A Physics-Informed Neural Network Framework for Elastodynamic Wave Propagation in Bimaterial Systems Sonal Ankush Chibire1 , Jenn-Terng Gau1 , Bo Zhang1* 1
Department of Mechanical Engineering, Northern Illinois University, 1425 W Lincoln Hwy, DeKalb, 60115, IL, USA.
arXiv:2607.06479v1 [cs.AI] 7 Jul 2026
*Corresponding author(s). E-mail(s): [email protected]; Contributing authors: [email protected]; [email protected]; Abstract Physics-informed neural networks (PINNs) provide a promising framework for solving partial differential equations while embedding the underlying physical laws directly into the learning process. This study presents a PINN-based framework for modeling transient elastodynamic wave propagation in bimaterial systems governed by the axisymmetric equations of linear elasticity. A steel-aluminum specimen representative of a Split Hopkinson Pressure Bar configuration is considered, and the governing elastodynamic equations, together with the corresponding initial, boundary, and interface conditions, are incorporated directly into the network through a physics-informed loss function. High-fidelity finite-element simulations performed using ANSYS Workbench Explicit Dynamics are used for validation and as supplementary data constraints during training. The proposed framework accurately predicts wave transmission and reflection across the bimaterial interface and reproduces axial and radial displacement histories, face-averaged responses, and the dominant stress and strain evolution with close agreement to the finite-element solutions. The trained network further demonstrates the ability to predict wave responses at previously unseen time instants and for modified material properties without requiring additional finiteelement simulations, providing a continuous surrogate model for elastodynamic analysis. Mesh-sensitivity studies confirm numerical robustness, while additional material combinations demonstrate the generality of the proposed methodology. The results show that integrating physics-informed neural networks with explicit finite-element analysis provides an accurate and computationally efficient framework for elastodynamic wave propagation in heterogeneous solids, offering an
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effective surrogate modeling approach for high-rate solid mechanics and impact engineering applications. Keywords: Physics-Informed Neural Networks, Elastodynamic Wave Propagation, Bimaterial Systems, Split Hopkinson Pressure Bar, Finite Element Method, Surrogate Modeling
1 Introduction Recent advances in scientific machine learning [1–14] have created new opportunities for solving complex partial differential equations arising in computational mechanics and mechanical engineering. Among these developments, Physics-Informed Neural Networks (PINNs) [1, 4] have emerged as a promising computational framework by embedding governing equations, initial and boundary conditions directly into the neural-network training process. Unlike purely data-driven approaches, PINNs incorporate the underlying physical laws into the optimization procedure, enabling the solution of both forward problems governed by known equations and inverse problems in which unknown material properties or system parameters are inferred from limited experimental or numerical observations. Transient wave propagation in elastic solids plays a fundamental role in structural dynamics, impact engineering, and materials science. Accurate prediction of stress-wave transmission and reflection is essential for understanding the dynamic behavior of multilayer structures, protective systems, aerospace components, and other engineering systems subjected to high-rate loading. The problem becomes considerably more challenging in heterogeneous materials, where discontinuities in material properties and imperfect interfaces strongly influence wave propagation, stress redistribution, and energy transmission. Although conventional finite-element methods provide highly accurate solutions for elastodynamic wave propagation, repeated simulations required for parametric studies, optimization, and inverse analysis are computationally expensive, motivating the development of efficient surrogate modeling techniques. Extensive research has been devoted to analytical, numerical, and experimental investigations of elastic wave propagation. Barzkar and Adibi [15] proposed a unified viscoelastic framework connecting the Navier-Lamé and Navier-Stokes equations, providing a common description of elastic and viscous wave propagation, although their formulation was limited to one-dimensional problems. Ivanova et al. [16] investigated dynamic delamination in layered structures using a shear-lag analytical model and demonstrated the importance of interfacial stress concentrations while neglecting plastic deformation and time-dependent material behavior. Towfighi et al. [17] developed analytical models for wave propagation in anisotropic cylindrical plates, highlighting the influence of material anisotropy and geometric curvature on wave characteristics. Coker et al. [18] experimentally and numerically investigated frictional sliding under shear-impact loading, demonstrating transitions between crack-like and pulse-like sliding modes governed by interface friction and loading conditions. Ojha
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et al. [19] employed LS-DYNA to study high-velocity impact of Weldox 700E steel, revealing the coupled effects of strain rate, plastic deformation, and temperature on fracture behavior. Experimental studies by Bertholf and Karnes [20] established the theoretical foundation of Split Hopkinson Pressure Bar (SHPB) testing and quantified the influence of friction, inertia, and specimen geometry on stress-wave measurements. More recently, Shin [21] developed a numerical solution of the Pochhammer-Chree equation to improve dispersion correction for SHPB experiments, thereby enhancing the accuracy of experimentally measured wave responses. Parallel to these developments, Physics-Informed Neural Networks have rapidly emerged as an effective mesh-free methodology for solving partial differential equations in computational mechanics. Peng and Panesar [22] successfully applied PINNs to multilayer thermal simulations in additive manufacturing, demonstrating strong agreement with finite-element solutions while overcoming challenges associated with evolving geometries and material discontinuities. Wang and Thai, [23] integrated Classical Laminated Plate Theory with an energy-based PINN formulation to accurately predict the bending behavior of composite plates using limited training data. Margenberg et al. [24] proposed a hybrid deep neural network-multigrid framework that combines neural networks with finite-element solvers to achieve finite-element accuracy at substantially reduced computational cost. Despite these advances, relatively few studies have investigated PINN-based modeling of transient elastodynamic wave propagation in heterogeneous solids with realistic interface conditions. In particular, physics-informed formulations capable of accurately capturing stress-wave transmission and reflection across bimaterial interfaces while maintaining agreement with high-fidelity finite-element simulations remain limited. To address this gap, the present study develops a physics-informed computational framework for transient elastodynamic wave propagation in bimaterial systems. The governing equations of linear elastodynamics are formulated for a steel-aluminum specimen with partial interfacial slip to account for frictional effects during wave transmission and reflection. The coupled partial differential equations, together with the corresponding initial, boundary, and interface conditions, are directly incorporated into a PINN framework to predict transient displacement and stress fields throughout the specimen. The proposed methodology is validated against high-fidelity finiteelement simulations performed using ANSYS Workbench Explicit Dynamics under loading conditions representative of Split Hopkinson Pressure Bar experiments. The trained PINN accurately predicts displacement, stress, and strain responses, including wave propagation at previously unseen time instants, while substantially reducing the computational cost associated with repeated finite-element simulations. The proposed framework therefore provides an accurate and computationally efficient surrogate modeling approach for transient elastodynamic analysis in heterogeneous solids, with potential applications in impact-resistant structures, multilayer materials, aerospace engineering, and other high-rate solid mechanics problems.
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2 Methodology 2.1 Governing Equations Transient wave propagation in the bimaterial specimen is governed by the Navier– Lamé equations for a linear, isotropic elastic solid,
∂2u = (λ + µ)∇(∇ · u) + µ∇2 u, (1) ∂t2 where u is the displacement vector, ρ is the material density, and λ and µ are the Lamé parameters. To represent the cylindrical geometry of the Split Hopkinson Pressure Bar (SHPB) specimen, the governing equations are formulated in an axisymmetric coordinate system using the radial and axial displacement components, ur (r, x, t) and ux (r, x, t). This displacement-based formulation is employed consistently throughout the analytical model, the finite-element simulations, and the Physics-Informed Neural Network (PINN). Under the assumption of infinitesimal deformation, the non-zero strain components in cylindrical coordinates (r, θ, x) are ρ
∂ur , εrr = ∂r
ur εθθ = , r
∂ux εxx = , ∂x
1 εrx = 2
∂ur ∂ux + ∂x ∂r
.
(2)
The volumetric strain is tr(ε) = εrr + εθθ + εxx , where ε denotes the infinitesimal strain tensor. For an isotropic linear elastic material, the Lamé parameters are
(3)
Eν E , µ= , (4) (1 + ν )(1 − 2ν ) 2(1 + ν ) where E is Young’s modulus and ν is Poisson’s ratio. The corresponding Cauchy stress components are obtained from Hooke’s law, λ=
σrr = λ tr(ε) + 2µεrr , σθθ = λ tr(ε) + 2µεθθ , (5) σxx = λ tr(ε) + 2µεxx , σrx = 2µεrx . Substituting Eq. (5) into the linear momentum equations yields the governing axisymmetric elastodynamic equations for radial and axial wave propagation [25], ρ(x)
∂ 2 ur ∂σrr ∂σrx σrr − σθθ = + + , ∂t2 ∂r ∂x r
∂ 2 ux ∂σrx ∂σxx σrx = + + . 2 ∂t ∂r ∂x r Within the PINN framework, the second-order temporal derivatives, ρ(x)
4
(6) (7)
∂ 2 ur ∂ 2 ux , ux,tt = , (8) 2 ∂t ∂t2 are evaluated using automatic differentiation. The governing equations are enforced by minimizing the physics residuals, ∂σrx σrr − σθθ ∂σrr + + , (9) Resr = ρur,tt − ∂r ∂x r ∂σrx ∂σxx σrx Resx = ρux,tt − + + , (10) ∂r ∂x r which are incorporated into the PINN loss function to ensure that the predicted displacement field satisfies the governing elastodynamic equations throughout the computational domain. ur,tt =
2.2 Initial, Boundary, and Interface Conditions The governing equations are supplemented with appropriate initial, boundary, and interface conditions to ensure a well-posed elastodynamic problem. Initially, the specimen is assumed to be undeformed and at rest,
∂u (x, 0) = 0, (11) ∂t where u denotes the displacement vector. Axisymmetry is imposed along the cylindrical centerline (r = 0), while the outer cylindrical surface is assumed to be traction-free. At the incident-bar end, a prescribed axial impact velocity is applied to generate the stress wave, whereas the far end of the transmitted bar is treated as a traction-free boundary. At material interfaces, continuity of the axial displacement is enforced, u(x, 0) = 0,
+ u− (12) x = ux , where the superscripts (−) and (+) denote quantities evaluated on the two sides of the interface. The interface is assumed to be frictionless; consequently, the shear traction vanishes,
σrx = 0. (13) To ensure physically consistent wave transmission and reflection, continuity of the axial normal stress is imposed in a weak sense across both the steel–steel and steel–aluminum interfaces, − + σxx = σxx . (14) These initial, boundary, and interface conditions are incorporated into the PINN loss function through corresponding penalty terms, ensuring that the predicted displacement field satisfies both the governing elastodynamic equations and the prescribed physical constraints throughout the computational domain.
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Bimaterial specimen
Impact velocity V0
Steel (St)
Incident bar
Interface I (bar-steel)
(a)
Aluminum (Al)
Transmitted bar
Interface II Interface III (steel-aluminum) (Aluminum-bar) (b)
LI
LSt
Impact direction
Symmetry axis (axisymmetric model)
LAl Steel (elastic)
LT Aluminum (elastic)
Material Interfaces
Fig. 1 Schematic of the Split Hopkinson Pressure Bar (SHPB) assembly consisting of an incident bar, a bimaterial steel–aluminum specimen, and a transmitted bar. The incident impact generates a compressive stress wave that propagates through the specimen and across the material interfaces.
(a)
(b)
Fig 3: Meshing of a Specimen
Fig 2: 3D Quarter model of SHPB
Fig. 2 Finite-element model of the Split Hopkinson Pressure Bar (SHPB) assembly. (a) ThreeTable 2: Dimensions dimensional quarter-symmetry model consisting of The therefined incident bimaterial steel–aluminum mesh in bar, the specimen region enables accurate resolu,on of interfacial stress waves while maintaining stabilityspecimen, in the explicit ,me integra,on scheme. specimen, and transmitted bar. (b) Refined finite-element mesh in the numerical bimaterial showing Bar Specimen the discretization adopted near the steel–aluminum interface to accurately resolve stress-wave trans(in mm) 50 2.5 mission Length and reflection.
Diameter (in mm)
10
10
2.3 Finite-Element Model
4. Physics-Informed Neural Network Method
The analy,cal formula,ons for axisymmetric, bimaterial configura,on with interfacial constraints leads to substan,al mathema,cal complexity and limited analy,cal tractability [14]. To address this challenge, a Physics-Informed Neural Network (PINN) is employed as a physics-constrained surrogate model. The PINN is designed to:
TheStrategy finite-element model is developed in ANSYS Workbench Explicit Dynamics to 3.3 Meshing
i. approximate the spa,o-temporal displacement field (#" (0, 1, $), ## (0, 1, $)), provide a high-fidelity reference solution for validating thefullproposed Physics-Informed
Spa,al discre,za,on is selected to ensure consistency with the displacement-based analy,cal Neural Network (PINN). All components of theiii. Split Hopkinson Pressure Barequa,ons, (SHPB) enforce the governing axisymmetric elastodynamic together with all boundary, solu,on and to adequately resolve elas,c wave propaga,on governedini,al, by the axisymmetric and interface condi,ons, directly through the loss func,on, and assembly are modeled as linear, isotropic elastic materials, consistent with the assumpNavier–Lamé equa,ons. iii. incorporate displacement histories obtained from ANSYS Explicit Dynamics as addi,onal
tions of the governing Navier–Lamé equations. The incident and transmitted bars are soq constraints.
5 The incident and transmi?ed bars are discre,zed using a uniform elementofsize 1.010 mm, constructed from steel with a Young’s modulus 2.0of× MPa, a Poisson’s ratio this framework, the PINN serves as a bridge between the theore,cal elastodynamic −3 axial waveInpropaga,on which provides sufficient resolu,on for one-dimensional [13]. In consists of steel and of 0.30, and a density of 7850 kg m . The bimaterial formula,on and specimen the high-fidelity finite element (FE) solu,on. The resul,ng model is both contrast, aluminum, the bimetallic where specimenthe is discre,zed using refined element size of 0.25 mm to of 71and physics-consistent and data-consistent enables evalua,on of the displacement field at aluminum isaassigned a Young’s modulus 000 MPa, a Poisarbitrary spa,al loca,ons and ,me instants without the need for repeated FE simula,ons. accurately capture steep stress and displacement gradients near the material interfaces. −3
son’s ratio of 0.33, and a density of 2780 kg m
. These material properties define the
4.1 PINN Mapping and Network Architecture
ρ(for x),each λ(component x), and µis(xprovided ) in the governing elastodynamic equations, A detailedspatially summary ofvarying the meshfields sta,s,cs in Table 3, including The PINN is implemented as a fully connected feed-forward neural network that learns a the total enabling number of nodes and elements. material discontinuities across the steel–aluminum interface while preserving nonlinear mapping from space–,me coordinates to displacement components, displacement continuity. Table 3. Mesh sta0s0cs for SHPB components Component
Number of Nodes Number of Elements
6
Incident bar
1683
1150
Steel specimen
15,750
13,728
Aluminum specimen
15,750
13,728
Transmi?ed bar
1683
1150
(0, 1, $) → (#" (0, 1, $), ## (0, 1, $))
The numerical model consists of an incident bar, a bimaterial specimen composed of steel and aluminum, and a transmitted bar, as illustrated schematically in Fig. 1. The incident and transmitted bars each have a length of 50 mm and a diameter of 10 mm, while the bimaterial specimen has a total length of 2.5 mm with the same diameter, ensuring geometric compatibility throughout the SHPB assembly. Owing to geometric symmetry, a three-dimensional quarter-symmetry model is employed (Fig. 2(a)), which substantially reduces the computational cost while remaining fully consistent with the axisymmetric formulation presented in Section 2.1. Spatial discretization is designed to accurately resolve transient stress-wave propagation while maintaining computational efficiency. The incident and transmitted bars are discretized using a uniform element size of 1.0 mm, which provides sufficient resolution for one-dimensional axial wave propagation [26]. To accurately capture the steep stress and displacement gradients near the steel–aluminum interface, the bimaterial specimen is discretized using a refined element size of 0.25 mm. Figure 2(a) shows the three-dimensional quarter-symmetry finite-element model, while Fig. 2(b) presents the refined mesh adopted in the specimen region. The locally refined discretization enables accurate resolution of stress-wave transmission and reflection at the material interface while maintaining numerical stability during explicit time integration.
2.4 Physics-Informed Neural Network Adam Optimizer
Update network parameters Automatic differentiation
Neural network
t r x
σ
σ
σ
σ
ur
σ
σ
ux
σ
σ
Physics-based contributions
I
I
Initial conditions
∂t
∂t
Boundary conditions
∂x
∂x
Naiver-Lamé equations
∂r
∂r
Interface conditions
ℒBC
ℒInterface ℒIC
ℒPDE
Total Loss
Data-driven contributions
Inputs
(t, r, x)
ANSYS (explicit dynamics)
ℒdata =
1 d Nd ∑ i=1 N
uiPINN − uiANSYS
2 2
Fig. 3 Schematic of the proposed Physics-Informed Neural Network (PINN) framework for axisymmetric elastodynamic wave propagation. The neural network maps the spatial–temporal coordinates (r, x, t) to the radial and axial displacement fields (ur , ux ). Automatic differentiation is used to enforce the governing Navier–Lamé equations together with the initial, boundary, and interface conditions through physics-based loss terms, while displacement data obtained from ANSYS Explicit Dynamics provide additional data-driven constraints. The total weighted loss is minimized using the Adam optimizer to update the network parameters.
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The coupled axisymmetric elastodynamic equations, together with the prescribed initial, boundary, and interface conditions, constitute a high-dimensional constrained boundary-value problem for transient wave propagation in heterogeneous solids. Owing to the heterogeneous material properties and interfacial constraints, obtaining analytical solutions is generally intractable [1]. To address this challenge, a Physics-Informed Neural Network (PINN) is employed as a physics-constrained surrogate model. Unlike purely data-driven neural networks, the PINN incorporates the governing physical laws directly into the training process by embedding the elastodynamic equations and all associated constraints within a unified loss function. In addition, displacement data obtained from ANSYS Explicit Dynamics are incorporated as soft constraints to improve predictive accuracy while maintaining physical consistency. Consequently, the trained PINN provides a continuous approximation of the displacement field throughout the space–time domain and can be evaluated at arbitrary spatial locations and time instants without requiring additional finite-element simulations.
2.4.1 Network Architecture The PINN is implemented as a fully connected feed-forward neural network that learns the nonlinear mapping (r, x, t) −→ (ur (r, x, t), ux (r, x, t)) ,
(15)
where r, x, and t denote the radial coordinate, axial coordinate, and time, respectively, and ur (r, x, t) and ux (r, x, t) represent the corresponding radial and axial displacement components. The computational domain is defined by 0 ≤ r ≤ R,
0 ≤ x ≤ Lx ,
0 ≤ t ≤ T,
where R is the radius of the cylindrical specimen, Lx is its axial length, and T is the total simulation time. The network takes the spatial and temporal coordinates (r, x, t) as inputs and predicts the corresponding radial and axial displacement components, which directly correspond to the nodal displacement fields obtained from ANSYS Explicit Dynamics. All required spatial and temporal derivatives are computed through automatic differentiation, allowing direct evaluation of strains, stresses, and governing-equation residuals without numerical differentiation. Figure 3 illustrates the overall workflow of the proposed Physics-Informed Neural Network (PINN) framework. The spatial–temporal coordinates (r, x, t) are provided as inputs to a fully connected feed-forward neural network, which predicts the radial and axial displacement fields (ur , ux ). Automatic differentiation is then employed to compute the spatial and temporal derivatives required for evaluating the governing Navier–Lamé equations and constructing the residuals of the elastodynamic partial differential equations. The physics-based loss consists of contributions from the governing equations together with the prescribed initial, boundary, and interface conditions. In parallel, displacement data obtained from ANSYS Explicit Dynamics are incorporated as data-driven constraints through a data-loss term. The physics-based and data-driven contributions are combined to form the total weighted loss function, which
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is minimized using the Adam optimizer to iteratively update the network parameters until convergence.
2.4.2 Physics-Informed Loss Function The network parameters are obtained by minimizing a composite loss function consisting of physics-based and data-driven contributions,
L = wf LPDE + wb LBC + wi LIC + wΓ LInterface + wd LData , (16) where L denotes the total loss function; LPDE , LBC , LIC , LInterface , and LData represent the loss contributions associated with the governing partial differential equations, boundary conditions, initial conditions, interface conditions, and finiteelement displacement data, respectively. The weighting coefficients wf , wb , wi , wΓ , and wd control the relative importance of the corresponding loss terms during network training. All loss terms are evaluated at collocation points distributed throughout the computational space–time domain. The physics loss is constructed from the mean-squared residuals of the governing axisymmetric elastodynamic equations evaluated at interior collocation points (rf , xf , tf ). The stress components σrr , σθθ , σxx , and σrx are computed from the network-predicted displacement field using the constitutive relations presented in Section 2.1 together with the piecewise-constant material properties corresponding to steel and aluminum. All spatial and temporal derivatives required for evaluating the governing equations are computed through automatic differentiation [27]. The PDE loss is defined as the mean-squared value of the normalized residuals over all interior collocation points. Additional penalty terms enforce the prescribed initial, boundary, and interface conditions. The initial conditions constrain the displacement and velocity fields at t = 0, while the boundary-condition loss enforces axisymmetry, traction-free boundaries, and the prescribed impact loading. Interface conditions impose continuity of the axial displacement together with the corresponding traction conditions across the steel–steel and steel–aluminum interfaces. To further improve predictive accuracy, displacement histories obtained from ANSYS Explicit Dynamics are incorporated into the loss function as soft constraints. These observations include radial displacement histories on the steel and aluminum specimen faces together with axial displacement histories along the centerline (r = 0) at multiple spatial locations and time instants. For each data set, the corresponding spatial coordinates and time instants are supplied to the network, and the discrepancy between the predicted and finite-element displacement fields is minimized using a normalized mean-squared error. These data constraints anchor the learned solution to the high-fidelity finite-element simulations while preserving consistency with the governing physical laws.
2.4.3 Training Strategy The network parameters are optimized by minimizing the total weighted loss function using the Adam optimizer. To improve convergence and numerical stability, interior
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collocation points are periodically resampled during training to enhance spatial coverage and reduce overfitting. A progressive weighting strategy is adopted in which the governing-equation residual initially receives a relatively small weight, allowing the network to first satisfy the displacement observations and boundary conditions before gradually enforcing the governing physics. Furthermore, all physical variables are appropriately normalized using suitable reference scales to improve numerical At outer-radius nodes, axial predic,ons remain remain accurate accurate with with slight slight underes,ma,on underes,ma,on in conditioning and training stability. outer-radius nodes, axial predic,ons AtAt outer-radius nodes, axial predic,ons remain accurate with slight underes,ma,on in in aluminum [20], while radial responses agree well in phase and ,ming, with moderate After convergence, the trained PINN provides a smooth and differentiable approxAt outer-radius nodes, radial axial predic,ons remain accurate with slight underes,ma,on in aluminum [20],while while responsesagree agree well phase ,ming, moderate aluminum [20], radial responses well in in phase andand ,ming, withwith moderate imation of the displacement field, amplitude differences, indica,ng good capture of surface deforma,on. aluminum [20], while radial responses agree well in phase and ,ming, with moderate amplitudedifferences, differences, indica,ng good capture surface deforma,on. amplitude indica,ng good capture of of surface deforma,on. amplitude differences, indica,ng good capture of surface deforma,on. (ur (nodes, r, x, t), axial ux (r,agreement x, t)) , At inner-radius inner-radius and and interface-plane interface-plane excellent. Radial Radial displacement displacement nodes, axial agreement isis excellent. AtAtinner-radius and interface-plane nodes, axial agreement is excellent. Radial displacement Atwell inner-radius and interface-plane nodes, domain. axial agreement is excellent. Radialmodel displacement throughout theinentire computational The resulting surrogate accuis predicted steel, with modest late-,me devia,ons in aluminum due to higher higher wellpredicted predictedininsteel, steel,with withmodest modest late-,me devia,ons aluminum to isiswell late-,me devia,ons in in aluminum duedue to higher is well predicted in steel, with modest late-,me devia,ons in aluminum due to higher rately reproduces the finite-element solution while enabling efficient interpolation sensi,vity near near the the free free surface surface and and interface. interface. sensi,vity sensi,vity near theat free surface and interface. and prediction arbitrary locations and time instants without repeated sensi,vity near the free surfacespatial and interface. finite-element analyses. Overall, the PINN shows strong agreement with ANSYS across all loca,ons. loca,ons. Axial responses are Overall, the PINN shows strong agreement with ANSYS across all Axial responses Overall, the PINN shows strong agreement with ANSYS across all Axial responses are are Overall, the PINN shows strong agreement with ANSYS across allloca,ons. loca,ons. Axial responses are accurately captured, while radial responses show good qualita,ve and quan,ta,ve accurately captured, andand quan,ta,ve accurately captured, while radialresponses responsesshow showgood goodqualita,ve qualita,ve quan,ta,ve captured,while whileradial radial responses show good qualita,ve and quan,ta,ve 3accurately Results agreement, with larger discrepancies mainly in aluminum near the surface and interface. agreement, with larger ininin aluminum near the surface andand interface. agreement, with larger discrepancies mainly aluminum near surface interface. agreement, with largerdiscrepancies discrepanciesmainly mainly aluminum near thethe surface and interface. These results confirm the PINN as reliable physics-constrained surrogate for axisymmetric axisymmetric These results confirm the PINN asas aANSYS physics-constrained surrogate for for axisymmetric These results confirm the PINN as aareliable reliable physics-constrained surrogate 3.1 Validation Against Explicit Dynamics These results confirm the PINN areliable physics-constrained surrogate for axisymmetric elastodynamic wave propaga,on in bimaterial systems. elastodynamic wave propaga,on bimaterial systems. elastodynamic wave propaga,on in bimaterial systems. elastodynamic wave propaga,oninin bimaterial systems. (a)
(b)
(c)
(d)
Fig.4 Axial and Radial Displacements at Center node (0.0025,0.05017) and (0.0025, 0.05175)
Fig.4 and Radial Displacements at Center node (0.0025,0.05017) andand (0.0025, 0.05175) Fig.4 Axial and Radial Displacements at Center Center node (0.0025,0.05017) (0.0025, 0.05175) Fig. 4Axial Comparison between the displacement histories predicted by the proposed Physics-Informed Fig.4 Axial and Radial Displacements at node (0.0025,0.05017) and (0.0025, 0.05175) Neural Network (PINN) and ANSYS Explicit Dynamics at representative nodes located on the steel input face and aluminum output face. (a) Axial displacement at the steel input-face node. (b) Radial 5.2 Unseen at ,me displacement theinstants steel input-face node. (c) Axial displacement at the aluminum output-face node. Radial ,me displacement 5.2(d)Unseen instantsat the aluminum output-face node.
5.2 Unseen Unseen ,me ,me instants instants 5.2 For the 0.25 mm mesh, the trained PINN was evaluated over an unseen ,me window (200– For the 0.25 mesh, thethe trained PINN waswas evaluated over an unseen ,meshow window (200– 400 µs) atmm representa,ve steel and aluminum nodes. Axial displacements a smooth For the 0.25 mm mesh, trained PINN evaluated over an unseen unseen ,me window (200– For the 0.25 mm mesh, the trained PINN was evaluated over an ,me window (200– 10 400 µs) at representa,ve steel and aluminum nodes. Axial displacements show a smooth con,nua,on of the response, reaching a peak followed by gradual decay in both materials. 400 µs) µs) at at representa,ve representa,ve steel steel and and aluminum aluminum nodes. nodes. Axial Axial displacements displacements show show aa smooth smooth 400 con,nua,on of the response, reaching adue peak followed by gradual decay inzero bothand materials. Radial displacements, ini,ally nega,ve to contrac,on, increase toward slightly con,nua,on of of the the response, response, reaching reaching aa peak peak followed followed by by gradual gradual decay decay in in both materials. materials. con,nua,on Radial displacements, ini,ally nega,vephysically due to contrac,on, increase toward zero both and slightly posi,ve values by 400 µs, remaining consistent with prior wave mo,on. Radial displacements, ini,ally nega,ve due to contrac,on, increase toward zero and slightly Radial displacements, nega,ve due to contrac,on, toward zero and slightly posi,ve values by 400 µs,ini,ally remaining physically consistent with increase prior wave mo,on. posi,ve values by by 400 400 µs, remaining remaining physically consistent with prior prior wave mo,on. mo,on. These predic,ons indicate that the PINN captures the underlying elastodynamic behavior posi,ve values µs, physically consistent with wave rather than merely interpola,ng data. captures Once trained on ANSYS elastodynamic results, it serves as a These predic,ons indicate that the PINN the underlying behavior These predic,ons predic,ons indicate indicate that that the the PINN PINN captures captures the the underlying underlying elastodynamic elastodynamic behavior behavior These rather than merely interpola,ng data. Once trained on ANSYS results, it serves as a rather than merely interpola,ng data. Once trained on ANSYS results, it serves as aa rather than merely interpola,ng data. Once trained on ANSYS results, it serves as
The predictive capability of the proposed Physics-Informed Neural Network (PINN) was evaluated by comparison with high-fidelity finite-element simulations performed using ANSYS Explicit Dynamics. Representative monitoring nodes located on the steel input face and aluminum output face were selected to assess the transient elastodynamic response of the bimaterial specimen. The corresponding axial and radial displacement histories obtained from ANSYS Explicit Dynamics were incorporated into the data-loss term during training, enabling the network to simultaneously satisfy the governing elastodynamic equations while remaining consistent with the finite-element solution. Figure 4 compares the displacement histories predicted by the PINN with the ANSYS Explicit Dynamics results at the representative input- and output-face nodes. The axial displacement responses, shown in Figs. 4(a) and 4(c), exhibit excellent agreement throughout the transient loading process. The PINN accurately captures the wave arrival time, peak displacement, and subsequent elastic unloading in both the steel and aluminum regions. Minor deviations are observed at later times, particularly at the aluminum output face, where multiple wave reflections and material impedance mismatch produce increasingly complex wave interactions. The corresponding radial displacement histories, presented in Figs. 4(b) and 4(d), are also accurately reproduced by the proposed PINN. The network successfully captures the transient radial deformation associated with Poisson coupling, including the expansion, contraction, and overall temporal evolution of the displacement field. Slight discrepancies appear after the peak response, especially at the aluminum output face, owing to its lower elastic modulus and higher compliance, which increase the sensitivity of the radial response to reflected waves and interface interactions. Overall, the proposed PINN demonstrates excellent agreement with the ANSYS Explicit Dynamics solutions for both axial and radial displacement histories at the representative monitoring locations. The results confirm that the network accurately captures the essential characteristics of stress-wave propagation, transmission, and reflection in the steel–aluminum bimaterial specimen, demonstrating its effectiveness as a reliable physics-constrained surrogate model for axisymmetric elastodynamic wave propagation.
3.2 Generalization to Unseen Time Instants To evaluate the temporal generalization capability of the proposed Physics-Informed Neural Network (PINN), the trained model was tested over an unseen time interval of 200–400 µs without additional training. Figure 5 compares the PINN predictions with the corresponding ANSYS Explicit Dynamics results for representative monitoring nodes in the steel and aluminum specimens. As shown in Fig. 5(a) and (c), the PINN accurately reproduces the axial displacement histories throughout the unseen time interval. The predicted responses exhibit a smooth continuation beyond the training window, preserving the overall displacement evolution and demonstrating stable long-term prediction. Excellent agreement is observed for both the steel and aluminum nodes, indicating that the learned model successfully captures the underlying elastodynamic behavior rather than merely interpolating the training data. 11
con,nuous-,me surrogate thatthat cancan be be queried without con,nuous-,me surrogate queriedatatarbitrary arbitrary,mes ,mes and and loca,ons loca,ons without addi,onal simula,ons, enabling rapid evalua,on with while addi,onal simula,ons, enabling rapid evalua,on withimproved improvedtemporal temporal resolu,on resolu,on while preserving the essen,al physics. preserving the essen,al physics.
(a)
(b) Fig.6Unseen Unseen+me +meinstants: instants:Steel Steelnode noderadial radial Fig.6 displacement displacement
Fig.5Unseen Unseen+me +meinstants: instants:Steel Steelnode nodeaxial axial Fig.5 displacement displacement
Fig.5 Unseen +me instants: Steel node axial (c) Fig.5 Unseen +me instants: Steel node axial displacement
Fig.6 Unseen +me instants: Steel node radial
instants: Steel node radial displacement (d) Fig.6 Unseen +me displacement
displacement
Fig.8Unseen Unseen+me +meinstants: instants:Aluminum Aluminumnode noderadial radial Fig.7Unseen Unseen+me +meinstants: instants:Aluminum Aluminumnode nodeaxial axial Fig.8 Fig.7 displacement displacement displacement displacement Fig. 5 Comparison between the displacement histories predicted by the proposed Physics-Informed Neural Network (PINN) and ANSYS Explicit Dynamics over the unseen time interval of 200–400 µs. (a) Axial displacement of the representative steel node. (b) Radial displacement of the representative steel node. (c) Axial displacement of the representative aluminum node. (d) Radial displacement of the representative aluminum node. The PINN accurately the ANSYS predictions forradial both Fig.8reproduces Unseen +me instants: Aluminum node Fig.7 Unseen +me instants: Aluminum node axial materials, demonstrating excellent temporal generalization beyond the training interval. displacement displacement
Fig.8 Unseen +me instants: Aluminum node radial Fig.7 Unseen +me instants: Aluminum node axial Fig55shows showsthe thepost-training post-trainingaxial axialdisplacement displacementresponse responseofofthe thesteel steel node.The Thecurve curve Fig node. displacement displacementradial The corresponding displacement histories are presented in Fig. 5(b) and con,nuessmoothly smoothlyfrom fromthe thetraining trainingwindow windowatatabout about200 200µs, µs,indica,ng indica,ngstable stable predic,on con,nues (d). The PINN successfully captures the transient radial deformationpredic,on associated with beyondthe thetrained trainedregion. region.Since Sincethe thepeak peakdisplacement displacementoccurs occursearlier, earlier,around around20 20µs, µs,the thelater later beyond Poisson coupling, accurately reproducing the overall temporal evolution of the disresponse shows gradual elas,c unloading with a nearly flat, slightly decaying trend. The small response shows gradual elas,c unloading with a nearly flat, slightly decaying trend. The small placement field. Excellent agreement is observed for the aluminum node throughout residual varia,ons aremainly mainly duetotoreflected reflected wavesnode, anda?enua,on a?enua,on within thenoticeable bimaterial residual varia,ons due and within the bimaterial the5entire prediction interval. the waves steel however, a more Fig shows the are post-training axial For displacement response of the steel node. The curve deviaspecimen. specimen. tion develops nearfrom thethe endtraining of thewindow unseenattime where the PINN overestimates con,nues smoothly aboutwindow, 200 µs, indica,ng stable predic,on the magnitude of the radial displacement. This discrepancy is likely attributable to Fig Fi 5Fibeyond the post-training axial displacement response ofearlier, the steel node. The curve the trained region. Since the peak displacement occurs around 20200 µs, the later 6shows shows the unseen radial displacement response thesteel steel node between 200 and 400 6shows the unseen radial displacement response ofofthe node between and 400 the increased sensitivity of the radial response to late-time wave reflections, materesponse shows gradual elas,c unloading with a about nearly flat, slightly decaying trend.indica,ng The small con,nues from the training window at 200 µs, indica,ng stable predic,on µs. Thesmoothly radial displacement remains closeto tozero zero withsmall-amplitude small-amplitude varia,ons, indica,ng µs. The radial displacement remains close with varia,ons, rial impedance mismatch, and the accumulation of prediction errors during long-term residual varia,ons are mainly due to reflected waves and a?enua,on within the bimaterial beyond the trained region. Since the peak displacement occurs earlier, around 20 µs, the later extrapolation. specimen. response shows gradual elas,c unloading with a nearly flat, slightly decaying trend. The small
Overall, the proposed PINN demonstrates strong temporal generalization beyond residual mainly due to reflected wavescomplexity and a?enua,on within the bimaterial the training interval. Despite the increased of the late-time radial Fi 6varia,ons shows the are unseen radial displacement response of the steel node between 200 and 400response specimen. in The the radial steeldisplacement specimen, remains the network predicts thevaria,ons, dominant displacement µs. close toaccurately zero with small-amplitude indica,ng evolution in both materials while preserving the essential physics of transient wave Fi 6 shows the unseen radialtrained, displacement response of the steel node between 200 andmodel 400 that propagation. Once the PINN provides a continuous surrogate µs. The radial displacement remains close to zero with small-amplitude varia,ons, indica,ng can be efficiently evaluated at arbitrary spatial locations and time instants without additional finite-element simulations, thereby significantly reducing the computational cost of repeated elastodynamic analyses.
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(a)
(b)
Fig.9 Input faceface of steel: Axial face averaged Fig.9 Input of steel: Axial face averaged Fig.10 Input face ofof steel: Radial face averaged Fig.10 Input face steel: Radial face averaged Fig.9 Input faceface of steel: AxialAxial faceface averaged Fig.10 InputInput face face of steel: Radial face averaged Input of steel: averaged (d) (c)Fig.9 Fig.10 of steel: Radial face averaged
Fig.12 Input face aluminum: Axial face Fig.11 Input of aluminum: Axial face Fig.12 Input face ofof aluminum: Axial face Fig.11 Input faceface of aluminum: Axial face averaged averaged Axial face averaged averaged Fig.12 Input face of aluminum: Axial face Fig.11 Input face of aluminum: Fig.Fig.11 6 Comparison the face-averaged histories predicted by the proposed Fig.12 Input face of aluminum: Axial face Input face between of aluminum: Axial face displacement averaged Physics-Informedaveraged Neural Network (PINN) and ANSYS Explicit Dynamics. (a) Axial displacement at averaged averaged the steel input face. (b) Radial displacement at the steel input face. (c) Axial displacement at the
face-averaged responses show strong agreement between PINN and ANSYS bothaxial axial aluminum outputresponses face. (d) Radial displacement at the between aluminum output face. TheThe face-averaged show strong agreement PINN and ANSYS forforboth and radial displacements. The PINN accurately captures wave arrival, peak amplitude, and The face-averaged responses show strong agreement between andpeak ANSYS for both axial and radial displacements. The PINN accurately captures wavePINN arrival, and The face-averaged responses show strong agreement between PINN and amplitude, ANSYS for both axial 3.3in (# Axisymmetric Face-Averaged Responses decay in (# well as the contrac,on–relaxa,on behavior consistentwith with # ($), " ($)), and radial displacements. The PINN accurately captures wave arrival, peak amplitude, and decay ($), as as well as the contrac,on–relaxa,on behavior in in(#(# ($)), consistent # " and radial displacements. The PINN accurately captures wave arrival, peak amplitude, and Poisson effects. Poisson effects. decay (# ($), as well asthe the contrac,on–relaxa,on behavior in (# ($)), consistent with Toin further validate Physics-Informed Neural Network facedecay in# (## ($), as well as proposed the contrac,on–relaxa,on behavior in" (#" ($)),(PINN), consistent with averaged displacement histories were compared with the corresponding ANSYS Poisson effects. Minor late-,me differences, mainly in the radial response, arise from limited sensi,vity to Poisson effects. Minor late-,me differences, mainly in the radial response, arise from limited sensi,vity to Explicit Dynamics results. Face averaging is employed because Split Hopkinson transverse deforma,on and interface impedance mismatch. Overall, the results confirm that transverse deforma,on and interface impedance mismatch. Overall, the results confirm Minor late-,me differences, mainly the radialthat response, arise from limited sensi,vity to Pressure Bar (SHPB) theoryinassumes the measured stress and strainthat repMinor late-,me differences, mainly in the radial elastodynamic response, arise from limited sensi,vity to PINN reliably reproduces cross-sec,onal response, consistent with resent cross-sectional average quantities. mismatch. Accordingly, the PINN predictions are thethe PINN reliably reproduces thethe cross-sec,onal elastodynamic response, consistent transverse deforma,on and interface impedance Overall, the results confirmwith that transverse deforma,on andsame interface impedance mismatch. Overall, as the results confirm that SHPB theory FE predic,ons. post-processed using the axisymmetric averaging procedure finite-element SHPB theory andand FE predic,ons. the PINN reliably reproduces the cross-sec,onal elastodynamic response,the consistent with the PINN reliably reproduces thephysically cross-sec,onal elastodynamic response, consistent with solution to enable a direct and meaningful comparison. SHPB theory and FE predic,ons. 5.4 Stress–Strain Valida,on in Steel and Aluminum 5.4 Stress–Strain Valida,on in Steel and Aluminum For an and axisymmetric circular cross-section, the face-averaged displacement is SHPB theory FE predic,ons. computedthe asValida,on Following displacement valida,on, axial and radial stress–strain responsesininboth bothsteel steel 5.4 Stress–Strain invalida,on, Steel and Aluminum Following the displacement axial and radial stress–strain responses 5.4 Stress–Strain Valida,on in Steel and Aluminum Z Rand the PINN. This enables assessment of the and aluminum are compared between ANSYS and aluminum are compared between axial ANSYS andradial the PINN. This enables assessment the Following the displacement valida,on, and responses in bothofsteel u(r,radial x,stress–strain t) rstress–strain dr model’s ability to capture the cons,tu,ve elastodynamic behavior, wave transmission, andsteel Following thetodisplacement valida,on, axial and responses in both model’s ability capture the cons,tu,ve elastodynamic behavior, wave transmission, and 0 and aluminum are compared between the PINN. This enables assessment of the ū(x,ANSYS t) = and , (17) Z interface effects beyond kinema,c agreement. Par,cular a?en,on is given to peak stress R the PINN. This enables assessment of the and aluminum are compared between ANSYSPar,cular and interface effects beyond kinema,c agreement. a?en,on wave is given to peak stress model’s ability to capture the cons,tu,ve elastodynamic behavior, transmission, and r dr levels, wave arrival ,mes, and stress con,nuity across the bimaterial interface. model’s ability to,mes, capture cons,tu,ve behavior, wave transmission, and levels, wave arrival andthe stress con,nuity elastodynamic across 0 the bimaterial interface effects beyond kinema,c agreement. Par,cular a?en,on interface. is given to peak stress interface effects beyond kinema,c agreement. Par,cular a?en,on is given tothe peak stress where ū denotes the face-averaged displacement, u represents either The stress plots,mes, showand that bothcon,nuity ANSYS and PINNthe capture the interface. dominant stress peakaxial at levels, wave arrival stress bimaterial Thedisplacement stress plots show that both ANSYS and across PINNu capture the dominant stressradius. peak at u or the radial displacement , and R is the specimen The levels, wave arrival ,mes, and stress con,nuity interface. approximately 20xµs, which represents the arrivalacross ofr thethe firstbimaterial compressive stress wave in the approximately 20 µs, rwhich represents the arrival of the first compressive in the weighting factor arises from ANSYS the axisymmetric area element dA = 2stress πr stress dr.wave Thepeak integral The bimaterial stress plots show that both and PINN capture the dominant at system. This peak corresponds to rapid elas,c loading and transfer of mechanical is evaluated numerically using with 2048 sampling pointspeak dis- at The stress plots show that bothMonte ANSYS and integration PINN loading capture the dominant stress bimaterial system. This peak corresponds toCarlo rapid elas,c and transfer of mechanical approximately 20 µs,the which the regions. arrival ofThe thealuminum first compressive stress wave incloser the energy through steelarepresents and aluminum stress uniform response shows tributed according radial distribution to stress ensure area coverage energy through the andsquare-root aluminum regions. The aluminum response stress shows closerin the approximately 20steel µs, to which represents the arrival of the first compressive wave bimaterial system. This peak corresponds to rapid elas,c loading and transfer of mechanical agreement between ANSYS and PINN aqer the ini,al peak, while the steel stress response over thebetween circular cross-section. agreement ANSYS andcorresponds PINN aqer the ini,al elas,c peak, while theand steel stress response bimaterial system. This peak to The rapid loading transfer of mechanical energy through steel and aluminumInregions. aluminum response shows shows largerthe post-peak differences. steel, ANSYS predicts astress sharper decay aqer thecloser peak, shows larger post-peak differences. In steel,regions. ANSYS predicts a sharper decay aqer the peak,closer energy through the steel and aluminum The aluminum stress response shows agreement between ANSYS and PINN aqer the 13 ini,al peak, while the steel stress response agreement betweendifferences. ANSYS and In PINN aqer the predicts ini,al peak, while decay the steel response shows larger post-peak steel, ANSYS a sharper aqerstress the peak, shows larger post-peak differences. In steel, ANSYS predicts a sharper decay aqer the peak,
Figure 6 compares the face-averaged displacement histories predicted by the PINN with the corresponding ANSYS Explicit Dynamics results. The axial displacement histories shown in Fig. 6(a) and (c) exhibit excellent agreement for both the steel input face and the aluminum output face. The PINN accurately captures the wave arrival, displacement evolution, and subsequent elastic response throughout the loading process. The corresponding radial displacement histories are presented in Fig. 6(b) and (d). The PINN successfully reproduces the transient radial deformation associated with Poisson coupling and captures the overall temporal evolution of the cross-sectional response. Compared with the nodal predictions presented in Section 5.2, the faceaveraged responses exhibit smoother behavior and improved agreement with the finite-element solutions because local numerical fluctuations are reduced through crosssectional averaging. Minor discrepancies remain during the late stages of the steel response, where multiple wave reflections and material impedance mismatch produce increasingly complex transverse deformation. Overall, the face-averaged results demonstrate that the proposed PINN accurately reproduces the cross-sectional elastodynamic response of the bimaterial specimen. The excellent agreement with the ANSYS Explicit Dynamics simulations confirms the capability of the proposed framework to predict the response quantities most relevant to SHPB experiments while preserving the underlying transient wave-propagation physics.
3.4 Stress–Strain Validation Following the displacement validation, the stress and strain responses predicted by the proposed Physics-Informed Neural Network (PINN) were compared with the corresponding ANSYS Explicit Dynamics results to further assess the model’s ability to reproduce the transient elastodynamic behavior of the bimaterial specimen. In addition to the kinematic response, this comparison evaluates the constitutive response, stress-wave transmission, and material-interface effects. Figure 7 compares the stress and strain histories obtained from the PINN and ANSYS Explicit Dynamics. The stress responses shown in Fig. 7(a) and (b) indicate that both methods accurately capture the dominant compressive stress peak at approximately 20 µs, corresponding to the arrival of the first stress wave generated by the impact loading. Excellent agreement is observed for the aluminum specimen throughout the simulation. For the steel specimen, the initial peak is reproduced satisfactorily; however, larger discrepancies develop after the first wave passage, where the PINN predicts a smoother post-peak response than the finite-element solution. These differences are likely associated with the increased influence of wave reflections, stress redistribution, and interface interactions during the later stages of the transient response. The corresponding strain histories are presented in Fig. 7(c) and (d). Both the PINN and ANSYS accurately predict the initial transient strain peak associated with the arrival of the compressive wave. As expected, the aluminum specimen exhibits a larger strain amplitude than the steel specimen because its lower Young’s modulus results in greater deformation under comparable dynamic loading. Excellent agreement 14
dominant early-,me deforma,on response. Aqer the peak, the ANSYS strain gradually decays with small fluctua,ons caused by reflected waves and interface effects, while the PINN whereas the PINN response remains smoother and higher at later response ,mes. Physically, thisnearly flat. This indicates that the current PINN captures the main impactbecomes suggests that the PINN captures the primary wave loading event but does not fully reproduce induced strain response but underpredicts the secondary transient strain evolu,on at later the stress redistribu,on, wave reflec,on, and a?enua,on behavior,mes. aqer the first wave passage.
Fig.13 Stress in Steel
(a)
(b)
dominant early-,me deforma,on response. Aqer the peak, the ANSYS strain gradually decays with small fluctua,ons caused by reflected waves and interface effects, while the PINN response becomes nearly flat. This indicates that the current PINN captures the main impactinduced strain response but underpredicts the secondary transient strain evolu,on at later ,mes.
(c)
Fig.13 Stress in Steel
(d)
Fig.15 Strain in Steel Fig.14 Stress in Aluminum
The strain plots show a clear transient strain spike at approximately 20 µs for both steel and aluminum, indica,ng the first wave arrival and rapid deforma,on of the materials. The aluminum strain peak is higher than the steel strain peak because aluminum has a lower elas,c modulus and therefore experiences larger strain under comparable dynamic loading. The PINN matches the ini,al peak well, especially for aluminum, showing that it can learn the
Fig.15 Strain in Steel
Fig.16 Strain in Aluminum
Fig.14 Stress in Aluminum
Fig. 7 Comparison between the stress7.and strain histories predicted by the proposed PhysicsConclusion Informed Neural Network (PINN) and ANSYS Explicit Dynamics. (a) Stress history in the steel study an integrated analy,cal, and Physics-Informed Neural The strain plots show a(b) clear Stress transient strain spike at 20This µs for bothdeveloped steel and (c) specimen. history inapproximately the aluminum specimen. Strain history infinite-element, the steel specimen. (d) framework aluminum, indica,ng the first and rapid specimen. deforma,on Network of the materials. The for modeling transient wave propaga,on in a steel–aluminum bimaterial Strain history inwave thearrival aluminum
aluminum strain peak is higher than the steel strain peak because aluminum has a governing lower system. The axisymmetric elastodynamic equa,ons were formulated using elas,c modulus and therefore experiences larger strain under comparable dynamic loading. linear elas,city and implemented within a PINN framework, while ANSYS displacement-based The PINN matches the ini,al peak well, especially for aluminum, showing that it can learn the
is maintained for the aluminum specimen throughout simulation. For the steelshow that the Explicit Dynamics was used as thethe reference solu,on for valida,on. The results PINN can deformation successfully captureaccurately the dominant but wave-arrival behavior, peak response, and specimen, the PINN captures the initial predicts a more coupled axial–radial deforma,on trends in the bimaterial specimen. Strong agreement was rapid post-peak strain relaxation with reduced oscillatory behavior compared with the finite-element solution. Overall, the proposed PINN accurately reproduces the dominant stress and strain responses during the primary loading stage while maintaining excellent agreement for the aluminum specimen over the entire simulation. Although noticeable discrepancies Fig.16 Strain in Aluminum develop in the post-peak response of the steel specimen, the PINN successfully captures 7. Conclusion the principal constitutive behavior and transient wave propagation in the bimaterial SHPB system, demonstrating its effectiveness as a physics-constrained surrogate model This study developed an integrated analy,cal, finite-element, and Physics-Informed Neural for elastodynamic Network framework for modeling transientanalysis. wave propaga,on in a steel–aluminum bimaterial system. The governing axisymmetric elastodynamic equa,ons were formulated using displacement-based linear elas,city and implemented within a PINN framework, while ANSYS Explicit Dynamics was used as the reference solu,on for valida,on. The results show that the PINN can successfully capture presented the dominant an wave-arrival behavior, peak response, and This study integrated analytical, finite-element, and Physics-Informed coupled axial–radial deforma,on trends in the bimaterial specimen. Strong agreement was
4 Conclusion
Neural Network (PINN) framework for modeling transient wave propagation in a steel–aluminum bimaterial Split Hopkinson Pressure Bar (SHPB) specimen. The governing axisymmetric elastodynamic equations were formulated using displacementbased linear elasticity and embedded within a PINN framework, while ANSYS Explicit Dynamics provided high-fidelity reference solutions for training and validation.
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The proposed PINN accurately reproduced the transient displacement response of the bimaterial system. Excellent agreement with the finite-element results was obtained for the axial displacement histories at representative monitoring locations, including the centerline, off-axis regions, and material interface. The network also successfully captured the coupled axial–radial deformation associated with Poisson effects and reproduced the cross-sectional face-averaged responses required for SHPB analysis. Furthermore, evaluation at previously unseen time instants demonstrated that the trained PINN provides smooth and physically consistent predictions beyond the discrete training data, confirming its capability as a continuous space–time surrogate model. Stress and strain comparisons further demonstrated that the PINN accurately captured the primary compressive wave, including the wave-arrival time, peak response, and the material-dependent deformation behavior of both steel and aluminum. Although larger discrepancies were observed during the post-peak response, particularly in the steel specimen where multiple wave reflections and interface interactions become increasingly significant, the dominant elastodynamic behavior was reproduced with good accuracy. These results indicate that displacement predictions are generally more robust than derivative-based quantities such as stress and strain, highlighting the importance of validating PINN models using multiple physical fields rather than displacement alone. Overall, the proposed framework combines the physical consistency of analytical elastodynamics, the accuracy of explicit finite-element simulations, and the computational efficiency of physics-informed neural networks. Once trained, the PINN provides rapid, mesh-free prediction of displacement, stress, and strain throughout the space– time domain, substantially reducing the need for repeated finite-element simulations in parametric studies and engineering analyses. Future work will extend the proposed framework to other bimaterial combinations with different elastic properties and acoustic impedance contrasts in order to investigate wave transmission, reflection, and interface effects in heterogeneous materials. In addition, stronger physics enforcement, adaptive collocation strategies, and improved treatment of late-time wave reflections will be explored to further enhance prediction accuracy for complex transient elastodynamic problems.
Acknowledgements.
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