Dynamic equivalence of ultrasonic stress wave propagation in solids

Ultrasonics xxx (2017) xxx–xxx

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Ultrasonics journal homepage: www.elsevier.com/locate/ultras

Dynamic equivalence of ultrasonic stress wave propagation in solids Medhat Awad El-Hadek Department of Mechanical Design & Production, Faculty of Engineering, Port-Said University, Port-Said, Egypt

a r t i c l e

i n f o

Article history: Received 21 December 2016 Received in revised form 16 April 2017 Accepted 1 June 2017 Available online xxxx Keywords: Dynamic equivalence Numerical simulations Residual stresses Elasto-plastic modeling

a b s t r a c t Ultrasonic stress waves, generated during the dynamic impact on structures, were studied. A benchmark for the finite element analysis (FEA) was made to define the optimum geometrical factors, which were represented as, mesh distribution, analysis time, ultrasonic wave properties, element type, and shape to capture the dynamic phenomena compared to the theoretical exact solution. Comparison of three different dimensional finite element models was performed depending on the applied impact forces, the element size, and the structural geometry. A dynamic equivalence for these three different variables was established and found to be of a direct multiplication relation to the solid’s density with the Young’s modulus. The results demonstrated that the stresses in x, and y-directions, predicted by FEA simulations, matching well for the different materials under normalized time. Ó 2017 Elsevier B.V. All rights reserved.

1. Introduction Finite Element Analysis (FEA) is a numerical tool used independently from experimental work. Lately, FEA has been widely used in dynamic simulations to determine the stresses and deformations, the loads and forces movements, and the heat transfer, using variational matrices arrays and complex mesh diagrams [1]. Moreover, FEA has proven, in the last decade, to be a very successful tool for solving many partial differential equations and integral expression, for which no closed form solution exists. In addition, FEA minimizes the mismatch between experimental and analytical measurements; it alters the measurements to provide a closer agreement with experimental readings, using improved models which provide a close representation of the prototype problem [11]. On the other hand, FEA has a huge temptation of having a low cost relative to the experimental, which made it suitable for commercial and laboratory simulations; time dependent integration algorithms for dynamic implicit and explicit problems in FEA are some of the highly demanded applications [26]. Commonly, dynamic Loads as time dependent forces are imposed on structures by either natural earthquakes phenomena or as human activities. As more people started to use finite element analysis and compared their results with experimentally obtained results, a normalized comparison method became essential. FEA mode synthesis with simple coordinate reduction system was analyzed [28]. The procedure was capable of analyzing complex shapes, non-linear spatial mechanisms with irregularly shaped links in high detail. E-mail address: [email protected]

Du et al. [7] started dynamic FEA simulations using 3-D elastic beam with an arbitrary moving base. They used six degrees of freedom in the finite element structural dynamic model with a pre-twisted offset mass from the elastic center base. The results provided from the FE model showed comparability to some extent, as the base motion variables used in multi-body dynamics and the fundamental elements were approximated to solve the dynamic problem of rotating beamlike structures. Camacho and Ortiz [4] developed a FE model with Lagrangian deformations for fracture in brittle materials, where complicated rate-dependent boundary conditions as plasticity and thermal coupling were accounted for in the calculations. The calculated fracture histories in conical, lateral and radial directions where normalized with deferent coefficients to be compared with the experiment results. Zienkiewicz, and Taylor [34] explained the success of the FEA to capture the experimental phenomena dependents on the symmetrical stability of perturbations, such as, the geometrical complex charactering, accurateness of material property, the symmetry of applied load and the optimum application of the boundary conditions. Recently the strain rate effect was introduced to the FEA through the dynamic wave properties in solids by Bonet et al. [3]. The accurateness of the time dependent properties and applied boundary conditions is the key factor of having a successful FEA, as well as the minimization of the cost, and the reducing of the formation assembly effort [27]. Thus, comparing the FEA stress results with each other becomes a significant problem due to the numerous variable factors involved in the analysis, such as, different geometries [2], applied dynamic boundary conditions [13], bi normalized terms, or the

http://dx.doi.org/10.1016/j.ultras.2017.06.007 0041-624X/Ó 2017 Elsevier B.V. All rights reserved.

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usage of p-method rather than k-method in solids [14]. Recent efforts in normalizing different FEA outcomes to compare the stress results in different materials and models had limited success, as the dynamic equivalence was not clearly defined for comparing stress results [10,8,32,18]. The term dynamic equivalence was first proposed by Wheeler and Mura [30]. They used variational mechanics methods for determining the different displacement mode shapes and dispersion relations for a plane time-harmonic waves, propagating through an infinitely and periodically arranged composite material. The comparison was successful due to limited variables in their problem formation. Then, Nemat-Nasser and Yamada [24] studied the harmonic stress wave propagation effect on composite layering direction; exact solutions were compared with the experimental results showing difference of less than 10%, using normalized dynamic terms. The dynamically equivalence were used to express the SaintVenant’s principle of the strip symmetric stresses [17]. As the decaying load was deduced from the average power for the dynamic fields is required for the self-equilibrium conditions. Chaix et al. [5] studied the ultrasonic wave propagation in heterogeneous solid media and compared the theoretical analysis with experimental validation normalizing results and dynamic equivalence. The theoretical results was different than the experimental for cement based media, which was associated to the poor dynamic equivalent factor used in the analysis. Zhou and McDowell [33] introduced a dynamic equivalent for the deformation of the atomistic molecular particle systems. This dynamic equivalence was attenuated using several factors as, continuum couple stress fields, continuum body force and body moment fields, continuum kinetic fields and atomic deformation constants, and the linear and angular momenta distributions. Recent analysis for an accurate dynamic equivalence under ultrasonic stress wave propagation was introduced in literature; nonetheless, with limited success [10,6,13,21,9]. In this study, a benchmark FEA model was made to define the optimum geometrical factors such as, mesh distribution, analysis time, ultrasonic wave properties, element type and shape to capture the dynamic phenomena compared with the theoretical exact solutions. The analysis started with defining the dynamic problem in two different geometries with the same boundary conditions, resulting in different stress results. Subsequently, three different 3D finite element models have been developed with the same applied impact forces, element size and numbers, and structural geometry. During the dynamic analysis was based on p-method calculations of the numerical integrated equations of motion with respect to time. Finally, the stresses of the three different developed 3D models were used to propose a dynamic equivalence, which could be therefore used for comparing stress results.

modulus, bulk and shear moduli were introduced by Weng’s model using variational mechanics methods [25]. It takes into account the deformations, strain and stress state of the counterparts, interfacial stresses, and the elastic energy; in determining overall moduli of the composite. The elastic energy balance of the alloy mixture was employed, in terms of average strains and stresses divided into two main parts as hydrostatic and deviatoric. The stiffness and compliance of the mixture alloy tensors were used as Lijkl = (3K, 2G), Mijkl = (1/3K, 1/2G); respectively, in terms of shear, K, bulk, G, moduli, and c is Passion’s ratio. The bulk B and shear moduli m for the alloy two phase mixture are found as shown in Eq. (2),

Vf Bc ¼ 1 þ 3ð1V ÞBm f Bm þ 3Bm þ4lm

Bm Bf Bm

;

Vf lc ¼ 1 þ 6ð1V ÞðBm þ2l Þ f m lm þ l lml 5ð3B þ4l Þ m

m

f

ð2Þ

m

where Vf is the filers volume fraction, the subscripts ‘c’, ‘M’, and ‘f’ denotes the alloy two phase mixture, the main matrix material and the filler properties; respectively. The aM and bM are main matrix constant properties, which depend directly on the matrix bulk and shear moduli. The Young’s modulus is then calculated using the bulk and shear moduli as in Eq. (3),

Bc ¼

E ; 3ð1  2mÞ

lc ¼

E 2ð1 þ mÞ

ð3Þ

Crack Initiation Toughness Average Stress Approach was analyzed by Williams in homogeneous; as well as, nonhomogeneous materials [31]. Williams used a classical Airy’s stress function formulation in polar coordinates; whereas Irwin used for his analysis the complex variable method; following Westergaard, considering an infinite isotropic and homogeneous plate with infinite crack under uniform normal stresses r and shear stresses s. The crack tip was assumed as the origin for both Cartesian (x, y) and polar (r, h) coordinates [31,15,16]. Williams has assumed a variable separable stress function w in the polar coordinates as w = rk+1f(h). The values of the eigen parameter k for the free edged cracked plate is as sin2pk = 0. The stress function should satisfy the dimensional governing equation of r2 ðr2 wÞ ¼ 0 with no body forces. Where,

r2 is the harmonic (Laplacian) operator, r2 ¼ @r@2 þ 1r

@ @r

þ r12

@ @h2

Also,

the stress components can be defined using the stress function as presented in Eq. (4),

rhh ¼

@2w ; @r 2

rrr ¼ r2 w 

@2w ; @r 2

rrh ¼ 


 @ 1 @w @r r @h

ð4Þ

By replacing the Laplacian operator into the stress function the biharmonic equation becomes as in Eq. (5),

r2 ðr2 wÞ ¼ ðk  1Þðk  2Þrk3 ðf 00 ðhÞ þ ðk þ 1Þ2 f ðhÞÞ 2. Theory Ultrasonic techniques are commonly used to determine the different elastic properties of different types of materials such as, homogeneous, heterogeneous, isotropic, and anisotropic composites; as well as, detecting defects and voids within materials [20,12], as it has a wide range of applications. It is. Additionally, it is frequently adopted to measure velocity of waves, attenuation, density, and thickness. Using ultrasonic pulse echo techniques showed that the dynamic values of Young’s moduli were found to be correlated with the longitudinal Cl and shear Cs wave speeds in solids, as presented in Eq. (1):

C 2l ¼

E ð1  mÞ E 1 ; C 2s ¼ q ð1 þ mÞð1  2mÞ q 2ð1 þ mÞ

ð1Þ

where E, q and m refer to young’s modulus, apparent density, and Poisson’s ratio; respectively. Correlation between elastic young’s

1 1 00 00 þ ðk  1Þrk2 ðf ðhÞ þ ðk þ 1Þ2 f ðhÞÞ þ 2 r k1 ðf ðhÞ r r þ ðk þ 1Þ2 f ðhÞÞ ¼ 0

ð5Þ

The above equation is a fourth order ordinary differential equation as a function of h. The general solution of which is presented in Eq. (6),

f ðhÞ ¼ C 1 cosðk  1Þh þ C 2 sinðk  1Þh þ C 3 cosðk þ 1Þh þ C 4 sinðk þ 1Þh

ð6Þ

where C1  C4 are the unknown coefficients to be calculated from the body boundary conditions as the crack two faces are along h = ±p. For stress free crack faces the stress rhh = rrh = 0, at h = ±p, which leads to f(h) = f(h) = 0 along h = ±p. By applying the boundary conditions and separating the symmetric and the anti-symmetric parts, and obtaining a non-trivial solution in each case as k = n/2 resulting in Eq. (7) as,

Please cite this article in press as: M.A. El-Hadek, Dynamic equivalence of ultrasonic stress wave propagation in solids, Ultrasonics (2017), http://dx.doi. org/10.1016/j.ultras.2017.06.007

M.A. El-Hadek / Ultrasonics xxx (2017) xxx–xxx

 X n  ðn  2Þh n  2 ðn þ 2Þh w¼ r 2 C 1n cos  cos 2 nþ2 2 n¼1;3;...   ðn  2Þh ðn þ 2Þh þC 2n sin  sin 2 2    X n ðn  2Þh ðn þ 2Þ r 2 C 1n cos  cos þ 2 2 n¼2;4;...   ðn  2Þh n  2 ðn þ 2Þh þC 2n sin  sin 2 nþ2 2

ð7Þ

The terms with coefficients C1n and C2n correspond to the symmetric opening mode-I, and the asymmetric in-plane shearing mode-II for the crack deformations, respectively. Based on strain energy considerations, the absence of the negative values of n in the above equation should also be noted. The amplitude of the stress field at the crack tip, as represented by a parameter C, is the stress intensity factor that is the focus of attention throughout the fracture mechanics. The stress intensity factor is an engineering parameter, which describes the strength of the crack tip singularity. Using Eq. (7) the individual planar stress components can be expressed in following Eqs. (8)–(10),

rr ¼

rh ¼

 n ðn  2Þh ðn þ 2Þh 6  cos  2  cos 4 2 2 n¼1;3;...  ðn  2Þh ðn þ 2Þh n  cos þ n  cos 2 2  X n n ðn  2Þh ðn þ 2Þh 1 6  cos  2  cos r 2 C 1n þ 4 2 2 n¼2;4;...  ðn  2Þh ðn þ 2Þh n  cos þ n  cos 2 2 X

n

r 21 C 1n

 n ðn  2Þh ðn þ 2Þh n  cos  2  cos 4 2 2 n¼1;3;...  ðn  2Þh ðn þ 2Þh n  cos þ 2  cos 2 2  X n n ðn  2Þh ðn þ 2Þh þ n  cos  n  cos r 21 C 1n 4 2 2 n¼2;4;...  ðn  2Þh ðn þ 2Þh 2  cos þ 2  cos 2 2

rrh ¼

X

verified. A dynamically loaded cracked homogeneous PMMA beam studied by Lee and Freund [19] theoretically solved by Mason et al. [22] experimentally is chosen to benchmark the numerical model. The finite element model used in the study is shown in Fig. 1(a). A free-free beam is asymmetrically loaded by applying a velocity boundary condition on the beam face of the sample edge, as shown in Fig. 1(b). The crack tip stresses and deformations were computed for approximately 50 ms after impact. It should be pointed that both space and time must be defined systematically in dynamic analysis. As, the element size h is controlled by the corresponding wavelength having the highest frequency, fmax, and the lowest propagation speed (Vs); so the smallest element size is presented as h  0.25 ⁄ Cs/fmax [23]. The optimum time step is a function of the element size and the propagating wave of highest longitudinal propagation speed [23] through the smallest element size h, which can be expressed as Dt  h/Cl. Consequently, the selection of appropriate element size and integration time step anticipates the optimum uniform mesh size. In this study the smallest element size that ensures capturing the dynamic phenomena was 2 mm, as conducted in the next analysis.

3.2. Stress wave propagation in different geometries

ð8Þ

n

r 21 C 1n

 n ðn  2Þh ðn  2Þh n  sin  2  sin 4 2 2 n¼1;3;...  ðn þ 2Þh ðn þ 2Þh n  sin þ 2  sin 2 2  X n n ðn  2Þh ðn  2Þh 1 r 2 C 1n n  sin  2  sin þ 4 2 2 n¼2;4;...  ðn þ 2Þh ðn þ 2Þh þ2  sin  n  sin 2 2 X

3

ð9Þ

n

r 21 C 1n

Analysis of the geometrical effect on the stress waves is done after applying simple ultrasonic stress wave propagation on slightly different geometries. Ultrasonic stress wave propagation was studied using FEA, for two different impacted aluminum structural materials with the same boundary conditions. Half circle and triangle with diameter and base length of 37 mm, respectively, were subjected to ultrasonic impact velocity of 5 m/s for duration of 40 ms. Aluminum material is selected with 70 GPa of elastic modulus, 0.33 Poisson’s ratio, and 2700 kg/m3 with smallest element size of 3.3 mm to capture the stress phenomena as presented in Fig. 1. The stress dissipation trend at different points from the center shows compatibility, but different magnitudes. It should be noticed that in both Figs. 2(b) and (d) the y-stresses propagating in the exact center direction, as zero distance from the center, are the highest values as they dissipate towards the edges of the structure. Also, the measuring distances for the y-stresses were found to be the same for both structures the stress distribution pattern in the half circle is evenly smother than it in the rectangular one, having the same limiting stress values. So the comparison of the two dynamically FEA is possible due to the unification of the applied impact velocity, element modeling, diameter, and base length of the two structures. The next step is to try to find a dynamic equivalence for these three different variables.

3.3. Stress wave propagation in different materials properties

ð10Þ

These equations will be applied in the benchmark analysis in the next section, as the basis of the closed form theoretical solution to be compared with FEA results. 3. FE model results and discussions 3.1. Benchmark problem Benchmarking is a way of discovering what is the best performance being achieved, with the aim of attaining comparison advantages [8,29]. Reliability of the finite element model and SIF determination method under dynamic loading conditions was

Models I, II, III where adopted to be compared for dynamic equivalence approach. The structures of all models were kept constant with crack length a of 8 mm, beam span length l of 154 mm, beam width w of 42 mm, and beam thickness of 6 mm. The FE model is subjected to ultrasonic impact velocity of 5 m/s, for a time duration of 100 ms as presented Fig. 3. Model I was made of PMMA, having Elastic modulus E of 3000 MPa, Poisson’s Ratio c of 0.3, longitudinal wave velocity of 2010 m/s, and density q of 1000 kg/m3. Model II was made of Al-6060, having Elastic modulus E of 12,000 MPa, Poisson’s Ratio c of 0.3, longitudinal wave velocity of 2010 m/s, and density q of 4000 kg/m3. Model III was made of arbitrary light composite having Elastic modulus E of 8000 MPa, Poisson’s Ratio c of 0.3, longitudinal wave velocity of 5360 m/s, and density q of 375 kg/m3.

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M.A. El-Hadek / Ultrasonics xxx (2017) xxx–xxx

Fig. 1. (a) The asymmetrically impacted cracked PMMA beam numerical model boundary conditions configuration, and (b) the results of the normalized stress intensity factor varying with the normalized time.

(c)

(a) Y

Y X

0,0

0.024, 0

X

0.037, 0

Half circle FEA of

0

0

-4E+7

-4E+7

-8E+7 -1.2E+8

37mm from center

(b)

0, 0

4E+7

Stress y, (Pa)

Stress y, (Pa)

4E+7

0.012,0

12mm from center

0

4

8 12 16 20 24 28 32 36 40 44 48

Time t, (usec)

0.037, 0

-8E+7 -1.2E+8

37mm from center

(d)

24mm from center

-1.6E+8

12mm from center

at the center 0

-2E+8

0.024, 0

Triangle FEA of

24mm from center

-1.6E+8

0.012, 0

at the center 0

-2E+8

0

4

8 12 16 20 24 28 32 36 40 44 48

Time t, (usec)

Fig. 2. Ultrasonic stress wave propagation resulted from impact force of 5 m/s subjected for 40 ms on (a) half circle with diameter of 37 mm, (b) the stresses in Y-direction in different locations from the center, compared to wave propagation on (c) rectangle with base of 37 mm, and (d) the stresses in Y-direction in the same different locations from the center.

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M.A. El-Hadek / Ultrasonics xxx (2017) xxx–xxx

r crack

y B x θ a A

stresses resulted for models I and II of homogenous cracked structure measured at points A before crack tip, B after crack tip, and C at the far field locations. So the comparison of the two dynamically FEA was possible due to the unification of the applied impact velocity, element modeling, and subjected time. Mismatch between the results were noticed, as the stresses in x, y-directions for model II (with higher modulus of elasticity and density) at point A (prior to crack tip) in the cracked structure are higher than model I. However, at point B (above crack tip), the stresses in x, y-directions for model I were higher than model I. This finding could be attributed to the effect of crack tip singularity on the stress wave propagations for the heavy inertia models. Also, at point C (at far field) it oscillates for model II (with higher modulus of elasticity and density).

C

b Fig. 3. Ultrasonic stress wave propagation resulted from impact force of 5 m/s subjected for 100 ms on homogenous cracked structure and the stresses measured at points A before crack tip, B after crack tip, and C at the far field locations.

3.3.2. Comparing model I and model III as E⁄q = 1732.051 (constant) The ultrasonic stresses in the x, y-directions propagating at locations of points A before crack tip, B after crack tip, and C at the far field locations were measured for models I, and III. Figs. 6 and 7(a–c) represent the ultrasonic x, and y-direction compared to stresses resulted for models I and II of homogenous cracked structure measured at points A before crack tip, B after crack tip,

3.3.1. Comparing model I and model II as E/q = 3 (constant) The ultrasonic stresses in the x, y-directions propagating at locations of points A before crack tip, B after crack tip, and C at the far field locations were measured for models I, and II. Figs. 4 and 5(a–c) represent the ultrasonic x, and y-direction compared

At B

At A

-2E+6

5E+6

-3E+6

4E+6

-5E+6

Model 1

0

-4E+6

3E+6 Model 1

2E+6

-6E+6

Model 1

Sxx

Sxx

Sxx

5E+5

-1E+6

Model 2

6E+6

-5E+5

-7E+6 -8E+6

Model 2

-9E+6

1E+6

-1E+6

-1E+7

Model 2

-1.1E+7

0 -1E+6 0.00

At C

0

7E+6

-1.2E+7 1.00

2.00

3.00

4.00

5.00

-1.3E+7 0.00

1.00

2.00

3.00

-1.5E+6 5.00 0.00

4.00

1.00

2.00

3.00

t/ (Cl.W)

t/ (Cl.W)

t/ (Cl.W)

(a)

(b)

(c)

4.00

5.00

Fig. 4. Ultrasonic x-direction of compared stresses resulted from impact force of 5 m/s subjected for models I and II for a duration of 100 ms on homogenous cracked structure measured at points (a) A before crack tip, (b) B after crack tip, and (c) C at the far field locations.

At A

9E+5

Model 2

3E+6

3E+5

Syy

Syy

-1E+7

1E+6 0

0

Model 1

Model 1

-3E+5

-2E+7

-1E+6 -2E+6 -3E+6 0.00

Model 2

6E+5

Model 1

2E+6

Syy

At C

At B 0

4E+6

Model 2

1.00

2.00

3.00

t/ (Cl.W)

(a)

4.00

5.00

-3E+7 0.00

1.00

2.00

3.00

t/ (Cl.W)

(b)

4.00

5.00

-6E+5 -9E+5 0.00

1.00

2.00

3.00

4.00

5.00

t/ (Cl.W)

(c)

Fig. 5. Ultrasonic y-direction of compared stresses resulted from impact force of 5 m/s subjected for models I and II for a duration of 100 ms on homogenous cracked structure measured at points (a) A before crack tip, (b) B after crack tip, and (c) C at the far field locations.

Please cite this article in press as: M.A. El-Hadek, Dynamic equivalence of ultrasonic stress wave propagation in solids, Ultrasonics (2017), http://dx.doi. org/10.1016/j.ultras.2017.06.007

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M.A. El-Hadek / Ultrasonics xxx (2017) xxx–xxx

At A Model 3

-1E+6

Model 3

-2E+6 -3E+6

4E+6

2E+5

Model 1

-4E+6

2E+6

-5E+6

Sxx

3E+6

Sxx

Sxx

4E+5

0

5E+6

-6E+6

0

-7E+6 -8E+6

Model 1

1E+6

-9E+6

-2E+5

Model 3

-1E+7

0 -1E+6

At C

At B

6E+6

Model 1

-1.1E+7 0

2

4

6

8

10

12

14

-1.2E+7

0

2

4

6

8

10

12

14

-4E+5

0

2

4

6

8

t/(C l.W)

t/(C l.W)

t/(C l.W)

(a)

(b)

(c)

10

12

14

Fig. 6. Ultrasonic x-direction of compared stresses resulted from impact force of 5 m/s subjected for models I and III for a duration of 100 ms on homogenous cracked structure measured at points (a) A before crack tip, (b) B after crack tip, and (c) C at the far field locations.

At A

At B

4E+6

At C 4E+5

0

Model 3

-1E+6 3E+6

Model 3

Syy

Syy 1E+6

Syy

-3E+6

2E+6

-4E+6 -5E+6

Model 1

0

Model 1

Model 1

-6E+6

0

Model 3

2E+5

-2E+6

-2E+5

-7E+6 -1E+6

0

2

4

6

8

10

12

14

-8E+6

0

2

4

t/(C l.W)

6

8

10

12

14

-4E+5

0

2

4

6

t/(C l.W)

(a)

8

10

12

14

t/(C l.W)

(b)

(c)

Fig. 7. Ultrasonic y-direction of compared stresses resulted from impact force of 5 m/s subjected for models I and III for a duration of 100 ms on homogenous cracked structure measured at points (a) A before crack tip, (b) B after crack tip, and (c) C at the far field locations.

At A Model 3

-1E+6

Model 3

-2E+6 2E+5

-3E+6

4E+6

Model 11

-4E+6

2E+6

Sxx

-5E+6

3E+6

Sxx

Sxx

4E+5

0

5E+6

-6E+6

0

-7E+6

Model 11

-8E+6

Model 11

1E+6

-2E+5

-9E+6 -1E+7

0 -1E+6

At C

At B

6E+6

-1.1E+7 0

2

4

6

8

10

12

14

16

-1.2E+7

Model 3

0

2

4

6

8

10

t/(C l.W)

t/(C l.W)

(a)

(b)

12

14

16

-4E+5

0

2

4

6

8

10

12

14

16

t/(C l.W)

(c)

Fig. 8. Ultrasonic x-direction of compared stresses resulted from impact force of 5 m/s subjected for models I for a duration of 300 ms, and model III for a duration of 100 ms on homogenous cracked structure measured at points (a) A before crack tip, (b) B after crack tip, and (c) C at the far field locations.

Please cite this article in press as: M.A. El-Hadek, Dynamic equivalence of ultrasonic stress wave propagation in solids, Ultrasonics (2017), http://dx.doi. org/10.1016/j.ultras.2017.06.007

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M.A. El-Hadek / Ultrasonics xxx (2017) xxx–xxx

At A

At B

4E+6

At C

0

Model 3

4E+5

-1E+6 3E+6

-2E+6 Model 3

Syy

Syy 1E+6

Syy

-3E+6

2E+6

Model 3

2E+5

-4E+6

0

-5E+6 Model 11

Model 11 Model 11

-6E+6

0

-2E+5

-7E+6 -1E+6

0

2

4

6

8

10

12

14

16

-8E+6

0

2

4

6

8

10

t/(C l.W)

t/(C l.W)

(a)

(b)

12

14

16

-4E+5

0

2

4

6

8

10

12

14

16

t/(C l.W)

(c)

Fig. 9. Ultrasonic y-direction of compared stresses resulted from impact force of 5 m/s subjected for models I for a duration of 300 ms, and model III for a duration of 100 ms on homogenous cracked structure measured at points (a) A before crack tip, (b) B after crack tip, and (c) C at the far field locations.

and C at the far field locations. Therefore, the comparison of the two dynamically FEA was possible due to the unification of the applied impact velocity, element modeling, and subjected time. It should be noticed the stresses in x, y-directions for both models I, and III at points A (prior to crack tip), B (above crack tip), and C (at far field) are the same, for the normalized time duration of the models. However, the stresses in x, y-directions for model I have less time duration 5 compared to model III, which continues to normalized time duration 13. These results show that the dynamically equivalent constant of E⁄q is a potential candidate to be used for comparing stress propagation in different solid materials. 3.3.3. Comparing model I for time duration of 300 ms and model III for time duration of 100 ms as E⁄q = 1732.051 (constant) The ultrasonic stresses in the x, y-directions propagating at locations points A before crack tip, B after crack tip, and C at the far field locations were measured for models I for time duration of 300 ms, and model III for time duration of 100 ms. Figs. 8 and 9 (a–c) present the ultrasonic x, and y-direction of compared stresses resulted for models I and II of homogenous cracked structure measured at points A before crack tip, B after crack tip, and C at the far field locations. Therefore, the comparison of the two dynamically FEA was possible due to the unification of the applied impact velocity, element modeling, and subjected time. It should be noticed the stresses in x, y-directions for both models I for a duration of 300 ms, and model III for a duration of 100 ms at points A (prior to crack tip), B (above crack tip), and C (at far field) are the same for the normalized time duration of the models. The stresses in x, y-directions for model I compared to model III, with normalized time duration, are the same. These results shows that the dynamically equivalent constant of E⁄q is suitable for comparing stress propagation in deferent solid materials. 4. Summary and conclusion Ultrasonic stress waves generated during the dynamic impact on structures were studied. A benchmark FEA was introduced to define the optimum geometrical factors including, mesh distribution, analysis time, ultrasonic wave properties, element type, and shape; to capture the dynamic phenomena compared with the theoretical exact solution. The benchmark FEA was based on theoret-

ical dynamic stress asymptotic analysis. The benchmark FEA was prepared to define the smallest element size to ensure the capture of the dynamic phenomena, which was found to be at 2 mm. Comparison of different three dimensional (3D) finite element models has been performed depending on the applied impact forces, the element modeling, and the structural geometry. The ultrasonic stresses in the x, y-directions; propagating at different locations measured for models I, and II, mismatched the existed results. On the other hand, applying the same ultrasonic stresses for models I, and III matched he existed results. A dynamic equivalence for these three different variables was established and found to be of a direct multiplication relation of the solid’s density with the Young’s modulus. The results demonstrated that the stresses in x, and y-directions were matched with predicted FEA for different materials under normalized time.

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