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the nonlinear wave scattering problem is investigated by applying a frequency-domain analysis. Based on the
numerical results, the main properties of the scattered harmonic wave fields are revealed and compared withanalytical solutions.
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 DQND\#EDXXQL-VLHJHQGHregion inside the structure. Therefore, to describe and localize the real damage progress in more detail,it is important to analyze the processes and interactions of the elastic wave propagation in localized
damaged structures. In recent years, some studies have already examined this specific problem
analytically (22-24). In these previous works, a spherical nonlinear elastic inclusion within an
undamaged linear elastic host medium has been considered as the damage zone, and the analyticalexpressions for the scattered harmonic wave fields have been derived using the scattered far-field
solutions. Depending on the wavelength, the radius of the inclusion and the elastic wave mode
(longitudinal or shear wave), a characteristic distribution or shape of the associated scattered wave
field exists. Based on the above-mentioned analytical solutions, the wave propagation problem in a
3-D isotropic, linear elastic medium with a spherical nonlinear elastic region will be numericallyinvestigated in this study. In addition to the case of a quadratic nonlinear elastic inclusion, a hysteretic
nonlinear elastic inclusion is also analyzed by applying a suitable material law. Subsequently, thenumerical results for the two cases will be compared and discussed in the context of the analyticalsolutions.

2. NONLINEAR ELASTICITY
2.1 Classical quadratic nonlinear elasticity
Considering the third-order terms in the elastic strain energy of an isotropic solid the following
expression can be obtained

φ

λ

2μ
2

I12  2μ I 2

l2m 3
I1  2mI1 I 2
3

nI 3

(1)

Here, φ is the elastic strain energy, λ and μ are the second-order elastic constants, l, m and n are thethird-order elastic constants according to Murnaghan (1), and I 1, I 2 and I 3 represent the invariants ofthe strain tensor. By assuming infinitesimal strains and neglecting nonlinear contributions in the strain
tensor, the strain invariants in Eq. (1) can be expressed as follows
I1  ε xεy

I 2  (ε x ε y
I 3  ε xε y ε zεz,ε yε z1
4ε z ε x )  14 γ xy2

γ yz2

γ xy2 ,

(2)

γ xy γ yz γ xz  ε x γ  ε y γ  ε z γ ,where the infinitesimal strains are defined by
 su
su
sv
sw
εx  , ε y  , εz 
, γ xy  
 sy
sx
sy
sz

2
yz

2
xz

 sv
sv ¬­
­ , γ yz  
­
sx ®­
 sz

2
xy

 sw
sw ¬­
­ , γ xz   sxsy ®­­

su ¬­
­.
sz ®­

(3)

Here, u, v and w denote the displacements of the 3-D continuum in the x-, y- and z-directions.
Differentiating the strain energy given by Eq. (1) with respect to the strains defined by Eq. (3) yields
the following normal and shear stresses
σ x = E1ε x + λ ( ε y + ε z ) + E2ε x2 + l ( ε y2 + ε z2 ) + 2lε x ( ε y + ε z ) + ( 2 ( l − m ) + n ) ε y ε z +σ y = E1ε y + λ ( ε x + ε z ) + E2ε y2 + l ( ε x2 + ε z2 ) + 2lε y ( ε x + ε z ) + ( 2 ( l − m ) + n ) ε x ε z +σ z = E1ε z + λ ( ε x + ε y ) + E2ε z2 + l ( ε x2 + ε y2 ) + 2lε z ( ε x + ε y ) + ( 2 ( l − m ) + n ) ε x ε y +τ xy = μγ xy + mγ xy ( ε x + ε y + ε z ) +τ xz = μγ xz + mγ xz ( ε x + ε y + ε z ) +τ yz = μγ yz + mγ yz ( ε x + ε y + ε z ) +

n
4
n
4
n
4

m
2
m
2
m

(γ + γ + γ ) − n4 γ ,
2
xy

2
yz

2
xz

2
yz

(γ + γ + γ ) − n4 γ ,
2
xy

2
yz

2
xz

2
xz

(4)

(γ + γ + γ ) − 4 γ ,
2
2
xy

2
yz

2
xz

n

2
xy

(γ γ − 2ε γ ) ,
xz

yz

z

xy

( γ γ − 2ε γ ) ,
xy

yz

y

xz

(5)

(γ γ − 2ε xγ ) ,
xy

xz

yz

with E 1 =[λ + 2μ and E 2 =[l + 2m.

4747

2.2 Hysteretic nonlinear elasticity
As already mentioned in the introduction, that for an appropriate description of the nonlinear elastic
behavior of many engineering materials, it is often not sufficient to apply only the classical nonlinear
elastic theory. Therefore, non-classical nonlinear elastic theory has to be applied. Especially inheterogeneous materials, such as concretes, composites and damaged structures, a pronounced
non-classical hysteretic material behavior is often observed. To model the hysteretic nonlinear
behavior, the Nazarov-hysteresis model (25) is used in this study. In this case, the 1-D governing
constitutive equation is given by

σ x = E1 ( ε x − f ( ε x , εx ) ) ,(6)

where for the purely elastic case we have£¦γ 1ε nN ,¦¦¦ γ ε nNγ 1 γ 2 ε mnN 1ε ,1 ¦
2¦f ε , ε  ¤nN ¦¦γ 3ε nN ,¦¦¦¦γ 4ε nN  γ 3 γ 4 ε mnN1ε ,¦¥ε  0, ε  0,ε  0, ε  0,(7)ε  0, ε  0,ε  0, ε  0,with the hysteretic material parameters γ1 − γ 4 and the exponent n N. In Eq. (7), εm+ and ε m − denote themaximum and minimum strains in the current hysteresis loop. To transform the 1-D constitutive
equations to 3-D, the Kelvin decomposition method (26, 27) is used. Here, the linear elastic stiffness
matrix in the Voigt notation C V is transformed to the Kelvin notation CK by applying thetransformation relation CK =[TCVT, where the corresponding transformation matrix is defined byT  diag 1, 1, 1,

Accordingly,

thecorrespondingε  ε x , ε y , ε z , 2ε xy , 2ε xz , 2ε yzKstrain

2,

and

2,

2 .

(8)

stressvectorscanσ  σ x , σ y , σ z , 2τ xy , 2τ xz , 2τ yz
K

and

be

definedby. Subsequently, theeigenvalue problem of the Kelvin system is solved to obtain the respective constitutive relations in theˆ εˆ . For an isotropic linear elastic solid, the so-called eigenstiffness matrix caneigensystem as σˆ  C
be obtained via the corresponding eigenvalues as follows
ˆ  diag <3λC

2μ , 2μ , 2μ , μ , μ , μ > .

(9)

By means of the eigenvectors, the so-called eigenstrains or eigenstresses are defined byσˆ  <σˆ1 , σˆ 2 , σˆ 3 ,τˆ1 ,τˆ2 ,τˆ3 >  ¡ 13 σ xσyεˆ  <εˆ1 , εˆ2 , εˆ3 , γˆ1 , γˆ2 , γˆ3 >  ¡

εy

¢

¢

1
3

εx

σz ,
εz ,

1
6

1
6

2σ z  σ x  σ y ,2ε z  ε x  ε y ,

1
2

1
2

σ y  σ x , τ xy , τ xz , τ yz ¯° ,T±ε y  ε x , γ xy , γ xz , γ yz ¯° .
T

(10)

±
In the obtained system of equations, the associated eigenstresses are independent of each other anddetached from the original coordinate system. Thus, when it is stretched in one particular eigenstressdirection, no elastic energy is effectively involved in the other stress directions (invariant system) (27).
Therefore, the scalar hysteretic nonlinear stress-strain relationships can be applied for each of the six
independent directions. Finally, the stress components in the original Cartesian coordinate system canbe obtained by the inverse transform.

3. STAGGERED CHEBYSHEV-PSEUDOSPECTRAL COLLOCATION METHOD
Due to the advantages of the spectral methods regarding computing time and memory requirement,a numerical collocation method based on the Chebyshev polynomials is applied in this study. Thismethod has already been developed several decades ago (28) and has been successfully applied in thefields of engineering, geoscience or meteorology. In contrast to the Fourier spectral methods it can also
be applied to non-periodic boundary value problems. In the following, the key steps of the present
numerical method and the corresponding modifications are briefly explained. Firstly, the governing4748

equations of motion for the 3-D wave propagation problems are recast into a first-order hyperbolic
system of partial differential equations (PDEs), which can be written in the following formsτ xy sτ xz ¬­ sv ysτ yz sσ z ¬­sv x
1  sσ1  sτ xy sσ y sτ yz ¬­ svz
1  sτ­­ ,­­ ,­ . (11)  x   xzρ  sxρ  sxρ  sxstsysz ­® stsy
sz ­® stsy
sz ­­®
From Eq. (3), the strain rates are related to the particle velocities by the following kinematic equationssε y sv y
sε x svx
sε z sv z
,
,
,



st
sx
st
sy
st
sz
sγ xy  sv y svx ¬­sγ yz  svz
sγ xz  svz svx ¬­
­­ ,
 

 
­,
­

 sx
st
sy ®­
st
sz ®
st
 sx
 sy

(12)

sv y ¬­­.
sz ­­®

Finally, the nonlinear constitutive equations introduced in the previous sections are used to relate thestresses to the corresponding strains or strain rates.
To solve the obtained system of PDEs, complemented by the corresponding boundary and initialconditions, a staggered Chebyshev pseudo-spectral collocation is adopted, where the spatial
discretization is based on a Chebyshev polynomial expansion and the time discretization is applied by
explicit time integration. As a modification to the standard collocation method the field variables arelocated on a staggered non-equidistant grid according to Fig. 1. Here, N and N −1 collocation points
along one distinct direction of the computational grid are defined by the Gauss-Lobatto and Gauss
points (29). Applying the Chebyshev series on this grid the spatial derivatives of the field variables in
Eqs. (11) and (12) are obtained. To interpolate between the two different grids a Lagrangianinterpolation is applied. Furthermore, to evolve the velocities and strains in time a staggered leap frog
scheme (27) is used, which is second order accurate and based on a central difference approximation of
the corresponding time derivatives. According to the stability conditions, the dense meshing near theboundaries in the standard Chebyshev pseudo-spectral method leads to severe time-step restrictions.
Therefore, a mapping function is used here to achieve a more equidistant node distribution and relax
the time-step restriction (28). Furthermore, the spatial derivatives are determined by a direct matrix
multiplication of the predefined Chebyshev difference matrices with the field variable vectors. Due tothe highly parallel structure of the corresponding numerical scheme all computations are performed on
the graphical processing unit (GPU) which improves the computing time considerably compared to theconventional CPU based calculations.
1 jπ  
Gauss-Lobatto points: ξ j =  1 − cos ,2 N 
1
 2 j +1  
π  ,
Gauss points: ξ j +1/2 = 1 − cos 2 2N + 2  

j = 0,! , N
j = 0,!, N − 1

σ x ,σ y , σ z , ε x , ε y , ε zτ xy , γ xy
vx , u x
τ yz , γ yz
vy , u yvz , u z
τ xz , γ xzzyx

Figure 1 – Distribution of field variables at the 3-D staggered Gauss/ Gauss-Lobatto grid
The basic 3-D computational domain for the subsequent numerical simulations is illustrated in Fig. 2.
Here the physical domain including the linear elastic medium with the nonlinear spherical region issurrounded by convolutional perfectly matched layers (CPML’s) (30) absorbing the outgoing waves.
The physical domain under consideration is excited by an ultrasonic stress source (σx or τxy ) defined at
the free y-z-plane (x = 0) and prescribed by the following time harmonic function

T ( x = 0, y, z , t ) =



1
t 
 ⋅ H ( tsig − t ) ,Asig ⋅ sin ( 2π f sig t ) ⋅ 1 − cos  2π
 t  
2
sig  



(

)

(13)

where A sig is the stress amplitude, fsig is the excitation frequency, tsig is the signal duration and H is the4749

Heaviside step function. To visualize and analyze the associated wave scattering problem thefrequency-domain response is of particular interest and importance. For this purpose, the time-domain
displacement fields of selected planes within the considered 3-D continuum are transformed to thefrequency-domain by applying the fast Fourier transform (FFT). In order to separate the wave modes
scattered from the nonlinear spherical region, the displacements u x, uy and uz in the cartesian
coordinate system (obtained from the corresponding velocities by time integration) are transformed tothe spherical coordinate system (Fig. 3) as follows
ur = sin (θ ) ⋅ cos (ϕ ) ⋅ u x + sin (θ ) ⋅ sin (ϕ ) ⋅ u y + cos (θ ) ⋅ u z ,uθ = cos (θ ) ⋅ cos (ϕ ) ⋅ u x + cos (θ ) ⋅ sin (ϕ ) ⋅ u y − sin (θ ) ⋅ u z ,(14)

uϕ = − sin (ϕ ) ⋅ u x + cos (ϕ ) ⋅ u y ,

CPML

L/8

Nonlinear elastic spherey (damage region)

L/2

z

d

x

Free surfacez

100x100x100 nodes

θ
L/2

CPML

L/8
L/2

r

SphereLinear elastic medium(intact region)

L/2

P(r, ϕ, θ )CPMLIncident wave

Free surface

with the polar angle θ and the azimuthal angle ϕ. In Eq. (13), ur represents the longitudinal wave mode
whereas uϕ and u θ defining the two transverse wave modes of the scattered wave field.

y

ϕ
x

L/8

Figure 2 – 3-D Computational domain

Figure 3 – Spherical coordinate system

4. NUMERICAL RESULTS AND DISCUSSION
In this section, the propagation of longitudinal and shear waves in a linear elastic medium with alocalized damage region is investigated by applying the numerical method presented in the previous
section. The damage region is modeled by a nonlinear elastic spherical inclusion located at the centerof the computational domain inside the linear (undamaged) elastic host medium (Fig. 2). For
describing the nonlinear elastic behavior, the above-mentioned nonlinear constitutive relations are
separately implemented and considered. The main geometrical and material parameters used for thefollowing numerical examples are summarized in Table 1.
Table 1 – Simulation and material parameters
L
(mm)
20.00

d
(mm)
2.00

Asig
(MPa)
− 31.05

f sig
(kHz)
100

t sig
( μs)
60

λ

μ

ρ

(GPa)
9.75

(GPa)
15.00

(kg/m³)2350

4.1 Quadratic nonlinear elastic caseHarmonic generation in a quadratic nonlinear elastic medium is mainly associated with apronounced second harmonic wave field. Therefore, in this section only this second harmonic part of
the total scattered wave field will be considered. The third-order elastic constants are taken asl = m = n = − 2000 GPa. For the case of a longitudinal wave excitation ( σx ) the scattered longitudinal

4750

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