Thermomechanical Loading Conditions Research Articles

Comparative Analysis of Active Bonded Piezoelectric Repair Systems for Damaged Structures under Mechanical and Thermo-Mechanical Loads

Active repair systems employing piezoelectric (PZT) patches have emerged as promising solutions for mitigating crack propagation and enhancing structural integrity in various engineering applications. However, the existing literature predominantly focuses on the application of PZT patches for repairing structures under mechanical loading. In this study, a finite element analysis (FEA) is employed to investigate the repair of a centre-cracked aluminium plate under both mechanical and thermo-mechanical loading conditions. This study explores the influence of key parameters, including temperature, PZT patch thickness, type of PZT material, adhesive material, and adhesive thickness, on the structural integrity and crack propagation behaviour. The results reveal significant differences in stress distribution and crack propagation tendencies under varying loading conditions and parameter settings. These findings emphasize the necessity of considering thermo-mechanical loading conditions and parameter variations when designing effective active repair systems. In conclusion, this study provides valuable insights into optimizing PZT patch-based repair strategies for improved structural integrity and crack mitigation in aerospace and other engineering applications under diverse loading scenarios.

Effect of random variation in input parameter on cracked orthotropic plate using extended isogeometric analysis (XIGA) under thermomechanical loading

Abstract This research introduces the Stochastic Extended Isogeometric Analysis (XIGA) method to investigate fracture behavior of isotropic and orthotropic materials under mechanical, thermal, and thermomechanical loads. Employing knot spans from Isogeometric Analysis (IGA) for domain discretization, the study utilizes identical basis functions for geometry construction and solution discretization. Utilizing Extended Finite Element Method (XFEM) enrichment functions, accurate crack face displacement discontinuity and tip singularity within the stress field are characterized. Additionally, employing a second-order perturbation technique within XIGA framework, the research derives mean and coefficient of variance values for mixed-mode Stress Intensity Factors (SIF). Stochastic variations in material elastic properties, crack length, and crack angle are considered in this computation. Credibility and robustness of the study are confirmed through comparative analyses against available literatures and Monte Carlo Simulations (MCS). The observed exceptional agreement validates the precision and reliability of the proposed stochastic XIGA method for fracture analysis in orthotropic material systems under thermomechanical loading conditions.

Thermal protection technologies for solid propellant rocket engines

The article examines the thermal insulation characteristics of materials used in solid-fuel rocket engines manufactured in the United States, France, Italy, Japan, and other developed nations. The internal surfaces of combustion chambers in these rocket engines are subject to significant stress under thermo-mechanical loading conditions, necessitating specialized protection. The authors identify four categories of reinforced elastomeric materials that most effectively meet stringent requirements. Due to their adaptability, these materials may serve as versatile heat insulators and may be employed in a variety of high-temperature and corrosive surroundings.

A data-driven underground gas storage production system string failure prediction model for time-varying reliability analysis

In underground gas storage (UGS) systems, the production tubing string system undergoes complex thermo-mechanical loading conditions, leading to fatigue damage accumulation and potential failure risks. To accurately assess the fatigue reliability and mitigate risks, it is crucial to develop a data-driven physics-informed uncertainty modeling approach that captures the coupled thermo-mechanical effects and accounts for various uncertainties. This study proposes a data-driven physics-informed modeling method based on thermo-mechanical coupling for fatigue reliability assessment and risk mitigation in critical UGS systems. A quantitative analysis method is introduced, which relies on small-scale multi-stage loading experiments and combines the stochastic degradation process with time-series reliability theory. To analyze the fatigue life of joints under complex loading conditions, a modified S-N curve model is developed. This model takes into account the influences of the dimension coefficient, stress concentration factors, surface finish coefficients, and thermal load. It considers the uncertainties associated with different temperature loads and surface finish on fatigue life. By integrating physics-informed modeling, uncertainty quantification, and fatigue damage accumulation analysis, this approach provides a comprehensive framework for fatigue reliability assessment and risk mitigation in critical UGS systems. It enables informed decision-making for safe and reliable operations, as well as optimized maintenance strategies, ultimately enhancing the overall system performance and mitigating potential failures.

Temperature-dependent mechanical properties and crystal plasticity parameters for additively manufactured Haynes-214 alloy: Experiments and numerical modeling

Our experimental mechanical testing data demonstrated that the additively manufactured (AM) laser powder bed fusion (L-PBF) Haynes-214 alloy exhibits non-linear mechanical properties as the temperature rises from ambient to 870 °C. Crystal plasticity (CP) simulations provide an effective approach to gaining deeper insights into microstructure-property linkages under thermomechanical loading. This method can reduce the need for costly high-temperature mechanical testing while accounting for the effects of crystallographic texture and grain morphology on the mechanical behavior of AM materials. However, calibrating a CP model is time-consuming because individual simulations are computationally expensive and hundreds (or more) of iterations over parameter sets may be required. To address this issue, we have designed a machine learning-differential evolution (ML-DE) CP framework that can accurately interpolate the tensile properties of AM L-PBF Haynes-214 alloy across a wide temperature range from ambient to 870 °C, with minimal reliance on experimental data. The framework uses electron backscatter diffraction (EBSD) measurements to generate statistically equivalent microstructural volume elements to serve as inputs to the CP modeling framework. Stress–strain curves were generated from 1000 CP simulations, which serve as the training data set for the three ML regression algorithms explored: linear, extra-trees, and multi-layer perceptron. These three regression models were independently evaluated to compare their efficiency and identify the most suitable algorithm for the given problem. Results revealed that the extra-trees ML regressor outperforms the other models in both qualitative and quantitative aspects with an R2 of 0.98. Subsequently, the differential evolution optimization approach is employed to calibrate the ML-based CP material parameters with experimental results obtained at various temperatures. Finally, temperature-dependent CP material parameters are formulated. The effectiveness and efficiency of the designed framework are validated through comparison with experimental results, demonstrating a high degree of agreement. These calibrated parametric constitutive equations enable further use of the CP model to study the deformation behavior of this alloy under a wide range of thermo-mechanical loading conditions.

A thermodynamically consistent phase field model for brittle fracture in graded coatings under thermo-mechanical loading

This paper presents a thermodynamically consistent phase field formulation designed for investigating quasi-static brittle fracture under thermo-mechanical loading conditions in functionally graded materials. The governing equations for linear momentum and crack evolution are derived by minimizing the free energy functional, which includes elastic strain energy and surface energy, with respect to displacement field and the phase field variable, respectively. A thermodynamically consistent formulation is employed to establish the governing equation for heat conduction. The proposed formulation is converted into a set of finite element equations using the Ritz method and implemented in COMSOL Multiphysics, a commercial finite element software which allows for writing governing equations in their strong form. A segregated solver is utilized to solve iteratively the coupled problem. The proposed model is verified against existing mechanical and thermo-mechanical phase field models, demonstrating good agreement with the literature. To explore the model’s capabilities, a graded layer composed of Zirconia and a Titanium alloy with a surface crack undergoing thermo-mechanical loading is considered as a case study for this investigation. Additionally, the model incorporates a thermal conductivity degradation proposed. A parametric study is performed to investigate the effect of varying parameters like volume fraction, thermal loading rate, and the thermal conductivity degradation on the load–displacement curve of the graded layer. These findings contribute to the understanding of thermo-mechanical behavior of surface graded layers and serve as a foundation for addressing more complex thermo-mechanical problems.

Influence of carbon fibre orientation and heat flux on the thermomechanical response of CFRP using small-scale apparatus

This study uses a small-scale apparatus to investigate the influence of carbon fibre orientation and heat flux on the thermomechanical response of carbon fibre reinforced polymer (CFRP). The proposed test apparatus allows the thermomechanical behaviour, relating to the mechanical performance degradation with particular surface temperature and temperature gradient of a CFRP, to be investigated. The motivation of this study is to understand the mechanical response of load-bearing CFRP materials during heating using a cone heater. Experiments were performed on CFRP specimens produced in four unique lay-ups to study their thermomechanical bending behaviour in terms of failure times, displacement, temperature distribution and failure modes. The results from this study show that different fibre orientations respond differently to thermomechanical loads. For example, it is observed that CFRP containing uni-directional [90°] fibres demonstrate the worst overall load-bearing response to thermomechanical loading conditions. In contrast, woven bi-directional [0°, 90°] fibres demonstrate the best. Uni-directional [0°] fibres and unwoven multi-directional [0°, ±45°, 90°] fibres present a modest overall load-bearing response to thermomechanical loading conditions. These results demonstrate that it is possible to systematically evaluate the performance of different fibre lay-ups and relate the failures to heating of the solid to the glass transition temperature and decomposition temperature of the polymer matrix.

Couple-stress thermoelasticity of three-dimensional multilayered composites

This work presents a comprehensive analysis of field solutions for three-dimensional multilayered composites within the framework of the couple-stress thermoelasticity. Using double Fourier series expansion solutions, the powerful and elegant mathematical Stroh formalism is extended to include the thermal coupling with the couple-stress elasticity theory to capture small-scale microstructural effects. The size-dependent layered approach leads to concise and general eigenrelations for each homogeneous orthotropic plate, while the unconditionally stable dual-variable technique is used to propagate the extended displacement/temperature fields and force-/tangential couple-tractions through the internal interfaces of the multilayered structures. Closed-form expressions are derived for the recursive layer-to-layer relations, as well as for the temperature, heat flux, displacement, stress and couple-stress field components, with respect to simply-supported structures subjected to combined thermal and mechanical loading. Application examples, including highly anisotropic fiber-reinforced composites, show significant deviations from the classical thermoelasticity, demonstrating the critical importance of considering the microstructural characteristics of interface-dominated materials. The results result come from a series of scenarios, covering mechanical and thermomechanical loading conditions, changes in aspect ratios and stacking sequences within multilayered systems, and the consequences of varying the number of sandwich plates.

Grain refinement mechanism in chip and the machined subsurface during high-speed machining of Inconel 718 alloy

Microstructural defects on the machined surface of Inconel 718 alloy can be a significant source of fatigue cracks, potentially leading to failures of the structural components of aerospace engines. This work focused on the grain refinement mechanism induced by the dynamic recrystallization (DRX) behavior in both the chip and machined subsurface during high-speed machining (HSM) of Inconel 718 alloy. Multiscale metallurgical observations were conducted using electron backscatter diffraction (EBSD), transmission electron microscopy (TEM), and precession electron diffraction (PED) to characterize the crystal structures, including the grain size, grain boundary distribution, dislocation density, and grain orientation distribution. Finite element (FE) simulations were performed to obtain the strain, strain rate, and cutting temperature to analyze the microstructure evolution mechanism under the effect of severe deformation conditions in HSM. The results indicate that grain refinement in the chip primarily occurs in the shear band and secondary deformation zone. The primary cause of the variation in the final grain size is attributed to the distinct thermomechanical loading conditions experienced by the material. The microstructure observed in the machined subsurface reveals a significant gradient distribution, characterized by the presence of an amorphous structure, nanocrystallites, refined grains, and elongated grains along the depth direction. The equiaxed ultrafine nanocrystals observed in the machined near-surface result from both continuous DRX (cDRX) and discontinuous DRX (dDRX) mechanisms. In the deeper zone, the presence of numerous subgrains and a high geometrically necessary dislocation (GND) density reaching up to 1016/m2 within the grains indicate that the refined grains in this zone result from the cDRX mechanism. The variation in DRX mechanisms observed in the machined subsurface are closely related to the gradient distribution of the thermomechanical load generated in HSM.

A Single Microtip Shape Memory Polymer Surface with Reversible Dry Adhesion for Transfer Printing

Microfabrication technology is further being advanced with the assistance of the deterministic assembly of small‐scale materials or microobjects through transfer printing which utilizes a stamp to transfer microobjects. However, the growing trend of miniaturization in microfabrication technologies poses challenges in achieving precise and high‐yield transfer of microobjects with diverse compositions, sizes, and shapes. This work describes the study of a novel approach utilizing a geometrically simple single microtip shape memory polymer surface, as a facile stamp to transfer microobjects with almost any shape (down to tens of micrometer size). To showcase the unique features of this approach, the adhesive characteristics of the single microtip shape memory polymer surface are systemically tested in a variety of thermomechanical loading conditions and its exceptional adhesion strength (≈2 MPa) and reversibility (>1000:1) are found. Over these results, its functionality to manipulate diverse microobjects with planar and non‐planar geometries as well as with flat and rough surface profiles is highlighted, accordingly. Finally, its ability to correct transfer printing errors due to its strong adhesive capability is demonstrated. This work represents a promising avenue to advance transfer printing techniques for microfabrication technologies, particularly regarding achieving high precision and yield in the integration of heterogeneous materials.

Simulation of SMA-based engineering applications considering large displacement and rotation, thermomechanical coupling and partial phase transformation

Shape memory alloys (SMAs) undergo large recoverable deformation due to their inherent diffusionless martensitic phase transformation. Two factors need to be taken into account to incisively emulate the behaviour of SMA based structures: (i) consideration of large displacement and rotation (LDR) effect, and (ii) capturing the consequence of transformation induced material level coupling. In this study, both these effects are apprehended by extending the infinitesimal strain-based constitutive model of SMA as proposed by Lagoudas et al. (2012), assuming small elastic strain but with finite inelastic strain. To preclude any spurious stress generation out of large rotation, the Jaumann stress rate is used, and incremental objectivity for finite rotation between two successive time instants is conserved by rotating the strain increment and spin tensor to the rotation-independent mid-point configuration following Hughes–Winget (Hughes and Winget, 1980) algorithm. In addition, the contribution of thermoelastic and latent heat evolved during transformation is taken into consideration in the thermal equilibrium equation. Moreover, the effect of partial phase transformation yielding minor hysteresis loop has also been captured. The equilibrium equation is expressed in the current configuration following the Updated Lagrangian formulation. The mechanical and thermal equilibrium equations are solved concurrently in block matrix form using the Newton–Raphson (NR) iterative technique, considering the coupling terms resulting from the phase transformation. The efficacy and robustness of the developed finite element model are corroborated through various practical applications of SMA-based members, e.g., SMA ring, SMA-actuated beam, morphing of corrugated airfoil, orthodontic palatal expander, compliant gripper, undergoing LDR while subjected to different thermomechanical loading conditions. The combined effects of LDR along with thermomechanical coupling yield a stiffening behaviour during loading and sluggish response at the time of thermal recovery.

A large strain thermoplasticity model including recovery, recrystallisation and grain size effects

AbstractIn manufacturing, thermomechanical processes such as static annealing and hot working are commonly used to tailor the microstructure of metals to achieve favourable macroscopic material properties that meet specific application requirements. To improve sequential manufacturing processes and to accurately predict the microstructural changes of the material along the process chain, physically motivated constitutive models are required that simultaneously account for the effects of recovery and recrystallisation, as well as grain size dependencies. To this end, Cho et al. [Int. J. Plasticity 112, 123–157 (2019)] proposed a macroscopic hypo‐elasticity based large strain thermoplasticity model that aims at the unification of the effects of static and dynamic recovery and recrystallisation, as well as grain growth and refinement. In the present contribution, the hypo‐elasticity based large strain recrystallisation formulation proposed by Cho et al. is transferred to a hyper‐elasticity based large strain thermoplasticity framework to overcome the limitations typically accompanied with hypo‐elasticity based formulations. For this purpose, an isotropic temperature dependent hyper‐elastic Hencky type formulation defined in logarithmic strains is combined with a temperature dependent von Mises yield criterion. The recrystallisation modelling approach by Cho et al. is adopted, assuming a non‐associated temperature dependent proportional hardening rate in the form of an Armstrong‐Frederick type hardening minus recovery format, wherein the proportional hardening related internal variable is interpreted as a measure of dislocation density. It is shown that the hyper‐elasticity based format results in a thermodynamical consistency condition that effectively constrains the physically motivated evolutions of recrystallised volume fraction and average grain size. To investigate the capability of the model to predict the material response of unified recrystallisation thermodynamically consistently, representative thermomechanical sequential loading conditions including static annealing and hot working are studied.

Three-dimensional microstructure reconstruction for two-phase materials from three orthogonal surface maps

Introduction: A full three-dimensional (3D) microstructure characterization that captures the essential features of a given material is oftentimes desirable for determining critical mechanisms of deformation and failure and for conducting computational modeling to predict the material’s behavior under complex thermo-mechanical loading conditions. However, acquiring 3D microstructure representations is costly and time-consuming, whereas 2D surface maps taken from orthogonal perspectives can be readily produced by standard microscopic procedures. We present a robust and comprehensive approach for such 3D microstructure reconstructions based on three electron backscatter diffraction (EBSD) maps from orthogonal surfaces of two-phase materials.Methods: It is demonstrated that processing surface maps by spatial correlation functions combined with principal component analysis (PCA) results in a small set of unique descriptors that serve as a representative fingerprint of the 2D maps. In this way, the differences between surface maps of the real microstructure and virtual surface maps of a reconstructed 3D microstructure can be quantified and iteratively minimized by optimizing the 3D reconstruction.Results: To demonstrate the applicability of the method, the microstructure of a metastable austenitic steel in the two-phase region, where austenite and deformation-induced martensite coexist at room temperature, was characterized and reconstructed. After convergence, the synthetic 3D microstructure accurately describes the experimental system in terms of physical parameters such as volume fractions and phase shapes.Discussion: The resulting 3D microstructures represent the real microstructure in terms of their characteristic features such that multiple realizations of statistically equivalent microstructures can be generated easily. Thus, the presented approach ensures that the 3D reconstructed sample and the associated 2D surface maps are statistically equivalent.

Crack tip field analysis for thermo-mechanical fatigue loading

This study aimed to establish a computational strategy for the thermo-mechanical fatigue (TMF) simulations of crack tip fields considering both the in-phase (IP) and out-of-phase (OOP) loading histories. To this end, an algorithm for multi-physics numerical calculations was developed and implemented by incorporating Maxwell 3D, Fluent, and the transient structural modules of ANSYS 2021R1, to understand the mechanics of crack tip deformation under thermo-mechanical loading conditions. These computations are based on the coupled heat loss from magnetic field eddy currents and the forced convective air-cooling accounted for by turbulence model response, which provides the gradients of mechanical elastic–plastic deformations. The proposed algorithm was applied to replicate and analyse the crack growth test conditions of polycrystalline XH73M nickel-based alloys subjected to a triangular waveform of IP and OOP cycles and a temperature range of 400–650 °C. Further, multi-physics finite element (FE) modelling of the stress, strain, and displacement fields in uncracked and fractured single-edge-notch tension specimens was performed. The novelty of the results lies in the crack tip fields, which strongly influenced by the phase relation between mechanical loading and temperature, provided the electromagnetic and fluid dynamics characteristics and the elastic–plastic properties of the material are dependent on the current temperature and time across a particular deformation cycle. Notably, the complete three-dimensional multi-physics FE analysis presented herein is expected to contribute toward a better understanding of the different mechanisms driving TMF crack growth and also help address the related outstanding questions.

Size-dependent nonlinear thermomechanical in-plane stability of shallow arches at micro/nano-scale including nonlocal and couple stress tensors

In the present exploration, the unconventional tensors of nonlocal stress and couple stress are incorporated simultaneously in the in-plane nonlinear stability analysis of functionally graded (FG) multilayer shallow micro/nano-arches under thermomechanical loading conditions. For this purpose, the nonlocal couple stress (NCS) mechanics of continuum is implemented into the third-order shear flexible arch theory incorporating the von Karman kinematical nonlinearity. The nanocomposite material of FG multilayer shallow micro/nano-arches is reinforced with graphene nanofillers in accordance with different patterns of FG lamination. The generalized differential quadrature numerical strategy in conjunction with the pseudo arc-length continuation procedure are employed to deduce the roles of unconventional nonlocal and couple stress tensors in the NCS-based nonlinear stability paths of thermomechanical loaded FG multilayer shallow micro/nano-arches. It is found that for all patterns of the lamination, the contributions associated with the nonlocality and couple stress small scale effects on the value of the upper limit as well as the first bifurcation compressive loads are less than those on the lower limit and the second bifurcation ones. Also, it is demonstrated that by combining a temperature rise with the applied compressive lateral load, the upper limit lateral load increases, while the lower limit one decreases. Furthermore, it is seen that by applying a temperature rise, an initial lateral deflection is induced in the shallow micro/nano-arch before applying the compressive lateral load.

A constitutive model of liquid crystal elastomers with loading-history dependence

Liquid crystal elastomers (LCEs) are a class of active polymers of increasing interest, because of their excellent actuation performances. To quantify the actuation performances and understand the thermomechanical behavior, many theoretical models were developed, mainly based on the free energy framework with uniaxial order tensor, which, however, cannot well capture the biaxial liquid crystal alignment under biaxial stretching or shearing. The LCE with the shape memory cycle effect exhibits an evident loading-history dependence of the thermomechanical process, which cannot be described without introducing the evolution condition of the alignment. In this work, a constitutive model of LCE is established by introducing a triaxial order tensor to describe the complex alignment under uniaxial/biaxial loading conditions. Inspired by the classical theory of plasticity, an equivalent order parameter is proposed to characterize the order degree of the alignment, and its evolution condition is introduced to specify the condition of the alignment evolution. Experiments of LCE under a variety of thermomechanical loading conditions were carried out to validate the developed model, showing the capability of the model in capturing complex thermomechanical responses of the polydomain LCE with shape memory cycle effect under uniaxial/biaxial stretching. Furthermore, the constitutive model can be extended to predict nonlinear stress-strain curves and actuation performances of LCE lattices, which hold promising potential for applications in soft actuators and flexible robots.

Thermomechanical fatigue of additively manufactured 316L stainless steel

An important issue in energy conversion is the performance of materials under complex cyclic loading in a variable temperature field. The present study addresses a new field of research – thermomechanical fatigue of additively manufactured metallic materials, which is crucial for understanding the behaviour of this promising material class under real operating conditions. The material of interest – 316L austenitic stainless steel, commonly used for heat exchangers – was manufactured to bars using laser powder bed fusion. Cylindrical specimens with characteristic hierarchical, non-equilibrium cellular microstructure were machined out of the bars. Two orientations corresponding to the inclination of the building direction to the specimen axis were considered: 0° and 90°. The specimens were subjected to thermomechanical fatigue loading under in-phase (maximum tension coincides with maximum temperature) and out-of-phase (maximum compression coincides with maximum temperature) conditions. The cellular dislocation microstructure showed good stability despite gradual coarsening under the combined effect of thermal loading up to 750 °C and severe plastic deformation. Systematic electron microscopy observations further revealed that basic damage mechanisms – either creep or stress-assisted oxide cracking, the prevalence of which depends on thermomechanical loading conditions – correspond to the behaviour of conventional metallic materials. Under in-phase loading, intergranular creep damage is dominant, hence a key factor affecting the lifetime is the number of grain boundaries in the loading direction. Under out-of-phase loading, fatigue damage is dominant, and the lifetime is determined by transgranular propagation of a principal crack. Comparing the two orientations, the inherent microstructural texture was found to be a crucial factor, also determining the number of grain boundaries and cell walls in the loading direction. Hence, tailoring the microstructure for the service relevant loading conditions via additive manufacturing techniques enables to enhance the component performance in the important field of energy conversion.

Analysis of temperature gradients in thin-walled structures under thermomechanical fatigue loading conditions

In recent thermal gradient mechanical fatigue (TGMF) investigations, it is verified that the material fatigue lives are significantly affected by the temperature gradients. Determination of temperature gradients of thin-walled structures is always a difficult task. In the present work, tubular specimens were experimentally and computationally investigated under different conditions. The inverse heat conduction (IHCP) method was improved to predict the inner boundary from external measurements. Two models were finally developed, based on direct IHCP or modified Nusselt number estimation. The models are qualified to acquire temperature gradients in TGMF experiments with reasonable accuracy and versatility.

Design and analysis of angle-ply arrangement on fracture properties of FRP composite structure under thermo-mechanical loading conditions subjected to central circle cut-out

Design and analysis of angle-ply arrangement on fracture properties of FRP composite structure under thermo-mechanical loading conditions subjected to central circle cut-out

Fracture analysis of functionally graded material reinforced with MWCNT

Computational model based on extended isogeometric analysis is implemented for the fracture analysis of functionally graded material (FGM) reinforced with multi-walled carbon nanotube (MWCNT), i.e., FGM-MWCNT structure under mechanical and thermo-mechanical loading conditions. The FGM body taken for the study is composed of metal (Ti–6Al–4V) on the left side and ceramic (ZiO2) on right side. For the gradation of FGM properties that change throughout the domain length, exponential law is assumed. Moreover, the FGM is reinforced with 0–5% of MWCNT to obtain FGM-MWCNT composite. In order to perform this fracture analysis, first the equivalent mechanical and thermal properties of the FGM-MWCNT composite are evaluated using some of the micromechanics models cited in the literature. Using the interaction integral approach, SIFs are extracted. The results of the test show that when the volume percentage of MWCNTs increases, it enhances the failure resistance of the composite.