Nonlinear Behavior Research Articles

Stress transfer paths and damage modes of layered coal-rock mass under static loading

In order to explore the distribution characteristics of nonlinear mechanical behavior and the tendency of damage and failure of deep in situ coal-rock mass, a layered composite thin plate coal-rock mass model is established based on thin plate theory and finite element method. The continuity of interlayer stress and displacement is maintained by transfer matrix function, and the maximum shear stress is calculated by Mohr-Coulomb strength criterion. The distribution characteristics of stress, displacement, energy and maximum shear stress of coal-rock mass under static load are obtained. The analysis results are as follows : ① The torsional stress concentration areas of τxz and τyz are formed on the boundary of coal-rock interface, and the principal stress concentration areas of σx, σy and σz are formed in the center and center of the boundary, and the shear stress distribution of τxy is formed on the diagonal line. ② The maximum shear stress of the long axis has a strengthening effect on the shear failure, and each layer forms a stress state of alternating tension-shear-compression-shear. It first destroys in the middle of the coal-rock interface and boundary, and then destroys along the long side and short side. ③ The coal-rock mass presents a ′ X ′ -type shear zone along the radial and axial. The shear failure occurs first, followed by the principal stress compression failure. The cracks are preferentially developed in the direction perpendicular to the minimum principal stress, and the stepped ′ V ′ -shaped failure characteristics are formed under the action of shear and tensile stress.

Nonlinear dynamic analyses of sandwich porous FG cylindrical shells with dual-layered FG porous cores

This research examines the nonlinear vibrations and dynamic postbuckling (DPB) analyses of sandwich porous functionally graded (SPFG) cylindrical shells with dual-layered FG porous (FGP) cores exposed to external excitation, using a semi-analytical method. The study investigates two types of FG layers: one with evenly distributed porosities (FG-EPD) and another with unevenly distributed porosities (FG-UEPD). It also studies the FGP cores in four configurations: one featuring symmetric porosity distribution with stiffening in the surface areas (SPD-Stiff), another with softening in the surface areas (SPD-Soft), a third with nonsymmetric porosity distribution (NSPD), and a fourth with uniform porosity distribution (UPD). Therefore, the SPFG cylindrical shells with dual-layered FGP cores with eight different configurations are investigated. Utilizing the classical shell theory (CST) alongside the geometrical nonlinearity in von Kármán–Donnell framework, and Galerkin’s method, this study addresses the nonlinear dynamic problem. An approximate solution for the deflection shape using three terms is selected, and the relationship between frequency and amplitude in nonlinear vibration is clearly defined. The nonlinear dynamic behaviors including vibration and DPB responses are examined using the P-T method, which is named for its use of the piecewise constant argument in conjunction with the Taylor series expansion. The critical dynamic buckling load of SPFG cylindrical shells with dual-layered FGP cores is investigated via the Budiansky–Roth criterion.

Stability analysis of electrorheological (ER) lubricant operated multilobe hydrodynamic journal bearing

Purpose An electrorheological (ER) fluid consists of dielectric particles blended in a nonconducting oil. ER lubricants are often considered smart lubricants. This paper aims to examine the steady state and dynamic response of multilobe journal bearings using an ER lubricant. Design/methodology/approach Reynold’s equation has been used to describe the lubricant flow in the journal-bearing clearance space. The Bingham model is used to characterize the nonlinear behavior of the lubricant. The solution of the Reynolds equation is obtained using the Newton–Raphson method, with gaseous cavitation in the fluid film numerically addressed by applying a mass-conserving algorithm. The effects of lobe geometry and the applied electric field are investigated on film pressure profile, fluid film thickness, direct stiffness and damping parameters. The equation of motion for journal center coordinates is solved using the fourth-order Runge–Kutta method, to predict journal center motion trajectories. Findings Using ER lubricant combined with two-lobe journal bearing significantly improved the minimum film thickness by 49.75%, the direct stiffness parameter by 132.18% and the damping parameter by 206.3%. However, the multilobe configuration was found to negatively impact the frictional powerloss of the bearing system. In the case of multilobe configurations of journal bearings using ER lubricant, linear motion journal trajectories are observed to be reduced and exhibit increased stability. Originality/value This study presents the effect of an ER lubricant and multilobe configuration on the rotor-dynamic performance and stability analysis of hydrodynamic journal bearings. Peer review The peer review history for this article is available at: https://publons.com/publon/10.1108/ILT-06-2024-0201/

Newly formed solitary wave solutions and other solitons to the (3+1)-dimensional mKdV–ZK equation utilizing a new modified Sardar sub-equation approach

In this paper, we are going to investigate the (3+1)-dimensional nonlinear modified Korteweg–de Vries-Zakharov–Kuznetsov (mKdV–ZK) equation, which governs the behavior of weakly nonlinear ion acoustic waves in magnetized electron–positron plasma. By taking advantage of the newly proposed modified Sardar sub-equation method, we derive a comprehensive set of exact soliton solutions to the mKdV–ZK equation. Additionally, we provide graphical representations of the solutions, including 2D, 3D, and contour plots, to visualize the characteristics and features of these nonlinear wave structures. These solutions encompass a diverse range of wave patterns, including traveling waves, bright solitons, periodic waves, dark–bright solitons, lump-type solitons, and multi-soliton solutions. The obtained solutions provide valuable insights into the nonlinear behaviors and dynamics exhibited by the mKdV–ZK equation. The success of the new modified Sardar sub-equation method in obtaining a diverse range of solutions for the [Formula: see text]-dimensional mKdV–ZK equation highlights its potential for applications in the analysis of various nonlinear systems in plasma physics and beyond. Also, the study reviewed the superiority of the modified method compared to the Sardar sub-equation method.

Measurement of the Static Nonlinear Third‐Order Elastic Moduli of Rocks: Problems and Applicability

AbstractThe third‐order elastic (TOE) model has been used to describe the widely observed nonlinear mechanical behaviors of earth materials. In addition to linear elastic constants (λ, μ), three nonlinear elastic moduli (A, B, C) are required for isotropic rocks. Contrary to previous research on dynamic TOE moduli, this study followed the protocol to measure strain and stress under uniaxial and hydrostatic compressive tests statically, which were later used to invert for the full set of TOE moduli for four standard rock types with differing pore structures of a porous oolitic limestone, quartz‐rich sandstones, and a dense crystalline basalt. The applicability of the TOE model to characterize nonlinearity depends on the fulfillment of path‐independence and small‐strain assumptions. Using the measured static TOE moduli, the finite element model demonstrates that the stress in the vicinity of the wellbore is more amplified than the stress in the linear elastic case, which leads to a wider zone of rock failure around the wellbore. Due to the long‐standing discrepancy between static and dynamic moduli, the rarely reported full set of static TOE moduli is necessary and will benefit future research in understanding the effect of rock nonlinearity on geophysical and geomechanical applications, such as long‐term safe storage of CO2 and generating process of geohazards.

Exploring Nonlinear Dynamics In Brain Functionality Through Phase Portraits And Fuzzy Recurrence Plots.

Much of the complexity and diversity found in nature is driven by nonlinear phenomena, and this holds true for the brain. Nonlinear dynamics theory has been successfully utilized in explaining brain functions from a biophysics standpoint, and the field of statistical physics continues to make substantial progress in understanding brain connectivity and function. This study delves into complex brain functional connectivity using biophysical nonlinear dynamics approaches. We aim to uncover hidden information in high-dimensional and nonlinear neural signals, with the hope of providing a useful tool for analyzing information transitions in functionally complex networks. By utilizing phase portraits and fuzzy recurrence plots, we investigated the latent information in the functional connectivity of complex brain networks. Our numerical experiments, which include synthetic linear dynamics neural time series and a biophysically realistic neural mass model, showed that phase portraits and fuzzy recurrence plots are highly sensitive to changes in neural dynamics and can also be used to predict functional connectivity based on structural connectivity. Furthermore, the results showed that phase trajectories of neuronal activity encode low-dimensional dynamics, and the geometric properties of the limit-cycle attractor formed by the phase portraits can be used to explain the neurodynamics. Additionally, our results showed that the phase portrait and fuzzy recurrence plots can be used as functional connectivity descriptors, and both metrics were able to capture and explain nonlinear dynamics behavior during specific cognitive tasks. In conclusion, our findings suggest that phase portraits and fuzzy recurrence plots could be highly effective as functional connectivity descriptors, providing valuable insights into nonlinear dynamics in the brain.

Evaluating non-linear optical behavior in ionic liquids: Combining ab-initio polarizability with additivity rule

In the field of non-linear Optics, the additivity rule, which both experimental and theoretical researchers utilize, provides a simple and reliable method for calculating polarizability. However, this model neglects crucial factors, such as the solvent, the interaction distance, and external optical fields, leading to inaccurate results. In this study, the role of these parameters on the non-linear optical behavior of the ionic liquid BMIM+⋯BF4−, a system extensively studied for its intriguing non-linear optical properties, is investigated. To do that, first-principles calculations based on the density functional theory are developed; the Lennard-Jones potential is used as the reference model. The results of this study reveal the significant influence of the solvent on the second- and third-order non-linear optical responses, which are strongly affected by ion interactions, underscoring the limitations of the additivity rule in terms of molecular polarizability. The EFISHG values for each solvent are inversely proportional to the dielectric constant; for all considered solvents, it observed a clear dependence of polarizability values on the external field, ranging from 1.675 to 1.638× 10−23 esu. It was also found high sensitivity of the static polarizability to changes in the interaction distance and the solvent choice, ranging from 1.57 to 2.05× 10−30 esu, while considering the additivity rule, this parameter changed to 1.59× 10−30 esu. This information can serve as a reference for both theoretical and experimental researchers.

Mathematical modeling for the production performance of cyclic multi-thermal fluid stimulation process in layered heavy oil reservoirs

Multi-thermal fluid (MTF), with a unique thermophysical characteristics, has been widely applied in the enhanced heavy oil recovery processes. Especially for the layered heavy oil reservoirs, MTF shows more attractive than the conventional saturated steam. In this study, we propose an improved mathematical model for the production performance of cyclic MTF stimulation process in layered heavy oil reservoirs. In this model, based on a three-zone assumption, we first derive a productivity model for the thermal-fluids huff-n-puff process in layered heavy oil reservoirs. Thereafter, considering the typical EOR mechanisms of non-condensable gases in MTF, the viscosity reduction effect caused by CO2 dissolution and the effect of formation energy supplement caused by N2 are mathematically described. Simultaneously, because of the nonlinear temperature spreading behavior in the formation, an MTFs flooding experiment is also performed to characterize the effect. Meanwhile, a heating radius model with the consideration of steam overlap effect is also involved in our model. Then, the calculation results are compared against the results of a commercial simulator to confirm the accuracy, while a sensitivity analysis on the reservoir and operation parameters is performed. This paper provides a quick estimation approach for the productivity of cyclic MTF stimulation process in layered heavy oil reservoirs.

Coupled modes analysis of stimulated Brillouin scattering in the regimes of nonlinear ion acoustic waves

The nonlinear saturation of stimulated Brillouin scattering (SBS) in long scale length plasmas is studied in detail through coupled mode equations. Our model incorporates harmonic and subharmonic generation of ion acoustic waves (IAWs), as well as nonlinear Landau damping and the nonlinear frequency shift of IAWs induced by particle trapping. Numerical simulations are carried out across various IAW wavenumbers (k a λ De ) and electron-ion temperature ratios (Z i T e /T i ) within different SBS instability regimes. The results demonstrate that our model can distinguish the importance of each effect contributing to the nonlinear behavior in SBS under different plasma conditions. Furthermore, we examine the scaling of SBS reflectivity with laser intensity under conditions relevant to inertial confinement fusion.

Soliton molecules, bifurcation solitons and interaction solutions of a generalized (2 + 1)-dimensional korteweg-de vries system for the shallow-water waves

The central purpose of this paper is exploring the soliton molecules, bifurcation solitons and interaction solutions of the Korteweg–de Vries system based on the Hirota bilinear method. The studied system acts as an extension of the classic KdV system for the shallow-water waves, and is very useful to contribute in nonlinear wave phenomena. Firstly, the soliton molecules are obtained by adding resonance parameters in N-soliton. Then the interaction solutions between soliton/breather and soliton molecules are studied, as well as the interaction between two soliton molecules by using N-soliton. Moreover, a class of novel bifurcation solitons are derived, including Y-type bifurcation solitons, X-type bifurcation solitons and multiple-bifurcation solitons. In the end, the dynamic properties of soliton molecules, bifurcation solitons as well as the interaction solutions are presented graphically. The developed solutions of this research are all new and can enable us apprehend the nonlinear dynamic behaviors of the generalized (2+1)-dimensional Korteweg–de Vries system better.

Research on Rock Energy Constitutive Model Based on Functional Principle

The essence of rock fracture can be broadly categorized into four processes: energy input, energy accumulation, energy dissipation, and energy release. From the perspective of energy consumption, the failure of rock materials must be accompanied by energy dissipation. Dissipated energy serves as the internal driving force behind rock damage and progressive failure. Given that the process of rock loading and deformation involves energy accumulation and dissipation, the rock constitutive model theory is expanded by incorporating energy principles. By introducing the dynamic energy correction coefficient, according to the law of the conservation of energy, the total energy exerted by external loads on rocks is equal to the energy dissipated through the dynamic energy inside the rocks. A new type of energy constitutive model is established through the functional principle and momentum principle. To validate the model’s accuracy, a triaxial compression test was conducted on sandstone to examine the stress–strain behavior of the rock during the failure process. A sensitivity analysis of the parameters introduced into the model was conducted by comparing the model results, which helped to clarify the innate laws of significance of these parameters. The results indicated that the energy model more accurately captures the non-linear mechanical behavior of sandstone under high-stress loading conditions. The model curve fits the test data to a high degree. The fitting curve was basically consistent with the changing trend of the test curve, and the correlation coefficients were all above 0.90. Compared with other models, the model based on the energy principle not only accurately reflects the rock’s stress–strain curve, but also reflects the energy change law of rock. This has reference value for the safety analysis of rock mass engineering under loading conditions and aids in the development of anchoring and support schemes. The research results can fill in the blanks that exist in the energy method in terms of rock deformation and failure and provide a theoretical basis for deep rock engineering. Moreover, this research can further improve and extend the rock mechanics research system based on energy.

Model Predictive Control of a Stand-Alone Hybrid Battery-Hydrogen Energy System: A Case Study of the PHOEBUS Energy System

Model predictive control is a promising approach to robustly control complex energy systems, such as hybrid battery-hydrogen energy storage systems that enable seasonal storage of renewable energies. However, deriving a mathematical model of the energy system suitable for model predictive control is difficult due to the unique characteristics of each energy system component. This work introduces mixed integer linear programming models to describe the nonlinear multidimensional operational behavior of components using piecewise linear functions. Furthermore, this paper develops a new approach for deriving a strategy for seasonal storage of renewable energies using cost factors in the objective function of the optimization problem while considering degradation effects. An experimentally validated simulation model of the PHOEBUS Energy System is utilized to compare the performance of two model predictive controllers with a hysteresis band controller such as utilized for the real-world system. Furthermore, the sensitivity of the model predictive controller to the prediction horizon length and the temporal resolution is investigated. The prediction horizon was found to have the highest impact on the performance of the model predictive controller. The best-performing model predictive controller with a 14-day prediction horizon and perfect foresight increased the total energy stored at the end of the year by 18.9% while decreasing the degradation of the electrolyzer and the fuel cell.

Nonlinear Analysis of the Mechanical Response of an Existing Tunnel Induced by Shield Tunneling during the Entire Under-Crossing Process

The safety of existing tunnels during the entire under-crossing process of a new shield tunnel is critically important for ensuring the sustainable operation of urban transportation infrastructure. The nonlinear behavior of surrounding soils plays a significant role in the mechanical response of tunnel structures. In order to assess the mechanical response of the existing tunnel more reasonably, this study attempts to propose a novel theoretical solution and calculation method by simultaneously considering the nonlinear characteristics of surrounding soils and the tunneling effects of a new tunnel during its entire under-crossing process. Firstly, the additional stresses acting on the existing tunnel stemming from the tunneling effects of a new shield tunnel during different under-crossing stages are calculated using the typical Mindlin solution, as well as the Loganathan and Poulos solutions. The influences of the additional thrust, friction force, and grouting pressure and the loss of surrounding soils are taken into account. Then, the nonlinear Pasternak foundation model is introduced to characterize the behavior of surrounding soils, and the governing differential equation for the mechanical response of the existing tunnel is derived using the typical Euler–Bernoulli beam model. Subsequently, a novel theoretical solution and calculation approach are established using the finite difference formula and the Newton iteration method for assessing the mechanical response of the existing tunnel. Finally, one case study is performed to illustrate the mechanical behavior of the existing tunnel during the whole under-crossing process of a new shield tunnel, and the validity of the developed solution is verified against both the computed result of finite element simulation and the field measurements. In addition, the influences from the ultimate resistance and reaction coefficient of surrounding soils and those from the vertical distance and intersection angle between existing and newly constructed tunnels are analyzed and discussed in detail.

Comparative analysis of seismic response and vulnerability of laminated bamboo frame structure under near-field and far-field earthquake actions

The study offers an evaluation of seismic response and vulnerability for a five-story laminated bamboo frame structure, assessing the effectiveness of seismic intensity measures across various damage thresholds and the modeling uncertainties involved. A structural finite element model was developed in OpenSees software, utilizing the Pinching4 model to accurately represent the non-linear characteristics of T-shaped steel connections in beam-column joints. Two collections of earthquake records, one consisting of pulse-like near-field and the other of far-field events, were utilized to examine the structure's nonlinear behavior. Additionally, vulnerability curves for various damage states were generated using the multiple stripe analysis technique, applied to both ground motion datasets. The applicability of 24 seismic intensity measures was also analyzed across various damage limit states. The modeling uncertainty under different seismic actions was further estimated using a first-order second-moment method. Ultimately, the annual collapse probability of the structure are elucidated. Research indicates that due to the influence of joint stiffness, the overall structure demonstrates dynamic characteristics that are predominantly of the first mode shape. The spectral acceleration associated with the fundamental period achieves a reduced logarithmic standard deviation in the vulnerability analysis across various limit states. Other IMs considering higher mode effects or softening period actions no longer predominate. In most cases, the modeling uncertainty under far-field seismic action is higher than that under near-field, especially at the severe damage limit states by using an efficient intensity measure. The laminated bamboo frame structure faces a much higher collapse risk in near-field than in far-field conditions.

Spatiotemporal complexity analysis of a discrete space-time cancer growth model with self-diffusion and cross-diffusion

We investigate spatiotemporal pattern formation in cancer growth using discrete time and space variables. We first introduce the coupled map lattices (CMLs) model and provide a dynamical analysis of its fixed points along with stability results. We then offer parameter criteria for flip, Neimark–Sacker, and Turing bifurcations. In the presence of spatial diffusion, we find that stable homogeneous solutions can experience Turing instability under certain conditions. Numerical simulations reveal a variety of spatiotemporal patterns, including patches, spirals, and numerous other regular and irregular patterns. Compared to previous literature, our discrete model captures more complex and richer nonlinear dynamical behaviors, providing new insights into the formation of complex patterns in spatially extended discrete tumor models. These findings demonstrate the model’s ability to capture complex dynamics and offer valuable insights for understanding and treating cancer growth, highlighting its potential applications in biomedical research.

Wind-induced collapse mode and mechanism of super-large hyperbolic steel reticulated shell cooling tower with stiffening rings

With advancement of dry-air cooling technology, it has become feasible to use steel reticulated shell structures in design and construction of industrial cooling towers. Steel reticulated shell cooling towers (SRSCTs) with light weight, high strength, and rapid construction have development potential. Wind-induced collapse accidents in cooling towers frequently occur, characterized by pronounced nonlinear behavior involving significant deformation and material failure. Nevertheless, current stability and collapse analyses of cooling towers based on linear theory and elastic deformation are inadequate and unsafe. Thus, the wind-resistant stability of SRSCTs must be evaluated considering their inherent high flexibility and low damping levels. Considering a 220 m high hyperbolic SRSCT as a case study, a wind-induced collapse FEM simulation and analysis were performed. The average and fluctuating wind loads were measured in a reduced-scale model during wind tunnel tests and used as the time-variant wind pressure in a structural finite-element model. The dynamic performance and buckling collapse analyses of the cooling tower were conducted considering geometric and material nonlinearities. The entire wind-induced failure process of the SRSCT in extreme winds was systematically investigated. The weaknesses in the structural stiffness and the failure process were quantified, with five failure stages and four failure modes. The failure stages included linear elasticity, local member failure, local regional failure, overall structural failure, and collapse. The failure modes included circumferential plastic buckling zone failure, windward buckling failure of the shell, crosswind failure of the stiffening ring (hinged curved-beam mechanism), and windward failure of the stiffening ring (triangular arch-like mechanism). This study confirms that development of plastic buckling zones and formation of plastic hinges, considering geometric and material nonlinearity, are the dominant factor in wind-induced failure and collapse of SRSCTs.

Prediction of directional solidification in freeze casting of biomaterial scaffolds using physics-informed neural networks

Freeze casting, a manufacturing technique widely applied in biomedical fields for fabricating biomaterial scaffolds, poses challenges for predicting directional solidification due to its highly nonlinear behavior and complex interplay of process parameters. Conventional numerical methods, such as computational fluid dynamics (CFD), require adequate and accurate boundary condition knowledge, limiting their utility in real-world transient solidification applications due to technical limitations. In this study, we address this challenge by developing a physics-informed neural networks (PINNs) model to predict directional solidification in freeze-casting processes. The PINNs model integrates physical constraints with neural network predictions, requiring significantly fewer predetermined boundary conditions compared to CFD. Through a comparison with CFD simulations, the PINNs model demonstrates comparable accuracy in predicting temperature distribution and solidification patterns. This promising model achieves such a performance with only 5000 data points in space and time, equivalent to 250,000 timesteps, showcasing its ability to predict solidification dynamics with high accuracy. The study’s major contributions lie in providing insights into solidification patterns during freeze-casting scaffold fabrication, facilitating the design of biomaterial scaffolds with finely tuned microstructures essential for various tissue engineering applications. Furthermore, the reduced computational demands of the PINNs model offer potential cost and time savings in scaffold fabrication, promising advancements in biomedical engineering research and development.

Measuring Irreversibility from Learned Representations of Biological Patterns

Thermodynamic irreversibility is a fundamental characteristic of living matter, crucial for maintaining the complex spatiotemporal structures and functions that define biological systems. However, accurately quantifying biologically relevant irreversibility remains challenging due to noise and nonlinear interactions among many degrees of freedom. Here, we employ deep learning techniques to identify tractable, low-dimensional representations of phase-field patterns in a canonical protein signaling process—the Rho-GTPase system—as well as in complex Ginzburg-Landau dynamics. We demonstrate that factorizing variational autoencoder neural networks can robustly extract informative pattern features despite noise. These neural-network-derived representations reveal signatures of mesoscopic broken detailed balance and time-reversal asymmetry in both Rho-GTPase and complex Ginzburg-Landau wave dynamics. By applying the compression-based Ziv-Merhav estimator to these representations, we successfully recover irreversibility trends across complex Ginzburg-Landau patterns, which vary widely in spatiotemporal frequency and noise level. Our irreversibility estimates also recapitulate cell-activity trends in Rho-GTPase patterns subjected to metabolic inhibition. Furthermore, our framework distinguishes between stable and chaotic dynamical phases in these nonlinear systems. By leveraging advances in deep learning, our approach provides robust, model-free measurements of nonequilibrium and nonlinear behavior in complex living processes. Published by the American Physical Society 2024

Modeling of residual stiffness phenomenon in modified Iwan model of bolted joints and its application

Bolted joints have been widely used in various mechanical structures. Due to the presence of contact interfaces, the joints exhibit complex nonlinear behavior under dynamic loading. Effective prediction of the dynamic response of bolted structures requires the construction of appropriate dynamic models. This paper proposes a modified Iwan model which gives a more comprehensive description of joints than the previous Iwan models, especially for the phenomenon of residual stiffness in macro slip. The equations of the model's backbone curve, hysteresis curve, and energy dissipation are derived. The parameter identification procedure is also provided. Subsequently, connection elements based on the modified Iwan model are integrated into a single bolted joint and a thin-walled cylinder containing multiple bolted joints, the responses under quasi-static unidirectional loading, quasi-static cyclic loading and constant-frequency excitation are investigated. The physical interpretation of the parameters in the model is discussed, thus explaining the relationship between the bolted joint's physical parameters and some important variables. The results indicate that the model can effectively characterize the nonlinear mechanical behavior of the bolted joint for both micro and macro slip regime, with significant improvement in computational efficiency.

Measurement bias in self-heating x-ray free electron laser experiments from diffraction studies of phase transformation in titanium

X-ray self-heating is a common by-product of X-ray Free Electron Laser (XFEL) techniques that can affect targets, optics, and other irradiated materials. Diagnosis of heating and induced changes in samples may be performed using the x-ray beam itself as a probe. However, the relationship between conditions created by and inferred from x-ray irradiation is unclear and may be highly dependent on the material system under consideration. Here, we report on a simple case study of a titanium foil irradiated, heated, and probed by a MHz XFEL pulse train at 18.1 keV delivered by the European XFEL using measured x-ray diffraction to determine temperature and finite element analysis to interpret the experimental data. We find a complex relationship between apparent temperatures and sample temperature distributions that must be accounted for to adequately interpret the data, including beam averaging effects, multivalued temperatures due to sample phase transitions, and jumps and gaps in the observable temperature near phase transformations. The results have implications for studies employing x-ray probing of systems with large temperature gradients, particularly where these gradients are produced by the beam itself. Finally, this study shows the potential complexity of studying nonlinear sample behavior, such as phase transformations, where biasing effects of temperature gradients can become paramount, precluding clear observation of true transformation conditions.