An Asymptotic Approach for the Elastodynamic Problem of a Plate under Impact Loading
An approach is presented for analyzing the transient elastodynamic problem of a plate under an impact loading. The plate is considered to be in the form of a long strip under plane strain conditions. The loading is taken as a concentrated line force applied normal to the plate surface. It is assumed that this line force is suddenly applied and maintained thereafter (i.e., it is a Heaviside step function of time). Inertia effects are taken into consideration and the problem is treated exactly within the framework of elastodynamic theory. The approach is based on multiple Laplace transforms and on certain asymptotic arguments. In particular, the one-sided Laplace transform is applied to suppress time dependence and the two-sided Laplace transform to suppress the dependence upon a spatial variable (along the extent of the infinite strip). Exact inversions are then followed by invoking the asymptotic Tauber theorem and the Cagniard-deHoop technique. Various extensions of this basic analysis are also discussed.
This work introduces an asymptotic approach based on one- and two-sided Laplace transforms to deal with the transient problem of a plate under impact. The plate is taken in the form of a long strip under plane strain conditions. A Green’s function type of loading is taken here, that is, a concentrated line force (applied normal to the plate surface) with a Heaviside step function time-dependence. In this way, the present solution can be used as a fundamental solution for more general loadings. Inertia effects are taken into consideration and the problem is treated exactly within the framework of elastodynamic theory. Our main concern is on the stress and displacement field beneath the point of application of the impact load.
The present work deals with transient wave propagation in plate and strips. Related problems were examined previously by Mencher , Davids , Miklowitz [3, 4], Rosenfeld and Miklowitz , Lloyd and Miklowitz , Shmuely , and Dai and Wong , among others. Contrary to the involved analyses existing in the literature, we aim here at a simple asymptotic approach that can readily be applied to specific problems. This approach was effected by applying the Tauber theorem [9, 10] for asymptotic Laplace transform inversions and the Cagniard-deHoop technique [4, 11]. We should note that for static problems inversions using the Tauber theorem were provided by Georgiadis and Papadopoulos [12, 13] in dealing with problems in elastic strips. The present approach may have an advantage over more involved analyses in dealing with a nonstandard material response, such as anisotropic, viscoelastic, or thermoelastic.
Analyses of problems dealing with transient wave propagation in long strips usually aim at furnishing basic theoretical information for determining scabbing effects in plates. Since this type of damage is caused by the mode conversion of the initial compressional wave into a tensile one at the free boundary of the plate, it is necessary for the analysis to study, up to a certain time, the superposition effect of multiple wave reflections at the free boundaries of the strip. In addition, another application of the transient wave propagation problem in a strip is the so-called dynamic tear test of fracture specimens [14, 15]. This configuration involves a cracked strip, with the crack in the form of an initial notch normal to the lower side of the strip and with the impact load applied on the upper side. Experimental evidence shows that crack initiation occurs after several wave reflections (on the two sides of the strip) reach the tip of the notch . Of course, as a first step of an analysis it would be helpful for one to detect the stress and displacement field in the vertical cross-section of an uncracked plate. In fact, it is this field that would be removed by the initial notch.
2. Governing Equations and Problem Statement
Consider a 3D elastic body in the form of a plate or strip (see Figure 1). A Cartesian coordinate system is used with its origin taken beneath the position of the applied loading. The body has large dimensions along the and axes as compared to the strip height . Plane-strain conditions are assumed to prevail. The line load acts in the -plane on the upper side of the plate (), with an intensity (this intensity is expressed in dimensions of [force][length]-1) and the Heaviside step-function type dependence upon the time . This means that the loading is concentrated and is applied suddenly. Since we are concerned with a solution at relatively small times, the supports of the plate do not play a role. This is a standard assumption in this type of problems [2, 3, 5, 7, 8]. Indeed, experimental evidence with this configuration  shows that the strip or plate does not bend until the elastic waves in the body reach the supports.
The transient elastodynamic response of this body is governed by the following equations (Miklowitz ):
where all transformed quantities are functions of , and
The behavior of the functions (with ) defined above, in the cut -plane, is shown in Figure 4.
where , , , and are arbitrary functions, which should be determined in each specific problem of elastic wave propagation. Of course, instead of the hyperbolic functions, one could use exponential functions, but the former functions suit the present analysis better as it will become clear soon (when we discuss symmetric and antisymmetric modes). The general solution along with (6) and (7) will be utilized in subsequent analysis.
4. Solution Procedure
The solution procedure is based on superposition and asymptotic considerations.
4.1. Superposition Scheme
The solution of the problem described by (5)–(10) is greatly facilitated by decomposing the original problem into two auxiliary problems. The first (symmetric) problem and the second (antisymmetric) problem are shown in Figures 5 and 6, respectively. Clearly, the solution to the original problem results by superposing the two solutions of the auxiliary problems.
The two auxiliary problems will be analyzed separately. Both problems have the same initial conditions as the original one (cf. (10)).
The first (symmetric) problem has the following boundary conditions:
which in the multiple Laplace-transform domain become (cf. Table of transforms in )
In addition, due to the symmetry with respect to the line , the following properties prevail: and . Therefore, in view of the definition of the displacement potentials (cf. (5)), the transformed general solutions should have the following reduced forms:
which when solved provides the following solution for the transformed displacement potentials:
The second (antisymmetric) problem has the following boundary conditions:
In this case, anti-symmetry with respect to the line suggests the following reduced forms for the transformed general solutions:
By following the same procedure as before, we get
Then, the multiple transformed solution of the original problem follows by adding the solutions of the two auxiliary problems. Given the complicated forms of the above transformed expressions, it is seen that inverting the transforms is indeed a formidable task. In what follows, we concentrate on determining the field in the vicinity of the cross-section beneath the load. This is an area of intense deformation and potential fracture development.
4.2. Asymptotics and Preliminary Results
Here, we will provide an asymptotic analysis based on the Tauber theorem [9, 10] and present some results to check the validity of the procedure. According to the Tauber theorem, the limit of a transformed function for is directly related with the limit of its original function for . To perform such a correspondence, we will use the following asymptotic results for certain expressions entering the transformed solutions. Below, we provide details only for three typical cases and just present the remaining expressions. When , one obtains
where it is noticed that writing is consistent with the specific choice of branch cuts made in Figure 4.
Then, the stresses and at, respectively, the upper and lower side of the plate will be obtained. On the one hand, the former stress should tend to the expression given by the classical Flamant-Boussinesq problem (the static problem of an elastic half-plane under a concentrated force at the surface; [21, 22]). On the other hand, the latter stress should be zero because no bending of the plate exists in the infinite-strip configuration (this point was already explained in Section 2). Thus, these cases provide benchmarks for the present solution.
For the case , the transformed solution of the original problem provides
with the inversion (Churchill )
From the solution now of the Flamant-Boussinesq problem (see Figure 7) in a half-plane under a concentrated load, with an intensity , acting at the point of the surface (the load is directed downwards), the tangential stress (the superscript “FL” denotes that this stress follows from the Flamant-Boussinesq solution) is written as
Further, in the limit as (which corresponds to the case in the strip problem) and with the aid of the following representation of the derivative of the Dirac delta distribution (van der Pol and Bremmer )
(with being a positive real number), we write
Finally, by noting that (a result from the theory of distributions—van der Pol and Bremmer , Zayed ), we conclude that the static Flamant-Boussinesq stress expression at the surface is indeed recovered by the present solution.
For the case now, the transformed solution of the problem provides . Therefore, we get , as expected.
5. Exact Inversion: Results
In this section, we obtain asymptotic results by performing an exact inversion of the Cagniard-deHoop type [4, 11]. The procedure will be presented with some details for the stress . Then, we also obtain the displacement .
By the use of (26) and (33), we first obtain the asymptotic transformed expression for the tangential stress in the original problem (note that the antisymmetric mode gives no contribution, in this case, as expected on physical grounds):
Then, we consider the following series expansion, which is both absolutely and uniformly convergent:
where (with ), so that .
Using (43) allows us to perform the inversion term by term. Still, the inversion cannot be done readily since the transforms are complicated and not available in a Table. However, the Cagniard-deHoop technique can now be applied. In short, with this technique one performs an algebraic transformation of the path in the inversion integral of the two-sided Laplace transform so that this inversion integral takes eventually the form of a recognizable one-sided Laplace transform. This immediately leads to inversion by simple inspection. This technique has proven valuable in many problems of wave propagation [4, 11].
Indeed, in the other cases corresponding to the remaining terms in (43), will be simply replaced by , , , and so forth. To evaluate the above integral, we first perform the transformation
where , with . For , we get .
As Figure 8 shows, (45b) expresses a hyperbola in the -plane. The vertex of the hyperbola is determined to be by setting the imaginary part of the RHS of (45b) equal to zero. The broken lines in Figure 8 are the asymptotes of the hyperbola. These lines are defined by
Notice that in this case, the transition from the original Bromwich path to the Cagniard-deHoop path (given by (45b)) is simple because no poles and branch points are located inside the domain defined by these two paths. Further, if the variable is introduced along the Cagniard-deHoop path, the integral in (44) will take the form
Then, in the limit as , (47) provides
which is now in the form of a recognizable one-sided Laplace transform; that is, it is written as . The latter form immediately leads to the inversion  , where is a constant and is the Heaviside step function. Therefore, (49) provides the inversion
The above technique applies also to the remaining terms in (43). Thus, results can be obtained to any desirable accuracy.
In view of the above analysis, we obtain for the stress
From the above formulas, the results shown in Figure 9 are obtained by considering eight terms for in (52) (these terms correspond to eight arrivals of longitudinal waves in the location where the stress is calculated) and four terms for in (52) (these terms correspond to four arrivals of transverse waves). This choice of number of terms was made in view of the fact that the value of the velocity of longitudinal waves is roughly twice as much the value of the velocity of transverse waves, in most materials. The graph of Figure 9 depicts the variation of the normalized stress with the normalized time at the location . The material constants of Titanium were used here to obtain the numerical results: GPa (shear modulus), (Poisson’s ratio), and (mass density). The longitudinal wave speed is m/sec and the shear wave speed is m/sec. Finally, the strip height (plate thickness) is given a value of m.
One may observe in Figure 9 that the singularities correspond to the arrival of the initial longitudinal and transverse waves and their subsequent reflections at the strip boundaries. Each wave arrival at the location corresponds to the respective term in the series expansion in (52). Also, the sharp rises in Figure 9 correspond to the inverse square-root singularities in (52).
It is seen from (28) that the symmetric mode gives no contribution in this case. This is to be expected on physical grounds since the displacement at the strip mid-plane should be zero in the symmetric mode.
Again, the Cagniard-deHoop technique is applied. The term before the complex integral in (54) furnishes a little complication only. Indeed, one may handle this situation by the convolution theorem for Laplace transform (see, e.g., [10, 18]). Now, we end up with the form
which when inverted (by invoking the convolution theorem) provides
In the present case, we have
One can similarly obtain the integrals resulting from the remaining terms in (43). The final expression for the vertical displacement is given by
As before, representative numerical results are given for the material constants of Titanium and for a strip height . The graph of Figure 10 shows the variation of the normalized vertical displacement with the normalized time at the location . It is seen that since the displacement grows indefinitely with time, the vicinity of the cross-section is a site of potential fractures due to the accumulating effect of reflected waves.
6. Concluding Remarks
In this paper, an approach was presented for analyzing the transient elastodynamic problem of a plate in the form of a long strip under impact. Plane strain conditions were assumed and the loading was taken as a concentrated line force applied normal to the plate surface. This force was suddenly applied and maintained thereafter. Exact elastodynamic theory was utilized in an analysis accounting for inertia effects and an explicit time dependence. Our approach was based on multiple Laplace transforms and on certain asymptotic arguments. Exact inversions were performed by invoking the asymptotic Tauber theorem and the Cagniard-deHoop technique.
Since a Green’s function type of loading was taken here, the present solution can be used as a fundamental solution for more general loadings. Our analysis furnishes basic theoretical information for both determining scabbing effects in plates and studying the dynamic tear test of fracture specimens. In these applications, it is necessary for the analysis to study the superposition effect of multiple wave reflections at the free boundaries of the strip.
The asymptotic approach followed here may have an advantage over more involved analyses in dealing with a nonstandard material response such as anisotropic, viscoelastic, or thermoelastic response. Extensions of this kind are pursued by the present authors.
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