Fig. 21. Compressive Stress Pulse Approaching Specimen.
Fig. 22. Stress Pulse Transmitting into Specimen.
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Fig. 23. Stress Pulse Transmitting into Transmit Bar.
Fig. 24. Time History of Incident and Reflected Strains from Pseudo-Strain Gage
(Node 2728).
Strain Tensor Z-Z (2728) (in/in)
0.004
0.003
0.002
0.001
0
-0.001
-0.002
-0.003
-0.004
-0.005
0
0.0001
0.0002
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0.0003
Time (s)
0.0004
0.0005
0.0006
243
Fig. 25. Time History of Strain Rate on Specimen Calculated Pseudo-Strain Gages
and Eq. 46.
500
0
0
0,0001
0,0002
0,0003
0,0004
0,0005
0,0006
Strain Rate in/in-sec
-500
-1000
-1500
-2000
Avg Strain Rate = 1800 in/in-sec
-2500
-3000
-3500
-4000
Time sec
Fig. 26. Time History of Transmit Strain from Pseudo-Strain Gage (Node 731).
Strain Tensor Z-Z (731) (in/in)
5e-005
0
-5e-005
-0.0001
-0.00015
-0.0002
0
244
0.0001
0.0002
0.0003
Time (s)
0.0004
0.0005
0.0006
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Fig. 27. Time History of Specimen Stress Calculated by Pseudo Strain Gages and
Eq. 46
Strain Tensor Z-Z (731) (in/in)
5000
0
-5000
-10000
-15000
-20000
-25000
0
0.0001
0.0002
0.0003
Time (s)
0.0004
0.0005
0.0006
Fig. 28. Time History of Specimen Strain Directly from Specimen Node 280.
Strain Tensor Z-Z (280) (in/in)
0
-0.01
-0.02
-0.03
-0.04
-0.05
-0.06
-0.07
-0.08
-0.09
-0.1
-0.11
0
0.0001
0.0002
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0.0003
Time (s)
0.0004
0.0005
0.0006
245
Fig. 29. Time History of Specimen Strain Rate Directly from Specimen Node 280.
Derivative of Strain Tensor Z-Z (280)
Strain Rate in/in-sec
500
0
-500
-1000
Avg Strain Rate = 1900
in/in-sec
-1500
-2000
-2500
-3000
-3500
-4000
0.0001
0
0.0002
0.0003
Time (s)
0.0004
0.0005
0.0006
Fig. 30. Time History of Specimen Stress Directly from Specimen Node 280.
Stress Tensor Z-Z (280) (lbf/(in^2))
5000
0
-5000
-10000
-15000
-20000
-25000
0
246
0.0001
0.0002
0.0003
Time (s)
0.0004
0.0005
0.0006
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CONCLUSIONS
This chapter has reviewed the impact analysis of mechanical systems using stress wave
propagation methodology. The fundamental of stress wave propagation in solids for low
energy, medium and energy impact which are associated with elastic, plastic and shock
wave stress propagation. It was discussed that the response of an impacted solid depends
on the energy of the striking body.
For low impact velocities the initiation of elastic stress waves are formed at the impact
region. This elastic wave propagates through the body at the longitudinal wave speed.
When the wave reaches the solid’s boundary, a portion of the incident wave energy is
reflected inward within the solid and the remaining energy is transmitted through the
boundary. These trapped stress waves continue to reverberate in the solid setting up
wave interference thus producing large transient stresses. For impacts involving
moderate velocity that produce impact energies resulting in plastic deformation, it was
shown that an elastic initial stress wave is followed by a plastic wave. The plastic stress
wave travels slower than the elastic wave at the plastic wave speed which is dependent on
the stress-strain characteristics of the impacted material. For high intensity impacts of
ballistic speeds velocities, hydrodynamic behavior of solids has to be examined. The
basic wave equation was solved for a one-dimensional impact case along with the effect
of impacted medium’s properties on wave propagation.
Using this example,
developments of transmission and reflection coefficients were shown. An engineering
application of the stress wave theory for impact analysis was demonstrated on the use of
the Split Hopkinson Bar (SHB) test.
As discussed, the advantage of stress wave method was illustrated by describing an
accurate stress analysis for the impacted solid. The disadvantage of the stress wave
propagation was shown that it is highly mathematical and requires a large amount of
simplification of the impacted mechanical system to form a closed solution. In
engineering applications where complex geometries are involved, the stress wave method
can be captured by the use of Finite Element Analysis (FEA). The use of Finite Element
Methods (FEM) was discussed illustrating stress wave propagating in solids under impact
conditions. FEM parameters such as contact methodologies, element selection, material
constitutional equations, boundary conditions and proper incremental time step were
documented to help aid in the development of accurate engineering modeling of the
strains and stresses in wave propagation. Both a simple bar impact and the SHB test were
examined using FEM techniques and congruency between the FEA method and stress
wave theory were confirmed.
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