Suggestion of a framework of similarity laws for geometric distorted structures subjected to impact loading

A framework of similarity laws, termed oriented-density-length-velocity (ODLV) framework, is suggested for the geometric distorted structures subjected to impact loading. The distinct feature of this framework is that the newly proposed oriented dimensions, dimensionless numbers and scaling factors for physical quantity are explicitly expressed by the characteristic lengths of three spatial directions, which overcome the inherent defects that traditional scalar dimensional analysis could not express the effects of structural geometric characteristics and spatial directions for similarity. The non-scalabilities of geometrical distortion as well as other distortions such as different materials and gravity could be compensated by the reasonable correction for the impact velocity, the geometrical thickness and the density, when the proposed dimensionless number of equivalent stress is used between scaled model and prototype. Three analytical models of beam, plate and shell subjected to impact mass or impulsive velocity are verified by equation analysis. And a numerical model of circular plate subjected to dynamic pressure pulse is verified in more detail, form the Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 26 February 2020 doi:10.20944/preprints202002.0394.v1 © 2020 by the author(s). Distributed under a Creative Commons CC BY license. 2 view of point of space deformation, deformation history and the components of displacement, strain and stress. The results show that the proposed dimensionless numbers have attractively perfect ability to express the dimensionless response equations of displacement, angle, time, strain and strain rate. When the proposed dimensionless numbers are used to regularize impact models, the structural responses of the geometrically distorted scaled models can behave the completely identical behaviors with those of the prototype on space and time —not only for the directionindependent equivalent stress, strain and strain rate but also for the direction-dependent displacement, stress and strain components.


Introduction
It is well known that the similarity laws for structures subjected to impact loading were systematically summarized by Jones (1989). In this work, twenty-two dimensionless numbers, based on the classical mass-length-time (MLT) dimensional analysis, were proposed, and the basic geometric scaling factor ̅ was used to relate the physical quantities from scaled model to full-size prototype. The factor ̅ was defined as ̅ = ̅ / ̅ , where was scaling factor; ̅ was the characteristic lengths of structure; the subscripts and represented the scaled model and the prototype, respectively. However, it must be used with great care because the scaling factors of MLT would become invalid when the material strain-rate-sensitivity, the gravity and , , cylindrical coordinate system ( , , ) distance to the neutral plane of plates , , spherical coordinates system ( , , ) dimension of angle () scaled model dimension of length () prototype dimension of mass () modification for physical quantity ℝ dimension of radius () correction for physical quantity dimension of time dim ( ) dimension of physical quantity the fracture were taken into account (Jones, 1989;Lu and Yu, 2003). Many works had demonstrated the potential influence of these non-scalability for structural impact (Booth et al., 1983;Jones, 1989;Me-Bar, 1997;Jiang, et al., 2006a;Jiang, et al., 2006b;Li, et al., 2008;Fu, et al., 2018). A classic example was the work of Booth et al., which reported 13 drop tests on geometrically similar one-quarter-scale to full-scale structures for strain-rate-sensitive materials (Booth et al., 1983). The significant departures from similitude for the testing results revealed the nonnegligible influence of material strainrate-sensitivity for similarity. In addition, since the geometric scaling factor ̅ was used to equally scale the characteristic lengths of prototype configuration in different direction, the MLT system did not allow for the use of the scaled model with distorted geometric configuration.
It is vital that the non-scaling cases were more widespread than the perfect scaling ones due to the experimental difficulties in manufacturing scaled models where materials, geometry size and so on were often limited. Drazetic, et al. (1994) presented a non-direct scaling technique, through reasonably correcting the impact velocity of scaled model, to address the distortion case that the mass, impact velocity, geometrical thickness and materials of scaled model did not satisfy the scaling of MLT. By improving this technique, Oshiro and Alves (2004) proposed the initial impact velocity 0 -dynamic flow stress -impact mass (VSG) dimensional analysis to address the material distortion of strain-rate-sensitivity. In the VSG system, the dimensionless numbers were expressed as follows: where , , , ̇ and were acceleration, time, displacement, strain rate (for strain ) and stress, respectively. Based on this system, one more basic scaling factor ( was velocity) was used to scale the behaviors of prototype by reasonably correcting the impact velocity and impact mass of scaled model (Oshiro and Alves, 2004;2007;2009; Alves and Oshiro, 2006a;. The material distortion for the difference of density, yield stress and strain-rate-sensitivity between scaled model and prototype can also be compensated by increasing one more number and one more basic factor into the VSG system, where and were structure mass and density, respectively (Mazzariol, et al., 2016). The solutions of material distortion, further including material strain hardening effects, were verified by a group of transport equations of the continuum mechanics in the work of Sadeghi, et al. (2019a;. In addition, a new technique -adding extra mass to correct the density of structure, was proposed to compensate the non-scalability of gravity and material strain-rate-sensitivity (Jiang et al., 2016;Wei and Hu, 2019). The above studies formed a new similarity laws of 6 structural impact which using three basic scaling factors ̅ , and to relate physical quantities from scaled model to prototype. The main scaling factors of VSG system were listed in Table 1. Although the addressing ability for distortion problems was effectively extended, for the scaled model with geometric distorted configuration, the VSG system was still invalid. Based on the VSG scaling factors, Oshiro and Alves (2012) presented an indirect method for the geometric distortion structures, where the iterative scaling testing were used to determine the unknown parameter of distorted scaling factors.
However, because of its obvious defects that based on the experience of researchers and the numerous tests, the indirect method cannot be practical for a wide range of applications. In addition, since different unknown parameters need to be tested for different physical quantities, the indirect method was difficult to use for the comprehensive relations from scaled model to prototype.
In order to extend the application of dimensionless numbers for explanation of physical meaning, expression of response equations and scaling analysis, Wang,Xu and 7 Dai (2019) proposed the density-length-velocity (DLV) dimensional analysis, instead of the VSG system, to express the similarity laws of structural impact. In the DLV system, fifteen dimensionless numbers were expressed as follows: 2  2 2  3 2  2   22   3 2  3  3   ,  , , where , , , ̇ , ̈ , , and ̅ ′ were force, bending moment, angle, angular velocity, angular acceleration, energy, impulse and characteristic length of otherdirection (e.g., thickness and width ), respectively. Meanwhile the same scaling factors as VSG are used to address the distortion of different materials by the correction for the impact velocity and the density (or structure and impact mass) of scaled model.
In addition, some important improvements were that the Johnson's damage number = 0 2 0 (Johnson, 1972;Zhao, 1998a) (Zhao, 1998b;1999;Li and Jones, 2000;Shi and Gao, 2001;Hu, 2000;2009), are defined as where ′ and are elastic modulus and positive real number, respectively; ̅ and 8 ̅ are two characteristic lengths of different directions. It is evident from the DLV numbers that characteristic lengths of different directions were clearly expressed in some numbers, which reflected the influence of different geometric characteristics for structural scaled behaviors. Nevertheless, the DLV system also had no ability to directly express the geometric distorted structures due to the use of single characteristic length ̅ in most dimensionless numbers.
Recently, Mazzariol and Alves (2019a; presented a direct method for the geometric distorted structures. In the new method, the velocity-dynamic flow stressstructure mass (VSM) dimensional analysis, instead of the VSG system, was used to express dimensionless numbers as follows: 33  3  3  2  0  4  2  4   3   2   ,  ,  , where ′ 0 was unit width plastic bending moment. In the meantime, six basic scaling factors , , , , and formed the scaling factors of VSM, listed in Table 2, where was a general direction; = for cartesian coordinates ( , , ) and = 2 for cylindrical coordinates ( , ); was radius.

Assumptions
In order to re-derive the similarity laws of impact problems, we assume that the mechanical mechanism for geometric distorted impacted structures is the same as that of an initial flat thin-plate subjected to transverse impact loading. The thin-plate has the arbitrary shape and boundary conditions and is subjected to transverse impact loading, as shown in Fig. 1. The impact model of thin-plate is built in the cartesian coordinate system ( , , ). The subscripts , and represent the x-axis direction, the y-axis direction and the z-axis direction, respectively; the subscripts , and represent the x-y plane direction, the x-z plane direction and the y-z plane direction, respectively.  Reddy., 2007), where , and are the displacement of neutral plane elements on the -axis, the -axis and the -axis, respectively; is the distance to the neutral plane. The first two terms for , and represent the strain from neutral plane and the last term represents the bending strain from deflection. (4) The displacement components of thin-plate problem are expressed as = − ( ) , = − ( ) and = where the terms ( ) and ( ) represent two in-plane displacements from deflection (Reddy., 2007).
Although the above assumptions come from the thin-plate problem, the following derivation and analyses could be applied to more complex structures, such as the stiffened plates.

Dimensionless numbers
Different from previous authors, three notations , and are used to present three oriented dimensions of length in the spatial directions , and , respectively (Huntley., 1952;Chien., 1993 Table 3. The way to obtain these oriented dimensions is to reasonably analyze the definition of physical quantities. For example, the dimension  Based on the previous DLV system, the density , the characteristic lengths ̅ , ̅ , ̅ and the velocity are chosen as base to derive dimensionless numbers. When using the Buckingham Π theorem (Buckingham, 1914;1915), thirty basic physical quantities of Table 3 can be reduced to twenty-five dimensionless numbers, as follows: • six stress components Secondly, the numbers − (Eq.(5a-f)), the numbers − (Eq.(6a-f)), the numbers ̇−̇ (Eq.(7a-f)) and the numbers − (Eq.(8ac)) can be express as four tensor forms 2 ,, , , ̇ and represent stress tensor, strain tensor, strain-rate tensor and displacement vector, respectively; ̅ , ̅ and ̅ are three orthogonal characteristic lengths of structure in cartesian coordinate system; , = 1,2,3 ( , , ) , =

( ).
Thirdly, three spatial directions and structural geometric characteristics are explicitly expressed in these dimensionless numbers by the use of oriented characteristic lengths ̅ , ̅ and ̅ , which remedy the defects of previous scalar dimensionless system (i.e., MLT, VSG, DLV, VSM, etc.).
Finally, these numbers (Eqs.(5)- (11) impact mass is extended to the number = ̅ ̅ ̅ . Since the above analysis based on oriented-density-length-velocity dimensional analysis, these new proposed dimensionless numbers can be termed as the ODLV system (or the oriented-DLV system) in the following sections.

Dimensionless expression for equivalent stress, equivalent strain and equivalent strain rate
For structure subjected to impact loading, the similarity law of the equivalent stress , the equivalent strain and the equivalent strain rate ̇ needs to be further considered.
The establishment condition of Eqs. (19)-(21) is ≗ , which means the dimension and the dimension are not independent. In this equality, the operator '≗' is a defined sign of equality in which the directions of characteristic lengths in the x-y plane on both sides of the equation are not distinguished. Therefore, for the equivalent stress, strain and strain rate, the directions of the x-y plane are isotropic. In order to better describe this phenomenon, the notation = √ can be used instead of the length dimension of the x-y plane, where represents the dimension of area in the x-y plane; and the constraint relation ≗ can be expressed as the number − = ̅ ̅ , which means that the different directions of the x-y plane follow the same law of similarity.
When using the dimensional analysis of ODLV to express physical quantities , and ̇, three dimensionless numbers can be obtained as follows: , , eq eq eq xy xy eq z xy z eq eq z z z z where ̅ represents the characteristic length of the x-y plane. It can be learn from the above analysis that the number , the number and the number ̇ can be regard as the derivation forms for the isotropic x-y plane from the numbers , , , the numbers , , and the numbers respectively.
In what follows, the oriented dimension analysis for the relationships between stress components and strain components will be derived by the constitutive relation of the plastic increment theory.
In order to simplify the study, we consider only the plastic stress and strain (i.e., ignoring the elastic part) and the rigid-perfectly plastic materials. Then the ́− theory (Yu and Xue, 2010) is adopted as ( 2 + 2 + 2 )] 1/2 is the equivalent plastic strain increment.
When the stress component is ignored, the oriented dimensional analysis for Eq. (23a-f) will leads to the following relations eq xx eq eq eq eq y eq eq eq eq zx eq eq eq xy xy eq y y eq yz y x y z eq eq xz xz eq It is obvious that when the dimensions of , , , ; Eqs. (19) and (20)) are substituted, for the isotropic x-y plane, Eq. (24a,b,d) is automatically satisfied. While, when the dimensions of , , , which is different form the oriented dimension in Table 3 Based on the three modified dimensions, the numbers , , , ̇ , ̇ and ̇ are modified accordingly as where the superscript 'mod' represents modification for physical quantity. The difference between the numbers - Therefore, the number reflects the most essential dynamic similarity for the isotropic x-y plane by the ratio of the inertia force (or bending moment; or torque) is the dimensionless expression for equivalent stress following the same similarity laws in three geometric directions, while the response number is the dimensionless expression for equivalent stress following the same similarity laws only in the x-y plane.
(4) The effect of structure geometry exists not only in the numbers of stress, strain and strain rate components (Eqs. (5-7) and (26)) but also in the numbers of equivalent stress, strain and strain rete (Eq. (22)), which usually expressed by the ratio of characteristic lengths in two different directions. And the different power exponents of ratios indicate these dimensionless numbers follow different geometric similarity laws. (5) The characteristic length ̅ and the characteristic length ̅ are independent of each other in the thin-plate problem, which will be the foundation for the geometric distortion of thickness in the following content.

Scaling factors
The structural similarity means these dimensionless numbers are complete equality between scaled model and prototype, which can be used to derive scaling factors of physical quantity. For example, the equation Similarly, other dimensionless numbers (Eqs. (5)- (11), (22), (26)) can obtain their corresponding scaling factors. These scaling factors are listed in Table 4.  considering three spatial directions. Therefore, the ODLV system includes the same addressing ability for the material distortion with the VSG system. Compared with the VSM factors (Table 2), the factors of stress, strain, strain rate, displacement, and so on are reasonably modified.

Application of scaling factors on the constitutive equation
Based on the factor , and ̇ of Table 4, the non-scaling problems of structural impact can be addressed by reasonably correcting the input parameters of scaled model.
which gives a functional relation among four factors , ̅ , ̅ and .
For the scaling tests, the input parameters consist of three aspects: (1) where the factors ̅ , ̅ and are given according to the initial state of input parameters between the scaled model and the prototype. The superscript 'cor' represents the correction for physical quantity.
(2) The thickness of scaled model can be corrected to ( where the factors ̅ , and are given according to the initial state of input parameters between the geometrically similar scaled model and the prototype.
where the factors ̅ , ̅ and are given according to the initial state of input parameters between the scaled model and the prototype. Therefore, the structural mass It can be seen that the correction for the impact velocity, the geometrical thickness and the density is equivalent to the correction for the dimensionless number of equivalent stress. The above analysis indicated that the number is the dominant similarity condition that input parameters must be satisfied for structures subjected to impact loads.
Finally, from the above correction methods, the addressing ability for the nonscaling problems can be summarized as follows: (1) The material distortion has been included in the above correction methods since using the different density and constitutive equation between scaled model and prototype. Although the previous works (such as VSG and DLV) has been studied in detail through correcting the impact velocity and the density (or structural mass and impact mass), Eq. (31) provides a new correction method -correcting the geometric thickness of scaled model.
(3) The distortion of gravity can be compensated by the correction for geometrical thickness or density of scaled model (Eq. (31) or (32) The above analysis lays the foundation for the addressing ability of the non-scaling problems. These correction methods can be used in combination to compensate different distortion problems such as geometry, material and gravity distortion.

Transformation into cylindrical coordinate system
For the thin-plate problem, the derivations for similarity laws in above sections base on the cartesian coordinate system ( , , ) . When using the cylindrical coordinate system ( , , ), it is also very convenient.
In the cylindrical coordinate system ( , , ), the subscripts , , for physical quantity represent the -axis direction, the -axis direction and the z-axis direction, respectively; the notations ℝ and represent the dimension of radius and angle, respectively; and ̅ and ̅ represent characteristic radius and characteristic angle, respectively. Then, three oriented dimensions of length in the cylindrical coordinate system are = ℝ , = ℝ and , which similar to , and ; three oriented characteristic lengths in the cylindrical coordinate system are ̅ = ̅ , ̅ = ̅ ̅ and ̅ , which similar to ̅ , ̅ and ̅ ; and three subscripts in the cylindrical coordinate system are , and , which similar to the subscripts x , y and z of cartesian coordinate system. The transformation for dimensions, characteristic lengths and subscripts from the cartesian coordinate system to the cylindrical coordinate system are summarized as follows: ,, : : The dimensionless number of physical quantities in cylindrical coordinate system can be directly obtained. For example, the number

Analytical verification
In this section, three analytical models for beam, plate and shell are used to verify the effectiveness of the ODLV numbers for the geometric and material distortion.

The beams subjected to impact mass and impulsive velocity
The impact models of beams subjected to impact mass or impulsive velocity are studied as shown in Fig. 2. The beams are made of the rigid-plastic materials. For the clamped beams ( Fig. 2b and c), the effects of finite displacements are taken into account. a clamped beam subjected to impact mass at mid span.

Response equations
For a cantilever subjected to mass impact (Parkes, 1955;Yu and Qiu, 2018), as shown in Fig. 2a where is the mass per unit length; and = / is the mass ratio of impact mass to structure mass.
The final rotation angle ( ) at the root is given as ( ) For clamped beam subjected to impact mass (Fig. 2b), the maximum permanent transverse displacement ( ) at the mid-span is given as (Liu and Jones, 1988) where c is dimensional constant. The first term in radical sign of the right-hand side introduces the membrane strain and the bending strain. The second term introduces the transverse shear strain = 4 / .
For clamped beam subjected to impulsive velocity (Fig. 2c), the maximum permanent transverse displacement ( ) at the mid-span is given as (Jones, 1989) And the average strain rate (̇) is considered as (Jones, 1989)
(50) are the same, which leads to three scaling relations as (51a-c) can be rewritten as three scaling relations = , = The three basic scaling procedures actually include the non-scaling problems of the geometric distortion (i.e., at least two of three factors , and are not equal) and the distortion of different materials (i.e., ≠ 1 or ≠ 1).
It can be learned from the analysis of beams that the scaling relations of structural impact can be obtained directly by the use of the ODLV numbers. When we conduct a scaling test, the dimensionless numbers of input parameters determine the sufficient conditions of similarity, while the dimensionless numbers of output parameters determine the mapping relationship of structural response between scaled model and prototype. In this procedure, the ODLV bases provide five essential scaling factors, i.e. the density factor , three geometric factors ̅ , ̅ and ̅ and the velocity factor . When the geometric configuration in the x and y directions is scaled by the factors ̅ and ̅ , the velocity factor , the geometric thickness factor ̅ and the density factor can be corrected to address the non-scaling problems.

The rectangular plates subjected to impact mass and impulsive velocity
The impact models of a simple or clamp supported rectangular plates subjected to impact mass or impulsive velocity are studied as shown in Fig. 3. The plates are made of the rigid-plastic materials and already take into account the effects of the finite displacements. Fig. 3 A simple or clamp supported rectangular plate subjected to (a) impulsive velocity or (b) impact mass.

Response equations
For impulsive velocity loading, the maximum permanent transverse displacement ( ) for simple supports and clamped supports are given as (Jones, 1989) (2 )(2 )] is mass ratios; and is dimensionless moment resistance at supports ( ′ 00 and 1 for simply and fully clamped supports, respectively).

Dimensionless expression and scaling analysis
It is obvious that Eq. (52) (or Eq. (53)) can be changed into the following form where ′ is function relation about Eq. (52) (or Eq. (53)).
Similarly, Eq. (54) can be changed into the following form ( ) Equations (55) ). It is also significant to find that the dimensionless response equation of final deflection has been verified by more impact models of beams, square and rectangular plates, circular plates and circular membranes (Zhao, 1998b;Hu, 2009).

A spherical shell subjected to impulsive velocity
The impact model of a spherical shell subjected to a spherically symmetric radially outwards impulsive velocity is studied as shown in Fig. 4. The shell is made of the rigid-plastic materials. The impact model is built in spherical coordinate system ( , . Fig. 4 Segment of spherical shell subjected to impulsive velocity.

Response equations
The maximum permanent radial displacement ( ) is given as (Jones, 1989) where is radius of the sphere; ′ = is mass per unit surface area of shell; and 0 ′ = is the fully plastic membrane force for the shell cross-section.
The final response time is given as The biaxial membrane strains in the spherical shell are = = − / .
When substituting Eq. (57), the permanent membrane strain ( ) is derived as

Dimensionless expression and scaling analysis
respectively.
Equations (60) In a more general case, the numbers , and ̇ can be extended to the forms ( ) ( ) 2 , eq eq nn xy xy z eq eq z z  (Zhao, 1999) for more impact problems. It can 45 be learned from the above analysis that the case of = 2 is suitable for the beam and plate subjected to transverse impact; and the case of = 1 is suitable for the spherical shell and cylindrical shells subjected to radial impact. For more power exponent, further verification is needed in future work.

Numerical analysis
In this section, a numerical model of circular plate is used for more detailed validation of the ODLV system from the view of point of space deformation and deformation history.

A clamped circular plate subjected to dynamic pressure pulse
The impact model of a clamped circular plate subjected to two loading cases, as shown in Fig. 5, is now numerically analyzed.

Numerical modeling
The finite element model of circular plate is modelled in ABAQUS. The CAX4R axisymmetric elements are adopted to discretize structure with 450 elements in radius and 8 elements in thickness. The clamped external boundary is restrained in cylindrical coordinate system ( , , ). The rigid-perfectly plastic material (refering to 1006 Steel (Johnson and Cook, 1983) with = 350 is adopted to simplify the constitutive equation. The elastic modulus is set 1000 times larger than actual value ′ = 200 to eliminate elastic influence as much as possible. The density of the materials is 7.89 × 10 3 / 3 and Poisson ratio is 0.3. Two loading case, as shown in Fig. 5b, are adopted to study different deformation degrees. According to the impulse theorem  Table 5. The similarity of displacement fields is evaluated by the displacement components and of final time on neutral plane profile, with the results plotted in Fig. 6.
Obviously, the displacement components of the prototype and the scaled models are significantly different in the corresponding dimensionless spatial positions ( = ̅ ), as shown in Fig. 6a and c. While, it can be learned from Table 4   (

2) Stress fields
The similarity of stress fields is evaluated by the stress components , , and on neutral plane profile in time = (0.3/ ) (or 0 / = 1.2), with the results plotted in Fig. 7. Obviously, the in-plane stress components and of the scaled models are basically in accordance with these of prototype in the corresponding dimensionless spatial positions, as shown in Fig. 7a and c. While, it can be learned from Table 4  (

3) Strain fields
The similarity of in-plane strain fields is evaluated by the strain components and of final time on neutral plane profile, with the results plotted in Fig. 8.
Obviously, the in-plane strain components of the prototype and the scaled models are significantly different in the corresponding dimensionless spatial positions, as shown in Fig. 8a and c. While, it can be learned from Table 4   The similarity of transverse strain fields is evaluated by the strain components and in final time on neutral plane profile, with the results plotted in Fig. 10.
Obviously, the transverse strain components of the prototype and the scaled models are significantly different in the corresponding dimensionless spatial positions, as shown in Fig. 10a and c. While, it can be learned from are used to regularize structural response, the transverse dimensionless strain fields of scaled models behave good consistency with these of prototype on whole deformation profile, as expected - Fig. 10b and d.

(4) Equivalent strain fields
The similarity of equivalent strain fields is evaluated by the equivalent strain of final time on neutral plane profile, with the results plotted in Fig. 11. Obviously, the equivalent strain of the prototype and the scaled models are significantly different in the corresponding dimensionless spatial positions, as shown in Fig. 11a. It can be learned from are used to regularize structural response, the dimensionless equivalent strain fields of scaled models behave good consistency with these of prototype on whole deformation profile, as shown in Fig. 11b. • Loading Case II The similarity of the displacement, stress and strain components as well as the equivalent stress, strain and strain rate has been studied in detail under Loading case I, from space deformation and deformation history point of view. For a larger degree of deformation, the Loading case II is used to evaluate the similarity of circular plate. In order to simplify the study, the similarity of displacement components and equivalent strain will be further analyzed in what follows.
(1) Displacement fields The similarity of displacement fields is evaluated by the displacement components and of final time on neutral plane profile, with the results plotted in Fig. 13. It is evident from Fig. 13a, b to Fig. 13c, d that, after using the numbers = ̅ and to regularize structural response, the dimensionless displacement fields of scaled models become basically in accordance with these of prototype on whole deformation profile. In addition, some small deviations between scaled models and prototype can be observed for the final dimensionless displacements, as shown in Fig.   13b and d, which increase as distortion increases in the main. (

2) Equivalent strain fields
The similarity of strain fields is evaluated by the equivalent strain of final time on neutral plane profile, with the results plotted in Fig. 14. It is evident from Fig.  14a to Fig. 14b that, after using the number = ( ̅ ̅ ) 2 to regularize structural response, the dimensionless equivalent strain fields of scaled models become basically in accordance with these of prototype on whole deformation profile. In addition, the relatively obvious deviations can be discovered near the center and root of the circle plates, which increases as distortion increases in the main. However, for the relatively small distortion = 1.5 and 2, these deviations become smaller. (

3) Deformation history
The evaluation of displacement and equivalent strain fields indicates that the ODLV similarity laws are still relatively precise for Loading case II from the space deformation point of view. Some deviations for this large deformation case can be attributed to stress , , strain and strain rate ̇ that are not considered in the derivation for the number , the number and the number ̇ in Sect.
2.3. In order to further evaluate the similarity from the deformation history point of view, two representative physical quantities and are plotted in Fig. 15. It is evident from Fig. 15a to Fig. 15b that, after using the number = to regularize structural response, the dimensionless deflections of scaled models become basically in accordance with these of prototype during deformation history, which results in the maximum dimensionless deflection about ( Fig. 15d. Nonetheless, these deviations can be considered small in comparison to the results of Fig. 15c. However, for relatively small distortion = 1.5 and 2, these deviations are decreased significantly. In order to analysis the deviations of equivalent strain , the similarity of transverse shear strain at a quarter of the neutral plane of the circular plates (i.e., = ̅ = 1 2 ) is evaluated during deformation history, with the results plotted in Fig. 16.
It is evident from Fig. 16a to Fig. 16b that, after using the number = ( ̅ ̅ ) 3 to regularize structural response, the significantly different between scaled model and prototype become basically identical. After dimensionless time about 0 / = 3.0 , the dimensionless shear strain increases rapidly from very small values, as shown in Fig. 16b. And some degree of deviations increases with the deformation process. However, for relatively small distortion = 1.5 and 2, these deviations are decreased significantly. Compared with Loading case I, the dimensionless shear strain of final deformation history ( ) at this spatial position increases significantly -from 5.1 (Fig. 10d) to 951.6 (Fig. 16b). For Loading case I and Loading case II, the dimensionless equivalent strain of final deformation history ( ) at this spatial position are respectively 0.8 (Fig. 11b) and 35.2 (Fig. 15d). Therefore, for final deformation history and this spatial position, the ratio ( / ) is significantly increased from 6.4 of Loading case I to 27.0 of Loading case II, which indicates that the effects of shear strain for the similarity of equivalent strain are significantly increased. Fig. 16 Transverse shear strain at a quarter of the neutral plane during deformation history for circular plate and scaled models (Loading case II).
The above analysis for two loading cases shows that, when the ODLV numbers are used to regularize geometric distorted structures, the structural responses for the displacement, stress and strain components and the equivalent stress, strain and strain rate behave the perfect similarity on space deformation and during deformation history.
For Loading case I, the ODLV similarity laws are verified to be quite accurate, even for the serious geometrical distortion cases such as = 3, 4 . For Loading case II, the smaller distortion cases such as = 1.5, 2 have a more accurate similarity results while the larger distortion cases such = 3, 4 have the certain degree of deviations.
In addition, the similarities of stress components , , and on neutral plane shows that the proposed similarity laws for the membrane force, the shear force and the normal force are correct. Although stress , , strain and strain rate ̇ that are not considered in the derivations for the numbers , and ̇, the perfect similarity of all the physical quantities, especially for stress and strain components, indicates that the ODLV system can be used to precise and elaborate analysis for the geometric distorted structures subjected to impact loading.

Conclusions
A framework of similarity laws for the geometric distorted structures subjected to impact loading is presented here, which bases on the new prosed oriented-densitylength-velocity (ODLV) dimensional analysis. It can be considered as a new progress for similarity laws of structural impact since the influence of three spatial directions and geometric characteristics are expressed effectively. Compared with previous dimensional analysis systems such as MLT, VSG, VSM and DLV, six main aspects are reflected in the new proposed ODLV framework: (1) A group of oriented dimensions of physical quantity are defined by the oriented analysis for the basic physical definition of physical quantities, the equivalent stress, strain and strain rate and the constitutive relation of the plastic increment theory, which establish the foundation for directional dimensional analysis. Meanwhile, these oriented dimensions are the all-round consideration for all the basic physical quantities from thin-plate impact model that includes six stress components, six strain components, six strain rate components, three displacement components, equivalent stress, equivalent strain, equivalent strain rate, time, external loads and so on.
(2) Five bases, including three oriented characteristic lengths of the geometric space, are used to express the dimensionless number of physical quantities, which instead of three bases of past dimensional analysis system that used only one scalar characteristic length. Therefore, it extends the ability of dimensional analysis for expressing different spatial directions and geometric characteristics.
(3) Three correction methods for the impact velocity, the geometrical thickness and the density are proposed to address different distortion problems such as the geometrical distortion, different materials and gravity, which extends the capabilities of previous correction methods by further considering the effects of structural geometric characteristics.
(4) The most essential dynamic similarity for impact problems is represented by the proposed dimensionless numbers of equivalent stress, stress components and external loadings; for instance, the ratios of inertia moment to the dynamic bending moment (or the fully plastic bending moment), the ratios of inertia force to the structural dynamic force (or fully plastic membrane force or shear force), the ratio of the impact mass to the structural mass. Then, the similarity laws of membrane force, bending moment, shear force and normal force are expressed. In addition, the Zhao's response number is naturally derived by the proposed dimensionless number of equivalent stress; and the differences between the Johnson's damage number and the Zhao's response number are profoundly explained by oriented dimensional analysis of equivalent stress.
(5) The various response equations for impact problems of beam, plate and shell are perfectly expressed by the proposed dimensionless numbers, which improves the defective expression ability of the DLV system for response equations by further considering the direction of physical quantities. In addition, analysis for response equations indicates these dimensionless numbers are important and independent for structural impact.
(6) The precise and elaborate analysis ability for geometric distorted scaled structures is verified in detail by the regularized application of ODLV numbers for the numerical model of impacted circular plates, from the view of point of space deformation and deformation history. And the applicability of these numbers for the serious geometrical distortion and the large deformation are verified, especially for the dimensionless numbers of displacement, stress and strain components.
Since the framework of ODLV is based on the assumptions for thin-plate impact problem, for more complex structures and situations, it needs to be further verified and studied in the future.